Sampled-data consensus in multi-agent systems with asynchronous hybrid event-time driven interactions

Systems & Control Letters 89 (2016) 24–34

Contents lists available at ScienceDirect

Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Sampled-data consensus in multi-agent systems with asynchronous hybrid event-time driven interactions✩ Feng Xiao a,∗ , Tongwen Chen b a

Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150080, China

b

Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada

article

info

Article history: Received 14 December 2014 Received in revised form 13 May 2015 Accepted 10 December 2015

Keywords: Multi-agent systems Sampled-data consensus Asynchronous consensus Time delays

abstract This paper investigates sampled-data consensus in an undirected network of multiple integrators and characterizes the effectiveness of a hybrid event-time driven consensus protocol in different asynchronous scheduling schemes of event detection in terms of interaction topology, asynchronous matrix, and time delays. The proposed hybrid driven protocol has the benefit of guaranteed performance at reduced communication and computation costs and has robustness against interaction/event-detection time delays. Furthermore, the obtained results are still valid in many other practical situations, such as sampled-data consensus with measurement errors and quantized consensus. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Asynchronous individual dynamics is an important feature in large scale networks of multiple agents, and thus designing decentralized asynchronous coordinating protocols is of particular interest in both theoretical studies and engineering applications. Asynchronous sampled-data consensus, as such a research topic, has been an active research area for several years. Asynchronous sampled-data consensus was originally addressed by Lin et al. in the setup of multi-agent rendezvous [1]. Rendezvous control aims to drive all agents to meet at a specific point; in other words, it is to make all agents reach a consensus on positions. In [1], each agent was assumed to be able to continuously track the positions of all other agents within its sensing region and compute its way-points over a sequence of time intervals, uncorrelated with others. With several additional assumptions on registration intervals and sensing periods, the studied system attained a property similar to the symmetry of neighboring relationship in synchronous cases. The procedure of ‘‘analytic synchronization’’ was presented for convergence

✩ This work was supported by the Natural Sciences and Engineering Research Council of Canada, the National Natural Science Foundation of China (NSFC, Grant Nos. 61273030 and 61422302), and the Program for New Century Excellent Talents in University (Grant No. NCET-13-0178). ∗ Corresponding author. E-mail addresses: [email protected] (F. Xiao), [email protected] (T. Chen).

http://dx.doi.org/10.1016/j.sysconle.2015.12.006 0167-6911/© 2015 Elsevier B.V. All rights reserved.

analysis. Also by the concept of ‘‘analytic synchronization’’, Cao et al. investigated an asynchronous sampled-data version of the Vicsek model, where each agent sampled the headings of its neighbors at some discrete event times and changed its heading from one way-point to the other in a monotonic and piecewisecontinuous manner [2]. Based on nonlinear paracontractions theory, Fang and Antsaklis studied an asynchronous discrete-time consensus model and established a convergence result on state consensus under directional and time-varying topologies [3]. From the above results, it can be seen that all agents perform either periodic data sampling or aperiodic but time-driven data sampling with bounded periods. The same time-driven style of data sampling was also the basis of many results on double-integrator networks, see [4–6]; but it raises a problem on how to remove unnecessary data sampling at scheduled event times. The event-driven control is another technique widely used in scheduling data-sampling actions. It has favorable advantages over the pure time-driven control in applications with regard to communication costs [7–13]; but it is more theoretically challenging in convergence analysis and also has difficulty in ensuring a lower bound of inter-event times in protocol design [14]. For the singleintegrator consensus problem in undirected networks, Dimarogonas et al. designed several event-driven controllers, whose updates depended on the ratio of a certain measurement error with respect to the norm of a state function; to avoid continuous monitoring of measurement errors, these control laws were further revised by a self-triggering approach [15]. Also with the aim of relaxing the

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

requirement of continuous monitoring of neighbors’ states, Seyboth et al. proposed a consensus strategy based on exponentially decreasing event-triggering thresholds in time with nonnegative offset for networks of single-integrators with and without communication delays, and for networks of double-integrators [16]. In their strategy, each agent broadcasted its state whenever the difference between its current state and its latest broadcasted state exceeded its triggering threshold, and updated its controller whenever it sent or received a new measurement. The authors also analyzed the lower bound of inter-event times, which was related to the initial states of agents. In [17], the authors used the measurement error of a convex combination of neighbors’ states rather than the measurement error of agents’ own states to design an eventbased protocol for rendezvous and showed that lower bounds of inter-event times of each agent were state-dependent. Based on algebraic Riccati equations, event-based consensus control for general linear agents has also been studied recently [18,19]. In [18], the authors managed to define an event-triggering function, which avoided continuous communication between neighboring agents; and in [19], the authors tried the addition of a positive constant in event-triggering thresholds to guarantee positive inter-event times in some particular cases. This paper aims to solve the asynchronous consensus problem in the framework of edge-event based sampled-data consensus. Edge-event based consensus was previously addressed in fixed undirected networks in [20], and then revisited in both bidirectional networks and leader-following networks in the scenarios of continuous event detection and synchronous periodic event detection in [21]. In [22], the events on edges were considered in the synchronization of nonlinear dynamical agents with guaranteed lower bounded inter-event times. In the edge-event based framework, each information link is assigned a sequence of edge events, which activate the mutual data sampling and controller updates of the two linked agents. The idea of independent treatment of information links was previously indicated in [23]. It has the advantage of reduced communication costs and serves as an alternative to the traditional scheme in which data-sampling events are defined with respect to agents, and each event triggers the communication of its associated agent with all its neighbors [15–19]. In [21], event-detecting rules were developed to ensure a lower bound of inter-event times in the case of continuous event detection. In [24, 25], a class of edge-event based consensus protocols was given for undirected networks with synchronous periodic event detection and time-varying delays; and sufficient conditions in terms of maximum allowable time delay were given for consensus control. In this paper, we also study a time-driven fashion of eventdetection scheduling; but we focus on the case of asynchronous periodic and aperiodic event detection, and characterize the effectiveness and delay robustness of a class of hybrid event-time driven consensus protocols. The asynchronous feature of multiagent systems and aperiodic event detection differentiate this paper from [12] in basic model setups. In the latter paper, a strategy of periodic event-triggered control was presented for linear systems. The choice of time-driven scheduling of event detection relaxes the requirement of continuous state monitoring, and reduces the burden of event detectors; furthermore, it can naturally ensure a stateindependent lower bound of inter-event times and eliminate Zeno behavior of data sampling. The contributions of this paper are threefold. First, it presents several event-detecting rules and gives relaxed conditions in terms of interaction topology, asynchronous matrix, and time delays for consensus solvability in different asynchronous scheduling schemes of event detection. This paper also shows the consensus robustness against interaction time delays and the advantage in reducing communication and computation costs. Second, this paper provides a general theoretical approach to decide conditions,

25

under which we could revise continuously event-monitoring protocols with guaranteed effectiveness by adding a period of rest time after each data-sampling event. This revision brings a lower bound of inter-event times and makes considered protocols more applicable in engineering. Third, the obtained results and analysis techniques are valid in the traditional time-driven sampled-data consensus with asynchronous interactions and also applicable in other settings, such as in the presence of measurement errors or quantized communications [26,27]. This paper is organized as follows. The problem is formulated in Section 2. The asynchronous periodic and aperiodic event detection is studied in Sections 3 and 4, respectively. Application examples and simulations are given in Sections 5 and 6, respectively. Finally, the paper is concluded in Section 7. In the Appendix, some preliminary concepts and lemmas are given for reference. Notation. ⌊θ⌋ gives the largest integer not greater than θ ; 1 denotes the column vector [1 1 . . . 1]T with a compatible dimension; ◦ stands for the entrywise product (Hadamard product) of matrices; T = diag([ξ1 ξ2 . . . ξm ]) denotes the diagonal matrix with ξi in the (i, i)-th diagonal position; if T is nonnegative, T 1/2 = diag([ξ1 1/2 ξ2 1/2 . . . ξm 1/2 ]); and with an abuse of notation, T −1/2 = diag([ζ1 ζ2 . . . ζm ]), defined by

 1 −2 ζi = ξi , 0,

if ξi > 0 otherwise.

2. Problem formulation The studied multi-agent system is composed of n singleintegrators. They are labeled with 1 through n and take the following dynamics: x˙ i (t ) = ui (t ),

i = 1, 2, . . . , n,

where xi (t ) ∈ R denotes the state of agent i, and ui (t ) is a state feedback, called protocol, to be designed based on the local information received by agent i from its neighbors. The information links among agents are assumed to be bi-directional and modeled by the edges of an undirected simple graph G with n vertices v1 , v2 , . . . , vn . In G, vertex vi represents agent i, i = 1, 2, . . . , n; the existence of an edge (vi , vj ) implies an effective information link connecting agent i with agent j. Let m be the total number of edges in G and for notational simplicity, denote them by e1 , e2 , . . . , em . For each edge ep , there exists a pair of adjacent agents i and j, such that ep = (vi , vj ). In such a case, we say that vi and vj are incident to edge ep , denoted by i, j ∼ ep . For each link ep with i, j ∼ ep , agents i and j collectively generate p p p a sequence of time instants t0 , t1 , t2 , . . . , at which they check the triggering conditions of edge events of ep . If the conditions are satisfied, agents i and j sample the relative state between them and update their controllers. Let function kp (t ) index the most recent p p p edge-event time in t0 , t1 , t2 , . . . at time t; mathematically, p

kp (t ) = max k : tk ≤ t , i, j ∼ ep , an edge-event



p

between agents i and j occurs at tk .



Now, we propose the following protocol1 : ui (t ) =

 p,j:i,j∼ep

1 

p,j:i,j∼ep

=

  wp xj (tkpp (t ) ) − xi (tkpp (t ) ) ,



j∈{j:∃p, s.t. i,j∼ep }

singleton when i and j are given.



p∈{p:i,j∼ep } ,

i = 1, 2, . . . , n,

(1)

where {p : i, j ∼ ep } is in fact a

26

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

where wp is positive and called the weighting factor, measuring the contribution of link ep in local interactions [28,29]. For the data transmission delay over link ep and the common time delay of the incident agents in signal transmission from sensors to actuators, they are mathematically addable in the final form of inputs at actuator ends, see Fig. 1. In the presence of such time delays, denote the sum of them related to link ep by τp and assume that τp < p p tk+1 − tk for all k. Then the corresponding protocol is written as:



ui (t ) =

p,j:i,j∼ep

(B1) the edge-events over each link ep with i, j ∼ ep are collectively detected by both agents i and j; they initialize their first edgep event (as well as mutual data sampling) at time t0 ; p p (B2) event-detecting times t1 , t2 , . . . are generated periodically or aperiodically in a time-driven way; (B3) for k = 1, 2, . . . , an edge-event over ep is detected by agent i p at time tk if the following inequality is violated:

−



 wp xj (tkpp (t −τp ) ) − xi (tkpp (t −τp ) ) ,

i = 1, 2, . . . , n.

1 − α  p p xi (tkp (t p ) ) − xj (tkp (t p ) ) 2 k−1 k−1





   < sign xi (tkpp (t p ) ) − xj (tkpp (t p ) ) xi (tkp ) − xi (tkpp (t p ) ) k−1 k−1 k−1  β − 1  p  p < xi (tkp (t p ) ) − xj (tkp (t p ) );

(2)

2

Remark. (1) Note that this paper studies an asynchronous discontinuously event-detecting scheme and it is possible that p q tk ̸= tk for different links ep and eq . (2) This paper only studies time-invariant state transmission delays and assumes no event-actuating delays. In detail, when the triggering conditions of edge-events between any pair of p adjacent agents i and j (linked by ep ) are satisfied at tk , agents i and j get the knowledge of it immediately and then sample the relative state between them. The successful transmission p p p p of xi (tk ) − xj (tk ) may take a while. When xi (tk ) − xj (tk ) arrives at the actuators of agents i and j, it will be after time τp , which is defined as the time delay related to ep . (3) To illustrate protocol (2), assume that relative state xi − xj is p p p sampled consecutively at tk1 and tk2 , k2 > k1 . Then tkp (t −τp ) p

p

p

takes value tk1 over time interval [tk1 + τp , tk2 + τp ); in other words, the feedback part in terms of xj − xi in ui (t ) and uj (t ) is p updated at tk1 + τp when sampled data is available.

(4) Average consensus: Denote κ(t ) = 1/n i=1 xi (t ). By the bi-directional assumption of interaction topology G, it can be shown that dκ(t )/dt = 0; which says that the state average is time-invariant. So if agent states finally converge to a common value, then the final state must be κ . Consequently, the average consensus problem will be solved no matter how data sampling is driven [30].

n

The first set of event-detecting rules is given below for the case without time delays: (A1) the edge-events over each link ep with i, j ∼ ep are collectively detected by both agents i and j; they initialize their first edgep event (as well as mutual data sampling) at time t0 ; p p (A2) event-detecting times t1 , t2 , . . . are generated periodically or aperiodically in a time-driven way; p (A3) for k = 1, 2, . . . , an edge-event over ep occurs at time tk if the following inequality is violated:

k−1

k−1

agent j follows the same rule as agent i to detect edge-events and activate the mutual data sampling over ep . Remark. (1) The inequality in (B3) or (A3) has two parts. Violation of any part of the inequality will activate the corresponding data sampling. p p (2) Rule (A) and rule (B) ensure that xi (tk ) − xj (tk ) always shares p p the same sign with xi (t p p ) − xj (t p p ) and is located in between α(xi (t

p p kp (tk )

k (tk )

k (tk )

) − xj (tkpp (t p ) )) and β(xi (tkpp (t p ) ) − xj (tkpp (t p ) )). k

k−1

k−1

where α and β with 0 < α ≤ 1 ≤ β are protocol parameters. p p p Note that by the assumption that τp < tk+1 − tk , xi (t p p ) − xj (t

k (tk−1 )

p p

kp (tk−1 p

) is available to the event detectors of both agents i and j )

at tk . If each event detector can remember the state displacement of its host agent since the latest data sampling, then we can choose the following rules, which are applicable in the time-delay case and enforce rule (A) without information exchange in event detection:

k

The next two sections will present sufficient conditions for consensus convergence under protocols (1) and (2) under eventdetecting rules (A) and (B). We will study both of the periodic and aperiodic event detection in the presence and absence of time delays. 3. Asynchronous periodic event detection This section considers periodic event detection, and assumes that tki = t0i + kh for all links ei , where h is the common eventdetecting period, shared by all agents. Suppose that the event-detecting time sequences are properly labeled so that maxi {tki + τi } < mini {tki +1 + τi } for any k. Clearly, the asynchronous behavior of event detection plus time delays, is characterized by the difference of initial times t01 + τ1 , t02 + τ2 , . . . , t0m + τm . Define an Asynchronous Matrix H = [hij ] ∈ Rm×m j

by hij = max{t0i + τi − t0 − τj , 0}, i, j = 1, 2, . . . , m. Obviously, all its diagonal entries are zeros and hij < h for all i and j. Suppose that ′

    α xi (tkpp (t p ) ) − xj (tkpp (t p ) ) k−1 k−1    < sign xi (tkpp (t p ) ) − xj (tkpp (t p ) ) xi (tkp ) − xj (tkp ) k−1 k−1    p  p < β xi (tkp (t p ) ) − xj (tkp (t p ) ),

k

(3) We will see that agent states are bounded as time goes and our main results hold for any large β , so we can remove the righthand side of equation in rule (A3) or (B3) and the consensus is still reachable. However, the presence of β -term is to keep sampled data up-to-date to some extent and it is related to convergence speed.

′

j

t0i + τi′ = maxi {t0i + τi }. Then hi′ j > 0 for all j with t0 + τj ̸= t0i + τi′ . Define the m-by-m matrix H A = [hAij ] by

 3   min{hi′ i , hi′ j } , A hij = 2 max{hi′ i , hi′ j }   0,

if max{hi′ i , hi′ j } > 0 otherwise

and define T = diag([hi′ 1 hi′ 2 . . . hi′ m ]). To completely characterize time delays, introduce m-by-m L LL L LL matrices DM = [dM ij ], D = [dij ], and D = [dij ] as follows: j

j

i i L LL (1) dM ij = hij , dij = τj + (t0 − t0 ), dij = 0, if hij > 0 and τj > t0 − t0 ; j

i L LL (2) dM ij = τi , dij = dij = 0, if hij > 0 and τj ≤ t0 − t0 ;

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

27

Fig. 1. Interaction model.

j

j

i i L LL (3) dM ij = 0, dij = h − hji , dij = τj + t0 − t1 , if hij = 0 and τj > t1 − t0 ; j

i L LL (4) dM ij = 0, dij = τi , dij = 0, if hij = 0 and τj ≤ t1 − t0 .

To describe the structure of interaction topology G, we use an incidence matrix D of G, which is defined based on some given edge orientations; the matrix DT D, denoted by LE = [lEij ], is called Edge Laplacian and it has the same non-zero eigenvalues as the graph Laplacian DDT [31]. ¯ M , and σ LR be the largest eigenvalues of matrices Let λM , λ E L ◦ (1/2W 1/2 (h11T − H − H T )W 1/2 ), LE ◦ (W 1/2 H A W 1/2 ), and the largest singular value of matrix LE ◦ (1/2W 1/2 HW 1/2 ) respectively, where W = diag([w1 w2 . . . wm ]). Finally, let σ DM , σ¯ DM , σ DL , σ¯ DL , σ DLL , σ¯ DLL be the maximum singular values of the following matrices, respectively:

 E  1 M 1 L ◦ W 2D W 2     1 1   1 1    LE ◦ T 2 W 2 DM W 2 T − 2        1 1 L 1  E  2D W 2  W L ◦   2    1 1 1  LE ◦ (TW ) 2 DL W 2 √     2 h     1 1 LL 1  E 2D W 2  L ◦ W   2        1 1 1  LE ◦ √ (TW ) 2 DLL W 2 .

(3)

and not greater than r

and let σΞ denote the least upper bound of singular values of matrices in set Ξ (1). It is easy to check that LE ◦ (1/2W 1/2 (h11T − H − H T )W 1/2 ) ∈ Ξ (h/2), LE ◦ (W 1/2 H A W 1/2 ) ∈ Ξ (hasyn /2), LE ◦ (1/2W 1/2 HW 1/2 ) ∈ Ξ (hasyn /2) and thus λM ≤ hσΞ /2, λ¯ M , σ LR ≤ hasyn σΞ /2, where hasyn = maxij hij . Moreover, denote τmax = maxi τi . It is also easy to know that the matrices in (3) belong to matrix set Ξ (τmax ), Ξ (τmax ), Ξ (τmax /2), Ξ (τmax /2), Ξ (τmax /2), respectively. Then σ DM , σ¯ DM ≤ τmax σΞ , and σ DL , σ¯ DL , DLL DLL σ , σ¯ ≤ τmax σΞ /2. Therefore, we have the following corollaries: Corollary 1. Assume all edges share a common event-detecting period h and the interaction topology G is connected. If

+ hasyn + 3τmax <

α , σΞ

then the considered system under protocol (2) with rule (B) solves the average-consensus problem.

Theorem 1. Assume that the interaction topology G is connected. For the common event-detecting period h, asynchronous matrix H and time delays τ1 , τ2 , . . . , τm , if the following inequalities hold

Corollary 2. Assume all edges share a common event-detecting period h and the interaction topology G is connected and assume that there are no transmission time delays. If h 2

(4)

then protocol (2) under event-detecting rule (B) solves the averageconsensus problem; specially, in the delay-free case, if

 M λ + 2σ LR < α λ¯ M + σ LR < α

 1   1 Ξ (r ) = LE ◦ W 2 Θ W 2 : each entry of Θ is nonnegative 

h

With these symbols, we present the following theorem:

+ 2σ LR + σ DM + 2(σ DL + σ DLL ) < α + 2σ LR + σ DM + 2σ DL + σ DLL + σ¯ DLL < α + 2σ LR + σ DM + σ DL + σ DLL + σ¯ DL < α + σ LR + σ¯ DM + σ¯ DL + σ¯ DLL < α

Before proving Theorem 1, we study how event-detecting period h, asynchronous event detection and time delays affect the solvability of consensus problems. Define matrix set

2

2 h

 M λ    M λ λM  ¯M λ

is a necessary and sufficient condition for solving synchronous periodic sampled-data consensus problems [32].

(5)

then protocol (1) under event-detecting rule (A) or (B) solves the average-consensus problem. Remark. In the synchronous event-detecting case, we have that H = H A = 0 and thus if all weighting factors are 1, inequalities in (5) are reduced to h < 2α/λmax , where λmax is the largest eigenvalue of matrix W 1/2 LE W 1/2 . Recall that h < 2/λmax

+ hasyn <

α , σΞ

then protocol (1) under event-detecting rule (A) or (B) solves the average-consensus problem; in addition, we can find a maximum event-detecting period hmax = 2α/(3σΞ ), such that for any eventdetecting period h with h < hmax , protocol (1) under event-detecting rule (A) or (B) solves the average-consensus problem. Remark. (1) The definition of σΞ depends on the selection of D. In applications, we can employ Gershgorin Disk Theorem to estimate an upper bound of σΞ , independent of D. (2) For any given parameter α and any connected topology, by Corollary 2, we can always find hmax = 2α/(3σΞ ) and for any event-detecting period h < hmax , protocol (1) solves the average problem; and for such h, protocol (1) is robust against the time-delays, less than α/(3σΞ ) − h/2. In other words, inequalities (4) hold for the above h and time-delays.

28

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

Proof of Theorem 1. Denote δni (t ) = xi (t ) − κ and define Lyapunov function V (t ) = 1/2 i=1 δi (t )2 . Then dV (t ) dt

    = − wp xi (t ) − xj (t ) xi (tkpp (t −τp ) ) − xj (tkpp (t −τp ) ) , (6) p,j:i,j∼ep

where τp = 0 in the absence of time delays. Assume the consistent edge orientations in G with incident matrix D. For any edge ep with i, j ∼ ep , if it is oriented from vj to vi , p p define yp (t ) = xi (t ) − xj (t ) and yˆ p (t ) = xi (tkp (t ) ) − xj (tkp (t ) ); fur-

thermore, we define vectors x(t ) = [x1 (t ) x2 (t ) . . . xn (t )]T and y(t ) = [y1 (t ) y2 (t ) . . . ym (t )]T . Under protocol (1), the following equations are obtained:

 y(t ) = DT x(t )   m      y˙ i (t ) = − lEij wj yˆ j (t − τj )

(7)

j =1

  m   dV (t )    =− wi yˆ i (t − τi )yi (t ) dt

i=1

where t ≥ t0 = maxi {t0i + τi }. Since the interaction topology G is connected, the graph Laplacian DDT is of rank n − 1 [33] and it follows from y(t )T y(t ) = x(t )T DDT x(t ) that lim y(t ) = 0 ⇐⇒ lim x(t ) = κ 1.

t →∞

and yˆ i (tli )

yi (t ) =

( )−

lEij

wj

tki

j =1

−

m 

lEij wj

tki +τi

m 

wi

yˆ j (s − τj )ds

−

wi

tli +τi

l=3 i=1

−

m 

wi

i=1

tli+1 +τi



L ¯ M = [h¯ M H = [hLij ], H¯ L = [h¯ Lij ], H R = [hRij ], defined by the ij ], H following equations respectively:

(1)

h¯ M ij =

wi

yˆ i (t − τi )yi (t )dt

ˆ ( )yi (t )dt

tki +τi

yˆ i (tki )yi (t )dt ,

j=1

lEij wi wj yˆ i (tli )

j =1



t



tli +τi

tli +τi tli

s2

tli +τi

j

if hij > 0

j tk 2

  (sk − ) ,

otherwise

2

(3) hLij = hji 2 /2 + hji (h − hji );

(9)

+

m 

tli +τi tli

(10)

j

j

ˆ j (tl−2 ) + dLij yˆ j (tl−1 ) + dM ˆ j (tl ); yˆ j (s − τj )ds = dLL ij y ij y

t3i +τi

yˆ i (t − τi )yi (t )dt

t0

−hyˆ (l)T Wy(l)

+ hyˆ (l)T W (LE ◦ DM )W yˆ (l) + hyˆ (l)T W (LE ◦ DL )W yˆ (l − 1) + hyˆ (l)T W (LE ◦ DLL )W yˆ (l − 2) + yˆ (l)T W (LE ◦ H M )W yˆ (l) + yˆ (l)T W (LE ◦ H L )W yˆ (l − 1)  + yˆ (l)T W (LE ◦ H R )W yˆ (l + 1)

yˆ j (s − τj )ds

yˆ j (s1 − τj )ds1 ds2 .

k−1  

wi



l =3

In (10), we have



2

V (sk ) = V (t0 ) −

yˆ i (tli )yi (s)ds

m 

 (s − tki )2   k ,

Substituting Eqs. (10)–(12) into (9) gives that yi tli

= (t − tli − τi )wi yˆ i (tli )yi (tli )  m  i E i − (t − tl − τi ) lij wi wj yˆ i (tl )

−

otherwise

2

(5) hRij = hij 2 /2.

t tli +τi

if hij > 0

2   (h − hji ) ,

j (4) h¯ Lij = hji 2 /2 + hji (sk − tk );

in which for any t ∈ [tli + τi , tli+1 + τi ], l = 1, 2, . . . , k,



 (h − hij )2   + hij (h − hij ),

i =1

sk



yˆ j (s1 − τj )ds1 ds2

(2)

t0

i=1 k−1  m 

t3i +τi

tli +τi

2

yˆ j (s − τj )ds.



s2

To write Eq. (9) in a matrix form, by observing the above equations, we introduce the following m-by-m matrices H M = [hM ij ],

hM ij =

Given any k and time sk with sk = maxi {tki + τi }, by Eq. (7), we have V (sk ) = V (t0 ) −

tli +τi

t



j =1

tki +τi





  (h − h )2 hij 2 ij  yˆ i (tli )ˆyj (tli+1 ), + ( h − hij )hij yˆ i (tli )ˆyj (tli ) +   2 2   j i i    if t = tl+1 + τi and tl + τi ≥ tl + τj ;    2   hji (h − hji )2   + hji (h − hji ) yˆ i (tli )ˆyj (tli−1 ) + yˆ i (tli )ˆyj (tli ),   2 2   j   if t = tli+1 + τi and tli + τi < tl + τj ;   (sk − tki − τi )2 = (12) yˆ i (tki )ˆyj (tki ),  2   j i  if t = sk and tk + τi ≥ tk + τj ;    h 2    ji   + hji (sk − tkj − τj ) yˆ i (tki )ˆyj (tki −1 )    2  j 2    + (sk − τk − τj ) yˆ i (t i )ˆyj (t i ),  k k   2 j if t = sk and tki + τi < tk + τj .

In addition, Eq. (7) also implies that for t ∈ [tki + τi , tki +1 + τi ), m 

t

(8)

t →∞

yi tki



(11)

− yˆ (k)T TWy(k) + yˆ (k)T TW (LE ◦ DM )W yˆ (k) + yˆ (k)T TW (LE ◦ DL )W yˆ (k − 1) + yˆ (k)T TW (LE ◦ DLL )W yˆ (k − 2) + yˆ (k)T W (LE ◦ H¯ M )W yˆ (k) + yˆ (k)T W (LE ◦ H¯ L )W yˆ (k − 1),

(13)

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

where y(l) = [y1 (tl1 ) y2 (tl2 ) . . . ym (tlm )]T and yˆ (l) = [ˆy1 (tl1 ) yˆ 2 (tl2 ) . . . yˆ m (tlm )]T , l = 3, 4, . . . , k. By Lemma 1 and

By rule (A3) or (B3),

 R H + (H L )T = hH    R  H + (H¯ L )T = HT H M + (H M )T = h(h11T − H − H T )     1 (H¯ M + (H¯ M )T ) = T 12 H A T 21 

k→∞

lim y(k)T y(k) = 0.

(15)

Furthermore, by Eqs. (7) and (14), lim y˙ (t ) = 0,

t →∞

2

i = 1, 2, . . . , m,

which, together with Eq. (15), leads to limt →∞ y(t ) = 0. Therefore, by Eq. (8), for any i, limt →∞ xi (t ) = κ . 

we have that

 yˆ (l)W (LE  ◦ (H R + (H L )T ))W yˆ (l + 1)      LR ≤ hσ yˆ (l)T W yˆ (l) + yˆ (l + 1)T W yˆ (l + 1)  yˆ (l)T W (LE ◦ H M )W yˆ (l) ≤ hλM yˆ (l)T W yˆ (l)    ¯ M )W yˆ (k) ≤ λ¯ M yˆ (k)T WT yˆ (k) yˆ (k)T W (LE ◦ H

4. Asynchronous aperiodic event detection Suppose that the event-detecting periods are time-varying, and upper and lower bounded by hmax and hmin , respectively; and suppose that there exists a common time delay over all inter-agent links. Denote the time delay by τ and assume that τ < hmin . The main result of this section is given as follows:

and



29



¯ L )T ) W yˆ (k) yˆ (k − 1)T W LE ◦ (H R + (H   = yˆ (k − 1)T W LE ◦ (HT ) W yˆ (k)  1  1 1 1 1 1 = h 2 yˆ (k − 1)T W 2 W 2 (LE ◦ H )W 2 h− 2 W 2 T yˆ (k)   1 ≤ σ LR hyˆ (k − 1)T W yˆ (k − 1) + yˆ (k)T WT 2 yˆ (k) h   LR T ≤ σ hyˆ (k − 1) W yˆ (k − 1) + yˆ (k)T WT yˆ (k) .

Theorem 2. If the interaction topology G is connected and

 α  hmax <   λmax   2hmin (α − hmax λmax )   τ≤     √  hmax   hmax max mhmin + 4 + +2 (lEji )2 wi  hmin i j: j̸=i

Furthermore, we also have that for any l and k,

then protocol (2), driven by rule (B), solves the average consensus problem; in the delay-free case, if interaction topology G is connected, then protocol (1), driven by rule (A) or (B) with hmax < α/λmax , solves the average consensus problem.

 T yˆ (l) W (LE ◦ DM )W yˆ (l) ≤ σ DM yˆ (l)T W yˆ (l)   yˆ (l)T W (LE ◦ DL )W yˆ (l − 1)     ≤ σ DL yˆ (l)T W yˆ (l) + σ DL yˆ (l − 1)T W yˆ (l − 1)     yˆ (l)T W (LE ◦ DLL )W yˆ (l − 2)    ≤ σ DLL yˆ (l)T W yˆ (l) + σ DLL yˆ (l − 2)T W yˆ (l − 2) yˆ (k)T WT (LE ◦ DM )W yˆ (k) ≤ σ¯ DM yˆ (k)T WT yˆ (k)    yˆ (k)T WT (LE ◦ DL )W yˆ (k − 1)    ≤ σ¯ DL yˆ (k)T WT yˆ (k) + hσ¯ DL yˆ (k − 1)T W yˆ (k − 1)    T  (LE ◦ DLL )W yˆ (k − 2)  yˆ (k) WT DLL ≤ σ¯ yˆ (k)T WT yˆ (k) + hσ¯ DLL yˆ (k − 2)T W yˆ (k − 2).

Remark. Compared to the tight upper bound of allowable sampling periods, 2/λmax , given in [32] for solving synchronous periodic sampled-data consensus problems, Theorem 2 gives an upper bound of allowable sampling periods, 1/λmax , in a more general setting of asynchronous aperiodic sampled-data consensus; see Section 5.1.

Consequently, by Eq. (13) and the remarks after rule (B), V (sk ) ≤ V (t0 ) −

m 

wi

t3i +τi



Proof of Theorem 2. For analysis purpose, we combine all the event-detecting times into one single sequence in the increasing order, ignore the first several instants, and label them with t0 , t1 , t2 , . . . , satisfying t0 = maxi t1i . Given k, for any i and l with l < k, there exist li , li− and li+ , such that tli− = max{tlii , t0 } ≤ tl < tli+ = min{tlii +1 , tk }. Then we have that

yˆ i (t − τi )yi (t )dt

t0

i =1

+ yˆ (3)T W (LE ◦ H L )W yˆ (2) + hσ DLL yˆ (1)T W yˆ (1) + h(σ DL + σ DLL )ˆy(2)T W yˆ (2) − h(α − λM − σ LR − σ DM − 2σ DL − 2σ DLL )ˆy(3)T W yˆ (3) − h(α − λ − 2σ M

×

k−3 

LR

−σ

DM

− 2σ

DL

− 2σ

DLL

wi

)

= (tli+ − tli− )wi yˆ i (tli− )yi (tli− + τ ) i+ −1 m l t − t   q +1 q − (tq+1 − tq ) + tli+ − tq+1

l=3

× yˆ (k − 2) W yˆ (k − 2) − h(α − λ − 2σ

−σ

−σ

DL

−σ

DLL

× lEij wi wj yˆ i (tq )ˆyj (tq ),

− σ¯ ) DL

where (tq+1 − tq )/2 + tli+ − tq+1 < tli+ − tq ≤ hmax . In the above equation, by Eq. (7),

× yˆ (k − 1) W yˆ (k − 1) − (α − λ¯ M − σ LR − σ¯ DM − σ¯ DL − σ¯ DLL )ˆy(k)T WT yˆ (k). T

Since V (t ) is lower bounded by 0, by Eq. (4), the above inequality holds only when lim yˆ (k) W yˆ (k) = 0. T

k→∞

2

j=1 q=li−

T

DM

yˆ i (t − τ )yi (t )dt

l

− h(α − λM − 2σ LR − σ DM − 2σ DL − σ DLL − σ¯ DLL ) LR

l

t i− +τ

yˆ (l)T W yˆ (l)

M

t i+ +τ



(14)

(tli+ − tli− )wi yˆ i (tli− )yi (tli− + τ ) = (tli+ − tli− )wi yˆ i (tli− )yi (tli− )  m  − (tli+ − tli− )wi yˆ i (tli− ) lEij wj j=1

t i− l

t i− −τ l

yˆ j (s)ds,

30

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34 ij

and it can be shown that there exist nonnegative numbers τ1 and ij 2

ij 1

ij 2

τ for every pair of links i and j, such that τ + τ = τ and

(tli+ − tli− )wi yˆ i (tli− )

m 

lEij



wj

tli−

yˆ j (s)ds

tli− −τ

j=1

 √ = τ hmax tli+ − tli− wi yˆ i (tli− )  ij m   τ2ij tli+ − tli−  τ1 E × lij wj yˆ j (tli− ) + yˆ j (tli− − τ ) hmax τ τ j=1 √  τ hmax ≤ m(tli+ − tli− )wi 2 yˆ i (tli− )2 2

+

m 

( ) wj lEij 2

2

2  τ2ij yˆ j (tli− ) + yˆ j (tli− − τ ) . τ τ

 τ ij 1

j=1

(16)

1

τ

yˆ j (tli− ) +

2 τ2ij yˆ j (tli− − τ ) ≤ yˆ j (tli− )2 + yˆ j (tli− − τ )2 . τ j

For any event-detecting instant of link ej , j ̸= i, say, tl′ , the j

j

maximum event-detecting time of agent i on (tl′ , tl′ +1 + τ ) is ⌊(hmax + τ )/hmin ⌋ + 1. At each of these possible event-detecting j times, say, tli− , yˆ j (tl′ ) will appear as yˆ j (tli− ) or yˆ j (tli− −τ ) in Eq. (16). Thus for the given k, by Eq. (6) and the property that all the diagonal entries of LE are 2, we have V (tk ) ≤ V (t0 ) −

k−1 

 (tl+1 − tl )

l =0

m 

2

+



×

( )

lEij 2

ui (t ) =

   p wp xj (tkp (t ) ) − xi (tkpp (t ) ) + ∆(tkpp (t ) ) ,

p,j:i,j∼ep

(17) p xj t k

p xi tk

k (tk )



p

k (tk )

   p p p  xj (tk ) − xi (tk ) + ∆(tk ) 1+α    ≤ sign xj (tkp ) − xi (tkp ) + ∆(tkp ) xj (tkp ) − xi (tkp )   1  p p p  ≤ xj (tk ) − xi (tk ) + ∆(tk ). 1−α

 +1

hmin

 wj yˆ j (tl ) 2

.

2

i: i̸=j

Here, yˆ (tl ) = [ˆy1 (tl ) yˆ 2 (tl ) . . . yˆ (tl )]T and Θ l = diag([θ1l θ2l . . . θml ]), l = 0, 1, 2, . . . , k − 1, with 0 < θil < 1 given by

θil =

In the setup of time-driven sampled-data consensus, consider the following asynchronous sampled-data protocol with measurement errors:

has the property that for any k,

hmax + τ

 

5.2. Sampled-data consensus with measurement errors

1

 4+

hmin j=1

p

In rule (A) or (B), if we choose α = β = 1, then at each tk with i, j ∼ ep , the mutual data sampling between agents i and j is always triggered and thus the studied model becomes the purely timedriven sampled-data asynchronous consensus model. Our main results are valid in these special cases.

where |∆( )| ≤ α| ( ) − ( )|, 0 ≤ α < 1, and k (t ) = p p p p max{k : tk ≤ t }. The feedback component xj (tk ) − xi (tk ) + ∆(tk ) p p plays the same role as xi (t p p ) − xj (t p p ) in protocol (1) and it

E

i=1

m 1 

This section shows the application of our results by some examples.

p tk

− hmax yˆ (tl ) Θ WL W yˆ (tl )  √ m  τ hmax − m wi 2 yˆ i (tl )2 l

5. Application examples

i = 1, 2, . . . , n,

wi yˆ i (tli− )yi (tli− )

i=1

T

Remark. In the proof of Theorem 1, the complete asynchronous information of event detection and time delays, given by matrices H , DM , DL , and DLL , are used, while in the proof of Theorem 2, the asynchronous information is time-varying and thus we only use the upper and lower bounds hmax and hmin of event-detecting periods and consider the evolvement of the Lyapunov function V (t ) at the sequence of united event-detecting times.

5.1. Time-driven asynchronous sampled-data consensus j

In the above equation, (tli+ − tli− )/hmax < 1 and only when tkj (t ) li− is located in between tli− − τ and tli− , yˆ j (tli− ) and yˆ j (tli− − τ ) are not equal, and in such a case,

 τ ij

By the conditions provided in Theorem 2, we have liml→∞ yˆ (tl ) = 0, and by the same arguments as in proving Theorem 1, limt →∞ xi (t ) = κ, i = 1, 2, . . . , n. 

 1  tl+1 − tl + tli+ − tl+1 , hmax 2

i = 1, 2, . . . , m.

By a similar proof to that of Theorem 2, we have Corollary 3. If the interaction topology G is connected, then protocol (17) with the maximum data-sampling period hmax < 1/(λmax (1 + α)) solves the average consensus problem.

By the remarks after rule (B),

The above result can be further extended to the case with time delays.

yˆ i (tli− )yi (tli− ) ≥ α yˆ i (tl )2 ,

5.3. Log-quantized consensus

and thus by Lemma 2, for the given k, we have V (tk ) ≤ V (t0 ) −

+

×

1 hmin

α − λmax hmax − 

hmax hmin



τ hmax 2

 max m + i

   E 2 +2 (lji ) wi j:j̸=i

k−1  (tl+1 − tl )ˆy(tl )T W yˆ (tl ). l=0

Consider the following protocol:

√



4 hmin

ui (t ) =



  wp Qlog ε xj (tkpp (t ) ) − xi (tkpp (t ) ) ,

p,j:i,j∼ep

i = 1, 2, . . . , n,

(18)

where k (t ) has the same meaning as in protocol (17), and Qlog ε (·) is the Logarithmic quantizer with ε > 1 in the form that p

Qlog ε (ξ ) =



0, sign(ξ )ε ⌊logε |ξ |⌋ ,

if ξ = 0 otherwise.

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

31

5.4. Lower bounds of inter-event times Now, we go back to the event-driven setup. By observing the form of Eq. (6) with the aim to solve state consensus, we can design event-triggering conditions by continuously examining that p p xi (t ) − xj (t ) is located in between α(xi (tkp (t ) ) − xj (tkp (t ) )) and

β(xi (tkpp (t ) ) − xj (tkpp (t ) )), and thus dV (t )/dt ≤ 0 is preserved all

Fig. 2. Interaction topology.

Under protocols (18), it always holds that for any k,

   p p  Qlog ε (xj (tk ) − xi (tk ))    ≤ sign Qlog ε (xj (tkp ) − xi (tkp )) xj (tkp ) − xi (tkp )     < εQlog ε (xj (tkp ) − xi (tkp )). By Theorem 2, we have the following result: Corollary 4. If the interaction topology G is connected, then protocol (18), with the maximum data-sampling period hmax < 1/λmax , solves the average consensus problem.

(a) Rule (A).

the time [20]. However, such rules, as well as many other eventdriven protocols in multi-agent consensus, can hardly guarantee a lower bound of inter-event times for each link [14]. One may ask whether it is possible to add a period of rest time with a lower bound after each data sampling. This question will be answered in what follows. Consider the following continuously event-detecting rule (C) with a rest period h: (C1) the data sampling over each link ep with i, j ∼ ep is collectively activated by both agents i and j; they initialize their first edgep event (as well as mutual data sampling) at time t0 ; p (C2) for k = 0, 1, . . . , an edge-event occurs at time t ≥ tk + h and p set tk+1 = t if the following condition is violated:

       α xi (tkp ) − xj (tkp ) < sign xi (tkp ) − xj (tkp ) xi (t ) − xj (t )     < β xi (tkp ) − xj (tkp );

(b) Rule (B).

(c) Periodic data sampling. Fig. 3. State trajectories in the periodic event detection and periodic sampled-data control without time delays.

32

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

(a) Rule (A).

(b) Rule (B). Fig. 4. Edge-event numbers in the periodic event detection without time delays.

Fig. 5. Comparison of decreasing rates of V (t ) with respect to time t.

or (C2′ ) for k = 0, 1, . . . , an edge-event is detected by agent i at time p p t ≥ tk + h and set tk+1 = t if the following condition is violated:

−

1 − α p p  xi (tk ) − xj (tk ) 2





   < sign xi (tkp ) − xj (tkp ) xi (t ) − xi (tkp )  β − 1  p p  < xi (tk ) − xj (tk );

2 agent j follows the same rule as agent i to detect the edge-event over ep . The above two set of rules (C1, C2) and (C1, C2′ ) are the revisions of the event-detecting rule (A) proposed in [20] by adding a period of rest time h after each data sampling. Over the time interval [tkp + h, tkp+1 ], the two adjacent agents i and j continuously detect the activating conditions for data sampling. Not surprisingly, the same agent dynamics can be obtained by rule (A) if there are finite event-detecting times over each of such time intervals with event-detecting periods not greater than h, and the last eventp detecting time happens to be tk+1 . From this viewpoint, we can invoke Theorem 2 and obtain the following corollary: Corollary 5. If the interaction topology G is connected and maximum post-event rest period h < α/λmax , then protocol (1), driven by rule (C1, C2) or (C1, C2′ ), solves the average consensus problem.

Fig. 6. State trajectories and edge-event numbers in the periodic event detection under rule (B) with time delays.

6. Simulations In the simulations, the system consists of 9 agents and their interaction topology is depicted in Fig. 2. Let all weighting factors be 1, and let α = 0.5 and β = 1.5. Initial states are randomly distributed between 0 and 10. Initial edge-event instants of the 10 links are 0, 0.01, 0.02, 0.02, 0.015, 0.03, 0.05, 0.04, 0.06, 0.08, respectively.

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

33

Fig. 7. State trajectories and edge-event numbers in the aperiodic event detection under rule (B) without time delays.

Fig. 8. State trajectories and edge-event numbers in the aperiodic event detection under rule (B) with a common time delay.

In the periodic event-detecting case, we take the eventdetecting period h = 0.12. By Theorem 1, protocol (1) solves the average consensus problem in the absence of time delays. The state trajectories under rules (A) and (B) are shown in Fig. 3. The edge-event numbers in each period of h are shown in Fig. 4. It can be seen that rule (B) triggers more events than rule (A). This is consistent with the fact that rule (B) ensures the enforcement of rule (A). It also can be observed that our protocol under rule (A) or (B) has lower event frequencies than the traditional periodic datasampling protocol (given in Section 5.1), which triggers 10 edgeevents in each period of h. We give the state trajectories under the periodic data-sampling protocol in the same initial conditions in Fig. 3 and compare the decreasing speeds of V (t ) by our protocol under rules (A) and (B) and by the periodic data-sampling protocol in Fig. 5. These figures together show that our protocol has the advantage of higher convergence rates at reduced communication and computation costs. By Theorem 1, the system under rule (B) still solves the averageconsensus problem when there exist time delays 0.03, 0.02, 0.015, 0.012, 0.01, 0.011, 0.012, 0.007, 0.008, 0.02 over the 10 links respectively. The state trajectories and edge-event numbers in each period of h are shown in Fig. 6. In aperiodic event-detecting case, by Theorem 2, we have that the maximum allowable event-detecting period is bounded by 0.0927. In the simulations, we choose the event-detecting periods randomly from [0.4 × 0.0927, 0.8 × 0.0927], which allows a common time delay less than 0.0011. We take time delay τ = 0.0006. The state trajectories and edge-event numbers in each

period of 0.12 under rule (B) in the presence and absence of time delays are shown in Figs. 7 and 8, respectively. 7. Conclusions In this paper, we formulated the problem of asynchronous hybrid event-time driven sampled-data consensus and presented analysis results by studying a specific consensus protocol in several asynchronous scheduling schemes of event detection with and without time delays. We also gave simulation results to show the advantage of our protocol with respect to communication costs and convergence rates. Future research will include the extensions of these results to time-varying networks with fixed or timevarying delays. Appendix This appendix presents some preliminary notions and lemmas. Given an undirected simple graph G with vertex set V = {v1 , v2 , . . . , vn } and edge set E , a path in G from vi1 to vik is a sequence vi1 , vi2 , . . . , vik of finite vertices such that (vij , vij+1 ) ∈ E for j = 1, 2, . . . , k − 1. Graph G is called connected if for any two vertices, there always exists a path connecting them. Let m denote the number of edges in G and label these edges with 1 through m. Assign each edge an arbitrary orientation. For the assigned orientations, the n-by-m incidence matrix D = [dij ] is defined by

 dij =

−1, 1, 0,

if vi is the tail of the jth oriented edge, if vi is the head of the jth oriented edge, otherwise.

34

F. Xiao, T. Chen / Systems & Control Letters 89 (2016) 24–34

The graph Laplacian of G is defined by L = DDT , which is independent of the edge orientations or matrix D [33]. Lemma 1. For any column vectors ξ and ζ and real matrix A with compatible dimensions,

ξ T Aζ ≤

1 2

σmax (A)(ξ T ξ + ζ T ζ ),

where σmax (A) is the largest singular value of A. Proof. This lemma is obvious by observing that

ξ T Aζ =

1 2

[ξ T ζ T ]



0 AT

  ξ .  ζ

A 0

Lemma 2. For any column vector ξ , real positive-semidefinite matrix A, and nonnegative diagonal matrix Θ with compatible dimensions, if all the diagonal entries of Θ are not greater than 1, then

ξ T Θ Aξ ≤ λmax (A)ξ T ξ , λmax (A) is the largest eigenvalue of A. Proof. By Lemma 1, ξ T Θ Aξ ≤ 1/2σmax (A)(ξ T Θ 2 ξ + ξ T ξ ) ≤ σmax (A)ξ T ξ . Note that σmax (A) = λmax (A) by the positivesemidefinite property of A.  References [1] J. Lin, A.S. Morse, B.D.O. Anderson, The multiagent rendezvous problem—the asynchronous case, in: Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, pp. 1926–1931. [2] M. Cao, A.S. Morse, B.D.O. Anderson, Agreeing asynchronously, IEEE Trans. Automat. Control 53 (8) (2008) 1826–1838. [3] L. Fang, P.J. Antsaklis, Asynchronous consensus protocols using nonlinear paracontractions theory, IEEE Trans. Automat. Control 53 (10) (2008) 2351–2355. [4] Y. Cao, W. Ren, Sampled-data discrete-time coordination algorithms for double-integrator dynamics under dynamic directed interaction, Internat. J. Control 83 (3) (2010) 506–515. [5] Y. Gao, L. Wang, Sampled-data based consensus of continuous-time multiagent systems with time-varying topology, IEEE Trans. Automat. Control 56 (5) (2011) 1226–1231. [6] Y. Zhang, Y.P. Tian, Consensus of data-sampled multi-agent systems with random communication delay and packet loss, IEEE Trans. Automat. Control 55 (4) (2010) 939–943. [7] M. Zhong, C.G. Cassandras, Asymchronous distributed optimization with event-driven communication, IEEE Trans. Automat. Control 55 (12) (2010) 2735–2750. [8] K.J. Åström, B.M. Bernhardsson, Comparison of Riemann and Lebesgue sampling for first order stochastic systems, in: Proceedings of the 41st IEEE Conference on Decision and Control, 2002, pp. 2011–2016. [9] K.J. Åström, Event based control, in: Analysis and Design of Nonlinear Control Systems, 2008, pp. 127–147.

[10] M.D. Lemmon, Event-triggered feedback in control, estimation, and optimization, in: Networked Control Systems, in: Lecture Notes in Control and Information Sciences, vol. 405, Springer-Verlag, Berlin, Heidelburg, 2010, pp. 293–358. [11] D. Lehmann, J. Lunze, Event-based control with communication delays and packet losses, Internat. J. Control 85 (5) (2012) 563–577. [12] W.P.M.H. (Maurice) Heemels, M.C.F. (Tijs) Donkers, A.R. Teel, Periodic eventtriggered control for linear systems, IEEE Trans. Automat. Control 58 (4) (2013) 847–861. [13] B. Wang, X. Meng, T. Chen, Event based pulse-modulated control of linear stochastic systems, IEEE Trans. Automat. Control 59 (8) (2014) 2144–2150. [14] D.P. (Niek) Borgers, W.P.M.H. (Maurice) Heemels, Event-separation properties of event-triggered control systems, IEEE Trans. Automat. Control 59 (10) (2014) 2644–2656. [15] D.V. Dimarogonas, E. Frazzoli, K.H. Johansson, Distributed event-triggered control for multi-agent systems, IEEE Trans. Automat. Control 57 (5) (2012) 1291–1297. [16] G.S. Seyboth, D.V. Dimarogonas, K.H. Johansson, Event-based broadcasting for multi-agent average consensus, Automatica 49 (2013) 245–252. [17] Y. Fan, G. Feng, Y. Wang, C. Song, Distributed event-triggered control of multiagent systems with combinational measurements, Automatica 49 (2013) 671–675. [18] W. Zhu, Z.P. Jiang, G. Feng, Event-based consensus of multi-agent systems with general linear models, Automatica 50 (2014) 552–558. [19] E. Garcia, Y. Cao, D.W. Casbeer, Decentralized event-triggered consensus with general linear dynamics, Automatica 50 (2014) 2633–2640. [20] F. Xiao, X. Meng, T. Chen, Average sampled-data consensus driven by edge events, in: Proceedings of the 31st Chinese Control Conference, 2012, pp. 6239–6244. [21] F. Xiao, X. Meng, T. Chen, Sampled-data consensus in switching networks of integrators based on edge events, Internat. J. Control 88 (2) (2015) 391–402. [22] D. Liuzza, D.V. Dimarogonas, M. di Bernardo, K.H. Johansson, Distributed model-based event-triggered control for synchronization of multi-agent systems, IFAC NOLCOS, Toulouse, France, 2013. [23] D.V. Dimarogonas, K.H. Johansson, Stability analysis for multi-agent systems using the incidence matrix: quantized communication and formation control, Automatica 46 (4) (2010) 695–700. [24] F. Xiao, T. Chen, H. Gao, Synchronous hybrid event- and time-driven consensus in multiagent networks with time delays, IEEE Trans. Cybern. (2014) http://dx.doi.org/10.1109/TCYB.2015.2428056. [25] F. Xiao, T. Chen, H. Gao, Edge-event based consensus in networks with common time-varying delays, presented at the 19th IFAC World Congress, 2014, pp. 8299–8304. [26] A. Kashyap, T. Basar, R. Srikant, Quantized consensus, Automatica 43 (2007) 1192–1203. [27] D. Yuan, S. Xu, H. Zhao, Y. Chu, Distributed average consensus via gossip algorithm with real-valued and quantized data for 0 < q < 1, Systems Control Lett. 59 (2010) 536–542. [28] W. Ren, R.W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Automat. Control 50 (5) (2005) 655–661. [29] Z. Ji, Z. Wang, H. Lin, Z. Wang, Interconnection topologies for multi-agent coordination under leader–follower framework, Automatica 45 (12) (2009) 2857–2863. [30] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control 49 (9) (2004) 1520–1533. [31] M. Mesbahi, M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, Princeton, Oxford, 2010. [32] G. Xie, H. Liu, L. Wang, Y. Jia, Consensus in networked multi-agent systems via sampled control: fixed topology case, in: Proceedings of the 2009 American Control Conference, 2009, pp. 3902–3907. [33] C. Godsil, G. Royal, Algebraic Graph Theory, Springer-Verlag, New York, 2001.