Nonlinear Consensus Algorithms with Uncertain Couplings

NONLINEAR CONSENSUS ALGORITHMS WITHUNCERTAIN COUPLINGS

Anton V. Proskurnikov

ABSTRACT

Distributed algorithms for synchronization and consensus in multi-agent networks are considered. The agents are assumedto be linear of arbitrary order, the interaction topology may switch, and the couplings are uncertain, assumed only to satisfy certainquadratic constraints. Using the Kalman-Yakubovich-Popov lemma and absolute stability theory techniques, consensus criteriafor the networks of this type are obtained. These criteria extend a number of known results for agents with special dynamics andare close in spirit to the celebrated circle criterion for the stability of Lurie systems.

Key Words: Networks, consensus, synchronization, absolute stability, KYP lemma.

I. INTRODUCTION

The problem of multi-agent consensus (also calledagreement or controlled synchronization) has recentlyattracted considerable attention in different research commu-nities. The main subject of these projects is the synchronismachieved by means of local interactions between the agents(via communication or otherwise). Such a synchronismunderlies numerous natural phenomena and is the corner-stone of many technical designs and engineering approaches.Examples, just to mention a few, include numerous forms ofcollective behavior in complex biological and man-madesystems, such as flocking, swarming and rendezvous [1–3];synchronization in complex networks [4]; and oscillatorensembles [5–8]. The ideas of many consensus policies taketheir origin from distributed algorithms in computer sciences,averaging procedures in the theory of stochastic matrices, andagreement procedures in expert communities coming fromapplied statistics.

The most investigated consensus protocols are thosewith linear couplings [1,2,9,10], which are widespread andrecognized as being simple and effective. Nevertheless, aprogressively growing number of control and physical appli-cations require nonlinear consensus algorithms that cannot beanalyzed by standard linear tools (such as convergence crite-ria for infinite matrix products, Laplacian matrix decomposi-tion, and frequency-domain methods). Such algorithmscome from, for instance, oscillator networks with periodic

couplings [5,6], coordination with range-restricted sensoring[11–13], or nonlinear agreement procedures [14]. Under real-world conditions, linear protocols may become nonlinearbecause of analog-digital conversions, quantization effects,and nonlinear distortions in measurements. The mentionedchallenges stimulated the rapid development of nonlinearconsensus theory.

One of the first major breakthroughs in this area wasconcerned with the exploration of nonlinear averaging proce-dures for first-order agents [13–15]. The common property ofsuch algorithms is that the convex hull of the agents (consid-ered as the points in a vector space) is provided to be nesteddue to a special choice of the velocities. Namely, each agentmoves into the relative interior of the convex hull of itself andits neighbors. The celebrated paper by Moreau [15] estab-lishes the convergence of those contracting procedures in thediscrete-time case using the convex hull of the agents as aset-valued Lyapunov function (its diameter being commonLyapunov function). Analogous reasoning is applicable torendezvous algorithms on the plane [13] and continuous-timeconsensus protocols [14]. Another class of results for nonlin-ear consensus protocols deals with passive agents [16,17] andexploits Lyapunov functions constructed from individualstorage functions of the agents. For non-passive nonlinearlycoupled agents, e.g. double integrators, no common ways forestablishing synchronization seem to be known except forspecial types of nonlinearities [2] (e.g. hyperbolic tangents)explored by ingenious Lyapunov functions.

While most of the recent results on nonlinear consensusprotocols concern agents with special dynamics (passive,multiple integrators, etc.), in the present paper, we considernetworks of general multiple input multiple output (MIMO)agents with switching interaction topology. Unlike the previ-ously mentioned works, the topology is neither assumed tohave positive dwell-time, nor even be piecewise-continuous

Manuscript received February 11, 2012; revised November 4, 2012; acceptedAugust 18, 2013.

The author is with St.Petersburg State University, Department of Mathematics andMechanics and Institute for Problems of Mechanical Engineering RAS,Universitetsky pr. 28, St. Petersburg, 198504, RUSSIA (e-mail: [email protected]).

The paper was supported by RFBR, grants 14-08-01015 and 12-01-00808, 13-08-01014.

© 2014 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

Asian Journal of Control, Vol. 16, No. 5, pp. 1277–1288, September 2014Published online 20 F ebruary 2014 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.838

but supposed only to be a measurable function of time. Twoimportant kinds of consensus algorithms are considered.Those of the first type require the interaction graph to beundirected and have nonlinear couplings satisfying symmetryconditions resembling Newton’s Third Law. Algorithms ofthe second type are linear and satisfy a certain balance con-dition; they are applicable for directed topology. In bothcases, the full knowledge of couplings may be unavailable, sothey are proposed only to satisfy some quadratic constraints[18]. Using the absolute stability approach, a condition for therobust consensus (in the mentioned class of uncertainties) isobtained, which is closely related to the celebrated circlecriterion for stability of Lurie systems [18,19]. The proposedcriterion extends a number of known results on synchroniza-tion in nonlinearly coupled networks and is among the firstaddressing the issue of consensus robustness. It extends, forinstance, the results from [16,17] to the case of non-passiveMIMO agents, allows one to obtain synchronization criterionfor harmonic oscillators from [7], and provides new consen-sus results for double integrator agents and for strictlypassifiable [20] agents.

The paper generalizes results of previous works [21,22]in several directions. On one hand, we consider the case ofMIMO agents and general quadratic constraints whereas theearlier papers [21,22] were devoted to the case of single inputsingle output (SISO) agents and scalar couplings only. On theother hand, we consider not only undirected, but also directedinteraction graphs. Also, we consider new applications of theobtained criterion, concerning double integrator agents andstrictly passifiable agents.

The paper is organized as follows. Section II introducessome basic concepts from the graph theory. Section III isdevoted to the formulation of the problem, the main assump-tions are given in Section IV. Section V presents the mainresult of the paper (the sufficient condition for consensus),and Section VI illustrates some applications to agents withspecial dynamics. Section VII offers the proof of the mainresult.

II. PRELIMINARIES FROM THEGRAPH THEORY

A pair G = (V, E) of two finite set V (the set of nodes)and E ⊂ V × V (the set of arcs) is called a (directed) graph.The node v is connected to the node w in G if (v, w) ∈ E. Anysequence of nodes v1, v2, . . . , vk with (vi, vi+1) ∈ E for i = 1, 2,. . . , k − 1 is called a path between v1 and vk. The graph isstrongly connected if a path between any two differentnodes exists. Given a graph G = (V, E), the mirror graphˆ ( , ˆ )G V E= is obtained by inserting all inverted arcs, i.e.ˆ {( , ) : ( , ) }E E w v v w E= ∈∪ . A graph coinciding with its

mirror is said to be undirected. Throughout the paper, GN

stands for the class of all graphs G = (VN, E) with the node setVN = {1, 2, . . . , N} and set of arcs E containing no loops((v, v) ∉ E for any v ∈ VN). The adjacency matrix (ajk(G)) isdefined as follows: (ajk(G)) is 1 if (k, j) ∈ E and 0 otherwise.The Laplacian matrix of G is given by

L G

a a a

a a a

a a a

jj

N

N

jj

N

N

N N Njj

( ) =

− … −

− … −

− − …

=

=

=

∑∑

11 12 1

21 21 2

1 2 1

� � � �NN∑

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

. (1)

If G is undirected, then L(G) = L(G)T and λ1(G) (the leasteigenvalue of L(G)) is equal to 0, thus L(G) ≥ 0 [1]. Thesecond eigenvalue λ2(G), called the algebraic connectivity ofG, may be defined by [23]:

λ2

2

1

2

1

( ) min( )( )

( ).,

,

G Na G z z

z zz

ij j ii j

N

j ii j

N=−

−=

=

∑∑

(2)

The minimum in (2) is over the set of all z ∈ RN with zk ≠ zj

for some j, k. One has λ2(G) > 0 if and only if the undirectedgraph G is connected. Moreover, denoting the minimalnumber of arcs one has to delete to break the graphconnectivity by e(G) ≥ 0, the inequality holds [23]:

λ π2 2 1( ) ( ) cos .G e G

N≥ −⎛⎝⎜

⎞⎠⎟ (3)

Following [1], we define the algebraic connectivity of adirected graph G by (2) and denote it as λ2(G) despiteit no longer being an eigenvalue of L(G). Since

a G a G a G a Gjk kj jk kj( ) + ( ) ≥ ( ) = ( )ˆ ˆ , thus λ λ2 21

2G G( ) ≥ ( )ˆ , for

a strongly connected graph G N∈G , one has:

λ π π2 1 1G e G

N N( ) ≥ ( ) −⎛⎝⎜

⎞⎠⎟ ≥ −ˆ cos cos . (4)

The estimates (3),(4) and many others [24] may be usefulwhenever precise computation of λ2 is complicated (e.g. thegraph is unknown or too large).

III. PROBLEM FORMULATION

Throughout the paper, we deal with a team of identicalagents, indexed 1 through N ≥ 2 and governed by a commonMIMO state-space model:

�x t Ax t Bu t y t Cx tj j j j j( ) ( ) ( ), ( ) ( ).= + = (5)

Here, t ≥ 0, j = 1, 2, . . . , N and xj ∈ Rd, uj ∈ Rm, yj ∈ Rp standfor the state, control, and output of the jth agent, respectively.


Asian Journal of Control, Vol. 16, No. 5, pp. 1277–1288, September 20141278

The model (5) is assumed to be controllable and observable.The control inputs are affected by interactions (viacommunication or otherwise) between the agents.The interaction topology is described by a graphG t V E tN N( ) [ , ( )]= ∈G : the output of the kth agent exertsinfluence on the jth one the if and only if (k, j) ∈ E(t).Specifically, we examine distributed control protocols of thefollowing type:

u t t y t y tj jk k j

k k j E t

( ) ( , ( ) ( )).:( , ) ( )

= −∈∑ ϕ (6)

The mappings φjk:[0; +∞] × Rp → Rm are referred to ascouplings and determine interaction “strength” between theagents. The aim of the paper is to give conditions under whichsuch a protocol establishes consensus among the agents in thefollowing sense.

Definition 1. The protocol (6) provides the output consensusif the following two claims hold for all initial states (xj(0)) andsome constant M > 0:

lim ( ) ( ) , ;t

j ky t y t k j→+∞

− = ∀0 (7)

W W( ) : ( ) ( ) ( ) .t x t x t M tj k

j k

= − ≤ ∀ ≥≠∑ 2

0 0 (8)

If, additionally, W( )t → 0 as t → +∞, we say that theprotocol (6) provides the state consensus. We say thatthe exponential state consensus is established ifW W( ) ( )t Me t≤ −α 0 for some constants M, α > 0.

The state consensus implies the output consensus. The con-verse is true under the non-restrictive condition in practice ofuniform continuity at zero point.

Proposition 2. The output consensus implies the state con-sensus if limsup ( , )

y tjk t y

→ ≥=

0 00ϕ for any j, k.

Proof. Due to (6) and (7), the output consensus and theassumption limsup ( , )

y tjk t y

→ ≥=

0 00ϕ imply that |uj(t)| → 0 as

t → +∞. Taking by definition Xjk: = xk − xj, Ujk: = uk − uj, andYjk: = yk − yj, one obtains �X AX BUjk jk jk= + and Yjk = CXjk,where the pairs (A, B) and (A, C) are controllable and observ-able, respectively, Ujk(t) → 0 and Yjk(t) → 0 as t → +∞. Thus,Xjk(t) → 0 for any k, j q.e.d.

Remark 3. It may be noticed that we bound ourselves withstatic controllers (6) whereas consensus problems for high-order agents often require more general dynamic controllers[1,25,26] (observer-based, etc.)

�η η ξ η ξξ ϕ

j j j j j j

j jk k j

t t u t K t D t

t t y t y

( ) ( ), ( ) ( ) ( )

( ) ( , ( ) (

= + = += −

A B

ttk k j E t

)).:( , ) ( )∈∑ (9)

Actually, the closed-loop system (5),(9) may be reducedto one of the form (5),(6) by replacing xj(t) withx t col x t tj j j( ) : ( ( ), ( ))= η , uj(t) with ξj(t) and A, B, C withaugmented matrices A, B, C, as follows:

AA BK

BBD

C C: , : , : .= ⎡⎣⎢

⎤⎦⎥

= ⎡⎣⎢

⎤⎦⎥

= [ ]0

0A B

Such a reduction enables one to apply the consensus criteriadeveloped below to networks (5),(9), as illustrated by theexample from Section 6.3.

IV. MAIN ASSUMPTIONS

The main assumptions come to the connectivity ofthe interaction topology (Assumption 4), the symmetry orbalance condition on the couplings (Assumption 5), and thesector conditions for the couplings. We start with the assump-tion about the underlying graph.

Assumption 4. The graph G(t) is strongly connected for allt ≥ 0. The function G(·) is Lebesgue measurable, i.e.G−1(Γ) ⊂ R is a measurable set for any Γ ∈GN .

Preserving connectivity (in some sense) is necessary toprevent the agents from dissemination into separate clustersthat do not interact, thus cannot synchronize. Our nextassumptions concern the couplings φjk.

Assumption 5. For any pair j ≠ k and t ≥ 0, at least one of thefollowing two statements is valid:

(a) G(t) is undirected and φjk(t, y) = −φkj(t, −y);(b) The couplings are linear φjk(t, y) = wjk(t)y and the gains

wjk(t) ∈ Rp×m satisfy the balance condition

w t w t jjk

k k j E t

kj

k j k E t

( ) ( ) .:( , ) ( ) :( , ) ( )∈ ∈∑ ∑= ∀ (10)

The symmetry condition from (a) resembles Newton’s ThirdLaw, whereas the balance condition (b) is similar in spirit tothe mass or energy preservation law (the summary inflowat the node equals the cumulative outflow). In someapplications, e.g. oscillator networks [5], Assumption 5 holdsdue to exactly these laws.

Although the consensus condition (7) says nothingabout the asymptotics of outputs yj, under Assumption 5,the consensus implies that y t Ce xj

tA

t( ) − →

→∞�0 0 ∀j, where

A. V. Proskurnikov: Nonlinear Consensus Algorithms with Uncertain Couplings


1279

�xN

x jj

N

0 1

10=

=∑ ( ). Indeed, u jj

N

=∑ =1

0 due to Assumption

5. Summing up the equations from (5) yieldsy t NCe xjj

N tA( )=∑ =

1 0� ; thus, our claim is evident from (7).

In the present paper, we focus on the case where thecouplings may be unknown but satisfy some quadraticconstraint [18]. Precisely, ϕ jk ∈S( )F for all j, k whereF : � � �m p× → is a Hermitian form and S( )F stands forthe set of functions φ:[0; +∞) × Rp → Rm such that the claimshold:

(i) φ(t, 0) ≡ 0 and φ(t, y) is measurable in t for all y andcontinuous in y for almost all t;

(ii) For any compact subset K ⊂ R\{0},

inf ( ( , ), ) .,y K t

t y y∈ ≥

>0

0F ϕ (11)

Thus the consensus criterion should be given in terms of Fand the coefficients A, B, C, but not the couplings themselves.Such a criterion automatically ensures consensus for all cou-plings from the class S( )F that satisfy Assumption 5. Wenote that one of the simplest and most widespread classes ofquadratically constrained functions consists of scalar-valuedmaps satisfying the conventional sector condition with givenslopes [18,19], see Section 5.2 for details.

V. MAIN RESULTS

In Subsection 5.1, the main result of the paper is pre-sented, which establishes sufficient conditions for consensusin general MIMO case. Those conditions are especially trans-parent and convenient to use in the scalar case (k = m = 1),which is the case discussed in Subsection 5.2.

5.1 Consensus criterion for the MIMO agents case

We start by introducing some auxiliary notations. LetWx(λ) = (λI − A)−1B and Wy(λ) = CWx(λ) be the transfermatrices of the plant (5) from u to y, x, respectively, and let θstand for the minimal algebraic connectivity of the interactiontopology:

θ λ=≥

min ( ( ))t

G t0

2 (12)

Given a Hermitian form F ( , )u y as follows

F ( , ) ( ) ,ϕ ϕ ϕ ϕy Re qy y Qy R= − −* * * (13)

(where y k∈� , ϕ ∈�m and q ∈ Rm×p, Q = Q* ∈ Rp×p,R = R* ∈ Rm×m) and a matrix Λ∈ ×� p m, let

Π Λ Λ Λ Λ ΛF ( )( )

.= + + +−

q q QN

R* * *θ 1

2 1(14)

The following theorem is the main result of the paper.

Theorem 6. Suppose that Assumptions 4 and 5 holdand ϕ jk j k∈ ∀S( ) ,F where F is a quadratic form (13) withQ ≥ 0, Γ ≥ 0. Assume also that

(a) there exists matrix K ∈ Rm×p such that A − NBKC isHurwitz and F ( , )Ky y y> ∀ ≠0 0;

(b) for any frequency ω ∈ R such that det(iωI − A) ≠ 0,the following inequality is true

ΠF ( ( )) .W iy ω ≥ 0 (15)

Then, the protocol (6) provides the output consensus.Furthermore, if ε > 0 exists such that ∏ ≥F ( )( )W iy ω

( ) * ( )W i W ix xε ω ω for any ω, then the exponential stateconsensus is established.

The proof of this theorem will be given in Section VII.By Theorem 6, Conditions (a) and (b) imply the robust

consensus in the sense that (7) holds for all couplingsϕ jk ∈S( )F and time-varying interaction graphs G(t)satisfying Assumptions 4 and 5. Condition (b) is a frequency-domain inequality similar in flavor to the circle criterion forabsolute stability under quadratic constraints, where thesimilarity becomes more apparent in the scalar case (seeRemark 12).

The technical assumption (a) requires, first, existence oflinear mappings y ↦ Ky in the class S( )F , and, second, forsome of these maps the controller uj(t) = −NKyj(t) stabilizes theplant (6). The first of these claims is typically a non-restrictiveassumption that is fulfilled, for instance, in the scalar case wherethe quadratic constraint shapes into sector inequalities (see Sub-section 5.2). The second claim is more restrictive, for instance,not all MIMO systems may be stabilized by a static feedback.Nevertheless, this is unavoidable for robust consensus with theprotocol (6), as shown by the following.

Proposition 7. Under the robust consensus, (a) holds withany K ∈ Rm×p such that F ( , )Ky y y> ∀ ≠0 0.

Proof. The output consensus with complete graph G(t) ≡ G0

and φjk(t, y) = Ky implies the state consensus by Proposition2. Since N is an eigenvalue of the Laplacian L(G0), the matrixA − NBKC is Hurwitz [27, theorem 1].

In the case where (a) does not hold, more general pro-tocols of the form (9) may be put in use, as discussed inRemark 3 and illustrated in Subsection 6.3 for second-orderagents without velocity measurements.

Remark 8. In the conditions for consensus given byTheorem 6, the underlying topology of the network is con-cerned only by the multiplier θ in (14). Since, by assumption,Q ≥ 0, formal replacement of θ by its lower estimate retains



sufficiency for consensus. This observation is useful when-ever the exact computation of θ is complicated. Then, manyavailable constructive estimates of the algebraic connectivity,like (3) or (4), may be used; we refer the reader to [24] fortheir survey. Moreover, the value of θ is unimportant if Q = 0.

Remark 9. Assumption 4, together with the continuity of thecouplings φjk, implies, thanks to the celebrated Carathéodorytheorem, that, for any initial data x x j j

N0 10= =( ( )) a solution the

closed-loop system (5),(6) exists for t ∈ [0; t0), wheret0 = t0(x0) > 0. Along with the proof of Theorem 6, it will beshown (see Remark 21) that this solution is infinitelyprolongable. Notice that we do not state the uniqueness of thesolution.

5.2 Consensus for scalar sectorial couplings

In this paragraph, we interpret the result of Theorem 6for SISO agents and scalar couplings that satisfy the sectorinequalities with known slopes [18,19]. In other words,k = m = 1 and φjk ∈ S[α; β] where S[α; β] is a set of functionsφ:[0; +∞) × R → R such that φ(t, 0) ≡ 0, φ(t, σ) is measurablein t for all σ, continuous in σ for almost all t ≥ 0 and

α ϕ σσ

ϕ σσ

βσ σ

< ≤ <∈ ≥ ∈ ≥inf

( , )sup

( , ),

, ,K t K t

t t0 0

(16)

for any compact set K ⊂ R\{0}. It follows from (16) that thegraph of φ(t, ·) lies strictly between the lines ξ = ασ andξ = βσ everywhere except for the origin.

We note that, if α < 0 and β > 0, the protocol (6) doesnot provide consensus unless A is a Hurwitz matrix, eventhough all assumptions are satisfied. Since we are mostlyinterested in agents with unstable open-loop dynamics, weexclude the case α < 0, β > 0 from consideration, thus focus-ing on the cases where either 0 ≤ α < β or α < β ≤ 0. Moreo-ver, since the second of them is reduced to the first one by thesubstitution (α, β, B) ↦ (−β, −α, −B), we shall consider onlythe first case.

We first transform the inequalities (16) into quadraticconstraint. We introduce the constants

γβ α

δ ααβ

=+

≥ =+

≥−

10

10

1, , (17)

and the Hermitian forms as follows:

Fα β ϕ ϕ δ γ ϕ ϕ; ( , ) : , ,y Re y y y= − − ∈2 2 � (18)

Πα β λ λ θδ λ γ λ; ( )( )

, .= + +−

∈ReN

2

2 1� (19)

Proposition 10. S[ ; ] ( );α β α β=S F ; Π ΠFα β α β; ;≡ .

Proof. The sector inequalities (16) may be rewrittenas ( )( ) ( ) ( , );ϕ α β ϕ αβ ϕα β− − = + >− −y y y1 11 0F , where φ: =φ(t, y) and the latter inequality is uniform in t ≥ 0 and y ∈ K(for any compact K ⊂ R\{0}). The second claim is seen from(14),(19).

Proposition 10 allows one to obtain the following con-sensus criterion for the SISO agents case.

Theorem 11. Suppose that Assumptions 4 and 5 are satisfied,φjk ∈ S[α; β]∀j, k where 0 ≤ α < β ≤ ∞, and the following twoclaims hold:

(a) there exists μ ∈ (α; β) such that the matrix A − μNBCis Hurwitz;

(b) for any frequency ω ∈ R such that det(iωI − A) ≠ 0,the following inequality is true

Πα β λ λ ω; ( ) : ( ).≥ =0 for W iy (20)

Then, the protocol (6) provides the output consensus.If ε > 0 exists such that Πα;β(Wy(iω)) ≥ ε | Wx(iω)|2, theexponential state consensus is established.

This result immediately follows from Theorem 6 (applied forF F:= 0 , R: = γ, Q: = δ) and Proposition 10. Indeed, (a)coincides with (i) and (b) is nothing more than (ii) fromTheorem 6.

Inequality (20) is non-trivial since the Hermitian formΠα;β is not non-negative definite: its discriminant1

4

2

11 0

δγθN −

−⎡⎣⎢

⎤⎦⎥≤ since θ ≤ N by (2) and δγ ≤ 1/4.

Remark 12. Condition (b) means that the Nyquist curve{Wy(iω)} lies outside the set D, which is the open half-plane

z Re zN

∈ < −−

⎧⎨⎩

⎫⎬⎭

� :( )

γ2 1

for α = 0 and the disk

D = ∈ − <{ : }z z z� 0 0ρ for α > 0,

zN

0 01

2

1

21

2

1= − = −

−δθρ

δθθδγ

, .

Here, the first claim is trivial since α = 0 ⇒ δ = 0 by (17). The

second claim is valid since Re z zN

+ +−

≥ ⇔δθ γ| |

( )2

2 10

z z Re z z z− + − ≥ ⇔ /∈ρ20 0

2022 0 D.

The geometrical interpretation given by Remark 12highlights the similarity between the conditions presentedby Theorem 11 and the celebrated circle criterion forstability of Lurie systems [18,19]. Consider two agents(N = 2) applying the protocol (6) with two-directionalcommunication (E(t)={(1,2),(2,1)}) and the coupling func-tions φ12 = −φ21 ∈ S[α; β]. Taking X(t) = x2(t) − x1(t),



1281

Y(t) = y2(t) − y1(t), U(t) = u2(t) − u1(t) and Φ(t, Y) = −2φ12(t,Y) ⇒ Φ ∈ S[−2β; −2α], one has

�X t AX t BU t U t t Y t( ) ( ) ( ), ( ) ( , ( )).= + = Φ (21)

By the circle criterion [18] the equilibrium X = 0 of System(21) is exponentially stable if A + kBC is Hurwitz for somek ∈ [−2β; −2α] (which follows from (a) since θ = 2) andthe Nyquist curve { ( ) : }W iy ω ω ∈ ⊂� � lies strictly outsidethe disk based on the diameter [(−2α)−1 + 0i; (−2β)−1 + 0i](or the half-plane {z: Re z < (−2β)−1} in the case of α = 0).This disk (or half-plane) coincides with the mentioned set D.

VI. ILLUSTRATIVE EXAMPLES

Now, we illustrate that Theorems 6 and 11, applied toagents with special dynamics, provide improvement of recentresults in the area, as well as some new results.

6.1 Consensus among passive agents

Let the transfer matrix Wy be square (thus, k = m) andsatisfy the positive realness condition:

W i W iy y( ) ( )* .ω ω ω+ ≥ ∀ ∈0 � (22)

This implies their passivity [17,19], provided that (a) fromTheorem 11 holds. Consensus and synchronization of passivesystems have earned a considerable interest [16,17]. Most ofthe related results deal with nonlinear agents but the topologyis either time-invariant or with positive dwell time, and thecouplings are time-invariant. Although this paper isconcerned with only linear agents, on the positive side, itconsiders the most general case of measurable time-varyinginteraction topology and admits non-stationary couplings.

Taking the quadratic form F ( , ) :ϕ ϕy yT= andobserving that (22) implies (15) for any θ and

F ( , ) ( )Ky y y K K yT T= +1

22 ∀K ∈ Rm×m, we obtain the fol-

lowing corollary of Theorem 6.

Theorem 13. Let Assumptions 4 and 5 be valid, (22) hold,ϕ jk ∈S( )F with F(φ, y) = φTy, and there exists matrixK ∈ Rm×m such that K + KT > 0 and A − BKC is Hurwitz.Then, the protocol (6) establishes the output consensus.

In the scalar case the result of Theorem 13 becomes espe-cially simple.

Theorem 14. Let k = m = 1, Assumptions 4 and 5 be valid,(22) hold, φjk ∈ S[0; +∞], and A − μBC be Hurwitz for someμ > 0. Then, the protocol (6) establishes the output consensus.

Theorem 14 encompasses results from [17,28], con-cerning networks of first order-integrators with linear bal-anced and symmetric nonlinear couplings. It implies theresult on synchronization of Kuramoto oscillators with initialphases from (−π/2; π/2) [6]. Another example is the result of[7]. It concerns a network of identical harmonic oscillators� �q yj j= ∈ , �y q uj j j= − +ω0

2 and asserts that the y-outputconsensus is established by the linear balanced protocol(6) (Assumption 5b holds) with constant gains wjk > 0.Theorem 14 generalizes this result on time-varying gainsand nonlinear couplings. Indeed, W z z zy ( ) ( )= −2

02ω

and Wy(iω) + Wy(iω)* = 2ReWy(iω) = 0; thus, the feedbackuj = −μyj stabilizes the jth agent for any μ > 0.

6.2 Consensus among strictly passifiable agents

A natural generalization of passivity is passifiability: alinear agent is called passifiable if it may be made passive byan appropriate linear feedback. This is implied by the strictpassifiability [20] which implies, omitting some details, thatthe system has stable zero dynamics and a relative degree ofone. It is known that a strictly passifiable system may bestabilized by static high-gain feedback [20], so it is expect-able that a protocol (6) with sufficiently “strong” couplingsestablishes the output consensus between the agents, as wasshown in [25]. Unlike [25], dealing with linear time-invariantnetworks, we examine nonlinear consensus protocols withswitching topology that seem to have been unexplored in theliterature.

To simplify matters, we bound ourselves with SISOagents only: k = m = 1. Under such an assumption, strictpassifiability of the agent means that CB > 0 and the polyno-mial ψ(λ) = det(λI − A)Wy(λ) is Hurwitz (i.e., the agent isminimum-phase).

The following theorem shows that such a consensusholds whenever the couplings are strong enough.

Theorem 15. Suppose that the agents are strictlypassifiable, Assumptions 4 and 5 hold, and φjk ∈ S[α; +∞],where, for θ = mint≥0λ2(G(t)), one has

α αθ

ω ω ωωω

≥ = = −∈

*: sup ( ), ( ) :( )

( ).

12

�f f

ReW i

W i

y

y

(23)

Then, the output consensus is provided. Moreover, if α > α*and det(iωI − A) ≠ 0 for any ω ∈ R, then the exponential stateconsensus is provided.

The sup in (23) is finite since f(ω) = |ψ(iω)|−2

(ψ(iω))χ(−iω) + ψ(−iω)χ(iω)) with χ(z): = det(zI − A) is welldefined and continuous for all ω ∈ R, and f(ω) → −CAB/(CB)2

as ω → ±∞. The quantity α* is the coupling “strength” thresh-old, above which the consensus is necessarily achieved.



The first claim of Theorem 15 is immediate fromTheorem 11 with β = ∞ (which yields that δ = α, γ = 0) since(a) of Theorem 11 is valid for sufficiently large μ thanksto the strict passifiability [20], and (20) holds due to (23).Moreover, Πα;∞(Wy(iω)) ≥ (α − α*)θ | Wy(iω)|2. Since CB > 0and det(iωI − A) ≠ 0, a constant ξ > 0 exists such that|Wy(iω) | ≥ ξ | Wx(iω)| for all ω ∈ R. This proves the secondclaim since Πα;∞(Wy(iω)) ≥ ε | Wx(iω)|2 for ε: = ξ2θ(α − α*).

6.3 Consensus among double-integrator agents

The consensus problem for networks of double integra-tors has recently attracted considerable interest because ofvarious applications to multi-vehicle formation control[2,29,30]. We consider three types of consensus algorithmsfor the second-order agents as follows:

�� z u j Nj j= ∈ = …, , , , .1 2 (24)

All of these protocols assume relative positions to bemeasurable and differ by the type of the velocitymeasurements. Algorithms of the first type require access tothe agent’s absolute velocity, those of the second type userelative velocities only, and protocols of the third type do notmeasure the velocities at all.

We start with the protocols, using velocities of theagents in the global reference frame and analogous to thatfrom [30], involving, however, nonlinear couplings:

u pz t q z z q z zj j jk k j k j

k j E t

= − + − + −∈∑� � �ϕ ( , ( ) ( )).

( , ) ( )

0 1 (25)

The closed-loop system (24),(25) is the same as if thestandard protocol (6) were applied to agents:

�� z pz u y q z q zj j j j j j+ = = +, .0 1 (26)

Notice that, if p > 0, q0 > 0, q1 ≥ 0, then feedback uj = −μyj

stabilizes the plant (26) for any μ > 0. Since Wy(iω) = [q0 +q1iω][(iω)2 + p(iω)−1 and ReWy(iω) = (q1p − q0)(p2 + ω2)−1 forthe agent (26), it is seen that (20) holds for α = θ = 0 if

inf ( ) min ,( )

.ω

ω β∈

−

= −⎛⎝⎜

⎞⎠⎟≥ −

−�ReW i

q p q

p Ny

1 0

2

1

02 1

Theorem 11 implies the following result:

Corollary 16. Suppose that Assumptions 4 and 5 hold, p > 0,q1 ≥ 0, q0 > 0 and φjk ∈ S[0; β] with β ≤ +∞. Let alsop2β−1 + 2(N − 1)(q1p − q0) ≥ 0. Then, the protocol (25) estab-lishes the output consensus between double-integratoragents (24).

Corollary 16, in particular, guarantees consensus for q1p ≥ q0

and β = +∞ (notice that this does not follow from Theorem 14

as agents (26) are not passive) and for q1 = 0, β ≤ p2/(2q0(N − 1)) (the relative velocities are not used at all).Also, it can be seen from (26) that, in the assumptions ofProposition 2, the protocol (25) provides a condition�z tj ( )→ 0 besides the state consensus.

We turn now to the second situation, where the absolutevelocity is not accessible, so the agents apply protocol (25)with p = 0 (which is a standard protocol (6) withy t q z t q z tj j j( ) : ( ) ( )= +0 1 � ). Notice that the agents governed bythe model

�� z u y q z q zj j j j j= = +, ,0 1

are strictly passifiable if q0, q1 > 0 and have the transferfunction Wy(iω) = q0(iω)−2 + q1(iω)−1 so that (20) shapes intoKq Kq q0

2 412

02 0ω ω− −+ − ≥( ) , where K: = αθ. This permits us

to apply Theorem 15 to them, which gives rise to thefollowing.

Corollary 17. Suppose that q0, q1 > 0, Assumptions 4 and 5hold, and ϕ θjk S q q∈ +∞[ ]− −1

12

0; , where θ = inft≥0λ2(G(t)).Then, the protocol (25) with p = 0 establishes the exponentialstate consensus between the agents (24).

It should be noticed that relative positions and velocitieszj − zk and � �z zj k− usually are measured rather than theirlinear combinations yj − yk. By choosing appropriate q0,q1 > 0, one may ensure establishing the consensus for thecoupling class S[α; +∞] with arbitrarily small α > 0. Never-theless, unlike Corollary 16, Corollary 17 is not applicable tosaturation-like nonlinearities φjk, which belong to S[0; ∞] butnot to S[α; ∞] with α > 0.

Our last example concerns the situation when directobservation of the velocity is not possible and each of theagents (24) measures or receives only relative position zk − zj.For time-invariant topology, the following consensus protocolwas proposed in [31]:

�w w z zj j j j jk k j

k

N

= − + = −=∑ρ ξ ξ α1

1

, ( ) (27)

u t w j Nj j j( ) , , , , .= − − = …ρ ρ ξ2 3 1 2� (28)

where ρ1, ρ2, ρ3 > 0 are constant and where αjk are couplinggains regarding the constant topology (αjk > 0 if (k, j) ∈ E andαjk = 0 otherwise). The protocol (28) is a counterpart of asecond-order consensus algorithm uj = − ρ2ξj − ρ3ξj, whichprovides consensus for appropriate ρ2, ρ3 > 0 [2], where anunobservable term ξj is approximated via low-passdifferentiation (27). Below, we examine a protocol analogousto (27),(28), however, involving nonlinear and time-variantcouplings (caused, for example, by distortions inmeasurements):



1283

� �w t w t t u t w t t

t t z

j j j j j j

j jk k

( ) ( ) ( ), ( ) ( ) ( )

( ) ( ,

= − + = +=

ρ ξ ρ ρ ξξ ϕ

1 2 3

(( ) ( )):( , ) ( )

t z tj

k k j E t

−∈∑

According to Remark 3, one may consider this closedloop-system as a team of “augmented” agents

�� y w u w w uj j j j j j= + = − +ρ ρ ρ2 3 1, , (29)

applying the static protocol (6) (here, we presumed a littleabuse of notations, redesignating zj with yj and ξj with uj). Theagent (29) has the transfer function Wy(λ) = ((ρ3 + ρ2)λ +ρ1ρ3)/λ2(λ + ρ1)) and is found to be stabilized by anyfeedback uj = −Kyj, where K > 0. Nevertheless, such an agentis not strictly passifiable since its relative degree is 2.

One easily finds that

ReW iy ( )( ( ))

( ).ω ρ ρ ω ρ ρ

ω ρ ω= − + +

+3 1

2 22 3

212 2

Therefore, the frequency-domain condition (20) shapes into

γ ρ ρρ ωδθ ρ ρ ρ ρ ω ρ ρ ω

ω ρ ω

2 12 3

12 2

12

32

3 22 2

3 12 2

412

( )

[ ( ) ]

(

N −− +

++

+ + + −+ 22

0)

,≥

which is seen to be valid if

ρ ρδθ

ρ ρ γρ3 12

2 312

2 1≤ + ≤

−( ).

N(30)

Theorem 11 implies now the following corollary.

Corollary 18. Suppose that Assumptions 4 and 5 hold andφjk ∈ S[α; β], where 0 < α < β < ∞. Let ρ1, ρ2, ρ3 satisfy (30),where δ and γ are given by (17). Then, the protocol (27),(28)establishes the output consensus between the agents (24).

6.4 Simulations

To illustrate and confirm the results from Sections 6.1–6.3, we simulate teams of N = 4 identical agents with differentdynamics. We consider two types of switching topology. Theinteraction graph is constant between time instants tn = 0.1nand randomly switches at tn between three values that areeither undirected graphs G1, G2, G3 from Fig. 1 or directedgraphs ′G1 , ′G2 , ′G3 from Fig. 2. The corresponding valuesof θ = mint≥0λ2(G(t)) are λ2 3 2 2 0 586( ) .G = − ≈ andλ2 2 0 5( ) .′ =G , respectively.

The first test deals with four harmonic oscillators��z z uj j j+ =ω 2 , y zj j= � (where ω = 3) coupled via the

protocol (6) with φjk(t, y) = tanh y and G(t) ∈ {G1, G2, G3}.The result is displayed in Fig. 3 and confirms that the stateconsensus is achieved, as stated by Theorem 14 andProposition 2.

The next experiment deals with exponentially unstableagents � �z az uj j j= + ∈ , yj = zj (where a = 0.58) andlinear time-varying couplings φjk(t, y) = (2 + cost)yand G t G G G( ) { , , }∈ ′ ′ ′1 2 3 . Since Wy(iω) = 1/(iω − a) andReWy(iω) = −a/(a2 + ω2), (23) shapes into a ≤ αθ. This andφjk ∈ S[α; +∞] hold for α = 0.991. The result is depicted inFig. 4 and confirms consensus reaching predicted byTheorem 15.

Fig. 1. Undirected graphs G1 − G3. Minimal λ2 is 2 2− .

Fig. 2. Directed graphs ′ − ′G G1 3 . Minimal λ2 is 0.5.

0 1 2 3 4 5 6 7 8 9 10−6

−4

−2

0

2

4

6

Time (s)

z j(t)

0 1 2 3 4 5 6 7 8 9 10−10

−5

0

5

10

Time (s)

dz j(t

)/d

t

j=1j=2j=3j=4

j=1j=2j=3j=4

Fig. 3. Consensus between harmonic oscillators��z t z t u t( ) ( ) ( )+ =ω 2 , ω = 3, G(t) ∈ {G1, G2, G3},φjk(t, y) = tanh y.



Our last two simulation tests are to confirm Corollary17 and deal with double integrator agents (24), whereq0 = q1 = 1. The cases of both undirected and directed net-works were considered; they deal with nonlinear φjk(t,x) = 1.71x(2 + cost cos x) and linear φjk(t, x) = 2x couplings,respectively. The interaction graph G(t) switches between G1,G2, G3 and ′G1 , ′G2 , ′G3 respectively. Since φjk ∈ S[α;+∞]∀α < M, where M = 1.71,2 in the first and second cases,respectively, we have M q qθ ≥ 0 1

2 ; hence, the assumptions ofCorollary 17 hold. The simulation results are displayed inFigs. 5, 6. They show that not only is the state consensusestablished, in accordance with Corollary 17 and Proposition2, but also the entire states of the agents converge to acommon value.

VII. PROOF OF THEOREM 6

An outline of the proof of Theorem 6 is as follows.Using individual quadratic constraints for the couplings (11),we derive the “global” quadratic constraint on the solutions yj,uj of the closed loop system (5),(6) (Lemma 19). This enable

us to apply the Kalman-Yakubovich-Popov (KYP) lemma andestablish the existence of a quadratic Lyapunov function(Lemma 20). Theorem 6 is proved by means of thisfunction.

Throughout this section, the assumptions of Theorem 6are assumed to be valid. In particular, the Hermitian form Fdefined by (13) is given such that ϕ jk ∈S( )F .

To start with, we introduce the Hermitian form:

P y u Re u qy y QyN

u Ru

y u P y y u uj k j k

( , ) : ( )( )

,

( , ) : ( , )

= − − −−

= − −

* * *θ 1

2 1

Pjj k

N

,

.=∑

1

In the latter definition, we have y yNk

1, ,… ∈� ,u uN

m1, ,… ∈� and the matrices q, Q, R are the same as in

(13). For any sequence of column vectors ξ1, . . . , ξN letξ ξ ξ: ( , , )= …1

TNT T stand for their joint column vector, e.g.

y t y t y tTN

T T pN( ) : ( ( ) , , ( ) )= … ∈1 � and u t mN( )∈� .

Lemma 19. For each solution of (5), (6) one hasP( ( ), ( ))y t u t ≥ 0 ∀t ≥ 0. Moreover, for any ν > μ > 0,there exists ρ > 0 such that P( ( ), ( ))y t u t > ρ wheneverν μ> − >

≤ ≤max ( ) ( )

,1 j k Nk jy t y t .

Proof. We put ξjk(t): = αjk(G(t))φjk(t, yk(t) − yj(t)) andηjk(t): = αjk(G(t))[yk(t) − yj(t)]. By (6), u t tj jkk

N( ) ( )=

=∑ ξ1

,where ξjj: = 0. Therefore,

u t N tj jk

k

N

( ) ( ) ( )2 2

1

1≤ −=∑ ξ (31)

0 0.5 1 1.5 2 2.5 3−3

−2

−1

0

1

2

3

4

Time (s)

z j(t)

j=1j=2j=3j=4

Fig. 4. Consensus among unstable agents �y ay uj j j= + ,a = 0.58, G t G G G( ) { , , }∈ ′ ′ ′1 2 3 , φjk(t, y) = (2 + cost)y.

0 1 2 3 4 5 6−5

0

5

10

15

Time (s)

z j(t)

0 1 2 3 4 5 6−10

−5

0

5

10

Time (s)

dz j(t

)/d

t

j=1j=2j=3j=4

j=1j=2j=3j=4

Fig. 5. Consensus among double integrator agents (24),q0 = q1 = 1, φjk(t, x) = 1.71x(2 + cost cos x), G(t) ∈ {G1,G2, G3}.

0 1 2 3 4 5 6−10

−5

0

5

10

Time (s)

z j(t)

0 1 2 3 4 5 6−6

−4

−2

0

2

4

Time (s)

dz j(t

)/d

t

j=1j=2j=3j=4

j=1j=2j=3j=4

Fig. 6. Consensus among double integrator agents (24),q0 = q1 = 1, φjk(t, x) = 2x, G t G G G( ) { , , }∈ ′ ′ ′1 2 3 .



1285

by the Cauchy-Schwartz inequality, and

ξ jkT

j

j k

N

jT

j

j

N

qy u qy,

.= =∑ ∑=

1 1

(32)

Our next objective is to prove that

ξ jkT

k

j k

N

jT

j

j

N

qy u qy,

.= =∑ ∑= −

1 1

(33)

Under (a) in Assumption 5, this is immediate from (32) andthe equations ξjk = −ξkj. Under (b), one may put wjk(t): = 0 ifαjk(G(t)) = 0 and notice that, due to (10),

y w qy yw q q w

y

y w qy

kT

jkT

k

j k

N

kT

j k

NjkT T

jkk

kT

jkT

k

j

, ,

,

= =∑ ∑= +

=

=

1 1 2

1

2 kk

N

kT T

kj k

j k

N

jT

jkT

k

j k

N

jT

j k

N

y q w y

y w qy yw

= =

= =

∑ ∑

∑ ∑

+

=

1 1

1 1

1

2 ,

, ,

and

jjkT T

jkk

jT

jkT

k

j k

N

kT T

kj j

j k

N

q q wy

y w qy y q w y

+=

= += =∑ ∑

2

1

2

1

21 1, ,

.

Subtracting the above equalities, one obtains

ξ jkT

k

j k

N

k jT

jkT

k

j k

N

k jT

jkT

k

qy y y w qy

y y w qy y

, ,

( )

( )

= =∑ ∑= − =

= − +

1 1

1

2kkT T

kj k j

j k

N

jkT

k

j k

N

kT T

kj

j k

q w y y

qy y q

( ),

, ,

−[ ] =

= −

=

= =

∑

∑1

1 1

1

2

1

2ξ ξ

NN

∑ ,

which leads finally to the desired claim (33):

ξ ξjkT

k

j k

N

kT T

kj

j k

N

kT

k

k

N

qy y q u qy, ,

.= = =∑ ∑ ∑= − = −

1 1 1

By definition of S( )F , F �( ( , ), ) ( , )ϕ μ νjk t y y ≥ ≥ 0whenever 0 ≤ μ ≤ |σ| ≤ ν < ∞, where � ( , )μ ν > 0 if μ > 0.By putting y = yk − yj, αjk(t): = αjk(G(t)) and elementarytransformation, we see that

ξ jkT

k j

a

jk k jT

k j

b

q y y a t y y Q y y( ) ( )( ) ( )− − − − ��

−≥ξ ξ

μ νjkT

jk

c

jk kj kj

R

a t( ) ( , )�

whenever μkj ≤ |yk − yj| ≤ vkj. Here, a u qyjT

jj

N= −

=∑2 1by

(32), (33). From (2) and Q ≥ 0 one finds that b =∑a t Q y y N y y Q y yjkj k

N

k j k jT

k jj k

N− ≥ − −

=−

=∑ ∑( ) ( ) ( ) ( ),

/

,1

1 2 2 1

1θ,

and c N u jj

N≥ − −

=∑( )1 1 2

1by (31). Hence,

N y t u t a tjk kj kjj k

− ≥∑1P( ( ), ( )) ( ) ( , ),

� μ ν . By taking here

μkj: = 0 and vjk: = |yj − yk|, we arrive at the first claim of thelemma. To prove the second one, it suffices to note thatwhenever ν μ≥ − ≥max

,j kk jy y , first, |yk − yj| ≤ vjk: = v ∀j, k

and second, |yk − yj| ≥ μjk: = μ/(2N) for at leat one arc (j, k),whereas we put μjk: = 0 for all other pairs (j, k).

Lemma 20. Suppose that ΠF ( ( )) ( ) ( )W i W i W iy x xω ε ω ω≥ ∗for some ε ≥ 0 whenever ω ∈ R and det(iωI − A) ≠ 0. Thenthere exists a symmetric matrix H = HT > 0 such that�V x t y t u t x t( ( )) ( ( ), ( )) ( ( ))+ ≤ −P εW for any solution of

(5), where V x x x H x xk j k jj k

N( ) : ( ) ( )

,= − −

=∑ *1

and W( )x isfrom (8).

Proof. One can see by direct computation of �V that�V x y u x( ) ( , ) ( )+ + ≤P εW 0 follows from the LMI

2x*H(Ax + Bu) + P(Cx, u) + ε|x|2 ≤ 0∀x, u.

By the KYP lemma [18,32], the latter LMI hasa symmetric solution H = HT if and only ifP C i I A Bu u W i W iy x( ( ) , ) ( ( )) ( )ω ω ε ω− = − ≤ −−1 2ΠF for anyu m∈� and ω ∈ R (such that det(iωI − A)−1 ≠ 0). Since thelast requirement is nothing but (15), we see that the matrix Hexists. We are going to show its positive definiteness.

Let ( , , )x u yj j j+ + + be the solution of (5), (6), where

G(t) ≡ G0, the graph G0 is complete, φjk(t, y) = Ky, andK is taken from (i) of Theorem 6. By Lemma 19,P( ( ), ( ))y t u t+ + ≥ 0 , whereas lim ( ) ( )

tj kx t x t

→+∞

+ +− = 0 due to

(a). By integrating �V x t y t( ( )) ( ( )+ ++P , u t+ ≤( )) 0, we seethat V x x H( ( )) ( )+ +≥ ∀ ⇒ ≥0 0 0 0. Moreover, V x( ( ))+ =0

y u y y uN j( ( ) ( )) ( ) ( ) ( )+ + + +⇒ ⋅ ⋅ ≡ ⇒ ⋅ ≡…≡ ⋅ ⇒ ⋅ ≡0 0 01P andso x xN1 0 0+ +=…=( ) ( ) thanks to observability. Hence, H > 0.

Proof of Theorem 6. For the function V x x Hx( ) = * fromLemma 20 (with ε: = 0) and any solution of (5),(6),

0 00

≤ ≤ − ∫V x t V x y u dt

( ( )) ( ( )) ( ( ), ( )) ,P ξ ξ ξ (34)

where P( )… ≥ 0 by Lemma 19 and H > 0 by Lemma

20. It follows that W( ( )) ( ) ( ),

x t x t x tj kj k

N= − ≤

=∑ 2

1

W( )

( )( ( ))

M H

m Hx 0 , where M(H) and m(H) are the maximal and

minimal eigenvalues of H, respectively. Thus, (8) does hold.To prove (7), suppose the contrary. Then, there exist ζ > 0and a sequence tn ↑ +∞ such that max ( ) ( )

,j kj n k ny t y t− > 2ζ .

Since sup ( )t jx t≥ < ∞0 � , a number Δ > 0 exists such thatmax ( ) ( )

,j kj ky t y t− >ζ whenever |t − tn| ≤ Δ. By Lemma 19,

ρ > 0 exists such that P( ( ), ( ))y t u t > ρ if |t − tn| ≤ Δ, whichimplies violation of (34). This contradiction proves (7). Thisproves the first claim of Theorem 6, which concerns theoutput consensus.



To prove the second claim, we again use Lemma 20(with ε > 0 the same as in Theorem 6) and notice that, besides(34), �V x y u V( ) ( , )+ ≤ −P μ for sufficiently small μ > 0, thusV x t Me t( ( )) ≤ −μ . Since H > 0, we conclude that the exponen-tial state consensus is established.

Remark 21. The proof of Theorem 6 implicitly shows theexistence of a solution of (5) for all t ≥ 0. This follows fromthe classical facts of ODE theory since the solutions remainas t → ∞ due to (8) and (5).

VIII. CONCLUSION

Output consensus among identical high-order linearagents was examined in the case where the interaction topol-ogy is uncertain and switching, but the topology is assumed topreserve its connectivity. The couplings among the agents areuncertain as well and satisfy the conventional quadraticconstraint. A new criterion for robust output consensus isestablished.

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Anton Proskurnikov (born in 1982)received his M.Sc. (Specialist) degree in2003 and Ph.D. (Candidate of Science)degree in 2005 from St. Petersburg StateUniversity (supervised by Prof. VladimirA. Yakubovich). Now he works as a seniorresearcher in St. Petersburg State Univer-

sity (Department of Mathematics and Mechanics) and Insti-tute for Problems of Mechanical Engineering RAS. He alsooccupies position of the deputy chief R&D engineer in NavisEngineering OY, one of the leading manufacturers of dynamicpositioning and marine automation systems. His researchinterests include dynamics and control of complex networks,nonlinear control, robust control, optimal control and controlapplications to marine systems.



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Nonlinear Consensus Algorithms with Uncertain Couplings