Cooperative Event-Based Rigid Formation...

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS 1

Cooperative Event-Based Rigid Formation ControlZhiyong Sun , Qingchen Liu , Na Huang , Changbin Yu , Senior Member, IEEE,

and Brian D. O. Anderson , Life Fellow, IEEE

Abstract—This article discusses cooperative stabilization con-trol of rigid formations via an event-based approach. We firstdesign a centralized event-based formation control system, inwhich a central event controller determines the next triggeringtime and broadcasts the event signal to all the agents for con-trol input update. We then build on this approach to propose adistributed event control strategy, in which each agent can useits local event trigger and local information to update the con-trol input at its own event time. For both cases, the triggeringcondition, event function, and triggering behavior are discussedin detail, and the exponential convergence of the event-basedformation system is guaranteed.

Index Terms—Cooperative control, event-based control, graphrigidity, multiagent formation, rigid shape.

I. INTRODUCTION

FORMATION control involving multiagent networkedsystems has attracted increasing attention recently moti-

vated by its extensive applications in many civil and mil-itary areas. This article focuses particularly on formationshape control, i.e., designing distributed controllers to achieveor maintain a predefined formation shape for a multiagentsystem [1]. Via graph rigidity theory, the desired formation

Manuscript received June 8, 2019; accepted September 23, 2019. Thiswork was supported in part by the National Science Foundation of Chinaunder Grant 61761136005, Grant 61703130, and Grant 61673026, in partby the Key Projects of Science and Technology Plan of Zhejiang Provinceunder Grant 2019C04018, in part by the 111 Project under Grant D17019,and in part by the Australian Research Council’s Discovery Project underGrant DP-160104500 and Grant DP-190100887. The work of Q. Liu wassupported by the European Union’s Horizon 2020 Research and InnovationProgramme through the Marie Sklodowska-Curie under Grant 754462. Thisarticle was recommended by Associate Editor Z. Liu. (Corresponding author:Changbin Yu.)

Z. Sun was with the School of Engineering, Westlake University, Hangzhou,China. He is now with the Department of Automatic Control, Lund University,SE-221 00 Lund, Sweden (e-mail: [email protected]).

Q. Liu is with the Chair of Information-Oriented Control, TechnicalUniversity of Munich, Munich, Germany.

N. Huang is with the School of Automation, Hangzhou Dianzi University,Hangzhou 310018, China, and also with the School of Artificial Intelligence(Artificial Intelligence Institute), Hangzhou Dianzi University, Hangzhou310018, China.

C. Yu is with the School of Engineering, Westlake University,Hangzhou, China, and also with the Optus-Curtin Centre of Excellencein Artificial Intelligence, Curtin University, Perth, WA, Australia (e-mail:[email protected]).

B. D. O. Anderson is with the School of Automation, Hangzhou DianziUniversity, Hangzhou 310018, China, also with the Research School ofElectrical, Energy and Material Engineering, Australian National University,Canberra, ACT 2601, Australia, and also with Data61-CSIRO, Canberra, ACT2601, Australia.

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

Digital Object Identifier 10.1109/TSMC.2019.2945988

of the shape can be defined and controlled by certain intera-gent distances [2], [3] and there is no requirement on a globalor common coordinate frame having to be known to all theagents. This is in contrast to the linear consensus-based for-mation control approach, in which the target formation isdefined by certain vectors of relative positions and a globalor common coordinate frame is required for all the agents inthe formation group to implement the consensus-based for-mation control law (see detailed comparisons in [1]). Notethat such a coordinate alignment condition is a rather strictrequirement, which is undesirable for implementing forma-tion controllers in, e.g., a GPS-denied environment. Evenif one assumes that such coordinate alignment is satisfiedfor all agents, slight coordinate misalignment, perhaps aris-ing from sensor biases, can lead to a failure of formationcontrol [4], [5]. Motivated by all these considerations, in thisarticle, we focus on rigidity-based formation control.

There have been rich works on controller design and sta-bility analysis of rigid formation control (see [3], [6]–[8], andthe review in [1]), most of which assume that the control inputis updated in a continuous manner. The main objective of thisarticle is to provide alternative controllers to stabilize rigidformation shapes based on an event-triggered approach. AnEvent-based control is a promising strategy for real-world con-trol systems equipped with digital microprocessors, actuators,or sensors [9], [10]. A further motivation of using an event-based mechanism for updating control input, rather than usinga continuous updating strategy as discussed in [3] and [6]–[8], is that the formation system can save distributed processorresources and therefore can reduce much of the actuation orsensing/computation burden for each agent in the group. Werefer the readers to the recent survey papers [11], [12] whichprovide comprehensive and excellent reviews on event-basedcontrol for networked coordination and multiagent systems.

The event-based control strategies in multiagent forma-tion systems have recently attracted increasing attention inthe control community. Some recent attempts at applyingevent-triggered schemes in stabilizing multiagent formationsare reported (see [13]–[16]). However, these papers [13]–[16]have focused on the event-triggered consensus-based forma-tion control approach, in which the proposed event-basedcontrol schemes cannot be applied to solve the rathermore difficult formation control problem of stabilizing rigidformation shapes. We note that event-based rigid forma-tion control has been discussed briefly as an example ofteam-triggered network coordination in [17]. However, nothorough studies in the literature have been reported toachieve cooperative rigid formation control with feasible

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2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

and simple event-based solutions. This article is a firstcontribution to advance the event-based control strategy tothe design and implementation of rigid formation controlsystems.

Some preliminary results have been presented in conferencecontributions [18] and [19]. Compared to [18] and [19], thisarticle proposes new control methodologies to design event-based controllers as well as the event-triggering functions.The event controller presented in [18] is a preliminary one,which only updates parts of the control input and the con-trol is not necessarily piecewise constant. This limitation isremoved in the control design in this article. Also, the com-plicated controllers in [19] have been simplified. Moreover,and centrally to the novelty of this article, we will focus onthe design of distributed controllers based on novel event-triggering functions to achieve the cooperative formation task,while the event controllers in both [18] and [19] are cen-tralized. We prove that Zeno behavior is excluded in thedistributed event-based formation control system by provinga positive lower bound of the interevent-triggering time. Wealso notice that a decentralized event-triggered control wasrecently proposed in [20] to achieve rigid formation track-ing control. The triggering strategy in [20] requires eachagent to update the control input both at its own trigger-ing time and its neighbors’ triggering times, which increasesthe communication burden within the network. Furthermore,the triggering condition discussed in [20] adopts the positioninformation in the event function design, which results in verycomplicated control functions and may limit their practicalapplications.

In this article, we will provide a thorough study of rigidformation stabilization control with cooperative event-basedapproaches. To be specific, we will propose two feasible event-based control schemes (a centralized triggering scheme anda distributed triggering scheme) that aim to stabilize a gen-eral rigid formation shape. In this article, by a careful designof the triggering condition and event function, the aforemen-tioned communication requirements and controller complexityin [18]–[20] are avoided. For all cases, the triggering condi-tion, event function and triggering behavior are discussed indetail. One of the key results in the controller performanceanalysis with both centralized and distributed event-based con-trollers is exponential stability of the rigid formation system.The exponential stability is a favorable property for multiagentformation control system, which guarantees a certain level ofrobustness in undirected rigid formations, as discussed in therecent papers [21], [22].

The remaining parts of this article are structured as fol-lows. In Section II, preliminary concepts on graph rigiditytheory are introduced. In Section III, we propose a centralizedevent-based formation controller and discuss the convergenceproperty and the exclusion of the Zeno behavior. Section IVbuilds on the centralized scheme of Section III to developa distributed event-based controller design, and presents adetailed analysis of the triggering behavior and its feasibil-ity. The representative simulations are provided in Section Vto demonstrate the controller performance, followed by con-cluding remarks in Section VI.

II. PRELIMINARIES

A. Notations

We will use standard notations in this article. For a realmatrix M, its image, rank, and null space (kernel) are denoted,respectively, by Im(M), rank(M), and ker(M). The notation‖M‖ denotes the 2-norm of a vector M, or the induced 2-normof a matrix M, and ‖M‖F denotes the Frobenius norm for amatrix M. Note that there holds ‖M‖ ≤ ‖M‖F for any matrixM (see [23, Ch. 5]). We use diag{x} to denote a diagonalmatrix whose diagonal entries are the vector x. The notationspan{x1, x2, . . . , xk} denotes the subspace spanned by a set ofk real vectors x1, x2, . . . , xk. The notation ⊗ represents theKronecker product.

We recall Young’s inequality [24] for later analysis.Lemma 1: For given a, b ∈ R, it holds that

ab ≤ a2

2ε+ εb2

2where ε > 0.

B. Preliminary on Graph and Rigidity Theory

Consider an undirected graph with n vertices and m edges,denoted by G = (V, E). The vertex set V = {1, 2, . . . , n}represents the index of n agents in the group, and the edgeset E ⊂ V × V indicates the interconnection or neighboringrelationship of n agents. For agent i, its neighbor set Ni isdefined as Ni := {j ∈ V : (i, j) ∈ E}. For an undirected graph,its incidence matrix H = {hij} ∈ R

m×n is defined by (viaarbitrary orientations for all edges): hki = 1 if the kth edgesinks at node i, or hki = −1 if the kth edge leaves node i,or hki = 0 otherwise. Furthermore, its Laplacian matrix isdefined by L(G) = HTH. For any connected undirected graphG, it holds rank(L) = n − 1 and ker(L) = ker(H) = span{1n},where 1n denotes an n-tuple column vector of all ones. Notethat the Laplacian matrix does not depend on the arbitraryedge orientations in the undirected graph, nor does the rigidformation modeled by an undirected graph.

Graph rigidity theory has been a powerful tool in themodel and control of distance-based multiagent formationsystem. For the undirected graph G, we assign each vertexi an agent with its position pi ∈ R

d where d = {2, 3}.For the whole formation system, we use the stacked vectorp = [pT

1 , pT2 , . . . , pT

n ]T ∈ Rdn to represent a realization of G

in Rd. The framework, or a formation of G in R

d, is denotedby the pair (G, p). Now with H := H ⊗ Id ∈ R

dm×dn where Id

denotes the d × d identity matrix, and one can construct therelative position vector as

z = Hp (1)

where z = [zT1 , zT

2 , . . . , zTm]T ∈ R

dm, with zk = pi − pj beingthe relative position vector for the kth vertex pair defined bythe edge (i, j).

For distance-based rigid formations, the formation shape isdefined by a certain set of interagent distances, denoted by dkij

that links agents i and j. The squared distance error for edgek is defined as

ekij = ∥∥pi − pj

∥∥

2 − (

dkij

)2. (2)


SUN et al.: COOPERATIVE EVENT-BASED RIGID FORMATION CONTROL 3

(a) (b) (c)

Fig. 1. Examples on rigid and nonrigid formations. (a) nonrigid (flexible)formation (the dashed lines show a deformed formation shape, with the sameset of interagent distances); (b) minimally rigid formation; and (c) rigid (butnot minimally rigid) formation.

When no notational confusion arises, we may also use ek anddk occasionally to replace ekij and dkij , respectively, for nota-tional convenience. By choosing the same node-edge relationsas in the incidence matrix H, one can define the distance errorei for each edge i = 1, 2, . . . , m. The distance error vector isconstructed as e = [e1, e2, . . . , em]T .

In graph rigidity theory, a commonly used tool is the rigiditymatrix R ∈ R

m×dn, whose each row takes the following form:[

01×d, . . . ,(

pi − pj)T

, . . . , 01×d, . . . , (pj − pi)T , . . . , 01×d

]

.

(3)

Note that each edge in the graph G gives rise to a row of R.By defining Z = diag{z1, z2, . . . , zm}, a compact form of therigidity matrix is readily shown as (see [25], [26])

R(p) = ZTH. (4)

Definition 1 (See [27]): A framework (G, p) is infinitesi-mally rigid in the d-dimensional space if

rank(R(p)) = dn − d(d + 1)/2. (5)

If the framework is infinitesimally rigid and attains the min-imal number of edges, then we call it an infinitesimally andminimally rigid (IMR) framework. Fig. 1 shows several exam-ples on rigid and nonrigid formations. We focus on in thisarticle the design of event-based stabilization control laws forIMR formations.1

III. EVENT-BASED CONTROLLER DESIGN:CENTRALIZED CASE

This section focuses on the design of feasible event-basedformation controllers, by assuming that a centralized processoris available for collecting the global information and broad-casting the triggering signal to all the agents such that theircontrol inputs can be updated. The results in this section extendthe event-based formation control reported in [18] by propos-ing an alternative approach for the event function design,which also simplifies the event-based controllers proposedin [19] and [20]. Furthermore, the novel idea used for design-ing a simpler event function in this section will be usefulfor designing a feasible distributed version of an event-basedformation control system, which will be reported in the nextsection.

1With some complexity of calculation, the results extend to nonminimallyrigid formations (see [21], [26]).

A. Centralized Event Controller Design

We propose the following general form of event-basedformation control system:

pi(t) = ui(t) = ui(th)

=∑

j∈Ni

(

pj(th) − pi(th))

ek(th) (6)

for t ∈ [th, th+1), where h = 0, 1, 2, . . . , and th is the hth trig-gering time for updating new information in the controller.2

The control law is an obvious variant of the standard lawfor nonevent-based formation shape control [3]. Evidently, thecontrol input takes piecewise constant values in each timeinterval. In this section, we allow the switching times th tobe determined by a central controller. In a compact form, theabove position system can be written as

p(t) = −R(p(th))Te(th). (7)

Denote a vector δi(t) as

δi(t) = −∑

j∈Ni

(

pj(th) − pi(th))

ek(th)

+∑

j∈Ni

(

pj(t) − pi(t))

ek(t) (8)

for t ∈ [th, th+1). Then the formation control system (6) canbe equivalently stated as

pi(t) = ui(th) =∑

j∈Ni

(

pj(t) − pi(t))

ek(t) − δi(t). (9)

Define a vector δ(t) = [δ1(t)T , δ2(t)T , . . . , δn(t)T ]T ∈ Rdn.

Then, there holds

δ(t) = R(th)Te(th) − R(t)Te(t) (10)

which enables one to rewrite the compact form of the positionsystem as

p(t) = −R(t)Te(t) − δ(t). (11)

We now analyze the distance error system derived by theposition system with the proposed event-triggered controller.By noting that e(t) = 2R(t)p(t), one can derive a compactform of the distance error system with the event-triggeredcontroller (6) as

e(t) = 2R(t)p(t)

= −2R(t)R(p(th))Te(th) ∀t ∈ [

th, th+1). (12)

Note that all the entries of R(t) and e(t) contain the real-timevalues of p(t), and all the entries R(p(th)) and e(th) containthe piecewise-constant values p(th) during the time interval[th, th+1).

The new form of the position system (11) also implies anequivalent form of the distance error system

e(t) = −2R(t)(

R(t)Te(t) + δ(t))

. (13)

2The differential equation (6) that models the event-based formation controlsystem involves switching controls at every event updating instant, for whichwe understand its solution in the sense of Filippov [28].



Consider the following function:

V = 1

4

m∑

k=1

e2k (14)

as a candidate of Lyapunov-like function for the stability anal-ysis of the system (13). Similarly to the analysis in [18], weconstruct a sublevel set B(ρ) = {e : V(e) ≤ ρ} for somesuitably small ρ, such that when the distance error vector sat-isfies e ∈ B(ρ), the formation is IMR, and also the matrixR(p(t))R(p(t))T is positive definite. Before giving the mainproof, we record the following key result on the entries of thematrix R(p(t))R(p(t))T .

Lemma 2: When the formation shape is close to the desiredone such that the distance error e is in the set B(ρ), the entriesof the matrix R(p(t))R(p(t))T are continuously differentiablefunctions of e.

This lemma enables one to discuss the self-contained dis-tance error system (13) and thus a Lyapunov argument can beapplied to show the exponential convergence of the distanceerror system. The proof of Lemma 2 can be found in [21]or [25] and will not be presented here. From Lemma 2, onecan show that

V(t) = 1

2e(t)T e(t) = −e(t)TR(t)

(

R(t)Te(t) + δ(t))

= −e(t)TR(t)R(t)Te(t) − e(t)TR(t)δ(t)

≤ −‖R(t)Te(t)‖2 + ‖e(t)TR(t)‖‖δ(t)‖. (15)

If we enforce the following condition for the norm of δ(t):

‖δ(t)‖ ≤ γ∥∥R(t)Te(t)

∥∥ (16)

and also choose the parameter γ with 0 < γ < 1, then wecan ensure that

V(t) ≤ (γ − 1)‖R(t)Te(t)‖2. (17)

The norm condition in (16) suggests an option for designingthe event function, which takes the form as f := ‖δ(t)‖ −γ ‖R(t)Te(t)‖. Therefore, an event is triggered when

f := ‖δ(t)‖ − γ∥∥R(t)Te(t)

∥∥ = 0. (18)

The event time th is defined to satisfy f (th) = 0 forh = 0, 1, 2, . . . For the time interval t ∈ [th, th+1), the controlinput is chosen as u(t) = u(th) until the next event is trig-gered. We note that at the time instant that triggers an event,the state error vector δ(t) should be reset to zero. Then thenorm of the vector δ(t) (i.e., ‖δ(t)‖) grows from zero until itreaches the value of γ ‖R(t)Te(t)‖ again that triggers the nextevent.

We also show two key properties of the formation controlsystem (6) with the above event function (18).

Lemma 3: The formation centroid remains constant underthe control of (6) with the event function (18).

Proof: We denote the formation centroid by p(t) ∈ Rd, i.e.,

p(t) = (1/n)∑n

i=1 pi(t) = (1/n)(1n ⊗ Id)Tp(t). One can show

˙p(t) = 1

n(1n ⊗ Id)

T p(t)

= −1

n(1n ⊗ Id)

TR(p(th))Te(th)

= −1

n

(

Z(th)TH(1n ⊗ Id)

)Te(th). (19)

Note that ker(H) = span{1n} and therefore ker(H) =span{1n ⊗ Id}. Thus, ˙p(t) = 0, which implies a stationaryformation centroid.

The following lemma concerns the coordinate framerequirement and enables each agent to use its local coordi-nate frame to implement the control law, which is favorablefor networked formation control systems in, e.g., GPS deniedenvironments.

Lemma 4: To implement the controller (6) with the event-based control update condition in (18), each agent can use itsown local coordinate frame, which does not need to be alignedwith a global coordinate frame. In other words, a global orcommon coordinate frame is not required.

The proof for the above lemma is omitted here, whichfollows similar steps as in [25, Lemma 4].

We now arrive at the following main result of this section.Theorem 1: Consider a target formation with an IMR

shape. The proposed controller (6) and the event-triggeringfunction (18) will drive all the agents to reach the desiredrigid formation shape locally exponentially fast.

Proof: The above analysis relating to (15)–(18) establishesboundedness of e(t) since V is nonpositive. Now, we showthe exponential convergence of e(t) to zero will occur froma ball around the origin, which is equivalent to the desiredformation shape being reached exponentially fast. Note thatby the condition of infinitesimal rigidity, the matrix M(e) :=R(p)R(p)T is positive definite. Let λmin denote the smallesteigenvalue of M(e) when e(p(t)) is in the set B(ρ) (i.e., λmin =mine∈B(ρ) λ(M(e)) > 0). Note that the existence of λmin isguaranteed by the compactness of the set B(ρ) and the fact thatmatrix eigenvalues are continuous functions of matrix entries.By recalling (17), it further holds

V(t) ≤ (γ − 1)(

e(t)TR(t)R(t)Te(t))

≤ (γ − 1)λmin‖e(t)‖2. (20)

Noting the definition of the function V(t): V(t) =(1/4)

∑mk=1 e2

k = (1/4)‖e(t)‖2, it is immediate that

V(t) ≤ 4(γ − 1)λminV(t) (21)

and therefore

V(t) ≤ exp(

4(γ − 1)λmint)

V(0). (22)

The above inequality then implies

‖e(t)‖ = √

4V(t) ≤ exp(

2(γ − 1)λmint)√

4V(0)

= exp(

2(γ − 1)λmint)‖e(0)‖. (23)



Then one concludes that ‖e(t)‖ converges to zero expo-nentially fast, with the exponential decaying rate no less thanκ = 2(1 − γ )λmin.

The above theorem shows the convergence of the interagentdistance error, which does not directly imply the convergenceof agents’ positions p(t) to some fixed points. This is becausethat the desired equilibrium corresponding to the correct rigidshape is not a single point, but is a set of equilibrium pointsinduced by rotational and translation invariance (for a detaileddiscussion to this subtle point, see [29, Ch. 5]). A sufficientcondition for this strong convergence to a stationary forma-tion is guaranteed by the exponential convergence, which wasproved above. In fact, according to the position system (7),since the entries of the rigidity matrix R (which involve rela-tive positions) are bounded and the distance error vector e(t)is exponentially convergent as proved above, an integral of thesystem p(t) implies that p(t) converges to a point in the limit.Therefore, the convergence of the position system (7) can bestated as a consequence of Theorem 1.

Lemma 5: The event-triggered control law (6) and the eventfunction (18) guarantee the convergence of p(t) to a fixedpoint.

Remark 1: We remark that the above Theorem 1 (as well asthe subsequent results in later sections) concerns a local con-vergence. This is because the rigid formation shape controlsystem is nonlinear and typically exhibits multiple equilib-ria, which include the ones corresponding to correct formationshapes and those that do not correspond to correct shapes. Arecent paper [30] proves the instability of a set of degenerateequilibria that live in a lower-dimensional space. However,the stability property for more general equilibrium points isstill unknown. It is in fact considered as a very challengingopen problem to obtain an almost global convergence resultfor general rigid formations, except for some special formationshapes, such as two-dimensional (2-D) triangular formationshape, or 2-D rectangular shape, or three-dimensional (3-D)tetrahedral shape (see the review in [1]). We note that localconvergence is still valuable in practice, if one assumes thatinitial shapes are close to the target ones (which is a verycommon assumption in most rigidity-based formation controlworks (see [1], [3], [6], [25], [31]).

B. Exclusion of Zeno Behavior

In this section, we will analyze the exclusion of possibleZeno triggering behavior of the event-based formation controlsystem (6).

Definition 2 (See [32]): For agent i, a triggering is Zeno if

limh→∞ tih =

∞∑

h=0

(

tih+1 − tih) = ti∞ (24)

for some finite ti∞ (termed the Zeno time).Note that the triggering function (18) involves the evolution

of the term R(t)Te(t), whose derivative is calculated asd(

R(t)Te(t))

dt= R(t)Te(t) + R(t)T e(t)

= HT Z(t)e(t) − 2R(t)TR(t)(

R(t)Te(t) + δ(t))

.

(25)

According to the construction of the vector δ(t) in (10), therealso holds δ(t) = −(d(R(t)Te(t))/dt).

Before presenting the proof, we first show a useful bound.Lemma 6: The following bound holds:∥∥HT Z(t)e(t)

∥∥ ≤ √

d∥∥HT

∥∥∥∥H

∥∥‖e(t)‖‖p(t)‖

≤ √d∥∥HT

∥∥‖H‖‖e(0)‖∥∥R(t)Te(t) + δ(t)

∥∥.

(26)

Proof: We first show a trick to bound the term ‖Z(t)e(t)‖by deriving an alternative expression for Z(t)e(t)

Z(t)e(t) = diag{z1(t), z2(t), . . . , zm(t)}e(t)

=

⎡

⎢⎢⎢⎣

e1(t)z1(t)e2(t)z2(t)

...

em(t)zm(t)

⎤

⎥⎥⎥⎦

=

⎛

⎜⎜⎜⎝

⎡

⎢⎢⎢⎣

e1(t) 0 · · · 00 e2(t) · · · 0...

.... . .

...

0 0 · · · em(t)

⎤

⎥⎥⎥⎦

⊗ Id

⎞

⎟⎟⎟⎠

⎡

⎢⎢⎢⎣

z1(t)z2(t)

...

zm(t)

⎤

⎥⎥⎥⎦

=: (E(t) ⊗ Id)z(t) (27)

where E(t) is defined as a diagonal matrix in the form E(t) =diag{e1(t), e2(t), . . . , em(t)}.

Note that z(t) = Hp(t) and thus z(t) = Hp(t). Then one has

‖HT Z(t)e(t)‖ = ‖HT(E(t) ⊗ Id)z(t)‖≤ ∥∥HT

∥∥‖(E(t) ⊗ Id)‖‖Hp(t)‖

≤ ‖HT‖‖H‖‖(E(t) ⊗ Id)‖F‖p(t)‖≤ √

d‖HT‖‖H‖‖e(t)‖‖R(t)Te(t) + δ(t)‖ (28)

where we have used the following facts:

‖E(t)‖ ≤ ‖E(t)‖F

‖(E(t) ⊗ Id)‖F = √d‖E(t)‖F

‖E(t)‖F = ‖e(t)‖.The first inequality in (26) is thus proved. The second inequal-ity in (26) is due to the fact that ‖e(t)‖ ≤ ‖e(0)‖∀t > 0 shownin (23).

We now show that the Zeno triggering does not occur inthe formation control system (6) with the triggering func-tion (18) by proving a positive lower bound on the intereventtime interval.

Theorem 2: The time interval {th+1 − th} between twoconsecutive events is lower bounded by a positive value τ

τ = γ

α(1 + γ )> 0 (29)

where

α = √d∥∥HT

∥∥∥∥H

∥∥‖e(0)‖ + √

2λmax(

RTR(e))

> 0 (30)

in which λmax denotes the largest eigenvalue ofRTR(e) when e(p(t)) is in the set B(ρ) (i.e.,λmax = maxe∈B(ρ)λ(RTR(e)) > 0), and γ is the trigger-ing parameter designed in (18) which satisfies γ ∈ (0, 1).



Thus, Zeno triggering will not occur for the rigid formationcontrol system (6) with the triggering function (18).

Proof: We show that the growth of ‖δ‖ from 0 to thetriggering threshold value γ ‖RTe‖ needs to take a posi-tive time interval. To show this, the relative growth rateon ‖δ(t)‖/‖R(t)Te(t)‖ is considered. The following proof isinspired by the one used in [33]. In the following deriva-tion, we omit the argument of time t but it should be clearthat each state variable and the vector is considered as afunction of t

d

dt

‖δ‖‖RT e‖ ≤

(

1 + ‖δ‖‖RT e‖

)‖RT e + RT e‖‖RT e‖

=(

1 + ‖δ‖‖RT e‖

)‖HT Ze + RT e‖‖RT e‖

≤(

1 + ‖δ‖‖RT e‖

)

(√d‖HT‖‖H‖‖e‖‖p‖ + ‖2RT R(RT e + δ)‖

‖RT e‖

)

(appealing to Lemma 6)

≤(

1 + ‖δ‖‖RT e‖

)

(

(√

d‖HT‖‖H‖‖e(0)‖ + ‖2RT R‖)(

1 + ‖δ)‖‖RT e‖

))

≤ α

(

1 + ‖δ‖‖RT e‖

)2

(31)

where α is defined in (30), and we have used the inequal-ity

√d‖HT‖‖H‖‖e(0)‖ + ‖2RTR‖ ≤ √

d‖HT‖‖H‖‖e(0)‖ +√2λmax(RTR). Note that λmax always exists and is finite (i.e.,

upper bounded) because the set B(ρ) is a compact set withrespect to e and the eigenvalues of a matrix are continuousfunctions of the matrix elements. Thus, α defined in (30)exists, which is positive and upper bounded. If we denote(‖δ‖/‖RTe‖) by y we have the estimate y(t) ≤ α(1 + y(t))2.By the comparison principle there holds y(t) ≤ φ(t, φ0) whereφ(t, φ0) is the solution of φ = α(1+φ)2 with initial conditionφ(0, φ0) = φ0.

Solving the differential equation for φ in the time intervalt ∈ [th, th+1) yields φ(τ, 0) = [τα/(1 − τα)]. The interex-ecution time interval is thus bounded by the time it takesfor φ to evolve from 0 to γ . Solving the above equa-tion, one obtains a positive lower bound for the intereventtime interval τ = [γ /(α(1 + γ ))]. Thus, Zeno behavior isexcluded for the formation control system (6). The proof iscompleted.

Remark 2: We review several event-triggered formationstrategies reported in the literature and highlight the advan-tages of the event-triggered approach proposed in this section.In [18], the triggering function is based on the informationof the distance error e only, which cannot guarantee a purepiecewise-constant update of the formation control input. Theevent function designed in [19] is based on the information ofthe relative position z, while the event function designed [20]is based on the absolute position p. It is noted that event func-tions and triggering conditions, such as those in [19] and [20]are very complicated, which may limit their practical applica-tions. The event function (18) designed in this section involvesthe term RTe, in which the information of the relative position

z (involved in the entries of the rigidity matrix R) and of thedistance error e has been included. Such an event-triggeringfunction greatly reduces the controller complexity while at thesame time also maintains the discrete-time update nature of thecontrol input.

IV. EVENT-BASED CONTROLLER DESIGN:DISTRIBUTED CASE

A. Distributed Event Controller Design

In this section, we will further show how to design a dis-tributed event-triggered formation controller in the sense thateach agent can use only local measurements in terms of rela-tive positions to its neighbors to determine the next triggeringtime and control update value. Denote the event time foreach agent i as ti0, ti1, . . . , tih, . . . The dynamical system foragent i to achieve the desired interagent distances is nowrewritten as

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

tih)∀t ∈ [

tih, tih+1

)

(32)

and we aim to design a distributed event function with feasibletriggering condition such that the control input for agent iis updated at its own event times ti0, ti1, . . . , tih, . . . , based onlocal information.3

We consider the same Lyapunov function candidate as theone in Section III, but calculate the derivative as follows:

V(t) = 1

2e(t)T e(t) = −e(t)TR(t)(R(t)Te(t) + δ(t))

= −e(t)TR(t)R(t)Te(t) − eTR(t)δ(t)

≤ −‖R(t)Te(t)‖2 + ∥∥e(t)TR(t)δ(t)

∥∥

≤ −n∑

i=1

∥∥{

R(t)Te(t)}

i

∥∥

2 +n∑

i=1

∥∥{

R(t)Te(t)}

i

∥∥‖δi(t)‖

(33)

where {R(t)Te(t)}i ∈ Rd is a vector block taken from the (di−

d+1)th to the (di)th entries of the vector R(t)Te(t), and δi(t) isa vector block taken from the (di−d+1)th to the (di)th entriesof the vector δ(t). According to the definition of the rigiditymatrix in (3), it is obvious that {R(t)Te(t)}i only involves localinformation of agent i in terms of relative position vectors zkij

and distance errors ekij with j ∈ Ni. Based on this, the controlinput for agent i is designed as

pi(t) = ui(

tih) =

∑

j∈Ni

(

pj(

tih) − pi

(

tih))

ek(

tih)

)

= {R(tih)T

e(

tih)}i ∀t ∈ [

tih, tih+1

)

. (34)

Note that there holds

δi(t) ={

R(

tih)T

e(

tih)}

i− {

R(t)Te(t)}

i (35)

and we can restate (34) as pi(t) = −{R(t)Te(t)}i − δi(t), t ∈[tih, tih+1).

3Again, Filippov solutions [28] are envisaged for the differential equa-tion (32) with switching controls at every event updating instant.



By using the inequality ‖{R(t)Te(t)}i‖‖δ(t)i‖ ≤(1/2ai)‖δ(t)i‖2 + (ai/2)‖{R(t)Te(t)}i‖2 with ai ∈ (0, 1),one can further develop the above inequality (33) on V as

V(t) ≤ −n∑

i=1

∥∥{

R(t)Te(t)}

i

∥∥

2

+n∑

i=1

ai

2

∥∥{R(t)Te(t)}i

∥∥

2 +n∑

i=1

1

2ai‖δi(t)‖2

= −n∑

i=1

2 − ai

2

∥∥{

R(t)Te(t)}

i

∥∥

2 +n∑

i=1

1

2ai‖δi(t)‖2.

If we enforce the following condition on the norm of δi(t):

1

2ai‖δi(t)‖2 ≤ γi

2 − ai

2‖{R(t)Te(t)

}

i‖2 (36)

with γi ∈ (0, 1), we can ensure

V(t) ≤n∑

i=1

(γi − 1)2 − ai

2

∥∥{R(t)Te(t)}i

∥∥

2. (37)

Based on the above analysis, one can construct a local event-triggering function for agent i as

fi(t) := ‖δi(t)‖2 − γiai(2 − ai)∥∥{

R(t)Te(t)}

i

∥∥

2(38)

and the event time tih for agent i is defined to satisfy fi(tih) = 0for h = 0, 1, 2, . . . For the time interval t ∈ [tih, tih+1), thecontrol input is chosen as ui(t) = ui(tih) until the next eventfor agent i is triggered. At the time instant that agent i triggersan event, the local event vector δi should reset to zero. Notethat the condition ai ∈ (0, 1) will also be justified in lateranalysis in Lemma 9.

Now we show the following convergence result.Theorem 3: Consider the target formation with an IMR

shape. The proposed formation controller (34) and the dis-tributed event-triggering function (38) will drive all the agentsto reach a desired rigid formation shape locally exponentiallyfast.

Proof: The analysis is similar to Theorem 1 and we omitseveral steps here. Based on the derivation in (33)–(38), onecan conclude that

‖e(t)‖ ≤ exp(−κt)‖e(0)‖ (39)

with the exponential rate no less than κ = 2ζminλmin whereζmin = mini(1 − γi)[(2 − ai)/2].

The exponential convergence of e(t) implies that the abovelocal event-triggered controller (34) also guarantees the con-vergence of p to a fixed point, by which one can conclude asimilar result to the one in Lemma 5.

For the formation system with the distributed event-basedcontroller, an analogous result to Lemma 4 on the coordinateframe requirement is as follows.

Lemma 7: To implement the distributed formation con-troller (34), local coordinate frame for each agent can beused to perform relative position measurements. Furthermore,the local coordinate frame and local measurement for eachagent are sufficient to detect and check the distributed event

condition (36) and a common/global coordinate frame is notinvolved.

The proof of the first statement on the distributed con-troller (34) follows similar steps as in [25, Lemma 4]. Theoverall proof can be found in the arXiv version [34]. The abovelemma indicates that the distributed event-based controller (34)and distributed event function (38) still guarantee the SE(N)

invariance property and enable a convenient implementationfor the proposed formation control system without any coor-dinate alignment or any information of a global coordinateframe for each individual agent, which meets the requirementof distributed coordination control.

Differently to Lemma 3, we show that the distributed event-based controller cannot ensure a fixed formation centroid.

Lemma 8: The position of the formation centroid is notguaranteed to be fixed with the formation controller (34) andthe proposed triggering function (38).

Proof: The dynamics for the formation centroid can bederived as

˙p(t) = 1

n(1n ⊗ Id)

T p(t).

However, due to the asymmetric update of each agent’s controlinput by using the local event function (38) to determine alocal triggering time, one cannot decompose the vector p(t)into terms involving H and a single distance error vector asin (19). Thus, ˙p(t) is not guaranteed to be zero and thereexist motions for the formation centroid when the distributedevent-based controller (34) is applied.

Remark 3: We note a key property of the distributed event-based controller (34) and (38) proposed in this section. It isobvious from (34) and (38) that each agent i updates its owncontrol input by using only local information in terms of rel-ative positions of its neighbors (which can be measured byagent i’s local coordinate system), and is not affected by thecontrol input updates from its neighbors. Thus, such a localevent-triggered approach does not require any communicationbetween any two agents.

B. Analysis of Event-Triggering Behavior

In this section, we analyze the triggering behavior and eventfeasibility. We first consider two special cases: 1) singular trig-gering (i.e., after a feasible triggering event, no more triggeringexists) and 2) continuous triggering (i.e., events are triggeredcontinuously). For the definitions of singular triggering andcontinuous triggering, we refer the reader to [35].

Lemma 9 (Triggering Feasibility): Consider the distributedformation system with the distributed event-triggered for-mation controller (34) and the distributed event-triggeringfunction (38). If there exists tih such that {R(tih)

Te(tih)}i �= 0,then:

1) (no singular triggering) agent i will not exhibit singulartriggering for all t > tih;

2) (no continuous triggering) agent i will not exhibit con-tinuous triggering for all t > tih.

Proof: The proof is inspired by [35]. First note that due toai ∈ (0, 1), there holds ai(2 − ai) ∈ (0, 1) and because γi ∈(0, 1), there further holds γiai(2 − ai) ∈ (0, 1). For notational



convenience, define

�i := γiai(2 − ai). (40)

From (35) and (36), for t ∈ [tih, tih+1) one has

χi

:=∥∥∥

{

R(

tih)T

e(

tih)}

i

∥∥∥

1 + √�i

≤ ‖{R(t)Te(t)}i‖

≤∥∥∥

{

R(

tih)T

e(

tih)}

i

∥∥∥

1 − √�i

=: χ i. (41)

We first prove the statement on nonsingular triggering.According to the definition of the event-triggering func-tion (38), local events for agent i can only occur either when‖{R(t)Te(t)}i‖ equals χ

ior when ‖{R(t)Te(t)}i‖ equals χ i.

Note that according to the vector norm property, one hasthe fact of equalities: ‖{R(t)Te(t)}i‖2 ≤ ‖{R(t)Te(t)}‖2 ≤λmax(R(t)TR(t))‖e(t)‖2, where λmax is the largest eigenvalueof R(t)TR(t) which is bounded for any e ∈ B(ρ). Alsonote that ‖e(t)‖2 decays exponentially fast to zero as provedin Theorem 3. This implies that ‖{R(t)Te(t)}i‖ will eventu-ally decrease to χ

i. By assuming that {R(tih)

Te(tih)}i �= 0,the next event time tih+1 for agent i always exists with{R(tih+1)

Te(tih+1)}i �= 0.The second statement can be proved by using similar argu-

ments to those above, and by observing that {R(t)Te(t)}i

evolves continuously and a local event is triggered if and onlyif (38) is satisfied.

In the following, we will further discuss the possibility ofthe Zeno behavior in the distributed event-based formationsystem (34).

Theorem 4 (Exclusion of Zeno Triggering): Consider thedistributed formation system with controller (34) and trigger-ing function (38).

1) At least one agent does not exhibit Zeno triggeringbehavior.

2) In addition, if there exists ε > 0 such that‖{R(t)Te(t)}i‖2 ≥ ε‖e(t)‖2 for all i = 1, 2, . . . , n andt ≥ 0, then there exists a common positive lower boundfor any interevent time interval for each agent. In thiscase, no agent will exhibit Zeno triggering behavior.

Proof: Note that ‖δi(t)‖2 ≤ ‖δ(t)‖2 holds for any i. Inaddition, there exists an agent i∗ such that ‖{R(t)Te(t)}i∗‖2 ≥(1/m)‖{R(t)Te(t)}‖2. Then one has

∥∥δi∗(t)

∥∥

∥∥{R(t)Te(t)}i∗

∥∥

≤ √m

‖δ(t)‖∥∥{R(t)Te(t)}∥∥ . (42)

By recalling the proof in Theorem 2, we can conclude thatthe interevent interval for agent i∗ is bounded from below bya time τi∗ that satisfies

√m

τi∗α

1 − τi∗α=√

γi∗ai∗(

2 − ai∗)

. (43)

So that τi∗ = [(√

γi∗ai∗(2 − ai∗))/(α(√

m+√

γi∗ai∗(2 − ai∗)))]> 0. The first statement is proved.

We then prove the second statement. Denote λmax as themaximum of λmax(RTR(e)) for all e ∈ B(ρ). Since B(ρ)

is a compact set, λmax exists and is bounded. Then, there

holds ‖{R(t)Te(t)}‖2 ≤ λmax‖e(t)‖2. Under the condition that‖{R(t)Te(t)}i‖2 ≥ ε‖e(t)‖2, one can further show

∥∥{R(t)Te(t)}i

∥∥

2 ≥ ε‖e(t)‖2 ≥ ε

λmax

∥∥{R(t)Te(t)}∥∥2

. (44)

By following a similar argument to that above and using theanalysis in the proof of Theorem 2, a lower bound on theinterevent interval τi for each agent can be derived by

τi =√

�i

α

(√

λmaxε

+ √�i

) > 0, i = 1, 2, . . . , n. (45)

The proof is completed.Remark 4: The first part of Theorem 4 is motivated by

[36, Theorem 4], which guarantees the exclusion of Zenobehavior for at least one agent. To improve the result forall the agents, we propose a condition in the second part ofTheorem 4. The above results in Theorem 4 on the distributedevent-based controller are more conservative than the central-ized case. The condition on the existence of ε > 0 essentiallyguarantees that ‖{R(t)Te(t)}i‖ cannot be zero at any finite time,and will be zero if and only if t = ∞. By a similar analysisfrom event-based multi-agent consensus dynamics in [37], onecan show that if ‖{R(t)Te(t)}i‖ = 0 at some finite time instantt, then agent i will exhibit a Zeno triggering and the time t isa Zeno time for agent i. However, we have performed manysimulations with different rigid formation shapes and observedthat in most cases ‖{R(t)Te(t)}i‖ is nonzero. We conjecturethat this may be due to the property of the infinitesimal rigid-ity of the target formation shape. A simulation example withthe event function of Theorem 4 that does not show Zenotriggering behavior will be provided in Section V. In the nextsection however, we will provide a simple modification of thedistributed controller to remove the condition on ε.

C. Modified Distributed Event Function

The event function (38) for agent i involves the compari-son of two terms, i.e., ‖δi(t)‖2 and γiai(2−ai)‖{R(t)Te(t)}i‖2.As noted above, the existence of ε > 0 can guarantee‖{R(t)Te(t)}i‖ �= 0 for any finite time t. To remove this con-dition in Theorem 4, and motivated by [38], we propose thefollowing modified event function by including an exponentialdecay term:

fi(t) := ‖δi(t)‖2 − γiai(2 − ai)∥∥{R(t)Te(t)}i

∥∥

2

− 2aiviexp(−θit) (46)

where vi > 0 and θi > 0 are the parameters that can beadjusted in the design to control the formation convergencespeed. Note that viexp(−θit) is always positive and convergesto zero when t → ∞. Thus, even if {R(t)Te(t)}i exhibits acrossing-zero scenario at some finite time instant, the additionof this decay term guarantees a positive threshold value in theevent function which avoids the case of comparing ‖δi(t)‖2 toa zero threshold.

The main result in this section is to show that the above-modified event function ensures Zeno-free triggering for allagents, and also drives the formation shape to reach the targetone.



Theorem 5: By using the proposed distributed event-triggered formation controller (34) and the modified distributedevent-triggering function (46), the desired formation shapecan be reached locally exponentially fast, and Zeno-triggeringbehavior is excluded for all the agents.

Proof: We consider the same Lyapunov function as usedin Theorems 1 and 3 and follow similar steps as above. Themodified event function (46) with the new triggering conditionyields

V(t) ≤n∑

i=1

(γi − 1)2 − ai

2‖{R(t)Te(t)}i‖2 + viexp(−θit)

(47)

which follows that:

V(t) ≤ −4ζminλminV(t) +n∑

i=1

viexp(−θit) (48)

where ζmin is defined as the same to the notation in Theorem 3(i.e., ζmin = mini(1 − γi)[(2 − ai)/2]). For notational conve-nience, we define κ = 2ζminλmin (also the same to Theorem 3).By the well-known comparison principle [39, Ch. 3.4], itfurther follows that:

V(t) ≤ exp(−2κt)V(0)

+n∑

i=1

vi

2κ − θi(exp(−θit) − exp(−2κt)) (49)

which implies that V(t) → 0 as t → ∞, or equivalently,‖e(t)‖ → 0, as t → ∞.

In the following analysis showing exclusion of Zeno behav-ior we let t ∈ [tih, tih+1). Note that fi(t) ≤ 0 can be equivalentlystated as

(�i + 1)‖δi(t)‖2

≤ �i

(

‖δi(t)‖2 + ∥∥{

R(t)Te(t)}

i

∥∥

2)

+ 2aiviexp(−θit)

(50)

where �i is defined in (40). Note that∥∥∥

{

R(

tih)T

e(

tih)}

i

∥∥∥

2 = ∥∥δi(t) + {R(t)Te(t)}i

∥∥

2

≤ 2(

‖δi(t)‖2 + ‖{R(t)Te(t)}

i‖2)

. (51)

Thus, a sufficient condition to ensure the above inequality (50)(and the inequality fi(t) ≤ 0) is

‖δi(t)‖2 ≤ �i

(2�i + 2)‖{R(tih

)Te(

tih)}i‖2 + 2aivi

�i + 1exp(−θit).

(52)

Note that from (10) there holds δi = −(d/dt){R(t)Te(t)}i. Itfollows that:

d

dt‖δi(t)‖ ≤

∥∥δi(t)T

∥∥

‖δi(t)‖ ‖δi(t)‖

=∥∥∥∥

d

dt

{

R(t)T e(t)}

i

∥∥∥∥

=∥∥∥∥∥∥

∑

j∈Ni

(

pj(t) − pi(t))

ek(t) +∑

j∈Ni

(

pj(t) − pi(t))

ek(t)

∥∥∥∥∥∥

=∥∥∥∥∥∥

∑

j∈Ni

(

ekij (t) ⊗ Id + 2zkij (t)zkij (t)T)(pj(t) − pi(t)

)

∥∥∥∥∥∥

=∥∥∥∥∥∥

∑

j∈Ni

Qij(t)

(

{R(

tjh′)T

e(

tjh′)

}j −{

R(

tih)T

e(

tih)}

i

)∥∥∥∥∥∥

≤∑

j∈Ni

‖Qij(t)‖∥∥∥∥∥

({

R(

tjh′)T

e(

tjh′)}

j−{

R(

tih)T

e(

tih)}

i

)∥∥∥∥∥

:= αi (53)

where Qij(t) := ekij(t) ⊗ Id + 2zkij(t)zkij(t)T , and tjh′ =

arg maxh{tjh|tjh ≤ t, j ∈ Ni}. By a straightforward argumentsimilar to Lemma 9, it can be shown that agent i will notexhibit singular triggering, which indicates that the variableαi is not equal to zero at any time interval. Also note that αi

is upper bounded which implies that (d/dt)‖δi(t)‖ is upperbounded. Since the inequality condition in (52) is a suffi-cient condition to ensure the triggering condition fi(t) ≤ 0shown in (46), one concludes that the next interevent intervalfor agent i can be lower bounded by the solution τ i

h to thefollowing algebraic equation:

τ ihαi =

√

�i

(2�i + 2)‖{R(tih

)Te(

tih)}i‖2 + 2aivi

�i + 1exp

(−θi(

tih + τ ih

))

.

(54)

Note that αi is positive and upper bounded, the first term inthe square root in the right-hand side is non-negative, and thesecond term (the exponential function) is positive at any finitetime tih. Therefore, the solution τ i

h to (54) always exists andis positive at any finite time tih. Also note that the exponentialdecaying function approaches zero only when t → ∞, whichimplies that tih → ∞ with h → ∞ as t → ∞.4 Accordingto Definition 2, we conclude that no agents will exhibit Zeno-triggering behavior with the modified event function (46).

Remark 5: In the modified event function (46), a positiveand exponential decay term viexp(−θit) is included whichguarantees that, even if {R(t)Te(t)}i becomes zero at somefinite time, the interevent time interval at any finite time ispositive and thus Zeno triggering is excluded. These trigger-ing parameters will have direct impacts on the performancetradeoff between the event-triggering frequency and the con-vergence speed of the formation stabilization process. As isclear in the proof of Theorems 3–5, larger values for γ (and γi)and vi give rise to larger comparison values in the event func-tion, which lead to a slower formation convergence but willresult in larger interevent time intervals (i.e., less frequent trig-gering in a bounded time). Similarly, smaller values for θi

will slow the decaying of the exponential term vie−θit, whichin turn enlarges the interevent time intervals but will slow

4The solution τ ih may or may not tend to zero as t → ∞. We remark that

even if τ ih tends to zero as t → ∞, this does not contradict the statement of

Zeno-free triggering for all agents, as revealed by Definition 2. According tothe surveys [12], [40], there is a concept termed “strong Zeno-free triggering,”in which the Zeno-free condition is strengthened in the sense that there shouldexist a uniform positive lower bound ε > 0 for the interevent time interval,such that τ i

h > ε for all h. We note that the proposed distributed event-based function does not ensure a strong Zeno-free triggering. To guarantee astrong Zeno-free triggering behavior, additional conditions (such as the normcondition in Theorem 4) should be imposed, which will then give rise a lowerbound of τ i

h such as the one in (45).



Fig. 2. Stabilization control of a 3-D rigid formation (double tetrahedron)with centralized event controller. The formation centroid remains station-ary. Dashed lines and solid lines represent, respectively, the initial and finalformation shapes. The legends will be the same to other figures.

down the formation convergence, as evidenced by the proof ofTheorem 5. A more general strategy for designing a Zeno-freeevent function is to include a positive Lp signal in the eventfunction (46). By following a similar analysis to [38], one canshow that, if the term viexp(−θit) in (46) is replaced by ageneral positive Lp signal, the local convergence of the event-based formation control system (34) with Zeno-free triggeringfor all agents still holds. However, a general Lp signal in (46)may only guarantee an asymptotic convergence rather than anexponential convergence of the overall formation system (34).

Remark 6: In this article, we suppose that relativeinformation for each agent is accessed by local measurementsfrom onboard sensors instead of by interagent communica-tions, which is often the favored technique used in rigidformation control. For the proposed distributed event-basedformation controller, no continuous-time communication isrequired to detect the triggering control (although continuous-time measurement of relative positions is required). In order tofurther relax the requirement of continuous-time measurement,one can also design a self-triggering scheme for stabilizingrigid formations. In the self-triggering control, each agent isrequired to predict the future state of itself and its neigh-bors so as to determine the next triggering time. However,as noted in [41], the energy consumption from measurementsis often negligible compared to that from communications,and the overall energy consumption in an event-triggeredcontrol system can be greatly reduced if interagent communi-cation is avoided. Thus, we do not consider to further relaxthe continuous-time measurement requirement in the eventfunction design.

V. SIMULATION STUDIES

Four sets of numerical simulations on both 3-D and 2-Drigid formations are provided in this section to show thebehavior of event-based rigid formation systems with theproposed centralized event-based controller and distributedevent-based controller, respectively. In the simulations, the val-ues of the triggering parameters γ (and γi), vi, and θi shouldbe chosen to be consistent with the conditions proposed inthe theoretical results, i.e., the conditions for these param-eters are γ ∈ (0, 1) [and γi ∈ (0, 1)], ai ∈ (0, 1),

Fig. 3. Evolution of the distance errors with centralized event controller.

Fig. 4. Performance of the centralized event-based controller. Top: Evolutionof ‖δ‖ and ‖δ‖max = γ ‖R(t)T e(t)‖. Bottom: Event-triggering instants.

Fig. 5. Stabilization control of a 3-D rigid formation (double tetrahedron)with the distributed event controller (34). Note that the formation centroid isnot stationary.

vi > 0, and θi > 0. We first consider a double tetrahe-dron formation in R

3, with the desired distances for eachedge being 2. The initial conditions for each agent are cho-sen as p1(0) = [0,−1.0, 0.5]T , p2(0) = [1.8, 1.6,−0.1]T ,p3(0) = [−0.2, 1.8, 0.05]T , p4(0) = [1.2, 1.9, 1.7]T , andp5(0) = [−1.0,−1.5,−1.2]T , so that the initial formationis close to the target one. The parameter γ in the triggerfunction is set as γ = 0.6. Figs. 2–4 illustrate formationconvergence and event performance with a centralized event-triggering function. The trajectories of each agent, togetherwith the initial shape and final shape are depicted in Fig. 2.Exponential convergence of each distance error is depictedin Fig. 3. Fig. 4 shows the triggering time instant and theevolution of the norm of the vector δ in the triggering func-tion (18), which is obviously bounded below by γ ‖R(t)Te(t)‖as required by (16).



Fig. 6. Stabilization control of a 2-D rigid formation with the distributedevent controller (34).

TABLE ICOMPARISON OF THE NUMBER OF TRIGGERED EVENTS BETWEEN THE

TWO DIFFERENT EVENT FUNCTIONS FROM t = 0 TO t = 3 S

We then perform another simulation on stabilizing the sameformation shape by applying the proposed distributed event-based controller (34) and distributed event-triggering func-tion (38). Agents’ initial positions are set the same as for thesimulation with the centralized event controller. The parame-ters γi, i = 1, 2, . . . , 5 are set as 0.8 and ai, i = 1, 2, . . . , 5are set as 0.6. The trajectories of each agent, together withthe initial shape and final shape are depicted in Fig. 5. Theevent times for each agent and the exponential convergenceof each distance error are depicted in Fig. 7. Note that no‖{R(t)Te(t)}i‖ crosses zero at any finite time and thereforeZeno-triggering is excluded for all agents, which is consistentwith the results in Section IV-B and Remark 4. Furthermore,the distance error system is shown in Fig. 7 also demonstratesalmost the same convergence property as shown in Fig. 3.

Third, we show simulations with the same double tetrahe-dron formation shape by using the modified event-triggeringfunction (46). The exponential decay term is chosen asviexp(−θit) = exp(−10t) with vi = 1 and θi = 10 foreach agent. Fig. 8 shows event-triggering times for each agentas well as the convergence of each distance error. As canbe observed from Fig. 8, Zeno-triggering is strictly excludedwith the modified event function (46), while the distance errorsystem shows an exponential convergence and the formationshape converges exponentially fast to the target shape. In addi-tion, we provide Table I to numerically compare the number oftriggered events between the event functions (38) and (46) inthe bounded time interval t ∈ (0, 3 s]. One can further observethat with the additional exponential term viexp(−θit) in theevent function, the triggering events are less frequent than thatin the distributed event function (38). It should be noted thatin comparison with the controller performance and simulationexamples discussed in [18]–[20], the proposed event-basedrigid formation controllers in this article demonstrate equal

Fig. 7. Controller performance of the distributed event-based forma-tion system (34) with distributed event function (38). Left: Event-triggeringinstants for each agent. Right: Exponential convergence of the distance errorswith distributed event controller.

Fig. 8. Controller performance of the distributed event-based formationsystem (34) [event function (46) with an exponential decay term]. Left: Event-triggering instants for each agent. Right: Exponential convergence of thedistance errors with distributed event controller.

Fig. 9. Controller performance of the distributed event-based formationsystem (34) with distributed event function (46) (2-D formation case). Left:Event-triggering instants for each agent. Right: Exponential convergence ofthe distance errors with distributed event controller.

or even better performance, while complicated controllers andunnecessary assumptions in [18]–[20] are avoided.

Lastly, we present an additional simulation example with a2-D formation shape to further demonstrate the performanceof the distributed event-based controller (34) with the modi-fied event-triggering function (46). The rigid formation shapecomprises three triangles, for which the desired edge lengthsare selected as {3, 2, 3, 4, 3, 5, 3}. The initial positions for eachagent are chosen as p1(0) = [0, 1.0]T , p2(0) = [0.2,−1.8]T ,p3(0) = [1.6,−1]T , p4(0) = [4.6,−2.4]T , and p5(0) =[5.3, 1.2]T . The parameters γi and ai in the trigger func-tion (46) are selected as γi = 0.2 and ai = 0.15, respectively.In addition, the exponential decay function vi exp(−θit) in (46)is set to be equal to 2 exp(−10t). We note that γi, ai, andvi exp(−θit) are all identical to each agent. Fig. 6 shows theevolution and convergence of the formation from the initial



shape to the desired shape. Fig. 9 shows the event-triggeringtimes for each agent as well as the exponential convergenceof the distance errors. All simulations agree well with the the-oretical derivations that demonstrate the performance of theproposed event-based formation controller.

VI. CONCLUSION

In this article, we have discussed in detail the designof feasible event-based controllers to stabilize rigid forma-tion shapes. A centralized event-based formation control isproposed first, which guarantees the exponential convergenceof distance errors and also excludes the existence of Zeno trig-gering. Due to a careful design of the triggering error and eventfunction, the controllers are much simpler and require muchless computation/measurement resources, compared with theresults reported in [18]–[20]. We then further propose a dis-tributed event-based controller such that each agent can triggera local event to update its control input based on only localmeasurement. The event feasibility and triggering behaviorhave been discussed in detail, which also guarantees Zeno-freebehavior for the event-based formation system and exponen-tial convergence of the distance error system. A modifieddistributed event function is proposed, by which the Zeno trig-gering is strictly excluded for each individual agent. A futuretopic is to explore possible extensions of the current results onsingle-integrator models to the double-integrator rigid forma-tion system [42] with an event-based control strategy to enablevelocity consensus and rigid flocking behavior.

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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 June 2018, he joined the Department ofAutomatic Control, Lund University, Lund, Sweden,as a Post-Doctoral Researcher. His current researchinterests include graph rigidity theory, control ofautonomous formations, and networked systems.

Dr. Sun was a recipient of the Australian PrimeMinister’s Endeavour Postgraduate Award in 2013 from the AustralianGovernment, the Outstanding Overseas Student Award from the ChineseGovernment in 2016, the Springer Ph.D. Thesis Prize from Springer in 2017,and the Best Student Paper Finalist Award from the 54th IEEE Conferenceon Decision and Control in 2015, at Osaka, Japan.

Qingchen Liu received the Ph.D. degree in systemand control from Australian National University,Canberra, ACT, Australia, in 2018.

He is currently an EuroTec Research Fellowwithin the Chair of Information-Oriented Control,Technical University of Munich, Munich, Germany.His current research interests include networkedsystems, distributed computation, and multiagentsystems.

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.

Changbin (Brad) Yu (M’08–SM’10) received theB.Eng. degree (Hons.) in computer engineeringfrom Nanyang Technological University, Singapore,in 2004, and the Ph.D. degree in engineeringfrom Australian National University, Canberra, ACT,Australia, (ANU) Canberra, in 2008.

He obtained tenure with ANU in 2014 andstill holds an Honorary Professorship. In 2017,he founded the AI and Robotics Centre, WestlakeUniversity, Hangzhou, China. He was appointed asthe Optus Chair to establish the Optus-Curtin Centre

of Excellence in AI, Curtin University, Perth, WA, Australia.Dr. Yu was a recipient of the Endeavour Asia Award, APD Fellowship,

QEII Fellowship, and a 2019 John Booker Medal in Engineering. He receivedmultiple grants and visiting fellowships from AAS and ATSE. He is a fellowof Engineers Australia.

Brian D. O. Anderson (M’66–SM’74–F’75–LF’07)was born in Sydney, Australia. He received thedegree in mathematics and electrical engineering,from Sydney University, Sydney, NSW, Australia,and the Ph.D. degree in electrical engineering fromStanford University, Stanford, CA, USA, in 1966.

He is an Emeritus Professor with the 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 Data-61 CSIRO, Canberra. He holds honorary doctoratesfrom a number of universities, including Université Catholique de Louvain,Belgium, and ETH, Zürich. His current research interests include distributedcontrol, social networks, and econometric modeling.

Dr. Anderson was a recipient of numerous awards, including the IEEEControl Systems Award of 1997, the 2001 IEEE James H. Mulligan,Jr. Education Medal, and the Bode Prize of the IEEE Control System Societyin 1992, as well as several IEEE and other best paper prizes. He is aPast President of the International Federation of Automatic Control and theAustralian Academy of Science. He is a fellow of the Australian Academy ofScience, the Australian Academy of Technological Sciences and Engineering,and the Royal Society. He is a Foreign Member of the U.S. National Academyof Engineering.

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