Consensus in symmetric multi-agent networks with sector nonlinear couplings*

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Consensus in symmetric multi-agent networks with sector nonlinear couplings. ? Anton V. Proskurnikov ∗ ∗ Institute for Problems of Mechanical Engineering of RAS and St.-Petersburg State University, St.-Petersburg, Russia E-mail: [email protected], [email protected]

Abstract: We investigate consensus dynamics in multi-agent networked systems with timevarying undirected topology and uncertain scalar-valued nonlinear couplings, satisfying sector inequalities. Using the Kalman-Yakubovich-Popov lemma and techniques of the absolute stability theory, we obtain consensus criteria for the networks of the specified type, analogous to the celebrated circle stability criterion for Lurie systems. Keywords: Multiagent networks, consensus, synchronization, absolute stability 1. INTRODUCTION. Recent years the problems of cooperative control in networks of dynamic agents have attracted enormous attention of the research community, because of growing necessity to analyze and control complex and highly interconnected dynamical systems with limited communications between the agents. The problems of distributed consensus, referred also as agreement or averaging problem, constitute an important class of decentralized cooperative control problems. The consensus problem consists in designing decentralized control policies or protocols, that make the agents’ state vectors or some outputs, treated as individual ”opinions”, converge to some desired function (”consensus opinion”). The first known averaging algorithms seem to originate in Markov chains theory (see e.g. Seneta (1981)), decision making in expert communities (DeGroot (1974), Eisenberg and Gale (1959)) and distributed computing (see D.Bertsekas and Tsitsiklis (1989)). More recent applications of the consensus dynamics include coordinating the groups of mobile autonomous vehicles and analysis of biological populations motion (Gazi and Fidan (2007), Ren et al. (2007), R. Sepulchre and Leonard (2008), Tanner et al. (2007),Olfati-Saber (2006) etc.), synchronization in various physical phenomena (Vicsek et al. (1995),Strogatz (2000)), data processing and fusion in sensor networks and others. The most commonly used consensus protocols are linear ones, and a great deal of results for such protocols under different communication restrictions (fixed or time-varying communication topology, communication delays and asynchronous measurements, etc.) have been obtained, see Jadbabaie et al. (2003), Fax and Murray (2004),OlfatiSaber and Murray (2004),Moreau (2004),Ren and Beard (2005), Chopra and Spong (2006),Lin et al. (2007),Seo et al. (2009),Wieland et al. (2008). Nevertheless, a lot ? Supported by RFBR, grants 09-01-00245,11-08-01218 and the Federal Program ”Scientific and scientific-pedagogical cadres of innovation Russia for 2009-2013”, State Contract 16.740.11.0042

978-3-902661-93-7/11/$20.00 © 2011 IFAC

of applications need nonlinear consensus dynamics due to the nonlinear nature of the interactions (for instance the coupled oscillators networks where the couplings should be periodic, e.g. the sine function in the celebrated Kuramoto model Kuramoto (1984)) or limitations on the control input, disabling one to use linear control laws. Even dealing with linear protocols, one has to take unknown nonlinear errors in the measured data into account, which lead to an uncertain nonlinear dynamics. Most known results on the nonlinear assume special agents dynamics, for instance to be first or second order plants (Olfati-Saber and Murray (2003),Moreau (2005),Hui and Haddad (2008),Ren (2008)) or be passive (Chopra and Spong (2006),Arcak (2007)). Below we consider the case of identical linear agents with arbitrary state-space model. We assume only a scalar-valued output to be measured, and the couplings may be nonlinear and uncertain, but supposed to satisfy strict sector inequalities and some symmetry conditions. The undirected network topology may be fixed or switching. Using the absolute stability approach and the Kalman-Yakubovich-Popov lemma, we obtain frequency-domain consensus criteria, resembling the well-known circle criteria for Lurie systems.

2. SOME GRAPH-THEORETIC CONCEPTS. Throughout the paper GN stands for the set of all undirected graphs, having common set of vertices (nodes) VN = {1, 2, . . . , N } and without loops (arcs with coincident ends). For any G ∈ GN and 1 ≤ j ≤ N denote by Nj (G) ⊂ VN the set of all neighbors (adjacent nodes) of the node j in the graph G. Let aij (G) = aji (G) (1 ≤ i, j ≤ N , G ∈ GN ) be 1 if the nodes i, j if they are connected with an arc in G and 0 otherwise. The matrix (aij (G)) is called the adjacency matrix of the graph G. The matrix associated with the PN quadratic form QG (z) = 21 i,j=1 aij (G)(zj − zi )2 , that is

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 PN j=1

  L(G) =  

a1j

−a21

...

−a1N



a2j ...

−a2N

   ≥ 0, 

−a12

PN j=1

.. .

.. .

..

. . P .. N

−aN 1

−aN 2

...

j=1

(1)

Definition 1. We say that the output consensus in the network (3), (4) is achieved, or that the protocol (4) provides the output consensus, if for any set of initial data (xj (0))N j=1 one has

aN j

lim |yj (t) − yi (t)| = 0,

is called the Laplacian matrix of the graph G. Its lowest eigenvalue λ1 (L(G)) is easily seen to be 0 (with eigenvector z = (1, 1, . . . , 1)T ). The second eigenvalue λ2 (L(G)) is called the algebraic connectivity or Fiedler eigenvalue of the graph G. It is easily shown Fiedler (1973) that QG (z) λ2 (L(G)) = N min PN (2) 2 z i,j=1 (zj − zi ) N where PN minimum2 is taken over the set of all z ∈ R with (z − z ) > 0. i i,j=1 j

In particular, for the complete graph G with N vertices (aij = 1 for any i 6= j) one has λ2 (L(G)) = N . Actually, the spectrum of L(G) in this case consists of two eigenvalues that are 0 and N (with multiplicity N − 1). It is known that λ2 (G) > 0 if and only if G is connected graph (i.e. any two nodes may be connected by path). Moreover, as shown in Fiedler (1973), λ2 (G) ≥ 2η(G)(1 − π cos N ) for any connected graph G ∈ GN , where η(G) is the minimum number of edges sufficient to delete in order to break graph’s connectivity (the so called edge connectivity number). Other properties of graph spectra, including upper and lower bounds on the algebraic connectivity, as well as proofs of the aforementioned facts can be found in Fiedler (1973),Cvetcovi´c et al. (1980),Merris (1994) and others. 3. PROBLEM STATEMENT. We consider a group of SISO plants with identical dynamics indexed with j = 1, 2, . . . , N called agents whose state-space models are x˙ j (t) = Axj (t) + Buj (t), yj (t) = Cxj (t). (3) Here t ≥ 0, xj (t) ∈ Rd stands for the state vector of jth agent, and uj (t), yj (t) ∈ R are the input signal and the measured output signal respectively. Throughout the paper we assume that the state space model (3) is fully controllable and observable. Below we investigate stability of distributed control policies (or protocols) of the following commonly used type. We assume that possible directions of the information exchange between the agents for any t ≥ 0 are defined by a graph G(t), called the communication or underlying graph (or topology) of the network at time t ≥ 0. This means that j-th agent obtains information about the value the output signal yk (t) if and only if k ∈ Nj (G(t)). We assume that the graph-valued function G(·) : [0; +∞] → GN is Lebesgue measurable. The input function of any agent is determined by neighbors’ outputs as follows: X uj (t) = ϕjk (t, yk (t) − yj (t)) (4) k∈Nj (G(t))

Here (ϕij (t, σ)) (with 1 ≤ i, j ≤ N , i 6= j) is a family of functions called couplings, describing the mutual interactions between the agents.

1 ≤ i, j ≤ N.

t→+∞

(5)

Analogously, the protocol (4) is said to provide the state consensus, if for any set of initial data (xj (0))N j=1 one has lim |xj (t) − xi (t)| = 0,

1 ≤ i, j ≤ N.

t→+∞

(6)

The couplings ϕij are, in general, nonlinear functions and uncertain, we suppose only the following assumptions to be valid. Assumption 2. (Symmetry in the network). Each coupling ϕij (t, y) is odd with respect to y: ϕkj (t, −y) = −ϕkj (t, y) and ϕkj (t, y) = ϕjk (t, y). In particular, ϕkj (t, 0) = 0 and ϕjk (yk − yj ) = −ϕkj (yj − yk ). Thus for any solution xj , uj (with 1 ≤ j ≤ N ) of the closed-loop system (3),(4) one has uj (t) = 0 if y1 (t) = y2 (t) = . . . = yN (t). Also one has N X

uj (t) = 0,

j=1

N N 1 X 1 X xj (t) = etA xj (0) N j=1 N j=1

(7)

Assumption 3. (Sector inequality). There exist α, β such that 0 ≤ α < β ≤ +∞ and the inequalities hold: ϕjk (t, σ) α≤ ≤ β, ∀σ 6= 0, ∀j 6= k. (8) σ Equivalently, the graph {(σ, ξ) : ξ = ϕ(t, σ))} of each function ϕij (t, ·) lies between the lines ξ = ασ and ξ = βσ (degenerating into the vertical line σ = 0 for β = +∞). If β = +∞, we additionally assume that ϕjk (t, σ) is uniformly bounded whenever σ runs over bounded set: for any compact K ⊂ R one has sup{|ϕjk (t, σ)| : t ≥ 0, σ ∈ K} < +∞. Assumption 4. (Strictness of sector inequality). If σ remains bounded and separated from 0, then ϕij (t, σ)/σ is separated from α, β uniformly for t ≥ 0. More precisely, for any compact set K ⊂ R \ {0} there exist α0 = α0 (K) > α, β 0 = β 0 (K) < β such that ϕij (t, σ) α0 ≤ ≤ β 0 , ∀t ≥ 0, σ ∈ K. σ Remark 5. Note that due to (7) the state consensus condition (6) implies that lim |xj (t) − etA x ˜| = 0,

t→+∞

x ˜=

N 1 X xj (0), N j=1

and the output consensus condition (5) implies that lim |yj (t) − CetA x ˜| = 0 for any j = 1, 2, . . . , N . t→+∞

Lemma 6. Let the couplings ϕij (t, σ) be uniformly continuous at σ = 0 (with respect to t ≥ 0), i.e. supt≥0 |ϕij (t, σ)| → 0 as σ → 0 (e.g. β < +∞). Then the output consensus (5) implies the state consensus (6). Indeed, the synchronization of outputs (5) implies that uj (t) → 0 as t → +∞ for any j. Since the functions Xij = xj −xi , Uij = uj −ui and Yij = yj −yi are evident to obey the controllable and observable state-space equations X˙ ij = AXij + BUij , Yij = CXij , one has Xij (t) → 0 as t → +∞. 

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The aim of the present paper is to give effective sufficient conditions for achievement of state or output consensus whenever the network (3), (4) satisfies the Assumptions 2-4. Such consensus criteria, involving the agent’s transfer functions and sector bounds only, are presented in the next section. Their proofs can be found in the Appendix. 4. MAIN RESULTS. For the sake of brevity we introduce some auxiliary notations. Introduce constants γ, δ as follows 1 αβ γ= , δ= (9) β+α β+α (by definition we take δ = α for β = +∞). Here α, β stand for the sector bounds from the Assumption 3. Assumptions 3 and 4 can now be reformulated as follows. Lemma 7. Suppose the couplings ϕjk (t, σ) satisfy Assumption 3. Then ηij (t, σ) := ϕij (t, σ)σ − δσ 2 − γϕij (t, σ)2 ≥ 0. (10) Assumption 4 being valid, ηij (t, σ) is separated from 0 (uniformly for t ≥ 0) if σ remains bounded and separated from 0. More precisely, for any compact set K ⊂ R \ {0} we have inf{ηij (t, σ) : t ≥ 0, σ ∈ K} > 0. The proof of Lemma 7 is obvious and omitted. Let Wx , Wy stand for the transfer functions of the plant (3) from u to x and y respectively Wx (λ) = (λId − A)−1 B, Wy (λ) = CWx (λ). (11) Define a function RN (iω, θ) as follows: γ RN (iω, θ) = Re Wy (iω) + δθ|Wy (iω)|2 + , (12) 2(N − 1) where θ ∈ R and ω ∈ R, det(iωId − A) 6= 0. 4.1 Fixed topology case. In this paragraph we deal with the case of fixed network topology G(t) = G0 with the graph G0 connected (for the disconnected network the protocol, evidently, can not guarantee synchronization between different connectivity domains). The class of all nonlinear protocols (4) satisfying Assumptions 2-4 contains linear protocols with the couplings ϕjk (t, σ) = µσ (µ ∈ (α; β)). Therefore one can expect that any of those nonlinear protocols guarantee consensus only if the consensus is achieved X in a linear network as follows: x˙ j = Axj + µBC (xk − xj ), 1 ≤ j ≤ N. (13) k∈Nj (G0 )

Introducing the joint vector x ¯(t) = col(x1 (t), . . . , xN (t)) ∈ RdN , the system (13) may be rewritten as follows: x ¯˙ = [IN ⊗ A − µL(G0 ) ⊗ BC] x ¯ (14) Note that accordingly to Lemma 6, the output and state consensus for the networked system (13) are equivalent. The achievement of state consensus for the linear network (13) is known to be easily verifiable, given the graph Laplacian eigenvalues (Fax and Murray (2004), Seo et al. (2009)): Lemma 8. Consider a linear networked system (13). The state consensus condition (6) is fulfilled for any initial data if and only if for all nonzero eigenvalues λ of L(G0 ) the matrix A − µλBC is Hurwitz.

The following result gives frequency-domain sufficient condition for the state consensus with exponential convergence rate in (6). Note that it does not require the couplings to satisfy Assumption 4. Theorem 9. Suppose that G0 is connected graph and for some µ ∈ (α; β) and any eigenvalue λ > 0 of L(G0 ) the matrix A − µλBC is Hurwitz. Suppose that a constant ε > 0 exists such that for any ω ∈ R one has RN (iω, λ2 (G0 )) ≥ ε|Wx (iω)|2 . (15) Then any protocol (4) with G(t) = G0 and couplings, satisfying Assumptions 2,3, provides the state consensus (6). Futhermore, a constant ν > 0 exists such that N N X X |xj (t) − xi (t)|2 ≤ M e−νt |xj (0) − xi (0)|2 . (16) i,j=1

i,j=1

Remark 10. Suppose the matrix A to have no eigenvalues on the imaginary axis. Then (15) may be rewritten as RN (iω, λ2 (G0 )) > 0 for all ω ∈ [−∞; +∞]. Geometrically the latter inequality may be treated as follows: if α > 0 and thus δ > 0, the Nyquist curve of the plant (3), i.e. the locus of points Wy (iω) on the complex plane, lies outside the circle centered at z0 with radius R0 , where r 1 1 2γδλ2 (G0 ) z0 = − , R0 = 1− 2δλ2 (G0 ) 2δλ2 (G0 ) N −1 For 2γδλ2 (G0 ) > N − 1 the circle vanishes, and the frequency domain inequality is always fulfilled. In the case of α = 0, the frequency domain inequality has the form Re Wy + 2(Nγ−1) ≥ 0, i.e. the Nyquist plot should lie on the right from the vertical line {− 2(Nγ−1) + iθ, θ ∈ R}. The result of Theorem 9 appears to be a generalization of the celebrated circle criterion in the absolute stability theory (Khalil (1996), Gelig et al. (2004)). Assume for that det(iωI − A) 6= 0 for simplicity and consider a group of N = 2 agents (3), coupled with a function ϕ12 = ϕ21 satisfying the Assumptions 2,3. Evidently λ2 (L(G0 )) = 2. Let X(t) = x2 (t) − x1 (t), Y (t) = y2 (t) − y1 (t) = CX(t), U (t) = u2 (t) − u1 (t) and Φ(t, Y ) = −2ϕ12 (t, Y (t)) so Φ(t, Y )/Y ∈ [−2β; −2α]. The achievement of exponential state consensus means the exponential stability of equilibrium point X = 0 of the nonlinear system as follows: ˙ X(t) = AX(t) + BU (t), U (t) = Φ(t, Y (t)) (17) The circle criterion provides the global stability under assumption that the Nyquist curve lies strictly outside the circle with the diameter [−(2α)−1 ; −(2β)−1 ]. A straightforward computation shows this circle to coincides with one from the Remark 10. Unfortunately conditions of the Theorem 9 are often violated, e.g. the frequency domain inequality (15) implies that γ > 0, excluding the case of infinite sector (β = +∞). Also it is usually violated for α = 0 (consider for example single integrator dynamics with A = 0 and B = C = 1). Under Assumption 4 it appears possible to weaken the frequency domain inequality, replacing the exponential state consensus condition with the output one. Theorem 11. Suppose that G0 is connected graph such that for some µ ∈ (α; β) and any eigenvalue λ > 0 of L(G0 ) the matrices A − µλBC is Hurwitz. Let a frequency domain inequality hold as follows: RN (iω, λ2 (L(G0 ))) ≥ 0, ∀ω ∈ R, det(iωId − A) 6= 0. (18)

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Then any protocol (4) with G(t) = G0 , satisfying Assumptions 2-4, provides that (1) deviations between the state vectors |xj (t) − xi (t)| remain bounded: for some constant M > 0 one has N N X X |xj (t) − xi (t)|2 ≤ M |xj (0) − xi (0)|2 (19) i,j=1

i,j=1

Theorem 15. Let G(·) ∈ GN (θ) for some θ ≥ 0. Assume that for some µ ∈ (α; β) the matrix A−µN BC is Hurwitz. Let a frequency domain inequality hold as follows: RN (iω, θ) ≥ 0, ∀ω ∈ R, det(iωId − A) 6= 0. (21) Then for any protocol (4) with given graph G(·) and couplings, satisfying Assumptions 2-4, the following statements are true: (1) deviations between the state vectors |xj (t) − xi (t)| remain bounded: for some constant M > 0 (19) holds; (2) if G(t) is connected for almost all t ≥ 0, the output consensus is achieved; (3) if G(t) is connected for almost all t ≥ 0 and supt≥0 |ϕij (t, σ)| → 0 as σ → 0 (e.g. β < +∞) the state consensus (6) is achieved;

(2) the output consensus (5) is achieved; (3) if supt≥0 |ϕij (t, σ)| → 0 as σ → 0 for all i, j (e.g. β < +∞) the state consensus (6) is also achieved. Remark 12. The frequency domain inequality (18) means that the points of the Nyquist curve should not go inside the circle (or half-plane) from Remark 10. 4.2 Time-varying network graph.

5. EXAMPLES

This paragraph is devoted to more complicated situation when the network topology is time-varying and may be unknown. Denote by GN (θ) the set of all graph-valued Lebesgue measurable functions G : [0; +∞] → GN such that λ2 (G(t)) ≥ θ for almost all t ≥ 0. Evidently GN (θ) consists of all measurable graph-valued functions G : [0; +∞] → GN for θ = 0, and is empty set for θ > N . Main results of this paragraph are Theorems 13 and 15 that are counterparts of Theorems 9 and 11 but concern the case of switching topology function G(·). The latter function may be uncertain, but belongs to GN (θ) for some θ ≥ 0. The less θ one takes, the more restrictive constraints on the agents’ dynamics and the sector [α; β] one has to pose in order to guarantee the consensus. For the case θ = 0 (the topology is completely unknown) we also need some connectivity assumptions. For any 0 ≤ θ ≤ N the set GN (θ) contains the graphvalued function G(t) ≡ CN , where CN is a complete graph. Therefore if any protocol (4) with G(·) ∈ GN (θ) and satisfying Assumptions 2-4 provides the consensus, the consensus is achieved for G(t) ≡ CN and ϕij (t, σ) = µσ, µ ∈ (α; β). Accordingly to the Lemma 8, the matrix A − µN BC should be Hurwitz. Theorem 13. Let G(·) ∈ GN (θ) for some θ ≥ 0. Assume that for some µ ∈ (α; β) the matrix A−µN BC is Hurwitz. Suppose that a constant ε > 0 exists such that for any ω ∈ R, det(iωI − A) 6= 0 one has RN (iω, θ) ≥ ε|Wx (iω)|2 .

(20)

Then any protocol (4) with given graph G(·) and couplings, satisfying Assumptions 2,3, provides the state consensus with exponential convergence rate (16). Remark 14. Note that for θ = 0 we do not need any connectivity assumptions on the graph G(·). In particular, for the totally disconnected graph and uj (t) ≡ 0 one has (6) as well. Therefore the assumptions of Theorem 13 for θ = 0 imply stability of agent’s dynamics: A is Hurwitz matrix. In most applications this is not the case so the proposition of Theorem 13 is of main interest for θ > 0. The next result is analogous to Theorem 11 and deals with the case of non-strict frequency domain inequality.

5.1 Single integrators and the Kuramoto model. Consider the first order agents’ dynamics given by X x˙ j = uj , uj = ϕ(xk − xj ), j = 1, . . . , N

(22)

k∈Nj (G(t))

where the function ϕ(σ) is odd, continuous and ϕ(σ)σ > α0 |σ|2 for σ 6= 0, where α ≥ 0 is some constant. It is evident that the protocol in (22) satisfies Assumptions 2-4 for α = α0 , β = +∞ corresponding to δ = α0 , γ = 0. 1 Since Wy = Wx = (iω) , one easily sees that the strict frequency-domain inequalities (15), (20) hold if and only if θ, α0 > 0, but (21) is fulfilled whenever θ, α0 ≥ 0. Note also that since A = 0, B = C = 1 the system x˙ = (A − λBC)x is exponentially stable for any λ > 0.

Combining the results of Theorems 9,11 to the networked system, one obtains the following convergence condition (proved by other methods in Olfati-Saber and Murray (2003),Chopra and Spong (2006)): Theorem 16. Consider a network (22) with coupling function ϕ(·) odd, continuous and satisfying ϕ(σ)σ > α0 |σ|2 for σ 6= 0. Then (1) the boundedness condition (19) holds; (2) if G(t) is connected almost everywhere, then the state consensus (6) is achieved, the convergence rate in (6) is exponential for α0 > 0; Among practical examples of the networked systems having dynamics (22) we mention the famous Kuramoto model of coupled oscillators dynamics. A population of N oscillators with phase angles ϑj is assumed to be governed by the following equations: N KX sin(ϑi − ϑj ) ϑ˙ j = ωj + N i=1

Here K > 0 is a coefficient describing the coupling strength. A natural generalization is the system of oscillators with time-varying interconnectedness topology G(·) : [0; +∞] → GN with pairs of oscillators not coupled: X K ϑ˙ j = ωj + sin(ϑi − ϑj ) N

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Assume that the oscillators are identical: ωj = ω for any 1 ≤ j ≤ N . Without loss of generality, we may assume that ωj = 0 (otherwise one can replace ϑj with ϑj − ωt). Suppose also that initial data satisfy inequalities |ϑj (0)| < π2 − ε for some ε > 0. It can be easily shown that |ϑj (t)| < π2 − ε for any t ≥ 0, therefore

the graph G. In particular, a2ij = aij . Due to (4) one PN 2 has uj (t) = k=1 ξjk (t) and therefore |uj (t)| ≤ (N − PN 1) k=1 |ξjk (t)|2 . Applying the inequalities (10) to σ = yj (t) − yi (t), one obtains

sin(ϑj − ϑi ) ≤1 ϑj − ϑi for some α = α(ε) > 0. From Theorem 16 the following result is immediate: Lemma 17. Suppose that ωj = 0 for any j. If |ϑj (0)| < π/2 and G(t) is connected for any t ≥ 0, the phase synchronization is achieved: lim (ϑj (t) − ϑi (t)) = 0, and

By summation due to Assumption 2 (ξij = −ξji ) and the definition of algebraic connectivity (2) one obtains that

α≤

2 ξij (yj − yi ) − δaij (G(t))|yj (t) − yi (t)|2 − γξij ≥ 0.

−2

N X

u j yj −

j=1

N N δθ X γ X 2 |yj (t) − yi (t)|2 − u ≥ 0. N i,j=1 N − 1 j=1 j

Multiplying P the latter inequality by N and using the identity uj = 0, one easily derives (A.1).

t→+∞

the convergence is exponential. 5.2 Second order dynamics. Suppose that agent model has a form z¨j = uj , yj = pz˙j + qzj with p, q > 0 and consider a protocol as follows: X uj = ϕ(yk − yj ), j = 1, . . . , N

(23)

k∈Nj (G(t))

with ϕ(σ) odd, continuous and satisfying ϕ(σ)σ ≥ α|σ|2 . Thus we consider a sector [α; +∞] for which δ = α, γ = 0. Transforming the system (23) into the state-space form by taking x = (z, z) ˙ T , A = ( 00 10 ), C = (q, p) and B = (0, 1)T , it is immediate that A − λBC is Hurwitz whenever λ > 0. Note that Wy (iω) = (piω + q)/(iω)2 . The function R from (12) is as follows: q p2 ω 2 + q 2 (θαp2 − q)ω 2 + θαq 2 RN (iω, θ) = − 2 + θα = ω ω4 ω4 2 In particular, for θ > q/(αp ) the strict frequency domain inequality (20) is fulfilled. Thus we obtain the convergence result as follows. Theorem 18. Suppose that λ2 (L(G(t))) > q/(αp2 ) for almost all t ≥ 0 (this is the case, e.g. if G(t) is connected π and q < 2αp2 (1 − cos N )). Then the state consensus with exponential convergence rate (16) is achieved.

Analyzing the proof of the previous lemma, the following result is easily shown: Lemma 20. Suppose the protocol (4) to satisfy AssumpPN tions 2-4 and let S(t) = i,j=1 |yi (t) − yj (t)|. If G(t) is connected and S(t) > 0 for some t ≥ 0, then the inequality in (A.1) is strict. Moreover, if G(t) is connected almost everywhere, S(t) is bounded and the set {t : S(t) > ε} has infinite Lebesgue measure for some ε > 0, then +∞ N Z X Fθ (xj (t) − xi (t), uj (t) − ui (t))dt = +∞. i,j=1 0

The next lemma is obvious consequence of the KalmanYakubovich-Popov lemma for the quadratic form Fθ : Lemma 21. Suppose that the frequency domain inequality (21) or, respectively, (20) is fulfilled for some θ. Then there exists a d × d-matrix Hθ = Hθ∗ such that 2xT Hθ (Ax + Bu) + Fθ (x, u) ≤ 0, or, respectively, 2xT Hθ (Ax + Bu) + Fθ (x, u) ≤ −ε|x|2 ,

A.1 Technical lemmas.

(A.3)

A.2 Proofs of Theorems 9,11,13,15. A scheme of the proof of all the four theorems may be outlined as follows. Suppose the frequency domain inequality (21) (respectively (20)) is fulfilled for some θ ≥ 0, which equals to λ2 (G0 ) in (18), (15). We introduce a Lyapunov function N X V (¯ x) = (xj − xi )T Hθ (xj − xi ),

Appendix A. PROOFS OF THE MAIN RESULTS.

(A.2)

x ¯ = col(x1 , . . . , xN ).

i,j=1

Denote by Fξ (x, u) a quadratic form of variables x ∈ Rd , u ∈ R as follows: Fξ (x, u) = −uCx − δξ|Cx|2 − 2(Nγ−1) u2 . Here δ, γ are defined by (9) and C is from (3). Lemma 19. Suppose that functions ϕij satisfy Assumptions 2,3 and uj (t) is given by (4) for any j = 1, . . . , N . Suppose also that λ2 (L(G(t))) ≥ θ. Then any solution of the system (3), (4) satisfies a quadratic constraint N X

Fθ (xj (t) − xi (t), uj (t) − ui (t)) ≥ 0.

Here Hθ is a matrix from Lemma 21. From (A.2) or, respectively from (A.3), one easily obtains that V˙ +

N X

F (xj − xi , uj − ui ) ≤ 0

(A.4)

i,j=1

or, respectively, N N X X V˙ + F (xj − xi , uj − ui ) ≤ −ε |xj − xi |2 . (A.5)

(A.1)

i,j=1

i,j=1

i,j=1

To prove the Lemma 19 introduce functions ξij (t) = aij (G(t))ϕij (t, yj (t) − yi (t)), ηij = aij (G(t))(yj (t) − yi (t)), i 6= j, where aij (G) stands for the adjacency matrix of

Assumptions of all Theorems 9,11,13,15 imply that the state consensus is provided by some (4) with ϕij (t, σ) = µσ (µ ∈ (α; β)) and G(·) ∈ GN (θ) being constant (G0 in Theorems 9,11 or complete graph in Theorems 13,15).

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

This protocol satisfy Assumptions 2-4. Substituting a correspondent solution of (3), (4) into (A.4) and integrating, one obtains due to (6) and V (¯ x(t)) → 0 as t → +∞ that +∞ N Z X V (¯ x(0)) ≥ F (xj (t) − xi (t), uj (t) − ui (t))dt (A.6) 0

i,j=1

Due to Lemma 20, we have V (¯ x(0)) > 0 whenever PN 2 |y (t) − y (t)| > 0 for some t ≥ 0. Thus V (¯ x(0)) ≤ j i,j=1 i 0 is possible only for yi − yj ≡ 0 and thus uj ≡ 0, which implies xj (0) − xi (0) = 0 for any i, j. Thus Hθ > 0. The remainders of proofs of Theorems 9,13 are trivial: x(t)) due to (A.5) one has that dV (¯ ≤ −νV (¯ x(t)) for some dt ν > 0, thus V (¯ x(t)) ≤ V (¯ x(0))e−νt , which is followed by the exponential rate state consensus (16). Inequality (19) is immediate from Lemma 19, (A.4) and Hθ > 0. To prove the output consensus, assume on the contrary that a sequence tn → +∞ and ε > 0 exist such PN that S(t) = i,j=1 |yi (t) − yj (t)| > 2ε for t = tk , k ≥ 1. Due to (19) and Assumption 3 one obtains that both yj −yi are y˙ j − y˙ i are globally bounded functions. Thus for some r > 0 one has |S(t)| > ε for t ∈ (tn − r; tn + r). But this contradicts Lemma 20 since (A.6) should be fulfilled. The contradiction shows that the output consensus is achieved. The last proposition follows from the Lemma 6.  REFERENCES Arcak, M. (2007). Passivity as a design tool for group coordination. IEEE Trans. Autom. Control, 52(8), 1380–1390. Chopra, N. and Spong, M. (2006). Passivity-based control of multi-agent systems. In S. Kawamura and M. Svinin (eds.), Advances in Robot Control:. Springer-Verlag, Berlin. Cvetcovi´c, D., Doob, M., and Sachs, H. (1980). Spectra of Graphs. Acad.Press, New York, San-Francisco, London. D.Bertsekas and Tsitsiklis, J. (1989). Parallel and Distributed Computation: Numerical Methods. PrenticeHall, Englewood Cliff, NJ. DeGroot, M. (1974). Reaching a consensus. Journal of the American Statistical Association, 69, 118–121. Eisenberg, E. and Gale, D. (1959). Consensus of subjective probabilities: The pari-mutuel method. Annals of Math. Statistics, 30(1), 165–168. Fax, J. and Murray, R. (2004). Information flow and cooperative control of vehicle formations. IEEE Trans. Automatic Control, 49(9), 1465–1476. Fiedler, M. (1973). Algebraic connectivity of graphs. Czech. Math. Journal, 23, 298–305. Gazi, V. and Fidan, B. (2007). Coordination and control of multi-agent dynamic systems: Models and approaches. In E.S. et al. (ed.), Swarm Robotics WS, LNCS 4433. Springer-Verlag, Berlin Heidelberg. Gelig, A., Leonov, G., and Yakubovich, V. (2004). Stability of stationary sets in control systems with discontinuous nonlinearities. World Scientific Publishing Co. Hui, Q. and Haddad, W. (2008). Distributed nonlinear control algorithms for network consensus. Automatica, 44, 2375–2381. Jadbabaie, A., Lin, J., and Morse, A. (2003). Coordination of groups of mobile autonomous agents using nearest

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