Lec 6: Time-Dependent Linear Iterations

CONTENTS 1 TIME-DEPENDENT AVERAGING ALGORITHMS

Lecture 6: time-dependent linear iterations

Sonia Mart´ınez

May 9, 2018

Abstract Iterations correspond to algorithm executions. Linear iterations are amenable to methods from linear algebra. We have seen this when analyzing of the flocking example with fixed topology. However, our study is no longer applicable if the topology is changing from one timestep to the next. The inherent richness of cooperative systems makes linear algebra insufficient – we need to invoke notions and tools from other areas, such as graph theory and stability analysis. This is what we do in this set of lecture notes. The treatment corresponds to selected parts from Chapter 1 in [1]. On the other hand, convergence factors of a (constant-time) linear iteration is introduced (see [1] and[2]) and we point out how this depends on the several properties of the underlying graph.

Contents

1 Time-dependent averaging algorithms1 1.1 Stability...... 3 1.2 Convergence for general sequence of stochastic matrices...... 3 1.3 Convergence for sequence of stochastic symmetric matrices...... 5 1.4 Final agreement value...... 6

2 Convergence speed of linear iterations7

A Toeplitz and tridiagonal circulant matrices 10

1 Time-dependent averaging algorithms

Linear distributed algorithms on synchronous networks are discrete-time linear dynamical systems whose evolution map is linear and has a sparsity structure related to the network. These algorithms represent an important class of iterative algorithms that ﬁnd applications in optimization, in the solution of systems of equations, and in distributed decision making; see, for instance [3]. In this section, we present some relevant results on distributed linear algorithms.

Example 1.1 (Flocking) In our previous lecture notes we have fully characterized the stability and convergence of the ﬂocking algorithm,

1 θ(` + 1) = F θ(`),F = (In + D)− (In + A),

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1 θ(` + 1) = F (`)θ(`),F (`) = (In + D(`))− (In + A(`)),

For instance, before it was good enough to determine whether F was semi-convergent. Now, if the topology is changing, even if individual F (`) are semi-convergent, how do we know that arbitrary products of them will be? Assuming they are and agents eventually ﬂock, can we determine the asymptotic common heading? • We study linear combination algorithms over time-dependent weighted directed graphs; we restrict our analysis to nonnegative weights. The averaging algorithm associated to a sequence of stochastic matrices n n F (`) ` Z 0 R × is the discrete-time dynamical system { | ∈ ≥ } ⊂

w(` + 1) = F (`) w(`), ` Z 0. (1) · ∈ ≥ In the literature, such algorithms are often referred to as agreement algorithms, or as consensus algorithms. As we have seen in previous lectures, there are useful ways to compute a stochastic matrix, and therefore, a time-independent averaging algorithm, from a weighted digraph.

Deﬁnition 1.2 (Adjacency- and Laplacian-based averaging) Let G be a weighted digraph with node set 1, . . . , n , weighted adjacency matrix A, weighted out-degree matrix Dout, and weighted Laplacian L. Then{ }

1 (i) the adjacency-based averaging algorithm is deﬁned by the stochastic matrix (In + Dout)− (In + A) and reads in components 1 n w (` + 1) = w (`) + a w (`) ; (2) i 1 + d (i) i ij j out j=1 X

(ii) given a positive scalar ε upper bounded by min 1/dout(i) i 1, . . . , n , the Laplacian-based averaging algorithm is defined by the stochastic matrix{I εL(G|) and∈ { reads in}} components n − n n w (` + 1) = 1 ε a w (`) + ε a w (`). (3) i − ij i ij j j=1,j=i j=1,j=i X6 X6 These notions are immediately extended to sequences of stochastic matrices arising from sequences of weighted digraphs. • We recognize both types of averaging algorithms from previous lectures. Adjacency-based averaging corresponds to the flocking example and Laplacian-based averaging corresponds to the discretization of the agreement example. Adjacency-based averaging algorithms arising from unweighted undirected graphs without self-loops are also known as equal-neighbor averaging rule or the Vicsek’s model [4]. Specifically, if G is an unweighted graph with vertices 1, . . . , n and without self-loops, then the equal-neighbor averaging rule is { } w (` + 1) = avrg w (`) w (`) j (i) , (4) i { i } ∪{ j | ∈ NG } where we adopt the shorthand avrg( x , . . . , x ) = (x + + x )/k. { 1 k} 1 ··· k 2

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Remark 1.3 (Sensing versus communication interpretation of directed edges) In the deﬁnition of averaging algorithms arising from digraphs, the digraph edges play the role of “sensing edges,” not that of “communication edges.” In other words, a nonzero entry aij, corresponding to the digraph edge (i, j), implies that the ith component of the state is updated with the jth component of the state. It is as if node i could sense the state of node j, rather than node i transmitting to node j its own state. •

1.1 Stability

Here, we present the main stability result for averaging algorithms associated to a sequence of stochastic matrices. We start by discussing equilibrium points and their stability. Recall that 1n is an eigenvector of any stochastic matrix with eigenvalue 1 and that the diagonal set diag(Rn) is the vector subspace generated n by 1n. Therefore, any point in diag(R ) is an equilibrium for any averaging algorithm. We refer to the points of the diag(Rn) as agreement conﬁgurations, since all the components of an element in diag(Rn) are equal to the same value. We will informally say that an algorithm achieves agreement if it steers the network state toward the set of agreement conﬁgurations.

Lemma 1.4 (Stability of agreement conﬁgurations) Any averaging algorithm (1) in Rn is uniformly stable and uniformly bounded with respect to diag(Rn).

1.2 Convergence for general sequence of stochastic matrices

Regarding convergence results, we need to introduce a useful property of collections of stochastic matrices. Given α ]0, 1], the set of non-degenerate matrices with respect to α consists of all stochastic matrices F with entries∈ f , for i, j 1, . . . , n , satisfying ij ∈ { } f [α, 1], and f 0 [α, 1] for j = i. ii ∈ ij ∈ { } ∪ 6 Additionally, the sequence of stochastic matrices F (`) ` Z 0 is non-degenerate if there exists α ]0, 1] { | ∈ ≥ } ∈ such that F (`) is non-degenerate with respect to α for all ` Z 0. We now state the main convergence result. ∈ ≥

n n Theorem 1.5 (Convergence for time-dependent stochastic matrices) Let F (`) ` Z 0 R × { | ∈ ≥ } ⊂ be a non-degenerate sequence of stochastic matrices. For ` Z 0, let G(`) be the unweighted digraph associated to F (`). The following statements are equivalent: ∈ ≥

(i) the set diag(Rn) is uniformly globally asymptotically stable for the averaging algorithm associated to F (`) ` Z 0 ; and { | ∈ ≥ } (ii) there exists a duration δ N such that, for all ` Z 0, the digraph ∈ ∈ ≥ G(` + 1) G(` + δ) ∪ · · · ∪ contains a globally reachable vertex.

We collect a few observations about this result.

Remarks 1.6 (Discussion of Theorem 1.5)

(i) The statement in Theorem 1.5(i) means that each solution to the time-dependent linear dynamical system (1) converges uniformly and asymptotically to the vector subspace generated by 1n. 3

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(ii) The necessary and sufficient condition in Theorem 1.5(ii) amounts to the existence of a uniformly bounded time duration δ with the property that a weak connectivity assumption holds over each collection of δ consecutive digraphs. We refer to Example 1.7 below for a counterexample showing that if the duration in Theorem 1.5 is not uniformly bounded, then there exist algorithms that do not 1.21.2converge. Convergence Convergence for for general general sequence sequence of of stochastic stochastic matrices matrices 1 1 AVERAGING AVERAGING ALGORITHMS ALGORITHMS 1.21.2 Convergence Convergence for forgeneral general sequence sequence of stochastic of stochastic matrices matrices 1 AVERAGING 1 AVERAGING ALGORITHMS ALGORITHMS (iii) Uniform convergence is a property of all solutions to system (1) starting at any arbitrary time, and not (iii)(iii)UniformUniform convergence convergence is is a a property property of of all all solutions solutions to to system system (1 ()1) starting starting at atanyany arbitrary arbitrary time time, and, and not not (iii)onlyUniform at time convergence equal to zero. is a property If we restrict of all solutionsour attention to system to solutions (1) starting that at onlyany arbitrary start at time time, zero, and not then (iii) Uniformonlyonly at atconvergence time time equal equal to is to azero. zero.property If If we we of restrict allrestrict solutions our our attention attentionto system to to(1 solutions) solutions starting that at thatany only only arbitrary start start at time at time time, and zero, zero, not then then Theoremonlyonly at time at 1.5 time equalshould equal to be zero. to modified zero. If we If restrictwe as follows: restrict our ourattention the attention statement to solutions to in solutions Theorem that that only 1.5 only(i) start implies, start at time at but time zero, is zero, not then implied then TheoremTheorem1.61.6shouldshould be be modified modified as as follows: follows: the the statement statement in in Theorem Theorem1.61.6(i)(i)implies,implies, but but is is not not implied implied by,Theorem theTheorem statement1.6 1.6shouldshould in be Theorem modified be modified 1.5 as(ii) follows: as. follows: the the statement statement in Theorem in Theorem1.6(i)1.6implies,(i) implies, but but is not is not implied implied by,by, the the statement statement in in Theorem Theorem1.61.6(ii)(ii). . by,by, the the statement statement in Theorem in Theorem1.6(ii)1.6.(ii). (iv)(iv)(iv) TheTheThe theorem theorem theorem applies applies applies only only only to to to sequences sequences sequences of of non-degenerate non-degenerate matrices. matrices. matrices. Indeed, Indeed, Indeed, there there exist exist sequences sequences sequences of of of (iv)(iv)TheThe theorem theorem applies applies only only to sequences to sequences of non-degenerate of non-degenerate matrices. matrices. Indeed, Indeed, there there exist exist sequences sequences of of degeneratedegeneratedegenerate stochastic stochastic stochastic matrices matrices matrices whose whose whose associated associated averaging averaging averaging algorithms algorithms algorithms converge. converge. converge. Furthermore, Furthermore, one one one does does does degeneratedegenerate stochastic stochastic matrices matrices whose whose associated associated averaging averaging algorithms algorithms converge. converge. Furthermore, Furthermore, one one does does notnotnot even even even need need need to to to consider consider consider sequences, sequences, sequences, because because it it it is is is possible possible possible to to to define define define converging converging converging algorithms algorithms by by by just just just notnot even even need need to consider to consider sequences, sequences, because because it is it possible is possible to define to define converging converging algorithms algorithms by just by just consideringconsidering a single a single stochastic stochastic matrix. matrix. Precisely Precisely when when the the stochastic stochastic matrix matrix is primitive, we we already already consideringconsideringconsidering a a single asingle single stochastic stochastic stochastic matrix. matrix. matrix. Precisely Precisely Precisely when when when the the the stochastic stochastic stochastic matrix matrix matrix is primitive, is is primitive, primitive, we alreadywe already knowknow that that the the associated associated averaging averaging algorithm algorithm will will converge, converge, cf. cf. [Lecture [Lecture 2, 2, Theorem 4.10]. Examples Examples knowknowknow that that that the the the associated associated associated averaging averaging averaging algorithm algorithm algorithm will will will converge, converge, converge, cf. cf. [Lecturecf. [Lecture [Lecture 2, Theorem 2, 2, Theorem Theorem 4.10]. 4.10]. 4.10]. Examples Examples of degenerateof degenerate primitive primitive stochastic stochastic matrices matrices (with (with converging converging associated associated averaging averaging algorithms) are are given given ofof degenerateof degenerate degenerate primitive primitive primitive stochastic stochastic stochastic matrices matrices matrices (with (with (with converging converging converging associated associated associated averaging averaging averaging algorithms) algorithms) algorithms) are givenare given in Examplein Example 1.81.9. . inin Examplein Example Example1.91.9.1.9. . • • • • Example 1.7 (Necessity of uniformly bounded duration) This example is a variation of [5, Exam- ExampleExampleExampleExample 1.8 1.8 1.8 1.8 (Necessity (Necessity (Necessity (Necessity of of uniformlyof uniformly uniformly bounded bounded bounded duration) duration) duration) duration)ThisThisThisThis example example example example is a is is variationis a a a variation variation variation of [6 of, of Exam-[ 6 [,6, Exam- Exam- pleplepleple 1].ple 1]. 1]. 1]. Consider 1].Consider Consider Consider Consider a a network network a networka network network composed composed composed composed composed by by by 3 by 3agents 3agents agents that that that that switches switches switches switches switches between between between between between the the thetopologies the the topologies topologies topologies topologies described described described described in Figure in in in Figure Figure Figure1. 1.1. LetLetLetx(0)x(0)x(0) = = (0= (0, (00, ,0,1),01), 1) be be be the the the initial initial initial condition. condition. condition. Let Let"ε1"1bebebe a a a small small small positive positive positive constant. constant. constant. Consider Consider the the the following following following LetLetx(0)x(0) = (0 =, 0 (0, 1), 0, be1) thebe the initial initial condition. condition. Let Let"1 be1"1 abe small a small positive positive constant. constant. Consider Consider the the following following

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This This This This is repeated is is is repeated repeated repeatedt1 times,tt11t1times,times,times, where where where wheret1 ist1 larget1isis large large agent forms the average of its own value and the received value. This is repeated t1 times, where t1 is large enoughenoughenoughenough so soso that so that that thatx1x(xtx(1t1)((t)t1))1 1 11"1".Thus,".Thus,1.Thus,.Thus,x(xt1(x)t(t)1)(1,(10(1(1, 1).0,,,001).,,1). After1). After After After that, that, that, that, according according according according to the to to to the thedigraph the digraph digraph digraph in Figure in in in Figure Figure1(b),1(b),1(b), enough so that x (1t1 1)1 1 ε1.1 Thus, x(t1⇡)1 ⇡(1, 0, 1). After that, according to the digraph in Figure1(b), agentagent 2 communicates 2 communicates1 1 to agent to agent1 3, t2 3,times,t times,1 where⇡ wheret2 ist largeis large enough enough so that so thatx3(tx1 +(tt2+) t )"1." In. particular, In particular, agentagent 2 2 communicates communicates≥ to to− agent agent 3, 3,t2t2times,times, where≈ wheret2t22isis large large enough enough so so that thatx3x3(3t(11t1++t22t)2) "11".1. In In particular, particular, agentx(t + 2t communicates) (1, 0, 0). After to agent that, 3,accordingt2 times, to where the digrapht2 is large in Figure enough1(c), so agent that 1x communicates3(t1 + t2) ε to1. agent In particular, 3, x(x1tx(1(t1t+1+2+t2t)t22)) (1(1,(10,,000).,,0).0). After After After that, that, that, according according to to the the digraph digraph digraph in in in Figure Figure Figure11(c),1(c),(c), agent agent agent 1 1 1 communicates communicates communicates≤ to to agent agent 3, 3, x(t + t ) ⇡⇡(1⇡⇡, 0, 0). After that, according to the digraph in Figure1(c), agent 1 communicates to agent 3, t3t 1ttimes,ttimes,3times,times,2 where where where wheret3t isttis3 largeisis large large large enough enough enough enough so so that so that thatx3x(tx1(3t+(tt1+t2++t+tt2t+3++)t tt)3)1) 11"11."" In"1... Inparticular, In In particular, particular, particular,x(t1xx+(x(tt(1tt2++++tt2tt3++)+tt3t))(1), 0(1, 1).(1, 0,,01)., 1). 3 3 ≈ 3 3 3 3 1 1 2 2 33 1 1 1 1 2 2 ⇡3 3 ⇡ t3Finally,times,Finally, according where accordingt3 tois the large to the digraph enough digraph in soFigure in that Figure1(d),x31(t(d),1 agent+ agentt2 2+ communicatest 23) communicates1ε1. to In agent to particular, agent 1, t4 1,times,tx(times,t1 where+ t where2 +t4t3is⇡)t⇡ largeis(1 large, 0, 1). Finally,Finally, according according to to the the digraph digraph in in Figure Figure1(d),1(d), agent agent 2 2 communicates communicates≥ − to to agent agent 1, 1,t44t4times,times, where wheret4≈t4isis large large Finally,enoughenoughenoughaccording so so that so that thatx1( toxtx11(+( thett1t+2+ digraph+tt2t3+++tt3t4+ in) t Figure4)"1." In11.(d), particular, In In particular, particular, agent 2x( communicatest1xx+((tt1t2+++tt2t3+++tt3t to4+)+ agentt4t)(0) , 0(0 1,,(01)., 0,t0,41)., We1).times, Wenow We now where repeat now repeat repeatt the4 is the large the enough so that x1(1t1 1+ t2 2+ t3 3+ t4)4 "1.1 In particular, x(t1 1+ t2 2+ t3 3+ t⇡4)4 ⇡(0, 0, 1). We now repeat the enoughaboveabove process, so process, that infinitelyx1( infinitelyt1 + t many2 + manyt3 times.+ t times.4) Duringε During1. In the particular, thekthk repetition,th repetition,x(t1 +"1t2is"+ replacedist3 replaced+ t4)⇡ by⇡" by(0k ,(and"0, 1).(andt1, Wett2,, nowtt3,, tt4 repeat,gett get the aboveabove process, process, infinitely infinitely many many times. times. During During the thekthkth repetition, repetition,"1"11isis replaced replaced by by"kk"k(and(andt11t,1t,22t,2t,3t,3t,4t4getget ≤ 1 ≈ aboveadjustedadjusted process, accordingly). accordingly). infinitely If we many If choosewe choose times. the the sequence During sequence the of " ofkkthso"k that repetition,so thatk=1 1"1kε1<"isk1/< replaced2,1 asymptotic/2, asymptotic by εk consensus(and consensust1, willt2, will nott3, t not4 get adjustedadjusted accordingly). accordingly). If If we we choose choose the the sequence sequence of of"k"ksoso that that k1k=1k=1=1"k"k<<1/12,/2, asymptotic asymptotic consensus consensus will will not not adjustedbe obtained.be obtained. accordingly). If we choose the sequence of ε so that ∞ ε < 1/2, asymptotic consensus will not bebe obtained. obtained. k P Pk=1 k • • be obtained. (Just notice that we can always find i and j so thatPPx ( K t ) x ( K t ) 1 K •ε• ExampleExample 1.9 1.9(Degenerate(Degenerate stochastic stochastic matrices matrices with with convergent convergentP i averagingk=1 averagingk algorithms):j algorithms):k=1 k ConsiderConsiderk=1 k forExampleExample all K, so 1.9 agreement 1.9(Degenerate(Degenerate will not stochastic stochasticbe possible.) matrices matrices with with convergent convergent| averaging averaging− algorithms): algorithms):| ≥ Consider−Consider thethe stochastic stochastic matrices matrices P P P • thethe stochastic stochastic matrices matrices 1100 011 110011001100 Example 1.8 (Degenerate stochastic1 011011 matrices with1 convergent0011 averaging algorithms): Consider A = 1011 and A = 11 0011. the stochastic matrices 1 A1 =1 101 and 2 A2 =21 2001100113 3 . AA1 1=2=22 1011013 andand AA2 2=2 =1100222110033. . 211022211033 2 2 110011100 1 0 0 0110110 1 1 60011600117 7 14 5 61660011600011 0 17 7 177 4 5 4 6 5 7 OneOne can can show show that that both both matrices matricesA1 = are are primitive41 primitive 0 1 (draw5 (drawand the the associatedA2 associated= 64 digraphs digraphsG715andG. andG2 andG and see thatsee that they they OneOne can can show show that that both both matrices matrices2 are are primitive primitive (draw (draw the the associated associated2441 digraphs 1 digraphs 05 0 5GG1 andandGG2 andand see see that that they they areare actually actually strongly strongly connected connected and and aperiodic).1 aperiodic). 1 0 Therefore, Therefore, the the averaging averaging algorithms algorithms1 1 associated associated2 2 to each to each one one areare actually actually strongly strongly connected connected and and aperiodic). aperiodic). Therefore, Therefore, the the averaging averaging0 0 1 algorithms algorithms 1 associated associated to to each each one one of themof them converge. converge. In fact, In fact, the the one one associated associated to A to2 convergesA converges in a in finite a finite number number of steps. of steps. On theOn theother other hand, hand, ofof them them converge. converge. In In fact, fact, the the one one associated associated to toAA2convergesconverges in in a a finite finite number number of of steps. steps. On On the the other other hand, hand, bothboth matrices matrices are are degenerate degenerate because because of the of the zero zero entries2 entries2 4 in their in their diagonals. diagonals. bothboth matrices matrices are are degenerate degenerate because because of of the the zero zero entries entries in in their their diagonals. diagonals. • • • • LetLet us particularize us particularize our our discussion discussion here here on adjacency- on adjacency- and and Laplacian-based Laplacian-based averaging averaging algorithms. algorithms. LetLetMAE247 us us particularize particularize – Cooperative our our Control discussion discussion of Multi-Agent here here on on adjacency- Systems. adjacency- Permission and and Laplacian-based Laplacian-based is granted to copy, averaging averaging distribute algorithms. algorithms. and modify this file, provided original sources5 5 are acknowledged. 5 5

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One can show that both matrices are primitive (draw the associated digraphs G1 and G2 and see that they are actually strongly connected and aperiodic). Therefore, the averaging algorithms associated to each one of them converge. In fact, the one associated to A2 converges in a ﬁnite number of steps. On the other hand, both matrices are degenerate because of the zero entries in their diagonals. • Let us particularize our discussion here on adjacency- and Laplacian-based averaging algorithms.

Corollary 1.9 (Convergence of adjacency- and Laplacian-based averaging algorithms) Let G(`) ` n n { | ∈ Z 0 R × be a sequence of weighted digraphs. The following statements are equivalent: ≥ } ⊂

(i) there exists δ N such that, for all ` Z 0, the digraph ∈ ∈ ≥ G(` + 1) G(` + δ) ∪ · · · ∪ contains a globally reachable vertex;

(ii) the set diag(Rn) is uniformly globally asymptotically stable for the adjacency-based averaging algorithm (2) (deﬁned with M aij ε > 0 for some M and ε) associated to G(`) ` Z 0 ; and ≥ ≥ { | ∈ ≥ } (iii) the set diag(Rn) is uniformly globally asymptotically stable for the Laplacian-based averaging algorithm (3) (deﬁned with ε < 1/n and M aij ε > 0 for some M and ε) associated to G(`) ` Z 0 . ≥ ≥ { | ∈ ≥ } The next result states that convergence is actually to a point, not to the set.

Proposition 1.10 (Convergence to a point in the invariant set) Under the assumptions in Theorem 1.5 and assuming that diag(Rn) is uniformly globally asymptotically stable for the averaging algorithm, each individual evolution converges to a speciﬁc point of diag(Rn).

1.3 Convergence for sequence of stochastic symmetric matrices

Theorem 1.5 gives a general result about non-degenerate stochastic matrices that are not necessarily symmetric. The following theorem presents a convergence result for the case of symmetric matrices (i.e., undirected digraphs) under connectivity requirements that are weaker (i.e., the duration does not need to be uniformly bounded) than those expressed in statement (ii) of Theorem 1.5.

Theorem 1.11 (Convergence for time-dependent stochastic symmetric matrices) Let F (`) ` n n { | ∈ Z 0 R × be a non-degenerate sequence of symmetric, stochastic matrices. For ` Z 0, let G(`) be the unweighted≥ } ⊂ graph associated to F (`). The following statements are equivalent: ∈ ≥

(i) the set diag(Rn) is globally asymptotically stable for the averaging algorithm associated to F (`) ` { | ∈ Z 0 ; and ≥ } (ii) for all ` Z 0, the graph ∈ ≥ G(τ) τ ` [≥ is connected.

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1.4 Final agreement value

In general, the final value upon which all wi, i 1, . . . , n , agree in the limit is unknown. This final value depends on the initial condition and the specific∈ { sequence} of matrices defining the time-dependent linear algorithm. In some cases, however, one can compute the final value by restricting the class of allowable matrices. We consider two settings: time-independent averaging algorithms and doubly stochastic averaging algorithms. First, we specialize the main convergence result to the case of time-independent averaging algorithms (this is a particular case of the general Theorem 2.1, which can be found in the lecture 4 notes). Note that, given a stochastic matrix F , convergence of the averaging algorithm associated to F for all initial conditions is equivalent to the matrix F being semi-convergent.

Proposition 1.12 (Time-independent averaging algorithm) Consider the linear dynamical system on Rn w(` + 1) = F w(`), ` Z 0. (5) ∈ ≥ n n n Assume that F R × is stochastic, let G(F ) denote its associated weighted digraph, and let v R be a left eigenvector of∈ F with eigenvalue 1. Assume either one of the two following properties: ∈

(i) F is primitive (i.e., G(F ) is strongly connected and aperiodic); or

n 1 (ii) F has non-zero diagonal terms and a column of F − has positive entries (i.e., G(F ) has self-loops at each node and has a globally reachable node).

T T Then every trajectory w of system (5) converges to (v w(0)/v 1n)1n.

Proof: From Theorem 1.5 we know that the dynamical system (5) converges if property (ii) holds. The same conclusion follows if F satisﬁes property (i) because of Perron–Frobenius [Lecture 2, Theorem 4.10 and Corollary 4.11]. To computing the limiting value, note that

vT w(` + 1) = vT F w(`) = vT w(`), that is, the quantity ` vT w(`) is constant. Because F is semi-convergent and stochastic, we know 7→ T that lim` + w(`) = α1n for some α. To conclude, we compute α from the relationship α(v 1n) = →T ∞ T lim` + v w(`) = v w(0). → ∞

Remarks 1.13 (Alternative conditions for time-independent averaging)

(i) The following necessary and suﬃcient condition generalizes and is weaker than the two suﬃcient conditions given in Proposition 1.12: every trajectory of system (5) is asymptotically convergent if and only if all sinks of the condensation digraph of G(F ) are aperiodic subgraphs of G(F ). We refer the interested reader to [6, Chapter 8] for the proof of this statement and for the related notion of ergodic classes of a Markov chain.

(ii) Without introducing any trajectory w, the result of the proposition can be equivalently stated by saying that ` T 1 T lim F = (v 1n)− 1nv . ` + • → ∞ Second, we focus on the case of time-dependent doubly stochastic averaging algorithms.

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Corollary 1.14 (Average consensus) Let F (`) ` Z 0 be a sequence of stochastic matrices as in { | ∈ ≥ } Theorem 1.5. If all matrices F (`), ` Z 0, are doubly stochastic, then every trajectory w of the averaging algorithms satisﬁes ∈ ≥ n n

wi(`) = wi(0), for all `, i=1 i=1 X X that is, the sum of the initial conditions is a conserved quantity. Therefore, if F (`) ` Z 0 is non- degenerate and satisﬁes property (ii) in Theorem 1.5, then { | ∈ ≥ }

1 n lim wj(`) = wi(0), j 1, . . . , n . ` + n ∈ { } → ∞ i=1 X Proof: The proof of the ﬁrst fact is an immediate consequence of

n n T T T wi(` + 1) = 1n w(` + 1) = 1n F (`)w(`) = 1n w(`) = wi(`). i=1 i=1 X X The second fact is an immediate consequence of the ﬁrst fact. In other words, if the matrices are doubly stochastic, then each component of the trajectories will converge to the average of the initial condition. We therefore adopt the following deﬁnition: an average-consensus averaging algorithm is an averaging algorithm whose sequence of stochastic matrices are all doubly stochastic.

2 Convergence speed of linear iterations

We know that any trajectory of a general averaging algorithm converges to the diagonal set diag(Rn) in infinite time; in what follows we characterize how fast this convergence takes place. We are interested in finding how this rate depends on the dimension of the underlying network and eigenvalues of an adjacency matrix. There will be a tradeoff between how sparse the interactions between agents are and how fast the algorithm converges. We begin with some general definitions for semi-convergent matrices.

Deﬁnition 2.1 (Convergence time, per-iteration convergence factor, and exponential convergence factor) n n ` Let A R × be semi-convergent with limit lim` + A = A∗. ∈ → ∞ n (i) For ε ]0, 1[, the ε-convergence time of A is the smallest time Tε(A) Z 0 such that, for all x0 R ∈ ∈ ≥ ∈ and ` Tε(A), ≥ ` A x A∗x ε x A∗x . 0 − 0 2 ≤ k 0 − 0k2

(ii) The per-step convergence factor of A, denoted as rstep(A) is

x(k + 1) A∗x0 2 rstep(A) = sup k − k , ∗ x(k)=A x0 x(k) A∗x0 2 6 k − k where the supremum is taken over any sequence.

(iii) The exponential convergence factor of A, denoted by r (A) [0, 1[, is exp ∈ ` 1/` A x0 A∗x0 2 rexp(A) = sup lim k − k . ∗ x0=A x0 ` + x0 A∗x0 2 • 6 → ∞ k − k

The exponential convergence factor is an asymptotic version of the per-step convergence factor and has the ` following interpretation: If the trajectory x(`) = A x0 maximizing the sup operator has the form x(`) = ` ρ (x x∗) + x∗, for ρ < 1, then it is immediate to see that r (A) = ρ. 0 − exp Lemma 2.2 (Exponential convergence factor of a convergent matrix) If A is a convergent matrix, then rexp(A) = ρ(A).

Next, we apply the notion of convergence time and exponential convergence factor to any non-degenerate stochastic matrix whose associated digraph has a globally reachable node.

Lemma 2.3 (Convergence factors and solution bounds) Let F be a stochastic matrix with strictly positive diagonal entries and whose associated digraph has a globally reachable node. Then

rexp(F ) = ρess(F ).

(Recall that ρ (F ) = max λ λ spec(F ) 1 .) ess {k kC | ∈ \{ }} In addition, if F is a doubly-stochastic and primitive, we have 1 r (F ) = F 11> , step k − n k2 1 r (F ) = ρ(F 11>). exp − n Moreover, r (F ) r (F ) and the equality holds if F is symmetric. exp ≤ step Proof: We only prove the ﬁrst part of this lemma, while the second part can be found in [2]. If v Rn is a left eigenvector of F , then, as in Proposition 1.12, ∈

` T 1 T lim F = F ∗ = (v 1n)− 1nv . ` + → ∞ T T Relying upon v F = v and F 1n = 1n, straightforward manipulations show that F ∗ = F ∗F = FF ∗ = F ∗F ∗ and in turn `+1 ` F F ∗ = (F F ∗)(F F ∗). − − − n ` For any w0 R such that w0 = F ∗w0, deﬁne the error variable e(`) := F w0 F ∗w0. Note that the error ∈ 6 − variable evolves according to e(` + 1) = (F F ∗)e(`) and converges to zero. Additionally, the rate at which ` − w(`) = F w0 converges to F ∗w0 is the same at which e(`) converges to zero, that is,

r (F F ∗) = r (F ). exp − exp Therefore, r (F ) = r (F F ∗) = ρ(F F ∗) = ρ (F ). exp exp − − ess

In what follows, we are interested in studying how the convergence time and the exponential convergence factor of a matrix depend upon ε and upon the dimension of the matrix itself.

n n Lemma 2.4 (Asymptotic relationship) Let A R × be a semi-convergent matrix and let ε ]0, 1]. In the limit as ε 0+, ∈ ∈ → 1 1 Tε(A) O log ε− . ∈ 1 rexp(A) −

Proof: By the deﬁnition of the exponential convergence factor and of lim sup, we know that for all η > 0, there exists N such that, for all ` > N,

` ` A x A∗x (r (A) + η) x A∗x . 0 − 0 2 ≤ exp k 0 − 0k2 ` The ε-convergence time is upper bounded by any ` such that (rexp(A)+η) ε. Selecting η = (1 rexp(A))/2, simple manipulations lead to ≤ − 1 1 ` log ε− . ≥ log((r (A) + 1)/2) − exp It is also immediate to note that 2 1 , for all r ]0, 1[. This establishes the bound in the 1 r ≥ log((r+1)/2) ∈ statement above. − − The following result establishes bounds on convergence factors and convergence times for stochastic matrices arising from the equal-neighbor averaging rule in equation (4).

Theorem 2.5 (Bounds on the convergence factor and the convergence time) Let G be an undirected 1 unweighted connected graph of order n and let ε ]0, 1]. Deﬁne the stochastic matrix F = (In +D(G))− (In + A(G)). There exists γ > 0 (independent of n) such∈ that the exponential convergence factor and convergence time of F satisfy

3 3 1 r (F ) 1 γn− , and T (F ) O(n log ε− ), exp ≤ − ε ∈ as ε 0+ and n + . → → ∞ Notes: An extension of the results provided in this set of notes is contained in the paper [7]. In this work, the authors are able to establish similar tight bounds on convergence for linear iterations over regular graphs which in particular contain circulant Toeplitz matrices. Other results include convergence bounds for linear iterations defined over undirected graphs, and for directed graphs associated with a general sequence of non- degenerate matrices for which convergence holds. The authors find that there are constants c1, c2 > 0 such that δ 1 n 1 − 1 c nδ − T (δ, ε) c δnnδ+1 log 1 2 ≤ n ≤ 2 ε and they describe an example where the convergence can be as bad as exponential in the number of agents. Several attempts have been made to design algorithms that converge to consensus in finite time which include using non-differentiable dynamics, and exploiting knowledge of the iteration matrix F . Let A be the matrix associated with a linear iteration. Note that in order to achieve consensus in a finite time K it is necessary K 1 T that A = n 1n1n . Only very special primitive matrices satisfy this property. A recent algorithm that achieves consensus for a tree provided by Morse et al employs the so-called ratio consensus:

xi(t) + j aijxj(t), t = 0, xi(t + 1) = a x (t) + (1 r )x (t 1), t 1. ( j ij Pj − i i − ≥ P zi(t) + j aijzj(t), t = 0, zi(t + 1) = a z (t) + (1 r )z (t 1), t 1. ( j ij Pj − i i − ≥ where here xi(0) = xi0 and zi(0) = 1. InP particular, for G that is a tree and a matrix A = (aij) which is an xi(t) 1 T unweighted adjacency matrix associated with the tree, it holds that 1 x(0)1n in a time no longer zi(t) → n n than the diameter of G.

MAE247 – Cooperative Control of Multi-Agent Systems. Permission is granted to copy, distribute and modify this ﬁle, provided original sources are acknowledged. A TOEPLITZ AND TRIDIAGONAL CIRCULANT MATRICES A Algorithms deﬁned by tridiagonal Toeplitz and tridiagonal circulant matrices

The convergence rates can be made more specific by considering a special type of graphs. This section presents a more detailed analysis of the convergence rates of linear distributed algorithms defined by tridiagonal Toeplitz matrices and by certain circulant matrices. Let us start by introducing the family of matrices under study. For n 2 and a, b, c R, define the n n matrices Tridn(a, b, c) and Circn(a, b, c) by ≥ ∈ × b c 0 ... 0 a b c . . . 0 . . . . . Tridn(a, b, c) = ...... ,   0 . . . a b c   0 ... 0 a b     and 0 ...... 0 a 0 ...... 0 0 . . . . . Circn(a, b, c) = Tridn(a, b, c) + ......   0 0 ... 0 0   c 0 ... 0 0     We call the matrices Tridn and Circn tridiagonal Toeplitz and tridiagonal circulant, respectively. The two matrices only differ in their (1, n) and (n, 1) entries. Note our convention that b a + c Circ (a, b, c) = . 2 a + c b Note that, for a = 0 and c = 0 (alternatively, a = 0 and c = 0), the synchronous networks defined by Trid(a, b, c) and Circ(a, b, c) are,6 respectively, the chain6 and the ring digraphs. If both a and c are non- vanishing, then the synchronous networks are, respectively, the undirected versions of the chain and the ring digraphs.

Remark A.1 (From Toeplitz to stochastic matrices): A tridiagonal Toeplitz matrix is not doubly stochastic unless its oﬀ-diagonal elements are zero, i.e, it is the identity matrix. A tridiagonal circulant matrix Circn is doubly stochastic as long as a, b, c 0 and a+b+c = 1. Our discussion below shows that the additional structure of these matrices allows us to≥ describe the eigenstructure of these matrices well beyond what the doubly stochastic characterization tells us. • Example A.2 (Cyclic pursuit in the “n-bugs problem”) Consider the scenario of n-bugs chasing each other on a circle of radius r. Assume that agents are ordered counterclockwise with identities i 1, . . . , n , ∈ { } where, for convenience, we identify n + 1 with 1. Denote by pi(`) = (r, θi(`)) the sequence of positions of bug i, initially at pi(0) = (r, θi(0)). Suppose that each bug is chasing the closest counterclockwise neighbor (according to the order we have given them on the circle).In other words, each bug feels an attraction toward the closest counterclockwise neighbor that can be described by the equation

θi(` + 1) = (1 k)θi(`) + kθi+1(`), ` Z 0, (6) − ∈ ≥ where k [0, 1]. Can you connect this example to a tridiagonal circulant matrix? Be careful with drawing conclusions∈ directly on the angles through (6), because this equation does not make sense without properly interpreting it “modulo 2π.” Instead, reason with inter-agent distances and justify that the equation for them is instead well-posed. • 10

Example A.3 (Cyclic balancing in the “n-bugs problem”) Consider the same scenario described in Example A.2 for the n-bugs problem. Suppose now that each bug makes a compromise between chasing its closest counterclockwise neighbor and the closest clockwise neighbor. In other words, each bug feels an attraction towards the closest counterclockwise and clockwise neighbors that can be described by the equation

θi(` + 1) = kθi+1(`) + (1 2k)θi(`) + kθi 1(`), ` Z 0, − − ∈ ≥ where k [0, 1]. Again, can you connect this example to a tridiagonal circulant matrix? ∈ •

Next, we characterize the eigenvalues and eigenvectors of Tridn and Circn.

Lemma A.4 (Eigenvalues and eigenvectors of tridiagonal Toeplitz and tridiagonal circulant matrices) For n 2 and a, b, c R, the following statements hold: ≥ ∈ (i) for ac = 0, the eigenvalues and eigenvectors of Trid (a, b, c) are, respectively, for i 1, . . . , n , 6 n ∈ { } a 1/2 iπ c sin n+1 2/2  a sin 2iπ  a iπ c n+1 n b + 2c cos C, C ; c n + 1 ∈  .  ∈ r  .     a n/2 niπ   c sin n+1      2π√ 1 (ii) the eigenvalues and eigenvectors of Circ (a, b, c) are, respectively, for i 1, . . . , n and ω = exp( − ), n ∈ { } n i2π i2π b + (a + c) cos + √ 1(c a) sin C, n − − n ∈ i (n 1)i T n and (1, ω , . . . , ω − ) C . ∈ Proof: Both facts are discussed, for example, in [6, Example 7.2.5 and Exercise 7.2.20]. Fact (ii) requires some straightforward algebraic manipulations. Figure2 illustrates the location of the eigenvalues of these matrices in the complex plane.

Remarks A.5 (Inclusion relationships for eigenvalues of tridiagonal Toeplitz and tridiagonal circulant matrices):

(i) The set of eigenvalues of Tridn(a, b, c) is contained in the real interval [b 2√ac, b + 2√ac], if ac 0, and in the interval in the complex plane [b 2√ 1 ac , b + 2√ 1 ac−], if ac 0. ≥ − − | | − | | ≤ (ii) The set of eigenvalues of Circn(a, b, c) is containedp in the ellipse onp the complex plane with center b, horizontal axis 2 a + c , and vertical axis 2 c a . | | | − | • Next, we characterize the convergence rate of linear algorithms deﬁned by tridiagonal Toeplitz and tridiagonal circulant matrices. As in the previous section, we are interested in asymptotic results as the system dimension n + and as the accuracy parameter ε goes to 0+. → ∞ Theorem A.6 (Linear algorithms deﬁned by tridiagonal Toeplitz and tridiagonal circulant ma- n n trices): Let n 2, ε ]0, 1[, and a, b, c R. Let x : Z 0 R and y : Z 0 R be solutions to ≥ ∈ ∈ ≥ → ≥ → x(` + 1) = Tridn(a, b, c) x(`), y(` + 1) = Circn(a, b, c) y(`), with initial conditions x(0) = x0 and y(0) = y0, respectively. The following statements hold: 11

MAE247 – Cooperative Control of Multi-Agent Systems. Permission is granted to copy, distribute and modify this ﬁle, provided original sources are acknowledged. 22 TOEPLITZ TOEPLITZ AND AND TRIDIAGONAL TRIDIAGONAL CIRCULANT CIRCULANT MATRICES MATRICES

Proof:Proof: BothBoth facts facts are are discussed, discussed, for for example, example, in in [7 [,7, Example Example 7.2.5 7.2.5 and and Exercise Exercise 7.2.20]. 7.2.20]. Fact Fact (ii) (ii) requires requires somesome straightforward straightforward algebraic algebraic manipulations. manipulations. Figure 44 illustratesillustrates the the location location of of the the eigenvalues eigenvaluesA TOEPLITZ of of these these AND matrices matrices TRIDIAGONAL in in the the complex complex CIRCULANT plane. plane. MATRICES

cc c c

(b,(b,0)0) (b,(0)b, 0)

(a) (b) (a)(a) (b) Figure 4: The eigenvalues of Toeplitz and circulant matrices (cf., Lemma 2.4) are closely related to the roots FigureFigure 2: 4: The The eigenvalues eigenvalues of of Toeplitz Toeplitz and and circulant circulant matrices matrices (cf., (cf., Lemma Lemma A.42.4)) are are closely closely related related to to the the roots roots of unity. Plotted in the complex plane, the black disks correspond in (a) to the eigenvalues of Trid13(a, b, c), ofof unity. unity. Plotted Plotted in in the the complex complex plane, the black disks correspond in (a) to the eigenvalues of Trid1313((a,a, b, b, c c),), and in (b) to the eigenvalues of Circ14(0,b,c). andand in in (b) (b) to to the the eigenvalues eigenvalues of Circ14(0,,b,c b, c).

Remarks 2.5 (Inclusion relationships for eigenvalues of tridiagonal Toeplitz and tridiagonal2 1 Remarks(i) if a = 2.5c =(Inclusion 0 and b + 2 relationshipsa = 1, then lim for` + eigenvaluesx(`) = 0n with of tridiagonalε-convergence Toeplitz time in Θ andn log tridiagonalε− ; circulant matrices):6 | | | | → ∞ circulant matrices): 1 (ii) if a = 0, c = 0 and 0 < b < 1, then lim` + x(`) = 0n with ε-convergence time in O n log n+logε− ; 6 | | → ∞ (i) andThe set of eigenvalues of Tridn(a, b, c) is contained in the real interval [b 2pac, b +2pac], if ac 0, (i) The set of eigenvalues of Tridn(a, b, c) is contained in the real interval [b 2pac, b +2 pac], if ac 0, and in the interval in the complex plane [b 2p 1 ac ,b+2p 1 ac ], if ac 0. (iii) ifanda in0 the, c interval0, 1 > in b > the0 and complexa + b plane+ c = [b 1, then2p lim1 | ac| ,by+2(`)p = 11|1Tac|y ],1 if acwith 0.ε-convergence time `| +| n |n |0 n  (ii) The≥ set2 of≥ eigenvalues1 of Circ (a, b, c) is containedp in the→ ∞ ellipse onp the complex plane with center b, in Θ n log ε− . n p p (ii) horizontalThe set of axis eigenvalues 2 a + c , of and Circ verticaln(a, b, axis c) is 2 containedc a . in the ellipse on the complex plane with center b, horizontal axis 2| a + c| , and vertical axis 2| c a| . • The proof of this result| can| be found in [1, Section| 1.8].| • Next, we characterize the convergence rate of linear algorithms defined by tridiagonal Toeplitz and tridiagonal Next,circulantNext, we matrices. characterize extend these As thein results the convergence previous to another section, rate ofinteresting welinear are algorithms interested set of tridiagonal in defined asymptotic by matrices. tridiagonal results as For Toeplitz then system2 and and dimension tridiagonala, b R, + ≥ ∈ definencirculant+ the matrices.andn asn thematrices As accuracy in the ATrid previous parametern (a, b) section, and" goes ATrid we to aren− 0(+a,. interested b) by in asymptotic results as the system dimension n ! +1 and× as the accuracy parameter " goes to 0+. ! 1 Theorem 2.6 (Linear algorithms defined by tridiagonala Toeplitz0 ...... and tridiagonal0 circulant ma- Theorem 2.6 (Linear algorithms defined by tridiagonal0n 0 Toeplitz...... and0 tridiagonaln circulant matrices): Let n 2, " ]0, 1[,anda, b, c R. Let x : Z 0 R and y : Z 0 R be solutions to 2 2 !. n. . . ! .n trices): Let n 2, " ]0,ATrid1[,and±(a,a, b b,) c = TridR. Let(a,x b,: aZ) 0 . R . and .y : Z .0 . R. be solutions to 2 n 2 n !. . . . !. x(` + 1) = Trid (a, b, c) x(`),y±(` + 1) = Circ (a, b, c) y(`), n 0 ...... n 0 0 x(` + 1) = Tridn(a, b, c) x(`),y(` + 1) = Circn(a, b, c)y(`), 0 ...... 0 a with initial conditions x(0) = x0 and y(0) = y0, respectively. The following statements hold: with initial conditions x(0) = x0 and y(0) = y0, respectively. The following statements hold: We refer to these matrices as augmented tridiagonal matrices. If we define 2 1 (i) if a = c =0and b +2a =1, then lim` + x(`)=0n with "-convergence time in ⇥ n log " ; 6 | | | | ! 1 2 1 (i) if a = c =0and b +2a =1, then lim` + x(`)=0n with "-convergence time in ⇥ n log " ; 6 | | | | 1 1! 1 0 0 ... 0 1 (ii) if a =0, c =0and 0 < b < 1, then lim` + x(`)=0n with "-convergence time in O n log n+log" ; 6 | | 1 !11 1 0 ... 0 1 (ii) andif a =0, c =0and 0 < b < 1, then lim−` + x(`)=0n with "-convergence time in O n log n+log " ; 6 | | 1 0! 1 1 1 ... 0 and − P+ = . . . .  , . .10. .. ..    1 0 ... 10 0 1 1   −  1 0 ... 0 0 1 MAE247 – Cooperative Control of Multi-Agent−  Systems Copyright c 2013 by Jorge Cortés.Permission is granted to copy, distribute and modify this file, provided that the original MAE247 – Cooperative Control of Multi-Agent Systems Copyright c 2013 by Jorge Cortés.Permission issource granted is acknowledged. to copy, distribute and modify this file, provided that the original source is acknowledged. 12

1 1 0 0 ... 0 1 1 1 0 ... 0  −1 0 1 1 ... 0 P =  . . . .  , −  ......   n 2  ( 1) − 0 ... 0 1 1  − n 1  ( 1) − 0 ... 0 0 1  −    then the following similarity transforms are satisﬁed:

b 2a 0 1 ATridn±(a, b) = P ± P − . (7) ± 0 Tridn 1(a, b, a) ± − Remark A.7 (From Toeplitz to stochastic matrices–revisited): An augmented tridiagonal matrix + ATridn is doubly stochastic as long as a, b 0 and 2a + b = 1. An augmented tridiagonal matrix ATridn− is not doubly stochastic unless it is the identity≥ matrix. • + To analyze the convergence properties of the linear algorithms determined by ATridn (a, b) and ATridn−(a, b), we will ﬁnd it useful to consider the vector

T n 2 n 1 T n 1n = (1, 1, 1,..., ( 1) − , ( 1) − ) R . − − − − ∈ In the following theorem, we will not assume that the matrices of interest are semi-convergent. We will establish convergence to a trajectory, rather than to a fixed point. For ε ]0, 1[, we say that a trajectory n n ∈ x : Z 0 R converges to xfinal : Z 0 R with convergence time Tε Z 0 if ≥ → ≥ → ∈ ≥ (i) x(`) x (`) 0 as ` + ; and k − final k2 → → ∞ (ii) T is the smallest time such that x(`) x (`) ε x(0) x (0) , for all ` T . ε k − final k2 ≤ k − final k2 ≥ ε Theorem A.8 (Linear algorithms defined by augmented tridiagonal matrices) Let n 2, ε ]0, 1[, n n ≥ ∈ and a, b R with a = 0 and b + 2 a = 1. Let x : Z 0 R and z : Z 0 R be solutions to ∈ 6 | | | | ≥ → ≥ → + x(` + 1) = ATridn (a, b) x(`), z(` + 1) = ATridn−(a, b) z(`), with initial conditions x(0) = x0 and z(0) = z0, respectively. The following statements hold:

1 T ` (i) lim` + x(`) xave(`)1n = 0n, where xave(`) = ( 1n x0)(b + 2a) , with ε-convergence time in → ∞ − n Θ n2 log ε 1 ; and − 1 T ` (ii) lim` + z(`) zave(`)1n = 0n, where zave(`) = ( n 1n z0)(b 2a) , with ε-convergence time in → ∞ − − − − Θ n2 log ε 1 . − Proof: We prove fact (i) and observe that the proof of fact (ii) is analogous. Consider the change of coordinates x (`) 0 x(`) = P ave0 = x (`)1 + P , + y(`) ave0 n + y(`) n 1 1 T where xave0 (`) R and y(`) R − . A quick calculation shows that xave0 (`) = n 1n x(`), and the similarity transformation∈ described in∈ equation (7) implies

y(` + 1) = Tridn 1(a, b, a) y(`), and xave0 (` + 1) = (b + 2a)xave0 (`). − 13

MAE247 – Cooperative Control of Multi-Agent Systems. Permission is granted to copy, distribute and modify this ﬁle, provided original sources are acknowledged. REFERENCES REFERENCES

Therefore, xave = xave0 . It is also clear that

0 0 0 1 x(` + 1) xave(` + 1)1n = P+ = P+ P+− (x(`) xave(`)1n). − y(` + 1) 0 Tridn 1(a, b, a) − − Consider the matrix in parentheses determining the trajectory ` (x(`) x (`)1 ). This matrix is 7→ − ave n symmetric, its singular values are 0 and the singular values of Tridn 1(a, b, a), and its eigenvectors are 1n − and the eigenvectors of Tridn 1(a, b, a) (padded with an extra zero). These facts are suﬃcient to duplicate, − step by step, the proof of fact (i) in Theorem A.6. Therefore, the trajectory ` (x(`) xave(`)1n) satisﬁes the stated properties. 7→ − We conclude this section with some useful bounds.

n n 1 n 1 Lemma A.9 (Bounds on vector norms) Assume that x R , y R − , and z R − jointly satisfy ∈ ∈ ∈ 0 0 x = P+ , x = P . y − z Then 1 x y (n 1) x and 1 x z (n 1) x . 2 k k2 ≤ k k2 ≤ − k k2 2 k k2 ≤ k k2 ≤ − k k2 The proof of this result is based on spelling out the coordinate expressions for x, y, and z.

References

[1] F. Bullo, J. Cortés,and S. Mart´ınez, Distributed Control of Robotic Networks, ser. Applied Mathematics Series. Princeton University Press, 2009, electronically available at http://coordinationbook.info. [2] F. Bullo, Lectures on Network Systems. Version 0.96, 2018, with contributions by J. Cortes, F. Dorfler, and S. Martinez. [3] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods. Athena Scientific, 1997. [4] T. Vicsek, A. Czirók,E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel type of phase transition in a system of self-driven particles,” Physical Review Letters, vol. 75, no. 6-7, pp. 1226–1229, 1995. [5] V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis, “Convergence in multiagent coordination, consensus, and flocking,” in IEEE Int. Conf. on Decision and Control and European Control Conference, Seville, Spain, Dec. 2005, pp. 2996–3000. [6] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. SIAM, 2001. [7] A. Olshevsky and J. N. Tsitsiklis, “Convergence speed in distributed consensus and averaging,” SIAM Journal on Control and Optimization, vol. 48, no. 1, pp. 33–55, 2009.

MAE247 – Cooperative Control of Multi-Agent Systems. Permission is granted to copy, distribute and modify this ﬁle, provided original sources are acknowledged.