Finite-Time Consensus for a Network of Perturbed Double Integratorsby Second-Order Sliding Mode Technique

Alessandro Pilloni, Alessandro Pisano, Mauro Franceschelli and Elio Usai

Abstract— This paper considers the problem of achievingconsensus in a network of agents whose dynamics consist ofperturbed double integrators. The considered class of perturba-tions is a bounded additive uncertainty, different for each agent,and such that the maximal magnitude is the only informationavailable a-priori. Agents are supposed to interact through anundirected, static and connected, communication topology. Themain contribution of the present work is a discontinuous localinteraction rule which is able to provide finite time consensuswhile completely rejecting the effect of the perturbations. Thelocal interaction rule, whose performance is investigated byLyapunov approach, is accompanied by a set of simple tuningrules for setting the algorithm’s parameters. Simulation resultsdemonstrate the effectiveness of the suggested scheme.

I. INTRODUCTION

Control applications in distributed and cooperative envi-ronments are studied with growing interest in recent years.The success of this paradigm can be attributed to the factthat many complex systems are naturally represented bya network of interacting subsystems or “agents”. Trans-portation networks, power grids, networks of mobile robots,localization systems are just few examples of engineeringapplications of networked systems [1], [2], [3]. The winningfeatures of networked structured applications comes fromtheir reduced cost, flexibility, scalability/reconfigurability androbustness to failures. An overview of the recent researchtrends in cooperative control can be found in [1], [4], [5]

In particular, Multi-Agent Systems deal with the analysis ofhow distributed interaction architectures affect the achieve-ment of a collective behavior through only local interaction[6]. A crucial problem in cooperative control is to achievea distributed agreement on parts of the networked system’sstate (i.e. barycenter, average phase, a common attitude)exploiting only local interaction and information exchange.This problem, referred to as consensus, has been widelystudied for both single and double integrators dynamics [6]-[15].

This work focuses on a consensus algorithm for perturbeddouble integrator dynamics. Since these systems are suitableto model networks of point-mass vehicles, they are of higherinterest as compared to the single integrator dynamics due

The research leading to these results has received funding from the Euro-pean Community’s Seventh Framework Programme FP7/2007-2013 underGrant Agreement n°257462 Project ”HYCON2-Network of excellence”.

A. Pilloni, A. Pisano, M. Franceschelli and E. Usai are withthe Department of Electrical and Electronic Engineering (DIEE),University of Cagliari, Cagliari 09123, Italy. E-mail addresses: alessandro.pilloni, pisano,mauro.franceschelli,eusai@diee.unica.it

to the possible applications in rendezvous, formation control,flocking and sensor networks [4], [7], [8], [9], [10], [11].

With reference to an undirected topology, asymptoticconsensus for double integrators dynamics is presented in [8],[12]. In [8] several nonlinear protocols have been presented,while in [12] linear interaction rules were dealt with byincluding in the analysis the effect of measurement delays.Sufficient conditions for achieving asymptotic consensusin presence of nonlinear second-order agents’ dynamics isconsidered in [13]. More recently, the finite-time consensusproblem for a network of double integrators is studied in [6]for an undirected static topology and in [10] for an undirectedstatic or switching topology using non-smooth protocols.However, none of the above works takes into account thepresence of perturbation terms in the agents’ dynamics bystudying “ideal” double integrators only.

In [14] finite time consensus with perturbation terms isinvestigated in the case of single integrators with undirectedand switching network topology.

Here, we propose a discontinuous consensus algorithm forachieving finite-time agreement in a network of perturbedagents with a static and undirected communication graphwhile completely rejecting the effect of the disturbances.The class of perturbations considered in the present workonly assume them to be bounded, in magnitude, by a-prioriknown constants. Complete rejection of such a wide classof disturbance was never achieved in the existing literaturequoted above. The proposed local interaction rule can bethought as a distributed version of the well-known TwistingSecond Order Sliding Mode Algorithm [15], [16], [17] witha non-trivial function of the neighbors states used as slid-ing manifold. All significant robustness properties againstuncertainties and disturbances typical of Variable StructureControl Theory are inherited [18].

The performance of the proposed protocol is investigatedby Lyapunov-based analysis, and simple tuning rules toadjust the algorithm parameters are provided.

The paper is organized as follows: in Section 2 prelimi-naries of graph theory and multi-agent systems are providedto clarify the notation that will be used in the paper. Theproblem statement and the proposed local interaction ruleare presented in Section 3. Section 4 presents a constructiveLyapunov analysis which demonstrates that the proposed dis-continuous protocol solves the finite-time consensus problemfor a network of double integrators affected by boundedunknown perturbations. To corroborate the theoretical results,simulation results are shown in Section 5. Section 6 providessome concluding remarks and hints for further investigation.

52nd IEEE Conference on Decision and ControlDecember 10-13, 2013. Florence, Italy

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II. PRELIMINARIES AND NOTATION

With reference to a network of N agents, we considerthe associated undirected communication graph G = (V ,E),where V = 1, . . . ,N is the set of agents and E ⊆ V2

represent the set of edges. The set of neighbors of the i-thagent is defined as N−

i = j ∈ V/i : (i, j) ∈ E. Topolog-ical information associated with graph G is encoded in theLaplacian Matrix L= [Li j] ∈ RN×N where

Li j :=

∣∣N−

i

∣∣ if i = j−1 if (i, j) ∈ E0 otherwise

(1)

and |N−i | is the cardinality of the i-th agent neighbor set.

For an undirected graph, L is a symmetric and positivesemi-definite matrix [5], [19], [20]. In addition, L has realeigenvalues that can be ordered sequentially as follows 0 =λ1 ≤ ·· · ≤ λN .

The null eigenvalue λ1 has multiplicity equal to the num-ber of connected components of G and the corresponding leftand right eigenvectors are respectively 1N = col(1, . . . ,1) ∈RN and 1T

N , where the expression col(.) denotes a columnvector. Thus, the following identities hold:

L ·1N = 0N , 1TN ·L= 0T

N (2)

In addition, the next property holds [21]

λ2‖ζ‖22 ≤ ζTLζ (3)

where λ2 is the smallest nonzero eigenvalue of L, known asalgebraic connectivity [22], and ζ ∈ RN is any vector suchthat 1T

Nζ = 0.

III. PROBLEM STATEMENT

Let us consider a connected network consisting of Nagents where each agent is governed by the followingperturbed double-integrator dynamics:

xi (t) =Axi (t)+B (ui (t)+ϑi (t)) , i ∈ V , (4)

withA=

(0 10 0

), B =

(01

), (5)

where xi = col(xi,1,xi,2) ∈ R2 represents the state of the i-th agent, ui(t) ∈ R is the control input, and ϑi(t) ∈ R is anunknown perturbation which corrupts the agent’s dynamics.A compact representation of the collective network dynamicscan be expressed as follows:

X = (IN×N⊗A)X +(IN×N⊗B)(U +Θ) (6)

where ⊗ denotes the Kronecker Product, IN×N ∈ RN×N isthe identity matrix of order N, and

X = col(x1, . . . ,xN) ∈ R2N

U = col(u1, . . . ,uN) ∈ RN

Θ = col(ϑ1, . . . ,ϑN) ∈ RN(7)

Then, let

Z1 = col(x1,1, . . . ,xN,1) ∈ RN

Z2 = col(x1,2, . . . ,xN,2) ∈ RN (8)

be two vectors which stack together, respectively, the positionand velocity of each agents. The collective dynamics (6) canbe expressed in the following regular form:

Z1 = Z2Z2 =U +Θ

(9)

We make the next assumption.Assumption 1: G is a connected undirected graph and

the disturbance vector Θ is bounded in accordance with:

|ϑi(t)| ≤Πi ≤Π with Π = maxi∈VΠi= ‖Θ‖∞ < ∞ (10)

The objective of the present work is to present a noveldiscontinuous local interaction rule guaranteeing the achieve-ment of the next finite-time consensus conditions:

|zi,1(t)− z j,1(t)|= 0|zi,2(t)− z j,2(t)|= 0 , t ≥ T, T < ∞, ∀ i, j ∈ V (11)

The next local interaction rule is suggested:

ui (t) =−a · sign(LiZ1)−b · sign(LiZ2) , i ∈ V (12)

where a and b are positive tuning constants, Li is the i-th rowof the Laplacian Matrix and sign(·) is defined as follows:

sign(S) =

1 if S> 0[−1,1] if S= 0−1 if S< 0

(13)

The proposed discontinuous protocol in (12) can be thoughtas a distributed version of the ”Twisting” second order slidingmode algorithm [15], where a non-trivial function of theneighbors states is used as sliding manifold. The argumentsof the sign functions take the next explicit form:

LiZk = ∑j∈N−

i

(zi,k− z j,k

), i ∈ V , k = 1,2 (14)

Define

σ (LZk) = col(sign(L1Zk) , . . . ,sign(LNZk)) , k = 1,2(15)

the local interaction protocol (12) can be rewritten at thenetwork level in the following compact form:

U (t) =−a ·σ (LZ1)−b ·σ (LZ2) (16)

Remark 1: Due to the proposed discontinuous local in-teraction rule, and the possibly discontinuous nature ofthe external disturbances (which are only supposed to beuniformly bounded), the closed loop network dynamics willbe discontinuous and the resulting solution notion needs tobe discussed and clarified. For a differential equation withdiscontinuous right-hand side, following [23], we understandthe resulting solution in the so-called Filippov sense as thesolution of an appropriate differential inclusion, the existenceof which is guaranteed (owing on certain properties ofthe associated set-valued map) and for which noticeableproperties, such as absolute continuity, are in force. Thereader is referred to [24] for a comprehensive account ofthe notions of solution for discontinuous dynamical systems.

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IV. CONVERGENCE ANALYSIS

In this section, we investigate the performance of protocol(16) by means of Lyapunov-Based Analysis. Simple tuningrules for the control gains a and b will be derived. Thefollowing disagreement vectors are defined:

δk(t) = Zk(t)−1N ·1T

NN

·Zk(t), k = 1,2 (17)

The achievement of the consensus condition (11) correspondsto the annihilation of vectors δ1(t) and δ2(t) in finite time.It is worth noting that both the disagreement vectors satisfythe following conditions [19]:

1TN ·δk = 0, Lδk =LZk k = 1,2 (18)

along with the property (3), which can be rewritten as:

δTk Lδk ≥ λ2‖δk‖2

2, k = 1,2 (19)

The dynamics of δ1 and δ2 can be easily derived by differ-entiating (17) and considering (9), (16) and (18) as follows:

δ1 = δ2

δ2 =−a ·σ (Lδ1)−b ·σ (Lδ2)+ Θ(t)+Ω(t) ·1N(20)

where

Θ(t) = Θ(t)− 1N ·1TN

NΘ(t) (21)

Ω(t) =a ·1T

Nσ (Lδ1)+b ·1TNσ (Lδ2)

N(22)

We are now in position to state the main result of the paper.Theorem 1: Consider the collective dynamics (9) and let

Assumption 1 be satisfied. Consider the local interaction rule(16) and let tuning parameters be selected according to

a > b+Π, b > Π. (23)

Then, the finite-time consensus property (11) is achieved.Proof: The proof is broken into three simple consecutivesteps.

. Equi-uniform stability: Consider the following candidateLyapunov function:

V (t) = a · ‖Lδ1‖1+12·δT

2 Lδ2 (24)

The considered Lyapunov function is a locally Lipschitzfunction and it is not differentiable when any entry of vectorLδ1 is zero. Thus, we refer for stability analysis to theLyapunov Generalized Theorem for non-smooth analysisreported in [25], which makes use of Clarke’s generalizedgradient [26] and involves a set-valued form for the resultingLyapunov function time derivative. In the remainder, werefer to the computation method illustrated in [25], where aLyapunov analysis involving an analogous sum-of-absolute-value Lyapunov function term was dealt with. We do notreport here all the necessary technicalities justifying thecorrectness of adopting the chain rule to compute the timederivative V (t), which exists almost everywhere in the formof an appropriate set valued map, by referring the reader to

the works [24], [25], [27] where discontinuous systems andnon-smooth Lyapunov tools analogous to those involved inthe present analysis were discussed in detail.

The corresponding time derivative takes the form

V (t) = aδT1 Lσ (Lδ1)+δ

T2 Lδ2 (25)

Considering (20) into (25), and taking into account thatΩ(t) is a scalar and that 1T

N ·L= 0 , the next simplificationcan be made after evaluating its time-derivative along thetrajectory of the perturbed dynamic (20)-(22)

V (t) = aδT2 Lσ (Lδ1)−aδT

2 Lσ (Lδ1)−b ·δT2 Lσ (Lδ2)

+ δT2 LΘ(t)+Ω(t)δT

2 L ·1N

= −b ·δT2 Lσ (Lδ2)+δ

T2 LΘ

= −b · ‖Lδ2‖1+δT2 LΘ (26)

Applying the Holder’s Inequality and taking into account(10), the next estimation, involving the last sign-undefinedterm in the right hand side of (26), can be made

|δT2 LΘ| ≤ ‖Lδ2‖1

∥∥Θ∥∥

∞≤Π · ‖Lδ2‖1 (27)

on the basis of which it yields

V (t)≤−(b−Π) · ‖Lδ2‖1 (28)

It follows by (23) that V (t)≤ 0, therefore the uncertain sys-tem (20)-(22) is equi-uniformly stable [28]. Indeed, initializ-ing the system in an arbitrarily vicinity of the origin such thatV (0)≤ R0, the uncertain system (20)-(22) cannot leave thisvicinity, regardless of whichever admissible uncertainty Θaffects the system. In other words, inequality (28) guaranteesthe boundedness of the perturbed disagreement vector’s statetrajectories in (20)-(22).

. Global equi-uniform asymptotic stability: To begin with,let us note that by virtue of (28) all possible solutions of(20)-(22), initialized at t0 ∈ R within the invariant compactset

DR =(δ1,δ2) ∈ R2N : V (δ1,δ2)≤ R

(29)

are a priori estimated by

supt∈[t0,∞]

V (δ1,δ2)≤ R (30)

which, considering the Lyapunov function definition (24),implies the following relations:

‖Lδ1‖1 ≤ R/a , δT2 Lδ2 ≤ 2R (31)

Furthermore, by virtue of (19), the following chain of in-equalities holds:

λ2‖δ2‖22 ≤ δT

2 Lδ2 ≤ λN‖δ2‖22 (32)

which allows to manipulate (31) as

‖Lδ1‖1 ≤ R/a , ‖δ2‖2 ≤√

2R/λ2 (33)

The idea behind the reminder of the proof is inspiredby the Extended Invariance Principle [28] and it is basedon constructing a parameterized family of local Lyapunovfunction VR(δ1,δ2), R > 0 such that each VR(δ1,δ2) is well

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posed on the corresponding compact set DR and its timederivative, initialized within DR is negative definite andyields the desired stability and convergence properties. Aparameterized Lyapunov function VR(δ1,δ2), R > 0 withthe properties above, can be obtained by augmenting theLyapunov function (24) as follows:

VR(t) =V (t)+κR ·U(t) (34)

where U(t) is the sign-indefinite function

U(t) = δT1 Lδ2 (35)

and κR is a proper positive constant. By Young’s inequalityand taking into account (33), it then follows that

U(t) = δT1 Lδ2 ≥− 1

2

(‖Lδ1‖2

2+‖δ2‖22

)≥

≥ − 12

(Ra · ‖Lδ1‖1+‖δ2‖2

2

) (36)

Taking into account (34), (36), and the left inequality in (32),it follows that

VR ≥[

a− κRR2a

]· ‖Lδ1‖1+

12[λ2−κR] · ‖δ2‖2

2 (37)

which, in accordance with the following restriction:

κR < min

2a2

R , λ2

(38)

guarantees that VR(t) is not negative in spite of having addedthe sign indefinite term U(t).

In turns, by differentiating (35) along the trajectory of theuncertain dynamic (20)-(22), it results:

U = δT2 Lδ2−a · ‖Lδ1‖1−b ·δT

1 Lσ (Lδ2)+δT1 LΘ (39)

In the right hand side of (39) there is a negative definiteterm (the second one) and three positive or sign-indefiniteterms. Bounds to the latter de-stabilizing terms are nowderived. As for the first entry, by the Holder inequality andthe right inequality in (33), it derives that

|δT2 Lδ2| ≤ ‖Lδ2‖2‖δ2‖2 ≤

√2 ·Rλ2· ‖Lδ2‖1 (40)

As for the term −b · δT1 Lσ (Lδ2) one can exploit

once more the Holder inequality, taking into account that‖σ(Lδ2)‖∞ = 1. to assess that

|b ·δT1 Lσ (Lδ2) | ≤ b‖Lδ1‖1‖σ (Lδ2)‖∞ ≤ b‖Lδ1‖1 (41)

The last term in the right hand side of (39) satisfies theinequality

|δT1 LΘ| ≤ ‖Lδ1‖1‖Θ‖∞ ≤Π · ‖Lδ1‖1 (42)

Therefore relation (39) can be estimated as

U ≤√

2·Rλ2· ‖Lδ2‖1−a · ‖Lδ1‖1+b · ‖Lδ1‖1+

+Π · ‖Lδ1‖1(43)

Then, by combining (28) and (43), we can upper-estimatethe time derivative of (34) as follows:

VR ≤−c1 · ‖Lδ1‖1− c2 · ‖Lδ2‖1 ≤≤−cR · (‖Lδ1‖1+‖Lδ2‖1)

(44)

where

c1 = κR (a−b−Π) , c2 = b−Π−κR√

2R/λ2 (45)

cR = min

c1,c2

(46)

and with the coefficient κR subject to the next inequalitywhich is stricter than (38)

κR < min

2a2

R ,λ2,√

λ22R · (b−Π)

(47)

The exponential convergence towards the manifold Lδ1 =Lδ2 = 0, can be shown taking into account the following

VR ≤ a · ‖Lδ1‖1+ 12 |δ

T2Lδ2|+κR · |δT

1 Lδ2| ≤≤ c1 · ‖Lδ1‖1+ c2 · ‖Lδ2‖1 ≤≤ cR · (‖Lδ1‖1+‖Lδ2‖1)

(48)

with

c1 = a+κR

√2Rλ2

, c2 =12

√2Rλ2

, cR = max

c1, c2

(49)

which can be derived applying the Holder Inequalities andsubstituting the bounds in (33), along with (44) which leadsto

VR(δ1,δ2)≤−ρR ·VR(δ1,δ2) (50)

with

ρR≤cR

cR=

min

κR (a−b−Π) ,b−Π−κR√

2R/λ2

max

a+κR√

2R/λ2,√

R/2λ2 (51)

which implies the exponential decay of VR(t).. Global equi-uniform finite-time stability: Consider the

uncertain disagreement vector dynamics (20)-(22) rewrittenin the following compact form:

δ = Φ(δ)+Ψ(t) , δ = col(δ1,δ2) ∈ R2N (52)

where

Ψ(t) = col(0N ,Θ+Ω ·1N

)= col(ψ1, . . . ,ψ2N)∈R2N (53)

is an uncertain vector of uniformly bounded functions, ac-cording to

|ψi(t)| ≤Mi, Mi =

0, i = 1, . . . ,N

Π+a+b, i = N +1, . . . ,2N (54)

and

Φ(δ) = col(δ2,−a ·σ (Lδ1)−b ·σ (Lδ2)) == col(φ1, . . . ,φ2N) ∈ R2N (55)

is a vector of piece-wise continuous functions.Since the proof of finite-time convergence will be based

on the properties of homogeneous systems, we introduce thedefinition of local homogeneity and the Quasi-homogeneityPrinciple, both taken from [16], upon which our reasoningwill rely on. Homogeneity concepts were applied for finite-time stability analysis of consensus protocols for first-orderagents in, e.g, [29].

Definition 1: A piece-wise continuous vector field

f(t) = col( f1(x), . . . , fn(x)) ∈ Rn with x ∈ Rn (56)

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is called locally homogeneous of degree q ∈ R with respectto the dilation vector r = (r1, . . . ,rn), ri > 0, if for all ε > 0

fi(ε

r1x1, . . . ,εrnxn,ε

−qt)= ε

q+ri fi (x, t) , i = 1, . . . ,n(57)

Theorem 2: [16] Quasi-homogeneity Principle. The un-certain system (52) is globally equi-uniformly finite-timestable if the following conditions are satisfied:

1) The right-hand side of (52) consists of a locallyhomogeneous piece-wise continuous function Φ(δ) ofdegree q < 0 with respect to dilation r and a piece-wise continuous function Ψ(t) whose components ψi,i = 1, . . . ,n are locally uniformly bounded by constantsMi ≥ 0 within a homogeneity ball;

2) Mi = 0 whenever q+ ri > 0;3) The uncertain system (52) is globally equi-uniformly

asymptotically stable around the origin.In accordance with the Definition 1, we observe that Φ(δ)is locally homogeneous of degree q =−1 with respect to thedilation vector r = [ri] ∈ R2N taking the form

ri =

ri = 2, with i = 1, . . . ,Nri = 1, with i = N +1, . . . ,2N (58)

Then, since system (52) is globally equi-uniformly asymp-totically stable, and the following conditions hold:

q+ ri > 0 ∀ i : Mi = 0 ⇒ i = 1, . . . ,Nq+ ri ≤ 0 ∀ i : Mi 6= 0 ⇒ i = N +1, . . . ,2N

(59)all the conditions of Theorem 2 are satisfied, and the pro-posed local interaction rule (12) guarantees the finite-timestability of system (52). As a consequence, the consensusproblem (11) is solved in finite-time. The proof of Theorem 1is completed.

Remark 2: Remarkably, the proposed controller, can pro-vide the global achievement of consensus by using a controlauthority that remains always bounded, thereby addressingissues of actuator saturation [30], [31], [8]. This is not thecase, e.g., with more classical consensus protocols, eitherlinear [4] or nonlinear [6], which imply an unboundedcontrol effort when the discrepancy between agent’s initialconditions tends to infinity.

Remark 3: Denoting δ = col(δ1,δ2) = [di] ∈ R2N , asdiscussed in Section 3 of [16], an upper bound on thesettling-time function after which the network of agents willbe synchronized (i.e., δ1 = δ2 = 0) is

T (t0,δ(t0))≤ τ (δ(0),ER)+1

1−2q

(ςR−1)q

s(ς) (60)

where

τ (δ(0),ER) = supδ(·,t0,δ(0))

infT ≥ 0 : δ(t, t0,δ(0)) ∈ ER

(61)for all t0 ∈ R, t ≥ t0 +T , and

s(ς) = supτ

(δ(0),E 1

2 ς

)(62)

11

2

3 4

5

6

7

89

10

Fig. 1. Communication topology for a network of ten agents.

are respectively the reaching-time functions and its upper-bound. ER denotes an ellipsoid of the form

ER =

δ ∈ R2N :

√2N

∑i=1

(di

Rri

)2

≤ 1

(63)

located within a homogeneity ball ς ≥ ε0R, and ε0 ≤ ε is apositive constant called lower estimate of the homogeneityparameter ε . For further details see ([16], Section 3).

V. NUMERICAL SIMULATIONS

To demonstrate the efficacy of the proposed protocol,a connected network of 10 agents, displayed in Figure 1,has been considered. Agents’ dynamics are governed byequations (4)-(5). The disturbances ϑi(t) are selected asharmonic signals with randomly time-varying chosen coeffi-cients guaranteeing that the upper-bound Π in (10) is Π = 2.The initial states are chosen as xi,1(0) = i, xi,2(0) =−5.5+ iwith i = 1, . . . ,10. For the computation of the control gains,according to the tuning rule (23), the values a = 15 andb = 10 were chosen. The continuous time network dynamics(9) has been simulated by using the Runge-Kutta 4-th orderintegration method with fixed step size equal to 10−3s.

Figure 2 shows the time evolutions of the disagreementvectors (17). It can be seen that consensus is achieved aftera transient of about 4 seconds. Figure 3 depicts a comparisonbetween the time evolution of the Lyapunov Function (34)during three tests where different control gain pairs (a,b)were adopted, according to the legend inserted in the plot.Comparing those curves it is seen how higher values ofthe control gains a and b can reduce the settling time toreach consensus. Overall, the collective network’s dynamicsbehaves as expected and the disturbance effect is rejectedin the sense that exact consensus is achieved in spite of thepresence of unknown perturbations corrupting the agents’dynamics.

VI. CONCLUSIONS AND FUTURE WORKS

This paper proposed a discontinuous distributed localinteraction rule for achieving finite-time consensus in anetwork of double integrators agents affected by boundeddisturbances. Agents are supposed to interact through anindirected, static and connected, communication topology. ALyapunov-Based analysis confirm effectiveness of the pro-posed algorithm to solve the finite-time consensus problemand provides a very simple set of tuning rules for adjustingthe algorithm parameters. Complete disturbance rejection has

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0 2 4 6 8 10−10

−5

0

5

10δ 1(t

)

0 2 4 6 8 10−10

−5

0

5

10

Time [s]

δ 2(t)

Fig. 2. Trajectories of the disagreement vectors δ1(t) and δ2(t).

0 2 4 6 8 100

100

200

300

400

500

600

700

800

900

Time [s]

VR

(t)

TEST1: a=15,b=10TEST2: a=7.5,b=5TEST3: a=5,b=2.5

Fig. 3. Transient evolution of the Lyapunov Function VR(t).

been provided. Numerical simulations confirm the theorydeveloped.

Further investigations will focus on relaxing the topolog-ical restriction for directed, possibly switching topologies.

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