Distributed average consensus: Beyond the realm of linearity

Transcription

1 Distributed average consensus: Beyond the ream of inearity Usman A. Khan, Soummya Kar, and José M. F. Moura Department of Eectrica and Computer Engineering Carnegie Meon University 5 Forbes Ave, Pittsburgh, PA 53 {ukhan, moura}@ece.cmu.edu, soummyak@andrew.cmu.edu Ph: (4)68-73 Fax: (4) Abstract In this paper, we present a distributed averageconsensus agorithm with non-inear updates. In particuar, we use a weighted combination of the sine of the state differences among the nodes as a consensus update instead of the conventiona inear update which just incudes a weighted combination of the state differences. The non-inear update comes from the theory of non-inear iterative agorithms that we present esewhere. We show the non-inear average-consensus converges to the initia average under appropriate conditions on the combining weights. By simuations, we show that the convergence rate of our agorithm outperform the conventiona inear case. I. INTRODUCTION Recenty, there has been a significant interest in inear distributed average-consensus (LDAC) probem []. In this probem, we are interested in computing the average of severa scaar quantities distributed over a sensor network. These scaar quantities become the initia conditions of the LDAC agorithm. In the LDAC agorithm, each sensor updates its state as a inear combination of the neighboring states. Under appropriate conditions [], [], the state at each sensor converges to the average of the initia conditions. So far, the focus in this research has been on inear updates, where the convergence rate ony depends on the network connectivity (second argest eigenvaue of graph Lapacian). In this paper, we introduce a distributed average-consensus agorithm with non-inear updates. The state update in the non-inear distributed average-consensus (NLDAC) consists of the sensor s previous state added to a inear combination of the sine of the state differences among the neighboring nodes. Due to the non-inearity introduced by the sine function, the convergence rate now depends on the actua states of the nodes. As wi be shown in the paper, this fact makes the convergence rate of NLDAC faster, by appropriate tuning the combining weights. The NLDAC using the sinusoids stems from our work on the theory of distributed non-inear iterative agorithms that we present esewhere. In fact, non-inear functions other than sinusoids can aso be empoyed, if the chosen noninear function has certain properties that we eaborate in the This work was partiay supported by NSF under grants # ECS-5449 and # CNS-4844, and by ONR under grant # MURI-N paper. The sine function can be used as a combining function in the distributed iterative agorithms since it is Lipschitz continuous with Lipschitz constant stricty ess than unity if the frequency and the domain of the sine function is chosen appropriatey. In this context, we note that our work can be tied to resuts on networks of couped osciators (see, for exampe, [3], [4], [5]). The aforementioned works concern quaitative properties of such networks. In contrast, we propose schemes for a design of agorithms with desirabe properties (in our case, average-consensus) and our methodoogy is different. The framework, presented here, goes beyond averageconsensus and is ikey to find appications in other areas of distributed signa processing, ike distributed phase-ocked oops [6], arge-scae power networks [7], where such form of dynamics arise naturay. For simpicity of the exposition, we assume that each sensor,, possesses a scaar quantity, y, such that y [ π/4 + ɛ, π/4 ɛ]. This is needed because we woud ike to operate in the domain of the sine function where the cos(y y j ) (as wi be shown, the cosine determines the convergence rate as it is the derivative of the sine) does not take the vaue, for any and j. It is noteworthy that this assumption does not put any restriction on the agorithm when y s are arbitrary as ong as they are bounded, i.e., y < M,. In such case, we can aways choose an appropriate frequency, ζ, such that cos(ζ(y y j )) does not vanish and the resuting convergence rate invoves an additiona factor of ζ. We now describe the rest of the paper. In Section II, we recapituate some reevant concepts from graph theory. Section III discusses the probem formuation and introduces the non-inear distributed average-consensus agorithm. We anayze the agorithm in Section IV and derive the conditions for convergence. In Section V, we present simuations and finay, Section VI concudes the paper. II. PRELIMINARIES Consider a network of N nodes where the N nodes are inter-connected through an undirected communication graph, G (V, A), where V {,..., N} is the set of vertices

2 and A {a j } is the adjacency matrix of the communication graph, G. Since the graph is undirected, the adjacency matrix, A, is symmetric. We define K() as the neighbors of node, i.e., K() {j a j }. () Simiary, we define D {} K(). () Let K be the tota number of edges in G. Let C {c k } k,...,k,...,n be the N K incidence matrix of G where its kth coumn represents the kth edge, (i, j) G, such that c ik and c jk. The Lapacian, L, of G is then defined as L CC T. (3) If w k is a weight associated to the kth edge in G, then the weighted Lapacian matrix is defined as L w CWC T, (4) where W is a K K diagona matrix such that the kth eement on its diagona (that represents the kth edge in G) is w k. Note that the Lapacian, L, is symmetric and positivesemidefinite. Hence, its eigenvaues are rea and nonnegative. If W (where denotes eement-wise inequaity), then L w is aso symmetric, positive-semidefinite, see [8] for detais. III. PROBLEM FORMULATION Consider a network of N nodes communicating over a graph G, where each node,, possesses a scaar quantity, y. We woud ike to consider distributed updates on G of the foowing form: at each sensor, we have x (t + ) x (t) + h (D ), x () y, (5) where h is some function such that the above agorithm converges to im (t + ) N x j () N y j, t N N, (6) j j i.e., to the average of the scaar quantities, y, the sensors possess. The conventiona average-consensus (LDAC) agorithm is inear where we choose h (D ) µ (x (t) x j (t)),. (7) j K() In this paper, we aow the functions, h, to be non-inear. In particuar, we choose h (D ) µ sin(x (t) x j (t)),. (8) j K() The above agorithm fas into a genera cass of non-inear distributed iterative agorithms that we present esewhere. We show that the iterative agorithm (5) converges to (6) when we choose the functions h to be of the form (8) under some conditions on µ and on the network connectivity. In the rest of the paper, we prove this resut and estabish the conditions required for convergence. IV. NLDAC ALGORITHM In this section, we derive some important resuts reating to our agorithm. This wi be hepfu in proving the convergence to the average of the initia conditions. Reca from the previous section, the NLDAC is given by the foowing update at sensor x (t + ) x (t) µ sin(x (t) x j (t)), (9) j K() with x () y. We now write the above agorithm in matrix form. Define x(t) [x,..., x N (t)] T, () f(x(t)) [f (x(t)),..., f N (x(t))] T, () where f (x(t)) is defined as f (x(t)) x (t) µ j K() sin(x (t) x j (t)). () With the above notation, agorithm (9) can be written compacty as A. Important resuts x(t + ) f(x(t)). (3) We now expore some properties of the function f : R N R N. We have the foowing emma. Lemma : The functions f are sum preserving, i.e., f (x(t)) x (t) y. (4) Proof: We start with the L.H.S of (4). We have f (x(t)) x (t) µ sin(x (t) x j (t)). j K() (5) In order to estabish (4), it suffices to show that µ sin(x (t) x j (t)). (6) j K() Since the communication graph is symmetric, we have j K() K(j). (7) Fix an and a j K(), then there are two terms in the L.H.S of (6) that contain both and j. For these two terms, we have µ sin(x (t) x j (t)) + µ sin(x j (t) x (t)), (8) due to the fact that sine is an odd function. The above argument is true, and hence, (4) foows. Ceary, from (4), we aso have x (t + ) x (t). (9) In the foowing emma, we estabish the fixed point of the agorithm (3).

3 Lemma : Let denote an N coumn-vector of s. For any c R, x c is a fixed point of (3). Proof: The proof is straightforward and reies on the fact that sin(). To provide the next resut, we et D(x) be a K K diagona matrix such that the kth eement on its diagona is cos(x i x j ), where i and j are those vertices that represent the edge described by the kth coumn of the incidence matrix, C. Lemma 3: Let the derivative of the function f(x) with respect to x be denoted by f (x), i.e., f (x) f(x) { } x fi (x), () x j f i (x) x j i,j,...,n then f (x) I µcd(x)c T. () Proof: Note that { f } (x) I µ j K(i) cos(x i x j ), i j, µ cos(x i x j ) i j, (i, j) G,, i j, (i, j) / G. () With the above, we note that µ ij j K(i) cos(x i x j ), i j, cos(x i x j ) i j, (i, j) G,, i j, (i, j) / G. (3) is a weighted Lapacian matrix with the corresponding weight for each edge, (i, j) G, coming from the matrix D(x). B. Error Anaysis In this subsection, we present the error propagation and provide an upper bound on the error norm. Let xavg N N x (). (4) Define the error in the iterations (3) as e(t + ) x(t + ) xavg. (5) The foowing emma provides an upper bound on the norm of e(t). Lemma 4: Let J T. Then, we have for some θ(t) N [, ] e(t + ) I µcd(θ(t)e(t))c T J e(t). (6) Proof: Note that Jxavg xavg, Jx(t) xavg, (7) where the first equation uses T N, and the second equation is a consequence of Lemma. Aso note from Lemma that f(xavg) xavg. From (5), (3), and (7), we have e(t + ) f(x(t)) f(xavg) J(x(t) xavg), g(x(t)) g(xavg), (8) where g : R N R N is defined as g(x) f(x) Jx. The error norm is, thus, given by e(t + ) g(x(t)) g(xavg), g (η(t)) ( x(t)) xavg ), g (η(t)) e(t), I µcd(η(t))c T J e(t), (9) where the second equation foows from the mean vaue theorem and the fact that g is continuousy differentiabe, so that there exists some θ(t) [, ] where η(t) θ(t)x(t) + ( θ(t))xavg. The ast equation foows from the definition of g and Lemma 3. Reca that the eements of D(η(t)) are of the form cos(η i (t) η j (t)). We have η i (t) η j (t) θ(t)x i (t) + ( θ(t))xavg (θ(t)x j (t) + ( θ(t))xavg), θ(t)(x i (t) x j (t)). (3) Hence, we can write D(η(t)) as D(θ(t)e(t)) and (6) foows. We introduce the foowing notation, which we wi use in presenting the main resut of this paper. Let L t CD(θ(t)e(t))C T. (3) Define L as the set of a possibe L t when the agorithm is initiaized with the initia conditions, x (), i.e., L {L t x() x ()}. (3) Reca that the weighted Lapacian, L t, is symmetric, positive-semidefinite, when the diagona matrix, D(θ(t)e(t)), is non-negative. Let Q t [q (t), q (t),..., q N (t)] be the matrix of N ineary independent eigenvectors of L t with the corresponding eigenvaues denoted by λ i (L t ), i,..., N. Without oss of generaity, we assume that λ λ... λ N, and q (t) with the corresponding eigenvaue λ (L t ). Define p inf (L t ), L (33) p N sup λ N (L t ). (34) L Lemma 5: The eigenvectors of the matrix I µl t J are the coumn vectors in the matrix Q t and the corresponding eigenvaues are and µλ i (L t ), i,..., N. Proof: The first eigenvector of the matrix I µl t J is q (t) with the eigenvaue. This can be shown as (I µl t J)q µl t J. (35) That the rest of the eigenvectors, q (t),..., q N (t), of L t, are aso the eigenvectors of I µl t J can be estabished by the fact that J is rank with eigenvector and the identity matrix can have any set of ineary independent vectors as its eigenvectors. Note that L t is symmetric, positive-semidefinite so its eigenvaues are positive reas.

4 The next emma estabishes that x (t) [ π/4+ɛ, π/4 ɛ],, t, when the initia condition, x(), ies in the same range. Lemma 6: Let the vector of network initia conditions, x(), be such that x () [ π/4 + ɛ, π/4 ɛ],, (36) where ɛ > is a sufficienty sma rea number. Then for µ in the range < µ π, we have d max x (t) [ π/4 + ɛ, π/4 ɛ],, t. (37) Proof: We use the foowing bound to prove this emma. π x sin(x) x, x π. (38) For any arbitrary node,, partition its neighbors, K(), into K L (, t) and K G (, t), where K L (, t) {j K() x (t) > x j (t)}, and K G (, t) {j K() x (t) < x j (t)}. We can write the NLDAC iterations (9) as x (t + ) x (t) µ sin(x (t) x j (t)) + µ j K G (,t) j K L (,t) x (t) µ + µ + µ π j K G (,t) sin(x j (t) x (t)), j K L (,t) (x j (t) x (t)), π (x (t) x j (t)) ( µ ) π K L(, t) µ K G (, t) x (t) j K L (,t) x j (t) + µ j K G (,t) x j (t),(39) where the inequaity foows from (37) and the bound in (38). Simiary, we can show x (t + ) x (t) µ (x (t) x j (t)) j K L (,t) + µ π (x j(t) x (t)), j K G (,t) ( µ K L (, t) µ ) π K G(, t) x (t) + µ j K L (,t) x j (t) + µ π j K G (,t) x j (t).(4) Combining (39) and (4), x (t + ) remains bounded above and beow by a convex combination of x j (t), j D, when µ π K L(, t) + µ K G (, t), (4) µ K L (, t) + µ π K G(, t). (4) The L.H.S is triviay satisfied in both of the above equations. To derive µ such that the R.H.S is satisfied for both of the above equations, we use µ π (K L(, t) + K G (, t) ), (43) π. (44) d max With µ satisfying the above equation, x (t + ) is bounded above and beow by a convex combinations of x j (t), j D(). So, if x j (t) [ π/4 + ɛ, π/4 ɛ], for j D(), so does its convex combinations and thus x (t + ) [ π/4 + ɛ, π/4 ɛ] and the emma foows. Using the above emma, we now have the foowing resut. Lemma 7: Let (37) hod. Let the network communication graph be connected, i.e., λ (L) >, then p >, for < µ π/d max. Proof: Consider L w as defined in (4) with W > (eement-wise inequaity, aso reca W is a K K diagona matrix with w k,...,k k w ij on its diagona denoting the weight of the kth edge, (i, j) G). We have z T L w z w ij (z i z j ), (45) (i,j) G for any z R N. Since w ij >, (i, j) G, the quadratic form (45) is if and ony if z c, for any c R. So L w has ony one eigenvaue of with eigenvector. Hence, λ (L w ) >. Aso, note that λ (L w ) is a continuous function of w ij s [9]. So the infimum of λ (L w ) is attainabe if the eements of W ie in a compact set. To this end, define C {W R K K w ii [cos(π/ ɛ), ], w ij (i j)}, (46) and note that D(θ(t)e(t)) C from Lemma 6. Since L t CD(θ(t)e(t))C T and D(θ(t)e(t)) C, we note that p inf L λ (L t ) inf W C λ (L w ). (47) We now use a contradiction argument to show p >. Assume on the contrary that p. Then inf W C λ (L w ) and there exists some W C such that λ (L w ). But, for a W C, we have λ (L w ) >, which is a contradiction. Hence, p >. We now present the convergence of NLDAC in the foowing theorem. Theorem : Let the vector of network initia conditions be denoted by x() such that (37) hods. Let the network communication graph, G, be connected, i.e., λ (L) >. If µ is such that < µ < p N, (48) then im e(t). (49) t Proof: From (34) and (48), we have for i,..., N µλ i (L t ) µp N > p N p N. (5) From (33), we have for i,..., N µλ i (L t ) µp <, (5)

5 π/ + ɛ Fig.. Cosine Range Data Range π/4 + ɛ π/4 ɛ π/ ɛ Figure corresponding to Remark (i). V. SIMULATIONS We consider a network of N nodes, shown in Fig. (a). We impement the conventiona inear distributed average-consensus agorithm with optima constant weights, i.e., we choose µ OPT LIN λ (L) + λ N (L), (55) in (7). The error norm in this case is shown in Fig. (b) as a red dotted curve. To show the performance of the NLDAC agorithm (9), we choose the foowing vaues of µ µ {.99 d max, d max, d max, 3d max and show the error norm in Fig. (b) and Fig. (c). } (56) from (48) the fact that p > from Lemma 7. Combining (5) and (5), we have < µλ i (L t ) <, i,..., N, (5) With the above, we have µλ i (L t ) < c <, i,..., N and for some c [, ). Thus, the error norm is given by e(t + ) max i N µλ i(l t ) e(t) < c e(t), (53) and (49) foows. We, further, have p N d max [8], where d max is the maximum degree of the graph generated by the Lapacian, L CC T. We now have convergence for < µ < d max p N. (54) Remarks: We now make some reevant remarks. (i) We expain our assumption in (37) with the hep of Fig.. When the data, x (),, ies in the interva [ π/4+ ɛ, π/4 ɛ], then x (t) x j (t) for a t and j must ie in the interva [ π/ + ɛ, π/ ɛ]. This is true for a t, since the choice of µ in (48) guarantees a contraction from Theorem. Hence, cos(x (t) x j (t)), t, must ie in the interva [cos(π/ ɛ), ], which is stricty greater than for ɛ >, as shown in Fig.. With cos(x (t) x j (t)) [cos(π/ ɛ), ], t, we note that L t does not ose the sparsity (zero-one) pattern of L and hence, λ (L t ) >, t. Ceary, if (37) does not hod but the initia data has a known bound, we can introduce a frequency parameter, ζ, in the sine function such that cos(ζ(x (t) x j (t))) [cos(π/ ɛ), ]. Hence, the assumption in (48) does not ose generaity. (ii) Note that p N may not be known or easiy computabe a priori.in that case, one may work with < µ < /d max as estabished in (54), which is readiy determined. (iii) Choosing µ away from the bound in (54), resuts into a divergence of the agorithm, as we wi eaborate in the simuations. This is a manifestation of the bifurcation phenomena as arise in the non-inear theory. VI. CONCLUSIONS In this paper, we present a non-inear distributed averageconsensus (NLDAC) agorithm that uses the sine of the state differences among the nodes instead of the conventiona inear update. The convergence rate of the NLDAC agorithm now depends on the cosine of the state differences (as cosine is the derivative of the sine) and, thus, depends on the actua state vaues. Due to this dependence, the convergence rate has an additiona degree of freedom as the convergence rate in LDAC ony depends on the network connectivity. We provide simuations to assert the theoretica findings. REFERENCES [] L. Xiao and S. Boyd, Fast inear iterations for distributed averaging, Systems and Contros Letters, vo. 53, no., pp , Apr. 4. [] R. Ofati-Saber, J. A. Fax, and R. M. Murray, Consensus and cooperation in networked muti-agent systems, Proceedings of the IEEE, vo. 95, no., pp. 5 33, Jan. 7. [3] Y. Kuramoto, Cooperative dynamics of osciator community, Progress of Theoretica Physics Supp., vo. 79, pp. 3 4, 984. [4] C. W. Wu and L. O. Chua, Synchronization in an array of ineary couped dynamica systems, IEEE Transactions on Circuits Systems I, vo. 4, pp , Aug [5] A. Mauroy and R. Sepuchre, Custering behaviors in networks of integrate-and-fire osciators, Chaos: Specia issue on Synchronization in Compex Networks, vo. 8, no. 3, pp , 8. [6] N. Varanese, O. Simeone, U. Spagnoini, and Y. Bar-Ness, Distributed frequency-ocked oops for wireess networks, in th IEEE Internationa Symposium on Spread Spectrum Techniques and Appications, Boogna, Itay, Aug. 8, pp [7] A. Bergen and D. Hi, A structure preserving mode for power system stabiity anaysis, IEEE Trans. on Power Apparatus and Systems, vo. PAS-, no., pp. 5 35, Jan. 98. [8] F. R. K. Chung, Spectra Graph Theory, Providence, RI : American Mathematica Society, 997. [9] F. R. Gantmacher, Matrix Theory (Voume I), Chesea Pubishing Co., USA, 959.

6 (a).8 Linear: Optima Constant Weights Non-inear: Constant weights (µ < dmax ) e(t) Iterations, t (b) e(t) µ dmax µ dmax µ 3dmax Iterations, t (c) Fig.. (a) An N node network. (b) Comparison of the NLDAC with LDAC using constant optima edge weights. (c) The error norm of NLDAC for different vaues of µ.