Adaldo, A., Alderisio, F., Liuzza, D., Shi, G., Dimarogonas, D. V., DiBernardo, M., & Johansson, K. H. (2015). Event-triggered pinning control ofswitching networks. IEEE Transactions on Control of Network Systems,2(2), 204-213. DOI: 10.1109/TCNS.2015.2428531

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204 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 2, NO. 2, JUNE 2015

Event-Triggered Pinning Controlof Switching Networks

Antonio Adaldo, Francesco Alderisio, Davide Liuzza, Guodong Shi, Dimos V. Dimarogonas, Member, IEEE,Mario di Bernardo, Fellow, IEEE, and Karl Henrik Johansson, Fellow, IEEE

Abstract—This paper investigates event-triggered pinning con-trol for the synchronization of complex networks of nonlineardynamical systems. We consider networks described by time-varying weighted graphs and featuring generic linear interactionprotocols. Sufficient conditions for the absence of Zeno behaviorare derived and exponential convergence of a global normed errorfunction is proven. Static networks are considered as a special case,wherein the existence of a lower bound for interevent times is alsoproven. Numerical examples demonstrate the effectiveness of theproposed control strategy.

Index Terms—Network analysis and control, networked controlsystems, switched systems.

I. INTRODUCTION

N ETWORKS of dynamical systems are a suitable modelfor many distributed phenomena in biology, social sci-

ences, physics, economics, and engineering [1], [2] and haveattracted much research interest in the last few decades [1]–[5].

Pinning control is a strategy to steer the collective behaviorof a multiagent system in a desired manner. In pinning controlproblems, the goal is for a set of interconnected dynamicalsystems to synchronize onto a given reference trajectory. Thereference trajectory is supposed to be a solution of the un-coupled agents’ dynamics, known a priori, and correspondingto some control objective. A small fraction of the agents isselected in order to receive direct feedback control. Such agentsare called pins or pinned agents. The remaining agents areinfluenced only through their connections with other agents.

Research on pinning control has been carried out from phys-ical and engineering perspectives. The focus is usually on the

Manuscript received September 26, 2014; revised February 16, 2015;accepted February 18, 2015. Date of publication April 30, 2015; date of currentversion June 16, 2015. This work was supported in part by the SwedishResearch Council and in part by the Knut and Alice Wallenberg Foundation.The first two authors would like to thank the EU for providing funding forvisiting KTH under the Erasmus programme. Recommended by AssociateEditor W. Ren.

A. Adaldo, D. Liuzza, G. Shi, D. V. Dimarogonas, and K. H. Johansson arewith ACCESS Linnaeus Centre and School of Electrical Engineering, RoyalInstitute of Technology, Stockholm 10044, Sweden (e-mail: adaldo@kth.se;liuzza@kth.se; guodongs@kth.se; dimos@kth.se; kallej@kth.se).

F. Alderisio is the with Department of Engineering Mathematics, Universityof Bristol, Bristol BS8 1UB, U.K. (e-mail: f.alderisio@bristol.ac.uk).

M. di Bernardo is with the Department of Engineering Mathematics,University of Bristol, Bristol BS8 1UB, U.K., and also with the Depart-ment of Electrical Engineering and Information Technology, University ofNaples Federico II, Naples 80125, Italy (e-mail: m.dibernardo@bristol.ac.uk;mario.dibernardo@unina.it).

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

Digital Object Identifier 10.1109/TCNS.2015.2428531

design of adaptive pinning controllers [6]–[8], or on finding cri-teria for the optimal selection of the agents to control [9]–[11],or on finding sufficient conditions for synchronization [12]–[14]. Such conditions usually relate to the agents’ dynamics,the network topology, and the pinning scheme.

In many scenarios of multiagent coordination, the assump-tion that the network topology is constant over time is unreal-istic. Topology variations are due to imperfect communicationbetween agents, or simply the existence of a proximity rangebeyond which communication is not possible. A large num-ber of papers investigate synchronization [15]–[18] or pinningcontrol [19]–[21] under time-varying interaction topologies.Note that communication failures can usually be regarded asswitching events. Therefore, a pinning control algorithm, whichis intended to be robust against such failures, can be designedby considering the controlled network as a switched system.

Pinning control algorithms have been traditionally designedunder the hypothesis of continuous-time communication. Inmany realistic network systems, however, such hypothesis isnot verified. Also, synchronized sampled communication ishard to obtain. Event-triggered control was introduced to limitthe amount of communication instances for feedback systems[22]. Recently, event-triggered control has been extended tomultiagent systems [23]–[29].

In a realistic multiagent control problem, several challengesare present at the same time: nonlinear dynamics, exogenousreference signals, limited communication capacity, and time-varying interaction topology. In [30], the authors addressed theproblem of event-triggered pinning synchronization consider-ing linear diffusive coupling and unweighted network topolo-gies. In this paper, a more general setup is considered, namely,weighted switching topologies with generic linear interactionsare investigated. A model-based and event-triggered pinningcontrol law is designed, which drives the agent states to ana priori specified common reference trajectory. We derive aset of sufficient conditions under which Zeno behavior [31]is avoided and the agents achieve exponential convergence tothe reference trajectory. Static networks are studied as a specialcase, for which we also prove that there exists a lower boundfor the interevent times in the sequences of updates of thecontrol signals. Differently from most existing works on event-triggered multiagent control, we envision an implementation ofthe control algorithm which does not require agents to exchangestate measurements at each update time. Agents exchange statemeasurements only when they establish their connection. Whenan agent updates its control signal to a new value, it is requiredto broadcast its value to its neighbors in the network. In this

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ADALDO et al.: EVENT-TRIGGERED PINNING CONTROL OF SWITCHING NETWORKS 205

way, it is possible for neighboring agents to predict each others’states consistently.

The rest of this paper is organized as follows. In Section II,we introduce some notation, formalisms, and properties that areused in later sections. In Section III, we define the mathematicalmodel adopted to describe the network to be controlled: westate the control objective and we give the expressions of theproposed event-triggered control law. In Section IV, we provethat the closed-loop system is well-posed and achieves thecontrol objective. In Section V, we provide some numericalexamples to illustrate the effectiveness of the proposed controlstrategy, and compare it to a time-triggered control strategy.Section VI presents some considerations on the robustness ofthe proposed algorithm. Section VII concludes this paper with asummary of our results and some possible future developments.

II. PRELIMINARIES

A. Notation and Mathematical Background

For x ∈ IRn, we denote x[N ] := [xT, . . . , xT]T ∈ IRnN . Fora symmetric square matrix A, A > 0 denotes that A is positivedefinite and A ≥ 0 denotes that it is positive semidefinite.

Definition 2.1: A function f : (t, x) ∈ IR× IRn → IRn isglobally Lipschitz with Lipschitz constant Lf if for all t ∈IR and all x, y ∈ IRn, it holds that ‖f(t, x)− f(t, y)‖ ≤Lf‖x− y‖.

The Kronecker product is denoted as ⊗. We recall someuseful lemmas, which can easily be derived from [32].

Lemma 2.1: Let A ∈ IRn×n have eigenvalues λ1, . . . , λn

and eigenvectors v1, . . . , vn, and B ∈ IRm×m have eigenvaluesμ1, . . . , μm and eigenvectors u1, . . . , um. Then, A⊗B haseigenvalues λiμj and eigenvectors vi ⊗ uj , i = 1, . . . , n, j =1, . . . ,m.

Lemma 2.2: Consider 0 ≤ A1, A2 ∈ IRN×N , 0 < B1, B2 ∈IRn×n, and 0 < c1, c2 ∈ IR. Then, c1(A1 ⊗B1) + c2(A2 ⊗B2) > 0 if and only if A1 +A2 > 0.

Proof: Preliminarly, note that from Lemma 2.1, wehave that A ≥ 0, B > 0 implies A⊗B ≥ 0 and A⊗B >0 ⇐⇒ A > 0, while from the equality (A⊗B)(C ⊗D) =(AC)⊗ (BD), we have (x⊗ y)T[c1(A1 ⊗B1) + c2(A2 ⊗B2)](x⊗ y) =c1(x

TA1x)(yTB1y) + c2(x

TA2x)(yTB2y) for

any x ∈ IRN , y ∈ IRn. Now suppose xT(A1 +A2)x = 0.Since A1, A2 ≥ 0, this implies xTA1x = xTA2x = 0 which,in turn, implies (x⊗ y)T[c1(A1 ⊗B1) + c2(A2 ⊗B2)](x⊗y) =c1(x

TA1x)(yTB1y) + c2(x

TA2x)(yTB2y) = 0. Vice-

versa, suppose that A1 +A2 > 0. Then, at least one ofA1, A2 ≥ 0 must be positive definite and, consequently, atleast one of c1(A1 ⊗B1), c2(A2 ⊗B2) ≥ 0 must be positivedefinite, which implies that their sum is positive definite. �

B. Graph Theory

We define a graph as a pair G = (V,W ) consisting of a setof nodes V = {1, . . . , N} and a time-varying matrix W (t) ={wij(t) ≥ 0} ∈ IRN×N . A graph is undirected if the weightwij(t) = wji(t) for all i, j ∈ V and at all times t; it is simpleif wii(t) ≡ 0 for all i ∈ V .

In a simple undirected graph, two nodes i and j are said to beneighbors or adjacent at time t if wij(t) >0. The value di(t) =

∑Nj=1 wij(t) is the degree of node i at time t. A path between

nodes i and j is a sequence of nodes, starting in i and endingin j or vice-versa, such that every two consecutive nodes areadjacent. A graph is connected if there exists a path betweenany two of its nodes. If a graph is not connected, then its nodescan be partitioned into subsets such that all resulting subgraphsare connected. Each such subgraph is called a component of theoriginal graph.

The Laplacian matrix L(t) = {lij(t)} ∈ IRN×N is defined as

lij(t) =

{di(t), if i = j,−wij(t), if i �= j.

The Laplacian of any simple undirected graph is symmetricwith zero row sum and, therefore, the vector 1[N ] is always aneigenvector with a zero eigenvalue. Also, it can be shown thatthe Laplacian of such graphs is positive semidefinite and thatit has as many zero eigenvalues as components of the graph.In particular, when the graph is connected, the Laplacian hasexactly one zero eigenvalue [5].

C. Pinning Control

For the pinning control problem, we extend the graph formal-ism as follows.

Definition 2.2: Consider a simple and undirected graph G =(V,W ) and a time-varying matrix P (t) = {pij(t) ≥ 0} withpij(t) ≡ 0 for all i �= j. The triple Ga = (V,W, P ) is an aug-mented graph. A node i for which pii(t) > 0 is pinned at time t.We say that a component of the graph is pinned if it contains atleast one pinned node. We also say that Ga is pinned if all of itscomponents are pinned. The matrix P (t) is the pinning matrixof Ga. For any two positive definite and unity-norm matricesC,K ∈ IRn×n and any two positive scalars c, k > 0, the matrix

La(t) = c (L(t)⊗ C) + k (P (t)⊗K) (1)

is called augmented Laplacian of Ga. The smallest eigenvalueof the augmented Laplacian is called augmented connectivityof Ga.

Note that by construction, the augmented Laplacian is pos-itive semidefinite and, therefore, its augmented connectivity isnon-negative. Since the augmented Laplacian is not necessarilypositive definite, the augmented connectivity may be zero.The following lemma shows that positive definiteness of theaugmented Laplacian is determined only by the structure of theaugmented graph, and not by C,K or the scalars c, k.

Lemma 2.3: The augmented Laplacian La of the augmentedgraph Ga is positive definite at time t if and only if Ga is pinnedat time t.

Proof: We are going to prove this property for c = k =C = K = 1, which means that n = 1 and La(t) = L(t) +P (t). Thanks to Lemma 2.2, the property extends automaticallyto generic values of c, k, C,K. Moreover, since the statementcan be applied at any generic time instant, in the proof, we aregoing to omit time-dependency.

Without loss of generality, suppose that the graph has m ≥ 1components and the nodes are ordered so that consecutivenodes belong to the same component. Namely, the first com-ponent contains nodes n0 = 1, . . . , n1, the second component

206 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 2, NO. 2, JUNE 2015

contains nodes n1 + 1, . . . , n2, and the last component containsnodes nm−1 + 1, . . . , nm = N . With such node ordering, theLaplacian is block-diagonal with m blocks L1, . . . Lm. Eachblock can be seen as the Laplacian of the corresponding com-ponent, which is connected by definition. Hence, each block hasexactly one zero eigenvalue. The corresponding eigenvector is1[�i], where �i := ni − ni−1 is the dimension of the ith blockor, equivalently, the number of nodes in the ith component.The pinning matrix P is diagonal by definition. Hence, the aug-mented Laplacian is itself block-diagonal with m blocks L1 +P1, . . . , Lm + Pm and, consequently, its eigenvalues are theunion of the eigenvalues of these blocks. Consider the genericith block Li + Pi. Since Li, Pi ≥ 0, we have xT(Li + Pi)x =0 ⇐⇒ xTLix = 0 and xTPix = 0. The first condition holdswhen x is a scalar multiple of 1[�i], while the second conditionholds when x has zero entries whenever Pi has nonzero entries.Hence, both of them are satisfied at the same time by a nonzerox only if Pi = 0, which means that the ith component is notpinned. On the other hand, if the ith component is pinned, thenthe ith block of La is positive definite. We can conclude that La

is positive definite if and only if Ga is pinned. �

III. PROBLEM STATEMENT

A. System Model and Control Objective

In this section, we define the multiagent system model,the control objective, and the event-triggered control law. Weconsider a network of N interconnected dynamical agents. Thestate of the ith agent is denoted as

xi(t) :=[x(1)i (t), . . . , x

(n)i (t)

]T∈ IRn

and the control input applied to that agent is denoted as

ui(t) :=[u(1)i (t), . . . , u

(n)i (t)

]T∈ R

n.

The state of each agent evolves according to the nonlinearcontrol system

xi(t) = f (t, xi(t)) + ui(t), xi(0) = xi,0 (2)

with t ≥ 0. It is desired that the agents converge to the referencetrajectory r(t) ∈ IRn defined by

r(t) = f (t, r(t)) , r(0) = r0 (3)

with t ≥ 0. We introduce the tracking errors ei(t) := r(t)−xi(t) and the mismatches eij(t) := xj(t)− xi(t) = ei(t)−ej(t). We also introduce the stack vectors

x(t) :=[x1(t)

T, . . . , xN (t)T]T

e(t) :=[e1(t)

T, . . . , eN (t)T]T

u(t) :=[u1(t)

T, . . . , uN (t)T]T

r[N ](t) :=[r(t)T, . . . , r(t)T

]Tall belonging to IRNn. Moreover, we define

F (t, x(t)) :=[f (t, x1(t))

T , . . . , f (t, xN (t))T]T

∈ IRNn.

For convenience, we denote η(t) := ‖e(t)‖. The control goalis to achieve convergence of the agents’ states to the referencetrajectory, in the sense that

limt→+∞

η(t) = 0.

B. Control Strategy

To solve the problem stated before, we propose the followingpiecewise-constant control signal for agent i in (2):

ui(t) = c

N∑j=1

wij

(t(i)ki

)Ceij

(t(i)ki

)

+ kpii

(t(i)ki

)Kei

(t(i)ki

), t ∈

[t(i)ki, t

(i)ki+1

)(4)

where the matrices C,K > 0 and scalars c, k > 0 are designparameters as in (1). Times t(i)ki

when signal ui(t) changes valueare events for agent i. Note that the control signal ui(t) ispiecewise-constant, since it is held constant over each interval[t(i)ki, t

(i)ki+1). Introduce the errors

eij(t) := eij

(t(i)ki

)− eij(t)

ei(t) := ei

(t(i)ki

)− ei(t) (5)

for t ∈ [t(i)ki, t

(i)ki+1). The sequence {t(i)ki

}+∞ki=0 is now defined

recursively as follows:

t(i)ki+1 :=inf

{t>t

(i)ki

:wij

(t(i)ki

)‖eij(t)‖≥ ς(t) for some j∈V

or wij(t) �= wij

(t(i)ki

)for some j ∈ V or

pii

(t(i)ki

)‖ei(t)‖ ≥ ς(t) or

pii(t) �= pii

(t(i)ki

)}(6)

where the threshold function ς(t) is defined as

ς(t) := ς0e−λςt (7)

with ς0 and λς being given positive design parameters. Allsequences are initialized at t = 0. Note that the events relatedto agent i include all of the instants when that agent establishesor loses a connection with another agent or with the reference.The control law (4) is now completely defined.

C. Control Implementation

Let us now discuss the implementation of the control law(4)–(7). We assume that each agent i ∈ V at each update timet(i)ki

computes the new control input ui(t(i)ki) according to (4),

given the values wij(t(i)ki) for all j ∈ V , eij(t

(i)ki) (with j being

a neighbor of i), pii(t(i)ki), and ei(t

(i)ki) (if i is a pin). We also

assume that each agent i is equipped with predictors that canlocally estimate the dynamics (2) of the agents themselves.When two agents i, j connect, they exchange their current statesxi, xj . With such information, they update their control inputsui, uj . According to (6), acquiring a new connection triggersa control update. After the update, the agents broadcast theirnewly computed control inputs to their respective neighbors.Similarly, when the reference node r connects to agent i, it

ADALDO et al.: EVENT-TRIGGERED PINNING CONTROL OF SWITCHING NETWORKS 207

sends its own current state to that agent, which updates itscontrol input ui, and broadcasts it to its neighbors. Hence,neighboring agents know each other’s state and control inputat each connection time. For each neighbor, j agent i runs astate prediction by integrating the equation

˙x(i)

j (t) = f(x(i)j (t), t

)+ uj(t)

where x(i)j denotes the state of agent j predicted by agent i.

A similar prediction is run by agent i for its own state and,if i is pinned, for the reference trajectory. Since the predictorsare based on the exact knowledge of the agent dynamics,the predicted states x

(i)j coincide with the real states xj(t).

Consequently, agent i estimates eij(t) according to (5) withoutcommunicating continuously with neighbors, but predictingxj(t) instead. Similarly, if agent i is pinned, it estimates ei(t)according to (5) without continuously querying the reference.Notice that having piecewise-constant control signals ui(t) im-plies that interagent communication is only necessary at updatetimes, when a newly calculated control input is broadcast.Hence, the control law (4) can be implemented locally, sinceeach agent relies only on information provided by neighboringagents. Similarly, each agent i does not need to be aware of allthe events, such as topology switches, but only of those relativeto pii(t) and wij(t), with j being its neighbor before or afterthe switch.

Remark 3.1: In the proposed implementation, we allowneighboring agents to have up-to-date estimates of each others’states. In principle, such estimates may be used to computetime-continuous control signals, resembling those that wouldbe applied in traditional continuous-time consensus algorithms.However, depending on the application, there can be reasonsto choose piecewise-constant control signals despite havingcontinuous state estimates at disposal. For example, it is not un-common that continuously varying control signals are avoidedin order to reduce actuator wear, or even that the availableactuators are technologically unable to exert a continuouslyvarying control input. Moreover, if the agents were to usecontinuously vary control inputs, the predictors embedded inan agent would need to continuously receive information fromthe neighbors of that agent.

IV. MAIN RESULTS

Consider here a special class of augmented graphs, calledswitching augmented graphs.

Definition 4.1: The augmented graph Ga = {V,W, P} issaid to be switching if the matrices W (t) and P (t) arepiecewise-constant, that is, wij(t), pii(t) ≥ 0 are piecewise-constant for all i, j ∈ V . A discontinuity point of wij(t) iscalled a switch for the pair (i, j) and a discontinuity point ofpii(t) is called a switch for node i. A switching-augmentedgraph is said to have a dwell time τd > 0 if two consecutiveswitches relative to the same pair or the same node are separatedby a time greater than or equal to τd.

Note that in a switching augmented graph, the augmentedLaplacian is a piecewise-constant matrix and the augmentedconnectivity is a piecewise-constant scalar.

We make the following assumptions.Assumption 4.1: The function f in (2) and (3) is globally

Lipschitz with Lipschitz constant Lf > 0.Assumption 4.2: The interactions among agents (2) and (3)

are described by a switching-augmented graph Ga = (V,W, P )with dwell time τd > 0. Moreover, there exists constant positiveupper bounds wij , pii such that

0 ≤wij(t) ≤ wij

0 ≤ pii(t) ≤ pii

for all t ≥ 0 and for all i, j ∈ V .Assumption 4.3: Let α(t) := λa(t)− Lf , where λa(t) is the

augmented connectivity of the augmented graph Ga defined asin Assumption 4.2. There exist two positive constants T and ψsuch that

0 < λς < ψ ≤ 1

T

∫ t+T

t

α(τ)dτ.

Remark 4.1: To satisfy Assumption 4.3, it is not necessaryfor the augmented graph to be always pinned. However, it isnecessary that there exists T > 0 such that within any finitetime window [t, t+ T ], the augmented graph is pinned for somenonzero time interval. The intuition behind Assumption 4.3 isthat in order to guarantee convergence without inducing Zeno[31], the control parameters should be designed in such a waythat the enforced convergence rate λς is slower than the naturalconvergence rate of the network, which is quantified as ψ. If afaster convergence rate is enforced, Zeno behavior may occur.

Theorem 4.1: Consider the pinning control system definedby the dynamics (2), reference (3), and control law (4)–(7). IfAssumptions 4.1, 4.2, and 4.3 hold, then the event sequencesdo not present accumulation points and the normed error η(t)converges exponentially to zero.

Proof: We first prove that no accumulation pointsof events occur. To do so, we note that Assumption 4.2excludes this possibility for events generated by switches.Still, we have to prove that there are no accumulation pointsof events generated by conditions wij(t)‖eij(t)‖ ≥ ς(t) orpii(t)‖ei(t)‖ ≥ ς(t).

Consider the closed-loop dynamics of the error

ei(t) = r(t)− xi(t)= f (t, r(t))− f (t, xi(t))

− c

N∑j=1

wij(t)C (eij(t) + eij(t))

− kpii(t)K (ei(t) + ei(t)) .

If we denote with li(t)T and pi(t)

T, the ith row of the Laplacianand the pinning matrix, respectively, we can rewrite the lastexpression as

ei(t) = f (t, r(t))− f (t, xi(t))−[(cli(t)

T ⊗ C)+(kpi(t)

T ⊗K)]

e(t)

− c

N∑j=1

wij(t)Ceij(t)− kpii(t)Kei(t). (8)

208 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 2, NO. 2, JUNE 2015

Denoting the sum of the last two terms of the above equationwith ξi(t), we have

ei(t) = f (t, r(t))− f (t, xi(t))

−[(cli(t)

T ⊗ C + kpi(t)T ⊗K

)]e(t)− ξi(t).

Denoting ξ(t) := [ξ1(t)T, . . . , ξN (t)T]

T, we can group the pre-

vious equations for i = 1, . . . , N as

e(t) = F(t, r[N ](t)

)− F (t, x(t))− La(t)e(t)− ξ(t)

where La(t) is the augmented Laplacian. From the trig-gering condition (6), we have wij(t)‖eij(t)‖ < ς(t) andpii(t)‖ei(t)‖ < ς(t). Therefore, taking into account that thematrices C,K are unity-norm, we have

‖ξi(t)‖ ≤ (cdi(t) + kpii(t)) ς(t)

where di(t) is the degree of node i. Considering this inequalityfor all i ∈ V , we can write

‖ξ(t)‖ ≤ Δ(t)ς(t) (9)

where

Δ(t) :=

√√√√ N∑i=1

(cdi(t) + kpii(t))2.

Then, we can write

e(t)Te(t) = e(t)T[F(t, r[N ](t)

)− F (t, x(t))

]−e(t)TLa(t)e(t)− e(t)Tξ(t).

Assumption 4.1 and the upper bound (9) yield

e(t)Te(t) ≤ Lfη(t)2 − λa(t)η(t)

2 + η(t)Δ(t)ς(t)

which, if we introduce α(t) as in Assumption 4.3, can berewritten as

e(t)Te(t) ≤ −α(t)η(t)2 + η(t)Δ(t)ς(t).

Hence

η(t) =d

dt‖e(t)‖ =

e(t)Te(t)

‖e(t)‖ =e(t)T ˙e(t)

η(t)

≤ − α(t)η(t) + Δ(t)ς(t). (10)

Applying the comparison lemma [33] to (10) over a timeinterval [t, t+ T ) yields

η(t+ T ) ≤ e−∫ t+T

tα(τ)dτ

η(t)

+

∫ t+T

t

e−∫ t+T

τα(σ)dσ

Δ(τ)ς(τ)dτ.

Under Assumption 4.2, we have

di(t) ≤ di :=

N∑j=1

wij

and consequently

Δ(t) ≤ Δ :=

√√√√ N∑i=1

(cdi + kpii)2

while under Assumption 4.3, we have∫ t+T

τ

α(σ)dσ =

∫ t+T

t

α(σ)dσ −∫ τ

t

α(σ)dσ

≥ψT − (τ − t)α

where

α := max0≤wij(t)≤wij , ∀i,j

0≤pii(t)≤pii, ∀i

α(t).

Therefore, we can bound η(t+ T ) as

η(t+ T ) ≤ e−ψT η(t) + Δe−ψT

∫ t+T

t

e(τ−t)ας(τ)dτ.

Substituting ς(τ) with its expression (7), we obtain

η(t+ T ) ≤ e−ψT η(t) +Δe−ψT

(e(α−λς)T − 1

)α− λς

ς(t).

Note that Assumptions 4.2 and 4.3 guarantee that α− λς > 0.For t = kT , we have

η ((k + 1)T ) ≤ aη(kT ) + bς(kT ) (11)

where a and b are the positive constants

a := e−ψT ,

b :=Δe−ψT

(e(α−λς)T − 1

)α− λς

.

From inequality (11), we can compute

η(kT ) ≤ akη(0) + b

k−1∑h=0

ς(hT )ak−1−h.

Substituting the expressions of a and ς(kT ), we obtain

η(kT ) ≤ e−ψkT η(0) + bς0e−ψ(k−1)T

k−1∑h=0

e(ψ−λς)hT . (12)

By explicitly computing the summation in (12), we obtain

η(kT ) ≤ e−ψkT η(0) + bς0e−ψ(k−1)T e(ψ−λς)kT − 1

e(ψ−λς)T − 1

≤ e−ψkT η(0) + bς0eψT

e(ψ−λς)T − 1e−λςkT .

Taking into account that λς < ψ yields

η(kT ) ≤(η(0) + bς0

eψT

e(ψ−λς)T − 1

)e−λςkT

= k′ης(kT ) (13)

ADALDO et al.: EVENT-TRIGGERED PINNING CONTROL OF SWITCHING NETWORKS 209

where

k′η :=η(0)

ς0+ b

eψT

e(ψ−λς)T − 1.

Observing that α(t) = λa(t)− Lf is lower-bounded by −Lf ,we can write

η(t) ≤ Lfη(t) + Δς(t)

which integrates both sides over an interval [kT, t] with kT ≤t < (k + 1)T , giving

η(t) ≤ eLf (t−kT )η(kT ) + Δς0

∫ t

kT

eLf (t−τ)eλςτdτ

≤ eLf (t−kT )η(kT )

+ Δς0eLf t

e−(Lf+λς)kT − e−(Lf+λς)t

Lf + λς

≤ eLfT

(η(kT ) +

Δ

Lf + λςς(kT )

).

Together with (13), the previous inequality yields

η(t) ≤ k′′ης(kT )

where

k′′η := eLfT

(k′η +

Δ

Lf + λς

).

As kT ≤ t < (k + 1)T , we have that ς(kT ) = eλςT ς((k +1)T ) ≤ eλςT ς(t), which leads to

η(t) ≤ k′′ηeλςT ς(t). (14)

The argument above is valid for all k = 0, 1, . . .; therefore,inequality (14) is valid at all times t ≥ 0. Consider now the dy-namics of ‖ei(t)‖. From (8), we apply the triangular inequality,which, considering Assumption 4.1 and that C,K are unity-norm, yields

‖ei(t)‖ ≤ (Lf ‖ei(t)‖+ (c‖li(t)‖+ kpii(t)) ‖e(t)‖+ cdi(t) + pii(t)k) ς(t). (15)

Under Assumption 4.2, we have pii(t) ≤ pii, di(t) ≤ di, and

‖li(t)‖ ≤ li :=

√√√√2

N∑j=1

w2ij .

Substituting these bounds into (15) and noting from (14) that‖ei(t)‖ ≤ ‖e(t)‖ = η(t) ≤ k′′ηe

λςT ς(t), we can write

‖ei(t)‖≤[(Lf+cli+kpii)k

′′ηe

λςT +cdi+kpii]ς(t)=Ωiς(t)

(16)where

Ωi := (Lf + cli + kpii)k′′ηe

λςT + cdi + kpii.

Now, observe that

ei(t) = −∫ t

t(i)

k

ei(σ)dσ

and therefore

‖ei(t)‖ ≤∫ t

t(i)

k

‖ei(σ)‖ dσ. (17)

Substituting (16) into (17) yields

‖ei(t)‖ ≤ Ωi

∫ t

t(i)

ki

ς(τ)dτ ≤ Ωiς(t(i)ki

)(t− t

(i)ki

).

Hence, the inequality pii(t(i)ki)‖ei(t)‖ ≥ ς(t) cannot be satisfied

as long as

Ωipii

(t(i)ki

)ς(t(i)ki

)(t−t

(i)ki

)<ς(t)= ς

(t(i)ki

)e−λς

(t−t

(i)

ki

)

that is

Ωipii

(t(i)ki

)(t− t

(i)ki

)< e

−λς

(t−t

(i)

ki

).

The above inequality is guaranteed in a nonempty inter-val [t(i)ki

, t(i)ki

+ τ ], where τ > 0 solves the equation Ωipiiτ =

e−λςτ . Therefore, there exists a positive lower bound on thetime needed to have pii(t

(i)ki)‖ei(t)‖ ≥ ς(t) after t(i)ki

.In the same way, it is possible to prove that there ex-

ists a lower bound on the interevent time needed to havewij(t

(i)ki)‖eij(t)‖ ≥ ς(t) after t(i)ki

, by considering

eij(t) = −∫ t

t(i)

ki

eij(σ)dσ =

∫ t

t(i)

ki

(ei(σ)− ej(σ)) dσ

and

‖eij(t)‖ ≤∫ t

t(i)

ki

(‖ei(σ)‖+ ‖ej(σ)‖) dσ.

Therefore, we conclude that event sequences {t(i)ki}+∞ki=0 present

no accumulation points.Exponential convergence of the error norm η(t) follows from

(14), and this concludes the proof. �Remark 4.2: Since events can be generated by switches,

which are exogenous with respect to the agents’ dynamics,two consecutive updates of signal ui—one caused by a switchand one caused by pii(t

(i)ki)‖ei(t)‖ or some wij(t

(i)ki)‖eij(t)‖

meeting the threshold function, may be arbitrarily close in time.For this reason, although we proved that no accumulation pointsof events exist, our algorithm can still generate control updatesthat are close. However, this would not be a Zeno behavior.

Definition 4.2: An augmented graph Ga = (V,W, P ) isstatic if W (t) and P (t) are constant.

Note that in a static-augmented graph, all of the entries wij

and pii, respectively, of W and P are constant scalars, and sois the degree di of node i for i = 1, . . . , N ; moreover, the aug-mented Laplacian is a constant matrix La and the augmentedconnectivity is a constant scalar λa. The following corollarydescends directly from Theorem 4.1.

Corollary 4.1: Consider the pinning control system definedby the dynamics (2), reference (3), control law (4)–(7), and astatic-augmented graph Ga with augmented connectivity λa. If

210 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 2, NO. 2, JUNE 2015

Fig. 1. Illustration of the augmented graph underlying the simulated network.The blue vertices represent the interconnected agents and the red vertexrepresents the reference. The blue edges represent interagent connections, whilethe red dashed edges represent the time-varying actions that the reference exertson two of the agents.

Assumption 4.1 holds and 0<λς <α :=λa−Lf , then the in-

terevent times t(i)ki+1−t(i)ki

are lower-bounded by a positive con-stant and the normed error η(t) converges exponentially to zero.

Remark 4.3: When the graph is static, less conservativebounds can be derived for the normed error and the intereventtimes. Namely, we find

η(t) ≤ kης(t)

with

kη :=η(0)

ς0+

Δ

α− λς

and

ωipii

(t(i)ki+1 − t

(i)ki

)≥ e

−λς

(t(i)

ki+1−t

(i)

ki

)

with

ωi := (Lf + c ‖li‖+ kpii) kη + cdi + kpii.

Here, Δ, li, di, and pii are defined as for a switching graph, butthey are all constant since the graph is static.

V. NUMERICAL EXAMPLES

In order to illustrate the effectiveness of the proposed controlalgorithm, we apply it to a simulated network of N = 5identical Chua oscillators [34]. The individual dynamics of eachoscillator is described by

f(x) =

⎡⎣ a (x2 − x1 − φ(x1))

x1 − x2 + x3

−bx2

⎤⎦ (18)

where

φ(y) := m1y +1

2(m0 −m1) (|y + 1| − |y − 1|) ∀y ∈ IR.

Choosing a = b = 0.9, m0 = −1.34, m1 = −0.73, the oscilla-tors are globally Lipschitz with Lf =3.54. See [27] for details.Let the controls be given by (4) with C = K = I3, interactiongain c = 5, and control gain k = 30. All of the agents areconnected to each other with interaction weight wij = 1.

Fig. 1 provides an illustration of the augmented graph un-derlying the simulated network. Our simulation is set on a time

Fig. 2. Second state variable x(2)i for all agents i = 1, . . . , 5 and reference

r(2), when no control input is applied. The state variables do not converge tothe reference.

Fig. 3. Same variables as in Fig. 2, but with the proposed control algorithmapplied. As predicted by our main result, all state variables converge to thereference.

interval [0, 30]s. At the beginning of the experiment, two agentsare pinned with pii = 1, which yields λa = 6.14. At t = 0.75,one pin is removed, so that λa = 2.88. At t = 0.90 s, the tworemaining pins are removed as well, which yields λa = 0. Att = 1.0, the original pinning scheme is restored and the cyclerepeats itself every second. It is easy to see that Assumption 4.2holds with τd = 1. If we set T = 1, we can calculate

ψ =1

T

T∫t

α(τ)dτ = 1.50.

For the threshold function, we pick ς0 = 1.0 and λς = 0.30, sothat Assumption 4.3 holds. For all of the agents, the initial statevalues are chosen in the domain of attraction of an uncontrolledChua’s oscillator with the given parameters.

Fig. 2 shows the trend of the second state variable of all ofthe agents and the reference when no control input is applied.Fig. 3 shows the trend of the same state variables when theproposed control input is applied. Fig. 4 shows in detail thesame state trajectories in the time interval [0.0, 1.0]. Fig. 5shows the control updates for each of the agents during the

ADALDO et al.: EVENT-TRIGGERED PINNING CONTROL OF SWITCHING NETWORKS 211

Fig. 4. Second state variable x(2)i for all agents i = 1, . . . , 5 and r(2) for the

reference, in the time interval [0.0,1.0], when the proposed control algorithm isapplied.

Fig. 5. Instants when a control update is triggered during the time interval[0.0, 1.0]. The vertical positions of the markers indicate which agent updates itscontrol signal.

TABLE IAVERAGE INTEREVENT TIME FOR EACH AGENT IN THE TIME INTERVAL

[0.0, 30.0], WITH THE PROPOSED CONTROL ALGORITHM APPLIED

first second of the simulation, while Table I shows the averageinterevent time exhibited by each agent during the simulation.It is possible to observe that the lowest of these values is above0.05 s, which means that the agent that updates its control inputmore often performs less than 20 updates/s on average.

To show the advantages of using an event-triggered controllaw instead of a time-triggered control law, we also ran aparallel simulation with time-triggered control updates. Weused the same network with the same initial conditions, but weexcluded the connection failures that characterized the originalsimulation. We chose a fixed updating period for all of thenodes, equal to 0.04 s. Hence, all of the nodes update their

Fig. 6. Second state variable x(2)i for all agents i = 1, . . . , 5 and r(2) for

the reference, when the control algorithm for robust bounded convergence isapplied.

control input more often, on average, than in the originalsimulation. Despite more conservative settings, the closed-loopsystem turned out to be unstable. Simulations are omitted herefor the sake of brevity.

VI. ROBUSTNESS

The previous sections focus on the scenario where exactknowledge of the agents’ model is available. It was shownthat Zeno behavior can be avoided even if perfect convergenceis required. However, if disturbances or modelling errors arepresent, we can modify the algorithm so that bounded conver-gence can be achieved with the absence of Zeno behavior. Wedefine bounded convergence as

lim supt→∞

η(t) ≤ ε

for some ε >0. What needs to be changed in the algorithm isthe expression of the threshold function ς(t). In the absenceof disturbances, we set ς(t) = ς0e

−λςt which forces all of theerror signals to eventually shrink to zero. If disturbances arepresent, we can set ς(t) = ς00 + ς0e

−λςt, which will force theglobal error e(t) to converge to a ball of radius proportional toς0. Proof of a similar convergence result is given in [27].

To corroborate the considerations before, we reconsider theexample proposed in Section V and assume that the predictorsembedded in the agents’ controllers rely on the model (18), withthe parameters given in Section V. However, we assume that thereal agents have parameter b equal to 0.84, 0.88, 0.92, and 0.96,respectively. The reference agent has b = 0.90. The thresholdfunction is modified as ς(t) = ς00 + ς0e

−λςt, with ς00 = 0.1.Figs. 6–8 and Table II illustrate the results of the simulation.We can see that bounded convergence is achieved and Zenobehavior does not occur.

VII. CONCLUSION

We proposed an algorithm for event-triggered pinning syn-chronization of complex networks with possibly switching

212 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 2, NO. 2, JUNE 2015

Fig. 7. Second state variable x(2)i for all agents i = 1, . . . , 5 and r(2) for the

reference, in the time interval [0.0, 1.0], when the algorithm for robust boundedconvergence is applied.

Fig. 8. Instants when a control update is triggered in the control algorithm forrobust bounded convergence, during the time interval [0.0, 1.0]. The verticalpositions of the markers indicate which agent updates its control signal.

TABLE IIAVERAGE INTEREVENT TIME FOR EACH OF THE AGENTS IN THE

TIME INTERVAL [0.0, 30.0], WHEN THE ALGORITHM FOR

ROBUST BOUNDED CONVERGENCE IS APPLIED

topologies. We found conditions for networked nonlinear sys-tems with event-triggered controllers under which Zeno behavioris excluded and the norm of the error signal vanishes expo-nentially. A constant lower bound on the interevent times hasalso been provided for the case of static networks. Numericalexamples have been presented to validate the theoretical results.

Some viable extensions of this work include the applicationof the proposed algorithm to more general classes of networks,such as networks with asymmetric couplings between theagents and networks where errors in the communication canoccur, such as delays and packet drops.

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Antonio Adaldo received the M.Sc. degree in au-tomation engineering from the Univerisity of NaplesFederico II, Naples, Italy, in 2013 and is currentlypursuing the Ph.D. degree in automatic control at theACCESS Linnaeus Centre in the School of ElectricalEngineering, KTH Royal Institute of Technology,Stockholm, Sweden.

His research interests include hybrid systems, non-linear control, and coordination in multiagent systems.

Francesco Alderisio received the M.Sc. degreein automation engineering from the Univerisity ofNaples Federico II, Naples, Italy, in 2013 andis currently pursuing the Ph.D. degree in engi-neering mathematics at the University of Bristol,Bristol, U.K.

Since 2014, he has been involved in the EuropeanProject AlterEgo. His current research interests in-clude human-robot interaction and modeling humancoordination in multiplayer games.

Davide Liuzza received the Ph.D. degree in au-tomation engineering from the Univerisity of NaplesFederico II, Naples, Italy, in 2013.

He was a visiting Ph.D. student in the Depart-ment of Applied Mathematics, University of Bristol,Bristol, U.K., in 2012 and at the ACCESS LinnaeusCentre, Royal Institute of Technology (KTH),Stockholm, Sweden, from 2012 to 2013. Since 2013,he has been a postdoctoral researcher at the Auto-matic Control Laboratory at KTH. His research inter-ests are in networked control systems, coordination

of multiagent systems, incremental stability of nonlinear systems, as well asnonlinear control and hybrid systems.

Guodong Shi received the Ph.D. degree in sys-tems theory from the Academy of Mathematics andSystems Science, Chinese Academy of Sciences,Beijing, China, in 2010.

From 2010 to 2014, he was a Postdoctoral Re-searcher at the ACCESS Linnaeus Centre, RoyalInstitute of Technology (KTH), Stockholm, Sweden.Since 2014, he has been with the College of Engi-neering and Computer Science, The Australian Na-tional University, Canberra, Australia, as a Lecturerand Future Engineering Research Leadership Fellow.

Dr. Shi was selected as the Triennial IFAC Young Author Prize Finalist in2011, and was a co-receipt of the Best Paper Award in Control Theory from the11th World Congress on Intelligent Control and Automation in 2014.

Dimos V. Dimarogonas (M’10) received the Diplomain Electrical and Computer Engineering in 2001 andthe Ph.D. degree in mechanical engineering from theNational Technical University of Athens (NTUA),Athens, Greece, in 2007.

Between 2007 and 2009, he was a PostdoctoralResearcher at the ACCESS Linnaeus Centre, RoyalInstitute of Technology (KTH), Stockholm, Sweden.Between 2009 and 2010, he was a Postdoctoral Asso-ciate at the Laboratory for Information and DecisionSystems (LIDS) at the Massachusetts Institute of

Technology (MIT), Cambridge, MA, USA. Currently, he is an AssociateProfessor at the ACCESS Linnaeus Centre, Royal Institute of Technology(KTH), Stockholm, Sweden. His current research interests include multiagentsystems, hybrid systems and control, robot navigation, and networked control.

Dr. Dimarogonas was awarded a Docent in Automatic Control from KTHin 2012. He serves on the Editorial Board of Automatica and the IET ControlTheory and Applications and is a member of the Technical Chamber of Greece.

Mario di Bernardo (SM’06–F’12) received thePh.D. degree in nonlinear dynamics and control fromthe University of Bristol, Bristol, U.K., in 1998.

Currently, he is Full Professor of Automatic Con-trol at the University of Naples Federico II, Naples,Italy. He is also Professor of Nonlinear Systems andControl at the University of Bristol.

In 2007, Prof. di Bernardo was bestowed the titleof Cavaliere of the Order of Merit of the ItalianRepublic for scientific merits from the Presidentof Italy, HE Giorgio Napolitano. In 2012, he was

elevated to the grade of Fellow of the IEEE for his contributions to the analysis,control, and applications of nonlinear systems and complex networks. In 2006and 2009, he was elected to the Board of Governors of the IEEE Circuits andSystems Society. In 2015 he became an appointed member of the Board ofGovernors of the IEEE Control Systems Society. From 2011 till 2014 he servedas Vice President for Financial Activities of the IEEE Circuits and SystemsSociety. He has authored or co-authored more than 200 international scientificpublications, including more than 100 papers in scientific journals, over 100contributions to refereed conference proceedings, a unique research monographon the dynamics and bifurcations of piecewise-smooth system, and two editedbooks. He serves on the Editorial Board of several international scientificjournals and conferences. He is Deputy Editor-in-Chief of the IEEE TRANS-ACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS. He is alsoAssociate Editor of the IEEE TRANSACTIONS ON CONTROL OF NETWORKSYSTEMS, Nonlinear Analysis: Hybrid Systems, and is Associate Editor of theConference Editorial Board of the IEEE Control System Society and the Euro-pean Control Association (EUCA). He is regularly invited as Plenary Speakersin Italy and abroad and has been organizer and co-organizer of several scientificinitiatives and events. He received funding from several agencies and industry.

Karl Henrik Johansson (F’13) received the M.Sc.and Ph.D. degrees in electrical engineering fromLund University, Lund, Sweden, in 1992 and 1997,respectively.

Currently, he is Director of the ACCESS LinnaeusCentre and Professor at the School of ElectricalEngineering, KTH Royal Institute of Technology,Stockholm, Sweden. He is a Wallenberg Scholar andhas held a six-year Senior Researcher Position withthe Swedish Research Council. He is also headingthe Stockholm Strategic Research Area ICT The

Next Generation. He has held visiting positions at University of California,Berkeley (1998–2000), Berkeley, CA, USA, and the California Institute ofTechnology (2006–2007), Pasadena, CA, USA. His research interests arein networked control systems; hybrid and embedded systems; as well asapplications in transportation, energy, and automation systems.

Dr. Johansson is currently on the Editorial Board of IEEE TRANSACTIONSON CONTROL OF NETWORK SYSTEMS and the European Journal of Control.He received the Best Paper Award of the IEEE International Conference onMobile Ad-hoc and Sensor Systems in 2009 and the Best Theory Paper Awardof the World Congress on Intelligent Control and Automation in 2014. In 2009,he was awarded Wallenberg Scholar, as one of the first ten scholars from allsciences, by the Knut and Alice Wallenberg Foundation. He was awarded anIndividual Grant for the Advancement of Research Leaders from the SwedishFoundation for Strategic Research in 2005. He received the triennial YoungAuthor Prize from IFAC in 1996 and the Peccei Award from the InternationalInstitute of System Analysis, Austria, in 1993.