event-based multiagent consensus control: zeno-free...

284 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 50, NO. 1, JANUARY 2020

Event-Based Multiagent Consensus Control:Zeno-Free Triggering via Lp Signals

Zhiyong Sun , Na Huang , Brian D. O. Anderson , Life Fellow, IEEE, and Zhisheng Duan

Abstract—In this paper, we develop some new event-triggeredalgorithms for achieving distributed consensus for a multiagentsystem that guarantees fully Zeno-free triggerings for all theagents. In the proposed framework, each agent updates its controlinput only at its own triggering instants by using local measure-ments (i.e., relative states) with respect to neighboring agents,and such local measurements can be done in a local coordinateframe. For all agents, a positive Lp signal function is embed-ded in the event detector functions, which aims to avoid thepossible comparison of an event error term to a zero thresholdthat may happen in a zero-crossing scenario. We further pro-pose a Zeno-free self-triggered algorithm to achieve multiagentconsensus, which enables discrete-time measurements and thusavoids continuous measurements between the neighboring agents.We also show that the proposed event-based consensus algo-rithms guarantee less frequent triggered events in a boundedtime interval compared to the conventional algorithm withoutLp signals. Simulations and comparisons are provided to validatethe performance and effectiveness of the proposed event-basedconsensus schemes.

Manuscript received February 27, 2018; revised June 30, 2018 and July 24,2018; accepted August 29, 2018. Date of publication September 26, 2018;date of current version October 22, 2019. This work was supported bythe Australian Research Council under Grant DP130103610 and GrantDP160104500. The work of Z. Sun was supported in part by the NationalScience Foundation of China under Grant 61703130, and in part by thePrime Minister’s Australia Asia Incoming Endeavor Postgraduate Award. Thework of N. Huang was supported in part by the National Science Foundationof China under Grant 61703130, and in part by 111 Project under GrantD17019. The work of B. D. O. Anderson was supported by 111 Project underGrant D17019. The work of Z. Duan was supported by the National ScienceFoundation of China under Grant 61673026. This paper was recommendedby Associate Editor Q.-L. Han. (Corresponding author: Na Huang.)

Z. Sun is with the School of Automation, Hangzhou Dianzi University,Hangzhou 310018, China, and also with the Research School ofEngineering, Australian National University, Canberra, ACT 2601, Australia(e-mail: [email protected]).

N. Huang is with the School of Automation, Hangzhou DianziUniversity, Hangzhou 310018, China, and also with the School ofArtificial Intelligence, Hangzhou Dianzi University, Hangzhou 310018, China(e-mail: [email protected]).

B. D. O. Anderson is with the School of Automation, HangzhouDianzi University, Hangzhou 310018, China, also with the ResearchSchool of Engineering, Australian National University, Canberra, ACT2601, Australia, and also with Data61-CSIRO, Canberra, ACT, Australia(e-mail: [email protected]).

Z. Duan is with the State Key Laboratory for Turbulence andComplex Systems, Department of Mechanics and Engineering Science,College of Engineering, Peking University, Beijing 100871, China(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCYB.2018.2868786

Index Terms—Event-based control, Lp signals, multiagentconsensus, Zeno-free triggering.

I. INTRODUCTION

A. Background

CONSENSUS in multiagent systems has been consid-ered as a fundamental and useful concept in the field

of distributed and cooperative control, which also servesas a basic component for many distributed algorithms andcoordination control tasks, including formation control, dis-tributed optimization, among others [1], [2]. In the last decade,multiagent consensus and coordination control has been a veryhot topic that attracts very active research activities in thecontrol community [3]–[5]. In practice, multiagent systemsare usually equipped with digital sensors and microproces-sors which typically have limited computation and/or actuationcapability. Therefore, a distributed and resource-aware frame-work control is very often preferred and desirable for practicalapplications. In recent years, there have been increasinginterests and advances on event-based control [6] that aimsto address such resource and computation challenges, whileseveral event-based control algorithms have been proposed,e.g., for linear/nonlinear control systems [7]–[10], for decen-tralized/networked control systems [11]–[15], and in particularfor multiagent control and distributed coordination systems(see [16]–[18]). This trend has been motivated by the factthat by using an event-triggered approach to update the con-troller input, instead of using a continuous updating strategy,the overall system can save resources in its sensors/processorsand thus can significantly reduce the computation and actu-ation burden. There has also been a growing amount ofrecent reports in the literature on event-based control for dis-tributed multiagent consensus, for which we refer the readersto [19]–[24] for some popular and typical event-triggeredconsensus algorithms, and the recent surveys [25] and [26]for some state-of-the-art developments in event-based andself-triggered multiagent consensus control. In particular, thepapers [27] and [28] provide a comprehensive survey on event-triggered control for multiagent systems from a sampled-datacontrol viewpoint. In the next section, we will explore someof the key ideas in these references in more detail.

The L2 stability or Lp stability of event-triggered networkedsystems have been discussed in [29] and [30]. In this paper,we adopt another viewpoint, that is, to include positive Lp

signals in the event-function design to achieve asymptotic (or

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SUN et al.: EVENT-BASED MULTIAGENT CONSENSUS CONTROL: ZENO-FREE TRIGGERING VIA Lp SIGNALS 285

exponential) consensus convergence with Zeno-free triggeringfor all the agents.

B. Related Papers: Revisits and Comparisons

In this section, we revisit several popular event-based con-sensus algorithms in the literature, and compare them interms of measurement/communication requirements, the con-sensus value (average consensus or not), local/global coordi-nate frame requirement, Zeno triggering behavior, and controlupdate frequency. In particular, we mention the coordinateframe requirement, an important issue that has largely beenignored in the literature for coordination control of multiagentsystems [31].

The paper [19] proposed a distributed event-triggered con-sensus algorithm, which enables all the agents to reach anaverage consensus. The control input for each agent is updatedat both its own triggering instants and the neighboring agents’triggering events. The authors also proved that at least oneagent does not exhibit Zeno behavior, but Zeno triggeringis not excluded for all the agents. We remark that the con-troller implementation in [19] requires a global coordinatesystem to be involved for each agent, and it is used for inter-preting the control message (i.e., neighbors’ states) receivedfrom other agents. The event-based consensus control in [19]has been generalized in [32] for multiagent consensus withnonlinear and state-dependent couplings, and been extendedin [33] for multiple linear systems with quantized relative statemeasurements.

The paper [20] improves the results in [19] and [34], byproposing a modified event-triggering scheme to achieve aver-age consensus while Zeno triggering for all the agents canbe excluded. However, the disadvantages of frequent controlupdates triggered by both an agent’s own and its neighbors’triggering instants, and global coordinate frame requirementstill remain in [20].

The paper [21] discussed a broadcast-based event-triggeredconsensus algorithm to achieve an approximate consensus (andaverage consensus under some stronger conditions). In [21],each agent updates its control input when it broadcasts orreceives a control message from its neighbors, which results infrequent triggering. Also, a global coordinate frame is requiredfor all the agents to interpret the state information sent by allother agents. In order to achieve an exact consensus, all theagents need to access some global information such as thealgebraic connectivity of the graph.

Another promising approach for event-triggered control isto employ a sampled-data control mechanism in the eventfunction design. A recent paper [35] proposed a sampled-data-based event-triggered scheme, which combines the benefitsof both sampled-data control and event-triggered control,and has been shown to be advantageous over the conven-tional periodic sampling scheme [3]. A key advantage ofsampled-data event-triggered control is that it is absolutelyZeno-free, due to the fact that a lower bound of the lengthof interevent time intervals is given explicitly by a samplingperiod [35]. We refer the readers to the survey papers [27], [28]for some recent developments on sampled-data-based event-triggered schemes. We also remark that in the sampled-data

event-triggered control [35], each individual agent needs aglobal coordinate system to interpret sampled data sent fromneighboring agents, and the control input is updated fromboth an agent’s own event instants and its neighbors’ sam-pling instants. This requirement may be burdensome in somecases as it increases control updating frequencies, which is notpresent in our scheme.

The paper that is closest to this paper is [24], whichdeveloped an event-based consensus scheme based on combi-national measurements. All agents need to measure the relativestates with their neighbors to implement the controller andto detect an event, and the measurements can be done vialocal coordinates (although Fan et al. [24] did not explic-itly show this). The consensus value is not necessarily theaverage one, as the average of initial states is not preserved.Furthermore, it has been shown in [24] that the triggering isless frequent than the algorithms in [19], since each agentis able to update its control input only at its own triggeringtime instants. The authors also claimed that Zeno triggeringfor all agents is excluded, while we will show in this paperthat this is not true. A follow-up work of [24] is [36], inwhich a truly Zeno-free self-triggered consensus algorithm isdeveloped, by employing a time regulation idea in the eventfunction design in which a complicated formula for minimuminterevent interval guarantee is involved.

C. Contributions

In the first part of this paper, we have revisited some popularand recent event-based consensus algorithms in the literature,and have also clarified some issues on their triggering perfor-mances and event function requirements. Their limitations andrequirements motivate us to propose a new event-based controlscheme for ensuring multiagent consensus, which is furtherextended to a self-triggered control strategy to avoid contin-uous measurements for each agent. The main contributionsof this paper include both a novel event-triggered and a self-triggered consensus algorithm which both guarantee Zeno-freetriggering for all the agents in a multiagent system that has amuch simpler triggering mechanism.

This paper builds on the results in [24] and [37], while somedrawbacks in the basic event-based consensus algorithm arepointed out. In particular, we have improved the triggeringbehaviors in two major aspects: 1) the exclusion of possi-ble Zeno triggering for all the agents at any time and 2) lessfrequent control updates via a flexible and also more gen-eral event-based control framework. These advantages in thenew event-triggered consensus algorithm will be achieved byproposing a modified event function and by including a pos-itive Lp signal in the event detector function (definitions willbecome clear in the context). In particular, the proposed Lp

signals in the current event-triggered control framework alsoinclude exponential functions for event function design dis-cussed in [21] and [38] as special cases. To summarize, in thefollowing we highlight some key advantages of the proposedevent-triggered consensus algorithm.

1) The proposed controllers (for consensus seeking andevent function) are totally distributed in the sensethat only local information from neighboring agents


is required. In particular, in measuring relative states(i.e., relative positions) each agent can use its own localcoordinate frame while the common knowledge of aglobal coordinate frame is not required.

2) Each agent can update its control input at its own trigger-ing time instants, which are independent to other agents’triggering instants. Furthermore, the interevent time canbe enlarged or adjusted by choosing different Lp func-tions in the event detector for each agent. This alsoensures a less frequent triggering as compared to someevent-triggered multiagent consensus schemes proposedin [19]–[21] and [24].

3) Different to the algorithms proposed in [19] and [21],the proposed event-triggered scheme does not requireany global information (such as graph Laplacian or alge-braic connectivity) or common parameters for each agentto implement the consensus controller and event updatelaw.

4) The proposed algorithm guarantees a truly Zeno-freetriggering for all the agents at any time, with simpler andmore flexible event functions and triggering algorithmas compared to some complex Zeno-free consensusmechanisms discussed in [20], [35], and [36].

5) A self-triggered consensus scheme is proposed basedon the modified event-triggered consensus scheme,which inherits the advantages and benefits of theevent-triggered consensus scheme, enables a discrete-time measurement of relative states for each individualagent, and also guarantees Zeno-free triggering for themultiagent group.

We remark that the consensus value under the proposedevent-triggered algorithm in this paper is not necessarily theaverage value of agents’ initial states, which would be a dis-advantage if average consensus or some value functions ofagents’ initial states should be preserved for the multiagentgroup. The discussions and comparisons for different event-based consensus algorithms are summarized in Table I. In thefirst line, “relative state” indicates continuous measurement (orcommunication) of relative states with respect to neighbors.

A preliminary version of this paper was presented in [39].The extensions in this paper are major, which include amore general framework on Zeno-free triggering control viaLp signals, complete proofs for all the key results whichwere omitted in [39], observations on measurement require-ments and global/local coordinate frame issues, a new sectionon Lp-signal-based Zeno-free self-triggered consensus con-trol, and new simulations to demonstrate the main technicalresults. We also prove some general results on Lp input–output stability of dynamical systems, which are of interestin their own rights in system stability analysis. Furthermore,the proposed event-based consensus schemes can be readilyextended and applied to other distributed control problems,e.g., the displacement-based formation control [40].

D. Paper Organization

The remaining parts of this paper are organized as follows.In Section II, we introduce essential preliminary conceptsand notations on graph theory and Lp function spaces, and

present the problem formulation. Section III analyzes a basicconsensus algorithm and some issues on Zeno-triggeringbehavior, followed by a new Zeno-free triggering consensusalgorithm and a detailed analysis on its triggering functionperformance. Simulations and comparisons will also be pro-vided in Section III to demonstrate the property and effective-ness of the proposed event-based consensus law. In Section IV,a self-triggered consensus algorithm is designed which alsoguarantees a Zeno-free triggering. Finally, Section V con-cludes this paper. Some proofs on key lemmas and theoremsare given in the Appendix.

II. PRELIMINARIES AND PROBLEM FORMULATION

A. Preliminaries on Graph Theory and Notations

Consider an undirected graph with m edges and n nodes,which is denoted by G = (V, E) with node set V ={1, 2, . . . , n} and edge set E ⊂ V × V . The neighboring setNi of node i is denoted as Ni := {j ∈ V : (i, j) ∈ E}. Wenow introduce several important matrices in graph theory. Thematrix which relates the nodes to the edges is called the inci-dence matrix, denoted by HG = {hki} ∈ R

m×n. By choosingarbitrary orientations for all edges for an undirected graph,the entries of its incidence matrix are defined as hki = 1 if thekth edge sinks at node i, or hki = −1 if the kth edge leavesnode i, or hki = 0 otherwise. Furthermore, for an undirectedgraph G, its Laplacian matrix, denoted by LG , can be definedas LG = HT

GHG . The adjacency matrix AG ∈ Rn×n is a sym-

metric matrix encoding the node adjacency relationships, withentries defined as Aij = 1 if {i, j} ∈ E and Aij = 0 otherwise.

For an undirected and connected graph, there holdsrank(HG) = n − 1 and null(HG) = span{1n}. For notationalconvenience in the analysis, we will also define H = HG ⊗ Id

and L = LG ⊗ Id, where ⊗ denotes the Kronecker product.Given a node element i ∈ V in the graph G, one can asso-

ciate to it a point pi of Euclidean space Rd. The column vector

p = [pT1 , pT

2 , . . . , pTn ]T ∈ R

nd collects the composite state vec-tor for all the n agents. For an edge k ∈ E with head j and taili, define the associated relative position vector as zk = pj−pi.Similarly, one can define z = [zT

1 , zT2 , . . . , zT

m]T ∈ Rdm as the

associated composite relative state vector for all the edges.Note that there holds

z = Hp. (1)

B. Preliminaries on Lp Function Spaces

We now introduce some notations in function norms andfunction spaces from [41]. For any fixed p ∈ [1,+∞), a func-tion f (t) : Rt≥0 → R belongs to Lp[0,+∞] iff f is locallyintegrable and

∫∞0 |f (t)|pdt < ∞. The function p-norm is

defined as

‖f‖p =(∫ ∞

0|f (t)|pdt

) 1p

. (2)

A function f (t) : Rt≥0 → R belongs to L∞[0,+∞] iffess supt≥0|f (t)| < ∞, where “ess sup” denotes essentialsupremum. The function ∞-norm is defined as

‖f‖∞ = ess supt≥0|f (t)|. (3)


TABLE ICOMPARISONS OF SOME TYPICAL EVENT-BASED CONSENSUS ALGORITHMS IN THE LITERATURE

It is well known that Lp is a linear space, in the sense thatif f1 ∈ Lp and f2 ∈ Lp, then f1 + f2 ∈ Lp. Similarly, onecan also define the function space Lp with p ∈ (0, 1), in thatLp := {f (t)| ∫∞0 |f (t)|pdt <∞, 0 < p < 1}. Note that Lp withp ∈ (0, 1) is also a linear space, i.e., if f1 ∈ Lp and f2 ∈ Lp

with 0 < p < 1 (or more generally, 0 < p ≤ ∞ ), thenf1 + f2 ∈ Lp (see [42]).

In this paper, we will frequently use the following result.Lemma 1 (Interpolation of Lp Space, [42]): Let that 0 <

p0 < p1 ≤ ∞ and f ∈ Lp0 ∩ Lp1 . Then f ∈ Lp for allp0 ≤ p ≤ p1.

In particular, if f ∈ L1 ∩L∞, then f ∈ Lp for ∀p ∈ [1,∞].

C. Problem Formulation

We consider in this paper the dynamics of each agentdescribed by a single integrator equation, modeled by pi(t) =ui(t), where ui(t) ∈ R

d is the control input for agent i.The aim is to design event-triggered updating functions andcontrollers to reach a state consensus. The well-known andstandard consensus protocol with continuous updating controlinput is described as

pi(t) = ui(t) =∑

j∈Ni

(pj(t)− pi(t)) = −{Lp(t)}i (4)

where {Lp}i is the ith column vector block [i.e., the vectorblock as taken from the (di− d+ 1)th to the (di)th entries] ofthe column vector Lp.

We first introduce several notations in event-based control.For each agent i, denote its event times as ti0, ti1, . . . , tih, . . .

1 Inthe framework of event-based control, the dynamic equationfor agent i should be written by

pi(t) = ui(t) = ui(tih

),∀t ∈ [

tih, tih+1

). (5)

In this paper, we aim to develop a novel triggeringscheme with Zeno-free property for the above multiagentsystems. A formal definition of Zeno triggering (which is alsotermed Zeno execution in the hybrid system study) is givenbelow [43]–[45].

Definition 1: For agent i, a triggering is Zeno if

limh→∞ tih =

∞∑

h=0

(tih+1 − tih) = ti∞ (6)

1The initial triggering time ti0 for agent i is not necessarily zero. Since eachagent is triggered at its own event time instants, a clock synchronization forindividual agents is not required.

for some finite ti∞ (termed the Zeno time).2

The proposed event-triggered consensus algorithm shouldsatisfy the following requirements.

1) For each individual agent, its event detector and con-sensus controller can be implemented using only localrelative state measurements to its neighboring agents.

2) For agent i, its control input should be updated at, andonly at, its own event time instants ti0, ti1, . . . , tih, . . .based on local information with its neighboring agents.

3) No agents exhibit a Zeno triggering at any time (i.e.,each agent i ensures a Zeno-free triggering).

4) The measurements of relative states for each agent toits neighbors should be done via that agent’s local coor-dinate frame, which should be independent of a globalcoordinate frame.

III. EVENT-BASED ZENO-FREE CONSENSUS SCHEME

A. Zeno Triggering Issue in Basic Event-Based ConsensusController

Consider the following event-based updating law:

pi(t) = ui(t) =∑

j∈Ni

(pj

(tih

)− pi(tih

))

= −{Lp

(tih

)}i,∀t ∈

[tih, tih+1

). (7)

We remark that similar forms of the above event-based con-troller have also been discussed in [24] and [37] (which havepossibly different event functions for event triggering). Notethat the right-hand side of (7) involves piecewise continuousterms, and we understand the solution of (7) in the sense ofFilippov [46].

By defining δi(t) = {Lp(tih)}i − {Lp(t)}i, the above (7)can be written as pi(t) = −{Lp(t)}i − δi(t). Let δ =[δT

1 , δT2 , . . . , δT

n ]T ∈ Rdn. Therefore, one can obtain a compact

form for the multiagent system in the following:

p(t) = −Lp(t)− δ(t). (8)

For the stability analysis, let us choose the followingLyapunov function candidate:

V = 1

2pTLp (9)

2Definition 1 implies that, for a Zeno-free triggering scheme, there shouldhold tih+1 − tih > 0 for all h. However, it may occur that tih+1 − tih → 0 ash→∞ (one typical example being that tih+1− tih = (1/h)). There have beensome disputes in the literature on whether the latter case should be consideredas a Zeno triggering at infinity (see the reviews in [25] and [26]). In this paper,by following standard concepts in hybrid systems [43]–[45], Zeno triggeringis only defined if there exists a finite ti∞.


whose derivative along (8) can be derived by

V(t) = p(t)TLp(t) = −p(t)TLLp(t)− p(t)TLδ(t)

≤ −p(t)TLLp(t)+ ‖p(t)TLδ(t)‖≤ −

n∑

i=1

‖{Lp(t)}i‖2 +n∑

i=1

‖{Lp(t)}i‖‖δi(t)‖. (10)

Notice that the inequality ‖{Lp(t)}i‖‖δi(t)‖ ≤(1/2ai)‖δ(t)i‖2 + (ai/2)‖{Lp(t)}i‖2 holds with ai ∈ (0, 2).Therefore, one can obtain the following formula according tothe above inequality (10) on V:

V(t) ≤ −n∑

i=1

‖{Lp(t)}i‖2

+n∑

i=1

ai

2‖{Lp(t)}i‖2 +

n∑

i=1

1

2ai‖δi(t)‖2

= −n∑

i=1

2− ai

2‖{Lp(t)}i‖2 +

n∑

i=1

1

2ai‖δi(t)‖2.

By enforcing the norm of δi(t) to satisfy

1

2ai‖δi(t)‖2 ≤ γi

2− ai

2‖{Lp(t)}i‖2 (11)

for a constant γi ∈ (0, 1), one can guarantee

V(t) ≤n∑

i=1

(γi − 1)2− ai

2‖{Lp(t)}i‖2 ≤ 0. (12)

Therefore, a local event triggering function for agent i canbe designed in the following form:

fi(t) := ‖δi(t)‖2 − γiai(2− ai)‖{Lp(t)}i‖2 (13)

and the event time tih for agent i is defined to satisfy fi(tih) = 0for h = 0, 1, 2, . . . Note that during the time interval t ∈[tih, tih+1), the control input remains as ui(t) = ui(tih) until anext event is triggered. When an event is triggered for agent i,its local event vector δi (or equivalently ‖δi(t)‖) should resetto zero.

Lemma 2: Consider the triggering function (13) associatedwith the event-triggered consensus controller (7). Then a stateconsensus can be reached exponentially fast. Also, for themultiagent group, there exists at least one agent that doesnot exhibit Zeno triggering behavior at any time instant t > 0.

For brevity we omit the proof here, and we refer the readersto [39] for a complete proof.

Remark 1: The design of the above event controller (7)follows similarly to that discussed in [37] (with a differentevent-triggered scheme) and [24] (with a scaled triggeringfunction). However, we remark that no rigorous analysis isactually available in [24] and [37] to ensure Zeno-free behav-ior for all the agents. Lemma 2 only ensures at least oneagent does not show Zeno triggering at any time, but doesnot guarantee Zeno-free triggering for all the agents at anytime. In practice, there may exist a finite time t∗ < ∞ suchthat {Lp(t∗)}i = 0 for some i, which leads to Zeno triggeringfor that agent i at the time instant t∗. In [47], a rigorous proofis provided to show that it is indeed a Zeno triggering.

B. Event-Triggered Consensus Scheme GuaranteeingZeno-Free Triggering

Motivated by the discussions in the above section, wepropose a modified event function in this section accordingto (13)

fi(t) := ‖δi(t)‖2 − γiai(2− ai)‖{Lp(t)}i‖2 − 2ai�i(t) (14)

where �i(t) is an additional function chosen by agent i in itsown event function. We consider the following cases for thefunction �i(t).

1) Case I: �i(t) ∈ Lp, with p ∈ [1,+∞) and �i(t) > 0for all t > 0.3

2) Case II: �i(t) ∈ Lp with p ∈ (0, 1), �i(t) > 0 for allt > 0 and is bounded.

3) Case III: �i(t) ∈ L∞, �i(t) > 0 for all t > 0, andlimt→∞�i(t)→ 0.

We remark that for different individual agents, they can choosedifferent �i(t) (satisfying one of the above conditions) intheir triggering functions, which improve the flexibility in thedesign and implementation.

The intuition behind the event comparison function (14) isthe following. There exists possible zero-crossing occurrencesof the term {Lp(t)}i, which is the main reason of Zeno-triggering behaviors. In the event comparison function (14),even if {Lp(t)}i shows a zero-crossing at some finite timeinstant, the inclusion of the additional Lp term �i(t) thereforecan guarantee a positive threshold value for comparison in theevent function value, which then avoids comparing ‖δi(t)‖2 toa zero threshold.

The following theorem shows that the convergence to aconsensus point is still guaranteed with the modified eventfunction (14).

Theorem 1: Consider the modified event function (14) withthe introduction of the Lp signals, and the multiagent systemwith the consensus controller (7). All agents’ states reach aconsensus point with t→∞.

Proof: In the proof, we consider the same Lyapunov func-tion as in (9). Note that V = (1/2)pT Lp = (1/2)pT HTHp =(1/2)zTz and V = 0 if and only if z = 0. With the modifiedevent function (14) one can derive

V(t) ≤n∑

i=1

(

(γi − 1)2− ai

2‖{Lp(t)}i‖2 +�i(t)

)

. (15)

Note the following relation holds:

n∑

i=1

‖{Lp(t)}i‖2 = pTLLp

= pTHTHHTHp = zTHHTz. (16)

The definition z = Hp indicates that z is in the range spaceof H. Then according to Lemma 9 and Corollary 1 (in theAppendix), one can show

zTHHTz ≥ λ+min(HHT)‖z‖2 (17)

3Note that limt→∞�i(t)→ 0 is not implied by these conditions.


where λ+min(HHT) denotes the smallest positive eigenvalue ofthe matrix HHT , whose value equals the graph algebraic con-nectivity λ2(LG). By defining ζmin = mini(1−γi)([2− ai]/2),one can obtain an inequality from (15) and (17) as follows:

V(t) ≤ −2ζminλ2(LG)V(t)+n∑

i=1

�i(t). (18)

For notational simplicity we define �(t) := ∑ni=1 �i(t).

From the properties of each �i(t) as listed after (14), and thefact that Lp is a linear space, we can obtain the correspondingthree cases for the composite function �(t).

1) Case I: �(t) ∈ Lp, with p ∈ [1,+∞) and �(t) > 0 forall t > 0.

2) Case II: �(t) ∈ Lp with p ∈ (0, 1) and �(t) > 0 for allt > 0. Since each �i(t) is bounded and there are a finitenumber of agents in the group, we also have �(t) ∈ L∞.According to the interpolation rule in Lemma 1, thisimplies �(t) ∈ Lp also holds for any p ∈ [1,∞], whichdegenerates to Case I.

3) Case III: �(t) ∈ L∞ (because of a finite number ofagents), �(t) > 0 for all t > 0, and limt→∞�(t)→ 0.

Now we consider a new dynamical system

W(t) = −2ζminλ2(LG)W(t)+�(t). (19)

By invoking the comparison method [48, Ch. 9.3], andassuming that V(0) = W(0), there holds V(t) ≤ W(t).Standard calculation shows the solution to (19) in the fol-lowing form:

W(t) = e−2ζminλ2(LG)tW(0)+∫ t

0e−2ζminλ2(LG)(t−τ)�(τ)dτ.

By the convergence result of Theorem 4 (shown in theAppendix), there holds W(∞) = 0, from which one concludesV(∞) = 0. In conclusion, there holds z(∞) = 0 and thusx1(∞) = x2(∞) = · · · = xn(∞), i.e., the multiagent stateconsensus is guaranteed when t→∞.

For a special case, if all the agents choose the functions�i(t) as some exponentially decaying functions, one can provethe exponential convergence of the consensus dynamics asshown in the following lemma.

Lemma 3: If each agent chooses the signal �i(t) as anexponentially decaying function, i.e., �i(t) = bie−ait, then theconvergence of the consensus is achieved exponentially fast.

Proof: Since the exponentially decaying function satisfiesthe conditions listed above, the asymptotic convergence tothe consensus value follows directly from Theorem 1. Wenow show the exponential convergence. Note that by fol-lowing the same arguments as in Theorem 1 we can againobtain V(t) ≤ −2ζminλ2(LG)V(t) + ∑n

i=1 �i(t). The factthat all �i(t) = e−ait implies that �(t) := ∑n

i=1 �i(t)can be upper bounded by an exponentially decaying func-tion, for which we write �(t) ≤ αe−βt := �(t) whereα and β are some positive constants chosen to bound thesum of all exponentially decaying function �i(t). SinceV(t) ≤ −2ζminλ2(LG)V(t)+�(t), by defining a new dynamicalsystem W(t) = −2ζminλ2(LG)W(t) + �(t) and assuming thatV(0) = W(0), there holds V(t) ≤ W(t). By solving the

differential equation W(t) one can show the following inequal-ity in (20) as shown at the top of the next page. Note thatthe case β = 2ζminλ2(LG) occurs if ai = β = 2ζminλ2(LG)

for all agents i, and therefore �(t) = ∑ni=1 bie−βt with

α = ∑ni=1 bi. From the construction of the Lyapunov func-

tion V = (1/2)‖z‖2, the exponential convergence of V → 0implies the exponential convergence of ‖z‖ → 0, which inturn implies that the multiagent state consensus is achievedexponentially fast.

C. Analysis on the Multiagent Triggering Performance

In this section, we discuss triggering properties of theproposed distributed triggering control strategy. We first showthe triggering feasibility by excluding singular triggering andZeno triggering. Generally speaking, singular triggering meansafter a feasible triggering event instant, there exists no moretriggering for any agent in the sequel. Zeno triggering meansthat the interevent interval tih+1 − tih becomes arbitrarily closeto zero at some finite time, or the limit of the sequencetih approaches a finite number (i.e., limh→∞ tih < ∞; seeDefinition 1). For the definition of singular triggering, we referthe readers to [24] and [49].

Theorem 2: (Feasibility of the Event-Based ConsensusAlgorithm): Consider the multiagent system with the proposeddistributed event-based controller (7) and the modified eventfunction (14).

1) (No singular triggering) Singular triggering is excludedfor all the agents for all t > 0.

2) (No Zeno triggering) Zeno behavior is excluded for allthe agents for all t > 0.

Proof: The proof for the first statement is omitted, as itfollows similar arguments as those in [24, Lemma 2]. We nowprove the second statement, that is, the Zeno-free triggeringproperty for all the agents. In the following analysis, we let t ∈[tih, tih+1). For notational convenience, define �i = γiai(2−ai).We first show a sufficient condition to guarantee fi(t) ≤ 0 withfi(t) defined in (14). Note that fi(t) ≤ 0 can be equivalentlystated as

(�i + 1)‖δi(t)‖2 ≤ �i

(‖δi(t)‖2 + ‖{Lp(t)}i‖2

)+ 2ai�i(t).

(21)

Note that

‖{Lp(tih

)}i‖2 = ‖δi(t)+ {Lp(t)}i‖2≤ 2

(‖δi(t)‖2 + ‖{Lp(t)}i‖2

). (22)

Thus, a sufficient condition to guarantee (21) (which furtherguarantees the inequality condition fi(t) ≤ 0) is that

‖δi(t)‖2 ≤ �i

(2�i + 2)

(‖{Lp

(tih

)}i‖2

)+ 2ai

(�i + 1)�i(t). (23)

We then calculate the change rate of ‖δi(t)‖d

dt‖δi(t)‖ ≤ ‖δi(t)T‖

‖δi(t)‖ ‖δi(t)‖

=∥∥∥∥

d

dt{Lp(t)}i

∥∥∥∥ =

∥∥∥∥∥∥

∑

j∈Ni

(pj(t)− pi(t)

)∥∥∥∥∥∥


V(t) ≤{

V(0)e−2ζminλ2(LG)t + αte−2ζminλ2(LG)t, if β = 2ζminλ2(LG)

V(0)e−2ζminλ2(LG)t + α2ζminλ2(LG)−β

(e−βt − e−2ζminλ2(LG)t

), if β �= 2ζminλ2(LG)

(20)

=∥∥∥∥∥∥

∑

j∈Ni

({Lp

(tihi(t)

)}

i−

{Lp

(tjhj(t)

)}

j

)∥∥∥∥∥∥

:= ‖ri(t)‖ (24)

where ri(t) := ∑j∈Ni

({Lp(tihi(t))}i − {Lp(tjhj(t)

)}j), tjhj(t)=

arg maxh{tjh|tjh ≤ t, j ∈ Ni}. Note that ri(t) cannot be zerofor all time intervals (i.e., ri(t) �= 0 for any t ∈ [tih, tih+1) andany h ∈ N+) since singular triggering is excluded in the firststatement. Thus, ‖ri(t)‖ is positive, bounded, and piecewiseconstant which implies that (d/dt)‖δi(t)‖ is upper boundedby a positive and piecewise constant. We refer the readers toSection IV (see Fig. 5) for detailed discussions on the piece-wise property of ri(t). Thus, following the similar proceduresin Section IV, we can obtain

∫ t

tih

‖ri(s)‖ds = ‖ri(τγ

)‖(t − τγ

)

or∫ t

tih

‖ri(s)‖ds =γ∑

k=1

‖ri(τk−1)‖(τk−τk−1)+‖ri(τγ

)‖(t − τγ

)

depending on the number of triggerings of agent i’s neighbors(see Fig. 5), where the notation τk is the same as the one inSection IV. From the above equations, one can further obtain

∫ t

tih

‖ri‖ds ≤ maxk=0,...,γ

‖ri(τk)‖(

γ∑

k=1

(τk − τk−1)+(t − τγ

))

= maxk=0,...,γ

‖ri(τk)‖(t − tih

).

From the sufficient condition given in (23) which guaranteesthe event triggering condition fi(t) ≤ 0, the next intereventinterval for agent i is lower bounded by

τ ih =

√�i

(2�i+2)

(‖{Lp(tih

)}i‖2

)+ 2ai(�i+1)

�i(tih + τ i

h

)

maxk=0,...,γ ‖ri(τk)‖ . (25)

Note that τ ih always exists and is positive, because

maxk=0,...,γ ‖ri(τk)‖ is positive and bounded and �i > 0. Thisimplies that there is no accumulation triggering point at anyfinite time. Also note that if �i → 0 with t → ∞, then thenumerator term in (25) approaches zero only when t → ∞,which implies that tih →∞ with h→∞ and t→∞. On theother hand, if �i does not converge to zero, then again onehas tih → ∞ with h → ∞ and t → ∞. In conclusion, Zenotriggering is excluded for all the agents at any time.

Remark 2: We mention that the above proof does not showwhether there exists a uniform lower bound of the intereventintervals of each agent, since such a bound would necessarilydepend on the property (e.g., the decaying rate) of the function�i(t) chosen by each agent. For some particular functions�i(t) (e.g., exponentially decaying functions), such a bound

may exist and it is possible to calculate an explicit lower boundon the interevent time (see [7] and [21]).

The following lemma shows that the multiagent system withthe modified event function (14) also enables an enlargedinterevent interval, as compared to the original one (13).

Lemma 4: Consider two identical multiagent groups withthe same event-based controller (7) and with the same dynam-ics. Suppose that they are triggered by the two different eventfunctions (14) and (13), respectively. Further suppose thatthey have the same {Lp(tik)}i at tik. Then the next event timetriggered by (14) is larger than that triggered by (13).

Proof: This is obvious from the proof of Theorem 2 and inparticular from (25). Note that the event function (13) can beconsidered as a special case of (14) with �i(t) = 0.

Lemma 4 also indicates that, by including the auxiliaryfunction �i(t) in the event function (14), the multiagent systemwill experience a longer interevent time interval (which istraded off by a slower convergence to consensus) as comparedto that when using (13).

We finally show a lemma regarding the coordinate framerequirement in measuring local relative positions. The coor-dinate frame is a particularly important issue in practice.Consider the case that consensus controllers are applied toachieve position rendezvous for a group of mobile robots. Therelative position measurements performed by a local coordi-nate frame (instead of a global coordinate frame) thereforeplay a key role to in a distributed rendezvous control.

Lemma 5: The implementation of the distributed consensuscontroller (7) does not involve a global coordinate frame, inthe sense that each individual agent could use its own localcoordinate frame to measure relative states to its neighbors.Furthermore, a local coordinate frame is sufficient for eachagent to implement the triggering condition (14).

Proof: We will divide the proof into two parts. In thefirst part, we show that the control input function for allagents is an SE(d)-invariant function,4 such that the controldirection remains invariant via measurements obtained in alocal coordinate frame. The second part is to show that theevent function remains unchanged with measurements fromany local coordinate frame.

From the consensus control dynamics (7) the control func-tion for agent i is ui =∑

j∈Ni(pj(tih)− pi(tih)),∀t ∈ [tih, tih+1),

and each agent measures the relative position (pj(tih)−pi(tih)) ofits neighboring agent j via its local coordinate frame. Given anarbitrary coordination rotation Ri ∈ SO(d) and a displacementof origin ϑi ∈ R

d, there holds

ui(Rip1 + ϑi, . . . , Ripn + ϑi)

=∑

j∈Ni

(Ripj

(tih

)+ ϑi −(Ripi

(tih

)+ ϑi))

4A function f is said to be SE(d)-invariant if for all R ∈ SO(d) and allx1, . . . , xn, ω ∈ R

d , there holds Rf (x1, . . . , xn) = f (Rx1 + ω, . . . , Rxn + ω).


= Ri

∑

j∈Ni

(pj

(tih

)− pi(tih

))

= Rifi(p1, . . . , pn). (26)

On the other hand, suppose that agent i’s position in a globalcoordinate system is measured as pg

i , while pij stands for agent

j’s position measured by agent i’s local coordinate system.Clearly, there exist a rotation matrix Ri ∈ SO(d) and a dis-placement vector ϑi ∈ R

d, such that pij = Rip

gj +ϑi. We rewrite

the controller designed in (7) in the global coordinate system,which holds

pgi = ug

i = R−1i ui

i(pij) = R−1

i Riuii(p

gj ) =

∑

j∈Ni

(pgj (t

ih)− pg

i (tih))

which takes the same form as the original controller in (7).Since Ri and ϑi are chosen arbitrarily, the above equationindicates that the controller (7) is independent of the globalcoordinate basis.

We now prove that the event function in (14) is independentof any global coordinate frame as well. By examining thefunction fi(t) in (14), one sees that the first term ‖δi(t)‖2 isthe norm of the relative measurements performed by agenti, which remains invariant under any rotation transformation.This is true also for the second term γiai(2 − ai)‖{Lp(t)}i‖2.The third scalar term 2ai�i(t) is chosen solely by agent i,which is independent of any coordinate frames. Therefore, thevalues of the event function fi(t) remain invariant under anycoordinate frame change. The statements are thus proved.

D. Numerical Simulation Examples and PerformanceComparisons

This section shows certain simulation examples to demon-strate the convergence behavior of a multiagent group underthe modified event-based controller. The simulation involvesa six-agent group, described by (7), in which the undirectedgraph is encoded by an adjacency matrix whose entries areset as A12 = A14 = A23 = A26 = A35 = A36 = 1 andother entries being zero. For ease of demonstration, the sim-ulation focuses on that each agent’s state is 1-D, but theresults and simulations can be well extended to any higher-dimensional space. The initial conditions for the six agentsare set as p1(0) = 3, p2(0) = −0.5, p3(0) = 2, p4(0) = −3,p5(0) = −1, p6(0) = 1. The event parameters are chosenas γ1 = 0.5, γ2 = 0.3, γ3 = 0.7, γ4 = 0.2, γ6 = 0.6,γ2 = 0.3; a1 = 0.3, a2 = 0.5, a3 = 0.6, a4 = 0.5, a5 = 0.5,a6 = 0.6. The functions �i(t) associated with each agent’sevent detector function are chosen as �1 = (3/[1+ t2]),�2 = (2/[1+ t2]), �3 = (1/[1+ t2]), �4 = (1/[2+ t2]),�5 = (3/[1+ t2]), and �6 = (1/[0.5+ t2]).5 Note that all thechosen functions �i(t) satisfy the conditions as proposed inTheorem 1.

Fig. 1 depicts the trajectories of {Lp(t)}i for the six agentsin the multiagent group. It can be observed from Fig. 1

5The magnitudes and parameters in these functions are deliberately chosenin order to facilitate a clear demonstration in simulation and comparison.Other Lp functions, such as the exponential decaying function or convergentand bounded functions like �(t) = (1/[t + 1]), are also good candidates.Simulation results with exponential decaying functions are provided in [39].

Fig. 1. Time evolutions of {Lp(t)}i with the event function (14).

Fig. 2. Event instants with the event function (14).

that, for each agent, {Lp(t)}i actually crosses zeros multipletimes, but all agents do not have Zeno triggering, thanks tothe term �i(t). Event time instants are shown in Fig. 2, inwhich no Zeno triggering behaviors or even dense eventsfor any agent are observed. As a comparison, the eventtime instants for all the agents under the original event-based function (13) are depicted in Fig. 3, where severaldense triggering instants occur in the time interval (0, 50)for agents 1, 2, 3, 4, and 6. Fig. 4 depicts the trajecto-ries of the six agents whose states reach the consensus asdesired.

For a better comparison, we also calculate the mini-mum/maximum interevent time and the event counts for allagents in the simulation time, which are shown in Table II.The table also illustrates the performance between eventfunctions (13) and the modified function (14). For a cleardemonstration and comparison in Table II, the number of trig-gered events is counted by choosing a threshold minimumtime (which is 5×10−5 in this case) while the default numer-ical accuracy setting in MATLAB is involved. However, withthe unmodified event function (13), around the zero-crossingtime, agents 1, 2, 3, 4, and 6 indeed exhibit accumulationtriggering points, at which the number of triggered events actu-ally approaches infinity (i.e., a Zeno-triggering point) from atheoretical viewpoint.

IV. EXTENSION TO SELF-TRIGGERED ZENO-FREE

CONSENSUS SCHEME

The proposed event-based consensus scheme requires eachagent to perform continuous-time measurements of relative


Fig. 3. Event instants with the event function (13).

Fig. 4. States trajectories for the six agents with the event function (14).

TABLE IICOMPARISONS OF EVENT TRIGGERING BEHAVIOR AND PERFORMANCE

BETWEEN THE EVENT FUNCTION (13) AND (14)

states to its neighboring agents in order to continuously detectthe event condition as shown in (14). This section proposes anextension of a self-triggered multiagent consensus approach toremove the assumption of continuous-time measurement. Wesuppose that at each triggering time instant tihi

, agent i mea-sures the relative state {Lp(tihi

)}i and sends it to its neighbors.Furthermore, the next event time tihi+1 for agent i is deter-

mined by {Lp(tihi)}i and the received message {Lp(tjhj

)}j fromits neighbors j ∈ Ni.

Recall that ri(t) := ∑j∈Ni

({Lp(tihi(t))}i − {Lp(tjhj(t)

)}j) from

the proof of Theorem 2, where hi(t) = arg maxhi∈N{tihi|tihi≤ t}.

This implies that tihi(t)for each agent is the latest event time

instant before t. From (7) one has

ri(t) = d

dt{Lp(t)}i. (27)

Fig. 5. Illustration of message updates in the self-triggered consensus scheme.

Note that the measurement error δi(t) = {Lp(tihi)}i−{Lp(t)}i

is continuous in [tihi, tihi+1) and δi(tihi

) = 0. Then one has

δi(t) = −∫ t

tihi

d

dt{Lp(t)}i = −

∫ t

tihi

ri(s)ds, t > tihi(28)

and

{Lp(t)}i ={

Lp(tihi)}

i+

∫ t

tihi

ri(s)ds, t > tihi. (29)

In order to determine the next event time instant tihi+1, oneneeds to obtain δi(t) and {Lp(t)}i from (28) and (29). Notethat ri(t) is piecewise constant and it updates only when anew message {Lp(tjhj(t)

)}j is received from any of its neighbors

according to (27). To calculate∫ t

tihiri(s)ds, by denoting t =

tihi+ ξi one has

�i

(tihi+ ξi

):=

∫ tihi+ξi

tihi

ri(s)ds

=∫ τ

tihi

ri(s)ds+ ri(τ )(

tihi+ ξi − τ

)(30)

where τ = maxj∈Ni{tjhj(t)≤ t} denotes the last update time

instant before t, which means that ri(t) keeps constant in [τ, t).We introduce a variable τγ with γ ∈ N to denote new

triggering update instants of neighboring agent j ∈ Ni in thetime interval [tihi

, tihi+1) for agent i, and let τ0 = tihi. For t > tihi

,there are two cases for ri(t).

Case 1: If during the time interval no agent in the neigh-boring set j ∈ Ni updates its control input, thenri(t) remains unchanged and one has

�i

(tihi+ ξi

)= ri

(τγ

)(tihi+ ξi − τγ

).

Case 2: If any message {Lp(tjhj(t))}j, j ∈ Ni is received,

then γ and τγ are updated as γ ← γ + 1 andτγ ← minj∈Ni{tjhj(t)

> τγ }. Then one can obtain

�i

(tihi+ ξi

)=

γ∑

k=1

ri(τk−1)(τk−τk−1)

+ ri(τγ

)(tihi+ξi−τγ

).

Then we can give a self-triggered scheme based on theevent-triggered counterpart (14) as follows:

tihi+1 = tihi+ inf{ξi > 0| fi(ξi) = 0} (31)

where fi(ξi) = ‖�i(tihi+ξi)‖2−γiai(2−ai)‖{Lp(tihi

)}i+�i(tihi+

ξi)‖2 − 2aiωi(tihi+ ξi). For each agent i, determining tihi+1


requires checking (31) and then obtaining ξi. Note that fi(ξi)

is associated with the latest measurements {Lp(tihi(t))}i and

{Lp(tjhj(t))}j, j ∈ Ni received before t (t = tihi

+ ξi) accordingto (27) and (30).

Remark 3: There are two cases for the integral computationof ri(t) in (30) as mentioned above. The illustration of theoccurrence of these two cases is shown in Fig. 5. Furthermore,one can refer to [36, Algorithm 1] for more details on how theabove self-triggered algorithm determines the next triggeringtime instant tihi+1. We will not repeat it in this paper.

Now, the corresponding consensus convergence for the self-triggered algorithm is concluded in the following theorem.

Theorem 3: Consider the multiagent system (7) and theself-triggered control scheme (31) with the introduction ofthe Lp signals. Then all agents’ states will achieve consensusasymptotically as t→∞. Furthermore, no agent will exhibitany singular triggering or Zeno triggering for all t > 0.

Remark 4: Note that the self-triggered algorithm (31) isobtained via replacing δi(t) and {Lp(t)}i in (14) by the equiv-alent calculations in (28) and (29). Thus, the event timeinstants of all the agents determined by the self-triggeredalgorithm (31) are exactly the same as those obtained by theevent-triggered scheme (14), so are the control inputs and statetrajectories.

Remark 5: In this paper, we focus on Zeno-free event-basedconsensus control for single-integrator multiagent systemsinteracted in an undirected graph. The proposed event-basedcontrol with Lp signals can be extended to consensus controlfor multiagent systems with more complex dynamics (lin-ear or nonlinear systems), with transmission delays, and/orwith directed graph interactions. For example, as shownin [50] and [51], since single-integrator consensus networkdynamics can tolerate some transmission delays, we antici-pate that the stability and convergence result of the proposedmultiagent network still hold if some small transmission delaysexist between neighboring agents. Furthermore, the proposedcontroller is totally distributed and does not involve any globalinformation such as graph size or algebraic connectivity; there-fore, an adaptive coupling control strategy is not required.For event-based consensus networks with linear or nonlineardynamics that involve global information in the control law,an adaptive control strategy (see [11], [52]) can be employedto alleviate the requirement of global information in controllaws. These ideas will be considered in the future research.

V. CONCLUSION

This paper has developed a new triggering control schemefor multiagent consensus seeking, which features Zeno-freetriggering and local updating for all the agents. We first dis-cuss in detail a basic event-based algorithm (similar to theconsensus schemes in [24] and [37]), and reveal its defects anddisadvantages associated with Zeno-triggering issues. Then, amodified and improved distributed triggering controller is thenproposed, which incorporates a Lp signal in agents’ eventfunctions and ensures the exclusion of any possible Zenotriggering for all the agents. The proposed event-based consen-sus control also possesses several benefits and advantageous

properties, such as no involvement of any global coordinateframe, independent, and local event updates for individualagents, and enlarged (and adjustable) triggering time intervals(and thus less frequent triggering) by trading off convergencespeed.

To remove the requirement of the continuous-time mea-surement in the proposed event-triggered controller, a self-triggered distributed consensus scheme has been furtherdesigned, which possesses all of the advantages from theproposed event-triggered consensus scheme, allows discrete-time updates on local measurements, and also guaranteesa truly Zeno-free update for all the agents in achievingconsensus.

APPENDIX

A. Convergence Results

Before giving the main proof, we first present some use-ful results regarding Lp space analysis. The following is ageneralization of the Barbalat’s lemma.

Lemma 6 (Generalized Barbalat’s Lemma): If f (t) ∈ L∞and f (t) ∈ Lp with 0 < p < +∞, then limt→∞ f (t) = 0.

The proof can be found in [53, p. 80]. The followinglemma characterizes some inequalities involving the convo-lution operations. Define the convolution of two functions(f , g) as

H(f , g) :=∫ +∞

0f (t − τ) g(τ ) dτ.

Lemma 7 (Young’s Convolution Inequality): Suppose that fis in Lp[0,+∞] and g is in Lq[0,+∞] and

1

p+ 1

q= 1

r+ 1 (32)

with 1 ≤ p, q, r ≤ +∞. Then there holds

‖H(f , g)‖r ≤ ‖f‖p‖g‖q. (33)

For the proof, see [42, pp. 525–526]. Lemma 7 indicates thatthe convolution is a continuous bilinear mapping from Lp×Lq

to Lr. We note several conditions of the above lemma. First, toensure r > 1, we should have p, q ≥ 1 and p, q ≤ r. Second,the condition (1/p) + (1/q) ≥ 1 is necessary, otherwise itwould result in r ≤ 0 which violates the condition. Third,the results can be extended to the case that either p, q, or requals ∞.

Lemma 8: Suppose f is an exponentially decaying func-tion, i.e., f := exp(−αt) where α > 0, and g ∈ Lq[0,+∞].Consider the convolution operation

H(f , g) :=∫ +∞

0e−α(t−τ) g(τ ) dτ. (34)

Then there holds

H ∈ Lr[0,+∞] (35)

with r ∈ [q,+∞].Proof: First note that for any p ∈ (0,∞]

∫ +∞

0|f (t)|pdt =

∫ +∞

0e(−αp)tdt = 1

αp<∞. (36)


Therefore f ∈ Lp, for every p ∈ (0,∞]. We now confineour attention to p ∈ [1,∞]. We first consider the case thatq ∈ [1,∞). From Lemma 8, letting p = 1 one obtains H ∈ Lq.Since p ∈ [1,∞], the equality (32) implies that r ∈ [q,+∞].We then consider the case q = ∞. The same reasoning indi-cates that H ∈ L∞ when q = ∞. As a summary, there holdsH ∈ Lr(0,+∞) with r ∈ [q,+∞].

Now we prove the following main convergence results. Notethat in the following theorem, the positivity of the Lq signalsis not required.

Theorem 4: Consider the following system:

y(t) = −αy(t)+ ρ(t) (37)

where y(t) ∈ R is the state variable, α > 0 is the coefficient,and ρ(t) ∈ Lq is an extra system signal satisfying either ofthe following conditions.

1) Case I: ρ(t) ∈ Lq, with q ∈ [1,+∞).2) Case II: ρ(t) ∈ L∞, and limt→∞ρ(t)→ 0.

Then there holds y(t) ∈ Lr for any r ∈ [q,+∞] andlimt→∞ y(t) = 0.

Proof: The solution to the system (37) with initial conditiony(0) can be described by

y(t) = e−αty(0)+∫ t

0e−α(t−τ) ρ(τ ) dτ

=: e−αty(0)+H(f (t), ρ(t)) (38)

where we have replaced the notation f (t) := e−αt for simplic-ity. For the first term, e−αty(0) ∈ Lp, for any p ∈ (0,∞] (inparticular p ∈ [q,∞]), and e−αty(0) → 0 as t → ∞. Wethen consider the convolution term H(f (t), ρ(t)), which sat-isfies H ∈ Lr for any r ∈ [q,+∞] according to Lemma 8.Therefore, by the linear space property of the Lp space, thereholds y(t) ∈ Lr for any r ∈ [q,+∞].

We then prove the convergence statement. According to thedifferentiation rule for convolution, one can obtain

H(f (t), ρ(t)) = H(f (t), ρ(t)

). (39)

Following similar arguments in Lemma 8 and (36), one canshow that f (t) ∈ Lp, with p ∈ (0,∞], and therefore H ∈ Lr forany r ∈ [q,+∞]. In particular, there holds H ∈ L∞. We thenconsider the following two cases of the function properties ofρ(t) as listed above.

1) Case I: In the above analysis, we have shown that H ∈Lr for any r ∈ [q,+∞] (with q < ∞) and H ∈ L∞.The generalized Barbalat’s lemma as stated in Lemma 6indicates that H(f (t), ρ(t))→ 0 as t→∞.

2) Case II: In the case of q = ∞, as shown above, thereholds that H ∈ L∞, H ∈ L∞, and limt→∞ρ(t) → 0.We now use the Final (Initial) Value Theoremin Laplace Transform (see [54]) to prove theconvergence. Let H(s), F(s), and G(s) denotethe Laplace transform of H(t), f (t), and ρ(t),respectively. Then there holds H(s) = F(s)G(s).Since f (t) is an exponentially decaying functionand ρ(t) is bounded, the Final Value Theoremapplies and enables one to obtain limt→∞H(t) =lims→0 sF(s)G(s) = (s/[s+ α])G(s)|s=0 =(1/α)sG(s)|s=0 = (1/α) limt→∞ ρ(t) = 0.

In summary, there holds H(f (t), ρ(t)) → 0 as t → ∞ andone concludes that limt→∞ y(t) = 0.

Lemma 8 and Theorem 4 can be seen as a generalizationof some classical Lp stability theories in [41], [48], and [53].

B. Matrix Inequality

Lemma 9: Suppose that A ∈ Rn×n is a symmetric, positive

semidefinite matrix. Denote null(A) as the null space of A,and Pnull(A) as the orthogonal projection to null(A). Then forevery x ∈ R

n, there holds

xTAx ≥ λ+min‖x− Pnull(A)x‖2 (40)

where λ+min denotes the minimum positive eigenvalue of A.Proof: Note that every x ∈ R

n can be decomposed as twoorthogonal parts x = Pnull(A)x+ (In − Pnull(A))x, i.e., one partin the null space of A and the other part in the orthogonalcomplement of null(A). Thus one has

xTAx = (Pnull(A)x+

(In − Pnull(A)

)x)T

A(Pnull(A)x+

(In − Pnull(A)

)x)

= ((In − Pnull(A)

)x)T

A((

In − Pnull(A)

)x)

≥ λ+min‖x− Pnull(A)x‖2. (41)

The last inequality is an application of the Courant–Fischertheorem [55, Th. 4.2.6].

The following corollary is an application of the abovelemma to the graph incidence or Laplacian matrix case.

Corollary 1: With the definitions of p, z, H, and L the samein Section II-B, there holds

pTLLp ≥ λ+min

(HHT)‖z‖2 (42)

where λ+min(HHT) is the smallest positive eigenvalue of HHT .Proof: First note that pTLLp = pTHTHHTHp = zTHHTz.

The definition z = Hp implies that z is in the range spaceof H, and thus Pnull(HHT )z = 0. Therefore Lemma 9 applieswhich gives (42).

Note that in the above corollary, λ+min(HHT) = λ2(HTH) =λ2(L) = λ2(LG), i.e., it is equivalent to the second smallesteigenvalue (the algebraic eigenvalue) of the Laplacian matrixLG if the undirected graph G is connected.

ACKNOWLEDGMENT

The first author would like to thank R. Suttner fromUniversität Würzburg for helpful discussions on topics offunction Lp spaces.

REFERENCES

[1] W. Ren and R. W. Beard, Distributed Consensus in Multi-VehicleCooperative Control. London, U.K.: Springer, 2008.

[2] Z. Zuo, Q. L. Han, B. Ning, X. Ge, and X.-M. Zhang, “An overview ofrecent advances in fixed-time cooperative control of multiagent systems,”IEEE Trans. Ind. Inf., vol. 14, no. 6, pp. 2322–2334, Jun. 2018.

[3] L. Ding, Q.-L. Han, and G. Guo, “Network-based leader–following con-sensus for distributed multi-agent systems,” Automatica, vol. 49, no. 7,pp. 2281–2286, 2013.

[4] Y. Cao, W. Yu, W. Ren, and G. Chen, “An overview of recent progressin the study of distributed multi-agent coordination,” IEEE Trans. Ind.Inf., vol. 9, no. 1, pp. 427–438, Feb. 2013.


[5] X. Ge and Q.-L. Han, “Consensus of multiagent systems subject to par-tially accessible and overlapping Markovian network topologies,” IEEETrans. Cybern., vol. 47, no. 8, pp. 1807–1819, Aug. 2017.

[6] W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada, “An introductionto event-triggered and self-triggered control,” in Proc. IEEE 51st Annu.Conf. Decis. Control, 2012, pp. 3270–3285.

[7] P. Tabuada, “Event-triggered real-time scheduling of stabilizing con-trol tasks,” IEEE Trans. Autom. Control, vol. 52, no. 9, pp. 1680–1685,Sep. 2007.

[8] A. Anta and P. Tabuada, “To sample or not to sample: Self-triggeredcontrol for nonlinear systems,” IEEE Trans. Autom. Control, vol. 55,no. 9, pp. 2030–2042, Sep. 2010.

[9] D. P. N. Borgers and W. P. M. H. Heemels, “Event-separation propertiesof event-triggered control systems,” IEEE Trans. Autom. Control, vol. 59,no. 10, pp. 2644–2656, Oct. 2014.

[10] K. Liu, P. Duan, Z. Duan, H. Cai, and J. Lü, “Leader–following consen-sus of multi-agent systems with switching networks and event-triggeredcontrol,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 65, no. 5,pp. 1696–1706, May 2018.

[11] D. Ye, M.-M. Chen, and H.-J. Yang, “Distributed adaptive event-triggered fault-tolerant consensus of multiagent systems with gen-eral linear dynamics,” IEEE Trans. Cybern., to be published,doi: 10.1109/TCYB.2017.2782731.

[12] X. Ge, Q.-L. Han, and F. Yang, “Event-based set-membership leader–following consensus of networked multi-agent systems subject to lim-ited communication resources and unknown-but-bounded noise,” IEEETrans. Ind. Electron., vol. 64, no. 6, pp. 5045–5054, Jun. 2017.

[13] M. Donkers and W. Heemels, “Output-based event-triggered control withguaranteed L∞-gain and improved and decentralized event-triggering,”IEEE Trans. Autom. Control, vol. 57, no. 6, pp. 1362–1376, Jun. 2012.

[14] P. Tallapragada and N. Chopra, “Decentralized event-triggering for con-trol of nonlinear systems,” IEEE Trans. Autom. Control, vol. 59, no. 12,pp. 3312–3324, Dec. 2014.

[15] M. Mazo and M. Cao, “Asynchronous decentralized event-triggeredcontrol,” Automatica, vol. 50, no. 12, pp. 3197–3203, 2014.

[16] W. Hu, L. Liu, and G. Feng, “Consensus of linear multi-agent systemsby distributed event-triggered strategy,” IEEE Trans. Cybern., vol. 46,no. 1, pp. 148–157, Jan. 2016.

[17] C. Nowzari and J. Cortes, “Team-triggered coordination for real-timecontrol of networked cyber-physical systems,” IEEE Trans. Autom.Control, vol. 61, no. 1, pp. 34–47, Jan. 2016.

[18] Q. Liu, M. Ye, J. Qin, and C. Yu, “Event-triggered algo-rithms for leader–follower consensus of networked Euler–Lagrangeagents,” IEEE Trans. Syst., Man, Cybern., Syst., to be published,doi: 10.1109/TSMC.2017.2772820.

[19] D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, “Distributedevent-triggered control for multi-agent systems,” IEEE Trans. Autom.Control, vol. 57, no. 5, pp. 1291–1297, May 2012.

[20] C. Nowzari and J. Cortés, “Distributed event-triggered coordination foraverage consensus on weight-balanced digraphs,” Automatica, vol. 68,pp. 237–244, Jun. 2016.

[21] G. S. Seyboth, D. V. Dimarogonas, and K. H. Johansson, “Event-basedbroadcasting for multi-agent average consensus,” Automatica, vol. 49,no. 1, pp. 245–252, 2013.

[22] A. Girard, “Dynamic triggering mechanisms for event-triggered control,”IEEE Trans. Autom. Control, vol. 60, no. 7, pp. 1992–1997, Jul. 2015.

[23] X. Yi, K. Liu, D. V. Dimarogonas, and K. H. Johansson, “Distributeddynamic event-triggered control for multi-agent systems,” arXiv preprintarXiv:1704.05434, 2017.

[24] Y. Fan, G. Feng, Y. Wang, and C. Song, “Distributed event-triggeredcontrol of multi-agent systems with combinational measurements,”Automatica, vol. 49, no. 2, pp. 671–675, 2013.

[25] J. Qin, Q. Ma, Y. Shi, and L. Wang, “Recent advances in consensus ofmulti-agent systems: A brief survey,” IEEE Trans. Ind. Electron., vol. 64,no. 6, pp. 4972–4983, Jun. 2017.

[26] L. Ding, Q.-L. Han, X. Ge, and X.-M. Zhang, “An overview of recentadvances in event-triggered consensus of multiagent systems,” IEEETrans. Cybern., vol. 48, no. 4, pp. 1110–1123, Apr. 2018.

[27] X. Ge, Q.-L. Han, D. Ding, X.-M. Zhang, and B. Ning, “A survey onrecent advances in distributed sampled-data cooperative control of multi-agent systems,” Neurocomputing, vol. 275, pp. 1684–1701, Jan. 2018.

[28] X.-M. Zhang, Q.-L. Han, and B.-L. Zhang, “An overview and deepinvestigation on sampled-data-based event-triggered control and filteringfor networked systems,” IEEE Trans. Ind. Inf., vol. 13, no. 1, pp. 4–16,Feb. 2017.

[29] V. S. Dolk, D. P. Borgers, and W. P. M. H. Heemels, “Output-based anddecentralized dynamic event-triggered control with guaranteed Lp-gainperformance and Zeno-freeness,” IEEE Trans. Autom. Control, vol. 62,no. 1, pp. 34–49, Jan. 2017.

[30] H. Yu and P. J. Antsaklis, “Event-triggered output feedback control fornetworked control systems using passivity: Achieving L2 stability in thepresence of communication delays and signal quantization,” Automatica,vol. 49, no. 1, pp. 30–38, 2013.

[31] C. Vasile, M. Schwager, and C. Belta, “Translational and rotationalinvariance in networked dynamical systems,” IEEE Trans. Control Netw.Syst., to be published, doi: 10.1109/TCNS.2017.2648499.

[32] Q. Jia and W. K. S. Tang, “Event-triggered protocol for the consen-sus of multi-agent systems with state-dependent nonlinear coupling,”IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 65, no. 2, pp. 723–732,Feb. 2018.

[33] Z.-G. Wu, Y. Xu, Y.-J. Pan, H. Su, and Y. Tang, “Event-triggeredcontrol for consensus problem in multi-agent systems with quan-tized relative state measurements and external disturbance,” IEEETrans. Circuits Syst. I, Reg. Papers, vol. 65, no. 7, pp. 2232–2242,Jul. 2018.

[34] E. Garcia, Y. Cao, H. Yu, P. Antsaklis, and D. Casbeer, “Decentralisedevent-triggered cooperative control with limited communication,” Int. J.Control, vol. 86, no. 9, pp. 1479–1488, 2013.

[35] G. Guo, L. Ding, and Q.-L. Han, “A distributed event-triggeredtransmission strategy for sampled-data consensus of multi-agent systems,” Automatica, vol. 50, no. 5, pp. 1489–1496,2014.

[36] Y. Fan, L. Liu, G. Feng, and Y. Wang, “Self-triggered consensusfor multi-agent systems with Zeno-free triggers,” IEEE Trans. Autom.Control, vol. 60, no. 10, pp. 2779–2784, Oct. 2015.

[37] D. V. Dimarogonas and E. Frazzoli, “Distributed event-triggered controlstrategies for multi-agent systems,” in Proc. 47th Annu. Allerton Conf.Commun. Control Comput., 2009, pp. 906–910.

[38] N. Huang, Z. Duan, and Y. Zhao, “Distributed consensus for multipleEuler–Lagrange systems: An event-triggered approach,” Sci. ChinaTechnol. Sci., vol. 59, no. 1, pp. 33–44, 2016.

[39] Z. Sun, N. Huang, B. D. O. Anderson, and Z. Duan, “A new distributedZeno-free event-triggered algorithm for multi-agent consensus,” in Proc.IEEE 55th Conf. Decis. Control, 2016, pp. 3444–3449.

[40] K.-K. Oh, M.-C. Park, and H.-S. Ahn, “A survey of multi-agentformation control,” Automatica, vol. 53, pp. 424–440, Mar. 2015.

[41] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input–OutputProperties. London, U.K.: Academic, 1975.

[42] B. Makarov and A. Podkorytov, Real Analysis: Measures, Integrals andApplications. London, U.K.: Springer, 2013.

[43] J. Zhang, K. H. Johansson, J. Lygeros, and S. Sastry, “Dynamicalsystems revisited: Hybrid systems with Zeno executions,” inProc. Int. Workshop Hybrid Syst. Comput. Control, 2000,pp. 451–464.

[44] A. D. Ames, P. Tabuada, and S. Sastry, “On the stability of Zeno equi-libria,” in Proc. Int. Workshop Hybrid Syst. Comput. Control, 2006,pp. 34–48.

[45] A. D. Ames, H. Zheng, R. D. Gregg, and S. Sastry, “Is there life afterZeno? Taking executions past the breaking (Zeno) point,” in Proc. IEEEAmer. Control Conf., 2006, pp. 2652–2657.

[46] J. Cortes, “Discontinuous dynamical systems,” IEEE Control Syst. Mag.,vol. 3, no. 28, pp. 36–73, Jun. 2008.

[47] Z. Sun, N. Huang, B. D. O. Anderson, and Z. Duan, “Comments on‘distributed event-triggered control of multi-agent systems with combi-national measurements,”’ Automatica, vol. 92, pp. 264–265 Jun. 2018.

[48] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ, USA:Prentice-Hall, 2002.

[49] J. Zhang, K. H. Johansson, J. Lygeros, and S. Sastry, “Zeno hybridsystems,” Int. J. Robust Nonlinear Control, vol. 11, no. 5, pp. 435–451,2001.

[50] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and coop-eration in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1,pp. 215–233, Jan. 2007.

[51] G. Xie, H. Liu, L. Wang, and Y. Jia, “Consensus in networked multi-agent systems via sampled control: Fixed topology case,” in Proc. Amer.Control Conf., 2009, pp. 3902–3907.

[52] B. Cheng and Z. Li, “Fully distributed event-triggered protocols for lin-ear multi-agent networks,” IEEE Trans. Autom. Control, to be published,doi: 10.1109/TAC.2018.2857723.

[53] G. Tao, Adaptive Control Design and Analysis, vol. 37. Hoboken, NJ,USA: Wiley, 2003.

http://dx.doi.org/10.1109/TCYB.2017.2782731

http://dx.doi.org/10.1109/TSMC.2017.2772820

http://dx.doi.org/10.1109/TCNS.2017.2648499

http://dx.doi.org/10.1109/TAC.2018.2857723


[54] R. J. Beerends, Fourier and Laplace Transforms. Cambridge, U.K.:Cambridge Univ. Press, 2003.

[55] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.:Cambridge Univ. Press, 2012.

Zhiyong Sun received the Ph.D. degree fromAustralian National University (ANU), Canberra,ACT, Australia, in 2017.

He was a Research Fellow/Lecturer with theResearch School of Engineering, ANU, from 2017to 2018. In 2018, he joined the Department ofAutomatic Control, Lund University, Lund, Sweden,as a Postdoctoral Fellow. His current researchinterests include graph rigidity theory, control ofautonomous formations, and networked systems.

Dr. Sun was a recipient of the Australian PrimeMinister’s Endeavor Postgraduate Award from the Australian Government in2013, the Outstanding Overseas Student Award from the Chinese Governmentin 2016, and the Springer Ph.D. Thesis Prize from Springer in 2017. Hewas the finalist of the Best Student Paper (BSP) Award in the 54th IEEEConference on Decision and Control (CDC 2015), a finalist of the BSP Awardin the 4th Australian Control Conference (AUCC 2014), and the Winner ofthe BSP Award in the 5th Australian Control Conference (AUCC 2015).

Na Huang received the B.S. degree in mathematicsfrom Inner Mongolia University, Hohhot, China,in 2011, and the Ph.D. degree in control theoryfrom the State Key Laboratory for Turbulenceand Complex Systems, Department of Mechanicsand Engineering Science, College of Engineering,Peking University, Beijing, China, in 2016.

Since 2016, she has been with the Schoolof Automation, Hangzhou Dianzi University,Hangzhou, China. Her current research interestsinclude cooperative control of multiagent systems,

event-triggered control, and sampled-data control.

Brian D. O. Anderson (M’66–SM’74–F’75–LF’07)was born in Sydney, NSW, Australia. He receivedthe degree in mathematics and electrical engi-neering from Sydney University, Sydney, and thePh.D. degree in electrical engineering from StanfordUniversity, Stanford, CA, USA, in 1966.

He is an Emeritus Professor with AustralianNational University, Canberra, ACT, Australia (hav-ing retired as a Distinguished Professor in 2016),a Distinguished Professor with Hangzhou DianziUniversity, Hangzhou, China, and a Distinguished

Researcher with National ICT Australia, Sydney. His current research interestsinclude distributed control, social networks, and econometric modeling.

Dr. Anderson was a recipient of the IEEE Control Systems Award of 1997,the 2001 IEEE James H. Mulligan, Jr. Education Medal, and the Bode Prize ofthe IEEE Control System Society in 1992, as well as several IEEE and otherbest paper prizes. He is a fellow of the Australian Academy of Science, theAustralian Academy of Technological Sciences and Engineering, the RoyalSociety, and a Foreign Member of the U.S. National Academy of Engineering.He holds honorary doctorates from a number of universities, including theUniversité Catholique de Louvain, Louvain-la-Neuve, Belgium, and ETHZürich, Zürich. He is the Past President of the International Federation ofAutomatic Control and the Australian Academy of Science.

Zhisheng Duan received the M.S. degree in math-ematics from Inner Mongolia University, Hohhot,China, in 1997 and the Ph.D. degree in control the-ory from Peking University, Beijing, China, in 2000.

From 2000 to 2002, he was a PostdoctoralFellow with Peking University, where he has been aFull Professor with the Department of Mechanicsand Engineering Science, College of Engineeringsince 2008. He obtained the Outstanding NationalNational Natural Science Foundation in China andhe is currently a Cheung Kong Scholar with Peking

University. His current research interests include robust control, stability ofinterconnected systems, flight control, and analysis and control of complexdynamical networks.

Prof. Duan was a recipient of the Guan-Zhao Zhi Best Paper Award at the2001 Chinese Control Conference and the 2011 First Class Award in NaturalScience from the Chinese Ministry of Education.

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