Continuous-time quantized consensus: convergence of Krasowskii solutions

Con tin uous-time quan tized consensus: con v ergence of Kraso vsk ii solutions P aolo F rasca ∗ No v em b er 11, 2018 Abstract This note studies a netw ork of agents ha ving conti nuous-time dynamics with qu an- tized interactio ns and time-va rying directed top ology . Due to the d iscontin uity of the dynamics, solutions o f the resulting ODE system are intended in the sense of K raso vskii. A limit connectivity graph is defined, whic h enco des p ersistent interactio ns b etw een nod es: if such graph has a globally reachable no d e, Krasovskii solutions reac h consensus (up to t he quantizer precision) after a finite time. U nder the additional assumption of a time-inv aria nt top ology , the conv ergence time is u pp er b ounded by a quantit y whic h dep ends on the netw ork size and t h e qu antizer precision. It is observed that the con- verge nce time can be very la rge for solutions whic h stay on a discontin uity surface. 1 In tro duction Problems of co nsensus a nd co ordination in netw ork s have been widely studied during the la st decade using a blend of to ols fro m control theor y and gr aph theory . While linear consensus systems based on time-inv ar ia nt netw orks are eas y to understand, thing s beco me harder when the net work top olog y dep ends on time, or when communication b et ween no de s is affected by limited precision due to bandwidth constr a int s. Consensus pr oblems have bee n studied on time-dep endent netw orks b y a v ast literature: we refer the r eader to the ear ly works [23, 22], as well as to the b o oks [5, 21] for an introduction and to [17] for re cent related results. On the other hand, co ordination and consensus hav e also bee n s tudied in systems sub ject to limited-precision effects, i.e. , to quantization. Mo st a uthors have fo cused on a v ariety of pr oblems for discr ete-time systems, including the analys is of conv er g ence assuming s ta tic qua ntizers [19][1][14][7][18] and the desig n of effective dynamic quantization schemes [6][20]. Quantized co ntin uous- time sys tems, instead, hav e attracted attention more recently . Controllers based o n qua nt izing the differences b et ween the states of connec ted no des ar e s tudied in [13] under the as sumption that the net work top ology is a tree, and in [1 1] using binar y quantizers in a leader-following framework. Quan tized co mm unication of s tates is ins tead co nsidered in [10] for static top ologies and in [24] for dissipative sy stems. In the analysis of contin uous-time quantized dynamics, the inher ent discontin uity of the system right-hand side entails some mathema tical difficulties, which ar e discussed in [1 2] and [10]. The latter pap er c onsiders a simple co n tinuous-time average consensus dynamics with time-inv ar iant top o logy and uniform static quantizers, and demonstrates that cho osing a suitable definition o f so lutio n is e s sential to ensure that solutio ns a re defined for all times ∗ P . F r asca is with the Dipartimento di Matematica, Po li tecnico di T ori no, T orino, Italy . paolo.fr asca@polito.it . The work of the author was partly supp orted by MIUR under grant PRIN- 20087W5P2 K. The author wi shes to thank F. Ceragioli and tw o anon ymous r eview ers for their remar ks. 1 and thu s to p ermit a mea ningful conv ergence analysis. A natura l a nd effective choice are Krasovskii so lutio ns, which indeed are co mplete for every initial condition and conv erg e to approximate co nsensus conditions under mild assumptions. Statemen t of contributions After this literature r eview, we are a ble to pr esent the contribution of this pap er. W e study a co ordination task for a netw ork of agents ha ving a sc a lar cont inuous-time dynamics, assuming that (i) the interaction b etw een the agents is weigh ted by time-dep endent co efficients whic h represent a dynamical co mm unication net work; and (ii) connected a gents can e xchange infor mation ab out their sta tes only throug h a (static) quantizer. Due to the quantization constraint, the goal of consensus b etw een states can only be a p- proximated up to the quantizer precis ion. Our ma in co n tribution consis ts in a sufficient condition for finite-time conv erg ence of Krasovskii solutions to the b est achiev able appr oxi- mation. This co ndition, presented in The o rem 1, is bas ed on the connectivity of a suitable limit gr aph. Compa r ed with the r eferenced literature, our conv ergence result holds (i) under milder ass umptions o n the net work co nnectivit y; and (ii) for a larger clas s of quantizers. Additionally , in Section 4 the conv ergence r esult is sp ecialized to uniform quantizers and to av erage - preserving dynamics. With the further ass umption o f time-inv ar iant top ology , w e also der ive an upp er b ound on the conv erg ence time, which is inv ersely prop or tional to the quantizer prec ision and is exp onentially incr easing with the net work s iz e. The tigh tness o f this b ound is dis cussed in view of ad ho c examples and of the evidences in the liter a ture. W e leave outside the scop e of this pap er the analysis of controllers based on quantization of differences, as well as the desig n of o ptimal controllers and quantizers, either dyna mic or static. 2 Mathematical to ols: Graphs and ODEs In this section w e pr ovide some background in differential equa tions and gr aph theory . F or our analys is it is necess a ry to recall fro m [16] a cer tain notion of s olution to a –p ossibly discontin uous– differential equation, which is bas e d on defining a suitable differential inclu- sion. Given 1 f : R > 0 × R N → R N and the differ ent ial eq ua tion ˙ x = f ( t, x ), we say that x : J → R N solves this differential eq ua tion in the Krasovskii s ense if x ( · ) is absolutely c o n- tin uous and for almos t every time t in the interv al J ⊂ R > 0 satisfies the differential inclus ion ˙ x ( t ) ∈ K f ( t, x ( t )), whe r e K f ( t, x ) = \ δ> 0 co f ( t, B ( x, δ )) , with co denoting the conv ex closure a nd B ( y , r ) the E uclidean ball o f radius r c en tered in y . Her e a nd elsew he r e in the pa pe r, “almo st every” means “e xcept in a set of zero Leb esque measure”. A solutio n is s aid to b e c omplete if J = (0 , + ∞ ) . No te that we will also apply the Kras ovskii op erator K to a utonomous functions f ( x ). An example of the co nv exification induced by K is provided later in Fig ure 1. A s imila r notion is that of Filipp ov solution [2, 1 The symbols Z , R , R ≥ 0 R > 0 denote the sets of i n teger, real, nonnegative and p ositive num b ers, r espec- tiv ely . R n denotes an n -dimensional Euclidean space. W riting R A , where A is a set of cardinality n , we are indexing the comp onen ts in the set A . Given r ∈ R , the set of the (inte ger) multiples of r i s denoted by r Z . 2 Definition 6 ], which is quite common in the liter ature but will not b e used her e. Indeed, every Filipp ov solution is also a Kras ovskii s olution, s o that the results in this pap er apply a fortiori to Filipp ov solutio ns . Our analysis also involv es gr aphs and w eighted g raphs. W e int ro duce here the ma in notions which we sha ll use later: the rea de r is re ferred to the liter ature, for instance to [8 ] or to the b o ok [21], for a mo re complete introductio n. Given a finite set of vertices (or no des) V , a (directed) gra ph G is a pair ( V , E ) where E ⊂ V × V is the set of edges (or a rcs). A weight ed g raph is triple ( V , E , A ) which includes a weigh ted adjacency matrix A ∈ R V × V ≥ 0 with the consistency condition tha t A uv > 0 if and o nly if ( u, v ) ∈ E . W e also a s sume that A uu = 0 for all u ∈ V . The L aplacian matrix asso c iated to A is a matrix L ∈ R V × V such that L uv = − A uv if u 6 = v and L uu = P v ∈ V A uv . A sink is a no de u with no o utgoing e dg e –that is, such that E do es no t contain any edge of the form ( u, v ) . A path (of length l ) from u to v in G is an or de r ed list of edg e s ( e 1 , . . . , e l ) in the form (( u, w 1 ) , ( w 1 , w 2 ) , ( w 2 , w 3 ) , . . . , ( w l − 1 , v )) . If such a path exists, we say that v can b e r eached from u . A cycle is a path fro m a no de to itself. A graph is sa id to b e co nnected if for every pair of no des ( u, v ), either v ca n be reached from u or u can b e r eached from v . Instead, a gra ph is said to b e stro ngly connected if every t wo no des can be r eached from each other . Giv en any direc ted g raph G = ( V , E ) we can consider its strongly connected comp onents, namely maximal strongly connected subgraphs G k , k ∈ { 1 , . . . , s } with set of vertices V k ⊂ V and set of ar cs E k = E ∩ ( V k × V k ) such that the sets V k form a partition of V . These c o mpo nent s may have c o nnections among ea ch other: in o rder to enco de these connections we define a directed g raph T ( G ) with s et o f vertices { 1 , . . . , s } such tha t there is an arc fro m h to k if ther e is an arc in G from a vertex in V h to a vertex in V k . W e obse r ve that (i) T ( G ) has no cycle; (ii) T ( G ) is connected a nd has one sink if and only if there exis ts in G a g lobally r eachable no de, i.e. , a no de which ca n be rea ched from every other no de. 3 Problem statemen t and main result In this sec tio n we introduce the dynamics of interest, and we state a nd prov e o ur main conv ergence r e sult. Let there b e N agents, indexed in a set I , and for any pa ir ( i, j ) ∈ I × I , let a ij ( · ) : R ≥ 0 → 0 ∪ [ a min , a max ] be a mea s urable function, with 0 < a min ≤ a max . These int era ction functions natura lly lead to the following definitions. F or every time t , we consider a weigh ted in teractio n g r aph G ( t ) = ( I , E ( t ) , A ( t )) , such that the i, j -th comp onent of the matrix A ( t ) is the v alue a ij ( t ) , and ( i, j ) ∈ E ( t ) if a nd only if a ij ( t ) > 0 . Given the function G ( t ), we define – following [17]– a n un b ounde d inter actions gr aph G ∞ = ( I , E ∞ ) by E ∞ = { ( i, j ) ∈ I × I : lim t → + ∞ Z t t 0 a ij ( s ) ds = + ∞ ∀ t 0 ≥ 0 } . W e obse r ve that G ∞ is the g r aph whos e edg es connect the no des which are connected in G ( t ) for an infinite duration o f time. F or i ∈ I , we let x i ( t ) b e a rea l v ariable a nd consider the dynamics ˙ x i = X j ∈ I a ij ( t )( q ( x j ) − q ( x i )) (1) where q : R → S is a quantizer mapping real num be rs into a discrete 2 set. System (1) can also b e rewr itten in vector form as ˙ x = − L ( t ) q ( x ) , 2 A subset S ⊂ R is said to be discrete if all its p oints are isolated. Examples i nclude the set of the intege rs and ev ery finite s ubset of R . Note that if S has no li mit poi n t in R , then S is discrete. 3 where x ( t ) ∈ R I is the s tate vector, L ( t ) is the La placian matr ix a sso ciated to the weight ed adjacency matrix A ( t ) and by a slight notational abuse, q is defined to op era te comp onent wise on vectors. W e consider for (1) solutions in the sens e o f K r asovskii, which we hav e defined in the prev ious section, and thanks to the linearity of the Kr asovskii op era tor K , we have that a Kras ovskii so lution to (1) is an abs olutely co n tinuous function of time which satisfies for almost every time the differe n tial inclusion ˙ x ∈ − L ( t ) K q ( x ) . By the current assumptions of b oundednes s o n the functions a ij , for a ny ¯ x ∈ R I there exists a complete Kr asovskii so lution x ( t ) to (1), such that x (0) = ¯ x. Note, how ever, that ther e ca n be mo r e than one o f such solutions. In the rest of this pap er, whenever we refer to a solution, we mea n a co mplete so lutio n. After these pr eliminary obse r v ations, we are ready to state and pr ov e that sys tem (1) reaches qua ntized co nsensus equilibria in finite time, provided the unbo unded in tera c tio ns graph ha s a g lobally reachable no de. Theorem 1 (Finite-time quantized consensus ) . L et S b e a subset of R with no limit p oint and q : R → S b e a non-de cr e asi ng function. L et x ( t ) b e a Kr asovskii solution to (1) . If T ( G ∞ ) is c onne cte d and has only one sink, then ther e exist T con ≥ 0 and s ∗ ∈ S such that, for every t ≥ T con , s ∗ ∈ K q ( x i ( t )) for every i ∈ I . Pr o of . Without los s of gener a lity , we may think of the elements of S as indexed in a s et A of consecutive integers, in s uc h a wa y that S = { s a : a ∈ A } a nd s a < s b if and only if a < b . Let ∆ min = inf {| s a − s b | : a, b ∈ A } . As there is no limit p o int of S , then ∆ min > 0 . Given the solution x ( · ) and z ∈ R , we define the following time-dep endent subset of indices I z ( t ) = { i ∈ I : z ∈ K q ( x i ( t )) } , and w e let m ( t ) = min i ∈ I min K q ( x i ( t )) and M ( t ) = max i ∈ I max K q ( x i ( t )). B y definitio n, M ( t ) a nd m ( t ) b elong to S and we denote m (0) = s m and M (0) = s M . The dynamics (1) implies that, at almost e very time t and for all i ∈ I , ˙ x i ( t ) ∈    X j ∈ I a ij ( t )( z j − z i ) : z k ∈ K q ( x k ( t ))    . (2) In particular , if i ∈ I m ( t ) ( t ), then ˙ x i ( t ) ∈ [0 , + ∞ ). Hence, m ( t ) ≥ m (0) fo r a ll t ≥ 0; similarly , we can deduce that M ( t ) ≤ M (0) . Our pro of aims a t showing that m ( t ) actually increases until the system reaches an equilibrium: there exist T con ≥ 0 and s ∗ ∈ S such that I s ∗ ( T con ) = I . The same conclusio n can b e reached by an analo gous ar gument ba sed on M ( t ). Note that for all t ≥ 0 it holds I = S s M s = s m I s ( t ), but sets of the for m I s h ( t ) ∩ I s h +1 ( t ) need no t to b e empty , in particular when some x k ( t ) is at a disco n tinuit y p o int o f q . In view of the last remark, w e deno te for br evity I ∂ s h ( t ) = I s h ( t ) ∩ I s h +1 ( t ) a nd ˚ I s h ( t ) = I s h ( t ) \ I s h +1 ( t ), and we sta rt our argument by consider ing the set ˚ I s m ( t ) and claiming that ˚ I s m ( t 1 ) ⊇ ˚ I s m ( t 2 ) for all t 2 ≥ t 1 ≥ 0. (3) W e show this fact by contradiction. Let x 0 ∈ R b e the discontin uit y p oint of q such that k ∈ I ∂ s m ( t ) if a nd only if x k ( t ) = x 0 . Assume, by contradiction, that there exists an ag ent i ∈ I such that x i ( t 1 ) > x 0 and x i ( t 2 ) < x 0 . Then there ar e thr ee co nsequences: (i) by contin uity , 4 there exis ts t ′ ∈ ( t 1 , t 2 ) such that x i ( t ′ ) = x 0 ; (ii) conse quent ly , x i ( t 2 ) = x 0 + R t 2 t ′ ˙ x i ( s ) ds ; (iii) x i ( t ) < x 0 for t ∈ ( t ′ , t 2 ), and s inc e i ∈ ˚ I s m ( t ), then ne c essarily ˙ x i ( t ) ≥ 0 . But (ii) implies that, for a s et of times of p ositive measure, ˙ x i ( t ) < 0 , which is a co n tra dic tio n. W e c o nclude that (3) holds and that if an ag e n t k r eaches I s m ( t ) (from the right), she necessar ily has to stop a t the b o r der of the cor resp onding interv al. Next, we wan t to prove that there e x ist times when the inc lus ion (3 ) is strict. W e define the set of the agents whose s tate is “strictly larger” than s m as I + s m ( t ) = M [ h = m +1 I s h ( t ) ! \ I s m ( t ) and T empty = inf { t ≥ 0 : I + s m ( t ) = ∅ } . If T empty is finite, then I + s m ( t ) = ∅ for e very t ≥ T empty . Then I s m ( T empty ) = I and we conclude that T con = T empty and s ∗ = s m , completing the pro of. Otherwise, we pro ceed with our ar gument and assume 3 by cont ra diction that ˚ I s m ( t ) = ˚ I s m (0) for all t > 0. W e a lso temp ora rily as sume that G ∞ is strong ly connected: the argument will be extended at the end of the pro of. Then, thanks to the stro ng connectivity of G ∞ , we ca n find an a rc ( i, j ) ∈ E ∞ such that i b elongs to ˚ I s m (0) a nd j do es not. As a consequence of (3 ), j / ∈ ˚ I s m ( t ) for all t ≥ 0 , and by contradiction we know that i ∈ ˚ I s m ( t ) for all t ≥ 0 . Notice that for almost e very t ≥ 0, ˙ x i ( t ) ≥ a ij ( t )  v j ( t ) − q ( x i ( t ))  , where v j ( t ) ∈ K q ( x j ( t )) is the r ealization of the inclusion in (2). Define J ∂ = { t ≥ 0 : j ∈ I ∂ s m ( t ) } a nd J + = { t ≥ 0 : j ∈ I + s m ( t ) } . Then x i ( t ) ≥ x i (0) + Z t 0 a ij ( s )  v j ( s ) − q ( x i ( t ))  ds (4) = x i (0) + Z J ∂ ∩ (0 ,t ) a ij ( s )  v j ( s ) − q ( x i ( t ))  ds + Z J + ∩ (0 ,t ) a ij ( s )  v j ( s ) − q ( x i ( t ))  ds ≥ x i (0) + ∆ min Z J ∂ ∩ (0 ,t ) a ij ( s ) α j ( s ) ds + ∆ min Z J + ∩ (0 ,t ) a ij ( s ) ds where in the la st inequality we hav e us ed the fact that if s ∈ J ∂ , then v j ( s ) = s m (1 − α j ( s )) + s m +1 α j ( s ) = s m + α j ( s )( s m +1 − s m ) , and α j ( s ) is a measura ble function taking v alues in [0 , 1] . Le t q − 1 ( s m ) denote the pre-imag e of s m under q . If sup q − 1 ( s m ) = + ∞ , then necessa rily s m = max S a nd the pr o of is completed since I s m (0) = I . Otherwise, we aim to show that the right-hand s ide of (4) is divergen t as t → ∞ . If J + has infinite measur e, divergence is clear from the as s umption a ij ( s ) ≥ a min . Otherwise, lim t →∞ R J + ∩ (0 ,t ) a ij ( s ) ds < ∞ and instead J ∂ has infinite measure: we want to use this fact, together with a low er b ound on α j ( s ). T o obtain such an estimate, we note that Equa tion (3 ) implies that ˙ x k ( t ) = 0 for almos t every t such that k ∈ I ∂ s m ( t ). Then, for almost every s ∈ J ∂ it holds ˙ x j ( s ) = 0 , and the equality ˙ x j ( s ) = X k a j k ( v k ( s ) − v j ( s )) = X k a j k  v k ( s ) − α j ( s )( s m +1 − s m ) − s m  3 If ˚ I s m (0) = ∅ , then there is nothing to pro ve: since in this case I s m (0) ⊆ ˚ I s m +1 (0), we can start our argumen t fr om ˚ I s m +1 (0) . 5 implies that α j ( s ) = P k a j k ( s )( v k ( s ) − s m ) ( s m +1 − s m ) P k a j k ≥ P k ∈ I ∂ s m ( s ) a j k ( s ) α k ( s ) + P k ∈ I + s m ( s ) a j k ( s ) P k a j k ≥ a min N a max β k ( s ) , where β k ( s ) is defined a s follows. Let l ∈ I b e such that ( k, l ) ∈ E ∞ and for all t in a set of times o f infinite mea sure either l ∈ I + s m ( t ) or l ∈ I ∂ s m ( t ). In the former case β k ( s ) = 1, in the latter β k ( s ) = α l ( s ) . By the connectivity assumption, there exists an infinite-meas ure set of times J such that for s ∈ J there is a pa th in G ( s ) from j to a no de in I + s m ( s ), a nd by a recursive reaso ning along this path, we conclude that for s ∈ J it holds α j ( s ) ≥  a min N a max  N . F rom (4 ) and the last inequality we ca n deduce x i ( t ) ≥ x i (0) + ∆ min Z J ∩ (0 ,t ) a min  a min N a max  N . (5) This inequality implies that x i ( t ) diverges a s t → + ∞ , which contradicts the fact that x i ( t ) ≤ M (0) for all t ≥ 0 . W e conclude that there exists T ′ > 0 s uch that for all t ≥ T ′ it holds i 6∈ ˚ I s m ( t ) and ˚ I s m ( t ) ( ˚ I s m (0) . Rep eating this argument for every element o f ˚ I s m (0), we obtain that there exists T 0 > 0 such tha t ˚ I s m ( T 0 ) = ∅ . Afterwards, the s ame r easoning which ha s b een applied to ˚ I s m can b e applied, with stra ightf or ward mo difications, to ˚ I s m +1 , ˚ I s m +2 , . . . , showing that ther e exis ts a sequence of times T k such that ˚ I s m + k ( T k ) = ∅ . Since M ( t ) ≤ s M , then the sequence of T k ’s must b e finite. This implies that there exist T con and s ∗ such that I s ∗ ( T con ) = I , under the assumption o f strong c onnectivity of G ∞ . In order to complete the pro o f, we still hav e to relax the connec tiv ity condition. I f G ∞ is not strongly co nnected, the ab ove arg ument may fail, b ecause at some time it may b e impo ssible to find an arc ( i, j ) co ming o ut of the se t of minima – say , the set ˚ I s m (0). But in such a case, nec essarily the sink comp onent is a subset of ˚ I s m (0). Then, since it is a ssumed that there is only one sink, it is still p ossible to conclude by applying the analog ous arg ument based on the maximal v alue M ( t ). Note that the as sumptions of Theorem 1 ab out S are satisfied, for instance, when S is a finite set or when S = ∆ Z . The latter imp ortant s pec ia l c ase is the topic of the next sectio n. 4 Uniform qu an tizers In this section, we a ssume that the states ar e communicated via a uniform quantizer, and we derive from T he o rem 1 a mo re precise conv erg ence result. After that, we study the c ase of av erage - preserving dynamics, and we estimate the conv erg ence time T con . Let then q b e the uniform quantizer with precis ion ∆ > 0, that is the ma p q : R → ∆ Z such tha t q ( z ) =  z ∆ + 1 2  ∆ . (6) The maps q and K q ( x ) are illustrated in Figure 1. 6 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 P S f r a g r e p la c e m e n t s x q ( x ) −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 P S f r a g r e p la c e m e n t s x K q ( x ) Figure 1: Visualizatio n of the map q ( x ) in (6) and the co rresp onding set-v alued map K q ( x ), when ∆ = 1 . Corollary 2 (Uniform quantizers) . L et x ( t ) b e a Kr asovskii solution to (1) and q b e define d as in (6) . If T ( G ∞ ) is c onne cte d and has only one sink, then ther e exist T con ≥ 0 and q ∞ ∈ ∆ Z such that for al l t ≥ T con , x i ( t ) ∈  q ∞ − ∆ 2 , q ∞ + ∆ 2  for al l i ∈ I . Pr o of . Since S = ∆ Z , Theo rem 1 implies that there exist a nonnega tive time T con and an int eger k s uch that for all t ≥ T con , it ho lds k ∆ ∈ K q ( x i ( t )) for a ll i ∈ I . This fac t is e q uiv alen t to the statement of the co rollary . 4.1 Av erage consensus In many applications one is concerned, r ather than with mere conv ergence , with convergence to a certain targ e t v alue, which is a function of the initial condition. F or instance, the ta rget can be the av er a ge of the initial sta tes: this pro blem is referr ed to as the aver age c onsensus pr oblem , and is studied in the next result. Corollary 3 (Average-preserving dynamics) . L et x ( t ) b e Kr asovskii solution to (1 ) and q as in (6) . D efine x av e ( t ) = 1 N P j ∈ I x j ( t ) . If T ( G ∞ ) is c onne cte d and has only one sink, and X j ∈ I a ij ( t ) = X i ∈ I a ij ( t ) for almost every t ≥ 0 , then x av e ( t ) = x av e (0) for every t > 0 and the c onclusion of Cor ol lary 2 holds. Mor e o ver, if x av e (0) 6 = ( k + 1 2 )∆ for every k ∈ Z , then q ∞ = q ( x av e (0)) , wher e as if x av e (0) = ( h + 1 2 )∆ for some h ∈ Z , t hen x i ( T con ) = x av e (0) for every i ∈ I . Pr o of . By linearity , for almo st every t > 0 d dt x av e ( t ) ∈ K   1 N X i ∈ I X j ∈ I ( a ij ( t ) − a j i ( t )) q ( x j ( t ))   . 7 By the assumption o n the a ij ’s, this implies that d dt x av e ( t ) = 0 for a lmost every t > 0 , so that the av erage is pr eserved. Corollar y 2 then implies that  q ∞ − ∆ 2 , q ∞ + ∆ 2  ∋ x av e ( T con ) = x av e (0). If in particular x av e (0) ∈  q ∞ − ∆ 2 , q ∞ + ∆ 2  , then it is clear that q ( x av e (0)) = q ∞ . Other wise, b eing x av e ( T con ) a t the b order of the in terv al, necessarily all x i ( T con ) must coincide. Note that Corollar y 3 provides a formula fo r the limit (quantized) v alue, and also a sufficient co ndition to a chiev e exact cons ensus b etw een the states. Corolla ry 3 impr ov es on earlier conv ergence results av ailable in the literature ab out av erage consensus o f Kr a sovskii solutions (cf. [1 0, Pro po sition 4]), a s it shows finite-time co nvergence fo r every initial condition and a llows for time-de p endent top olog ies. 4.2 Con v ergence time In o rder to estimate the con vergence time in Co r ollary 2 , we res trict our s elves to consider time-invariant top olo g ies, in the following s ense. W e as s ume that for every pair ( i , j ), e ither a ij ( t ) = 0 for all t ≥ 0 or a ij ( t ) ∈ [ a min , a max ] for all t ≥ 0, so that we may write G ( t ) = ( I , E , A ( t )) a nd G ∞ = ( I , E ). Prop ositio n 4 (Estimate of T con ) . L et x ( t ) b e Kr asovskii solution to (1) and q as in (6) . Assume t hat G ( t ) has time-invariant top olo gy , T ( G ∞ ) is c onne cte d and has only one sink. Then, T con ≤ 1 ∆ N a min  N a max a min  N max i,j ∈ I | q ( x i (0)) − q ( x j (0)) | . (7) Pr o of . The pro o f is based on sp ecializing the pr o of of Theorem 1 to the ca s e at hand: we refer to tha t pro of using the s ame notatio n. E quation (5 ) b ecomes , b eing the graph top olo g y time-inv aria nt , x i ( t ) ≥ x i (0) + ∆ Z J ∩ (0 ,t ) a min  a min N a max  N ds ≥ m (0) − 1 2 ∆ + ∆ a min  a min N a max  N t. Then, considering the seq uence o f T k ’s, we a rgue tha t T k − T k − 1 ≤ N a min  N a max a min  N for every k ≥ 1, a s every qua ntization interv al co nt ains a t most N a gents. On the other hand, k needs not to b e la r ger than ( M (0) − m (0)) / ∆. Thes e remar ks prov e the s tatement . Next, we wan t to discuss the tig htness of estimate (7), in terms of the dep endence on N and on ∆. The parameter ∆ repr esents the quantizer pr ecision and, in view o f Corollar y 2, also the accuracy which is a chiev able in a pproximating the conse ns us. T he b ound (7) allows for a conv ergence time whic h is p olyno mia l in ∆: the following example shows that there exist families of solutions which meet the b ound, exhibiting a conv ergence time prop ortiona l to ∆ − 1 . Indeed, for every N we can find a weight ed g raph G and an initial c o ndition ¯ x such that for a cer tain solution such that x (0) = ¯ x , T con ≥ 1 8 N a min ∆ max i,j ∈ I | q ( x i (0)) − q ( x j (0)) | . Example 1 (Slow conv erg ence: T con ∼ ∆ − 1 ) . W e let N ≥ 3, I = { 1 , . . . , N } a nd we assume the top olo gy to be a line gr aph, namely a ij =          1 if i = 1 and j = 2 1 if 2 ≤ i ≤ N − 1 a nd j = i − 1 , i + 1 1 if i = N and j = N − 1 0 otherwise . 8 Note that the res ulting dynamics (1) pr eserves the av era ge of the states. Regar ding the initial condition, w e ass ume x i (0) = ∆( i − 1 ) for all i ∈ I . In the analys is of the resulting system, we think o f the ag ent s as arra ng ed o n a line and we only descr ib e the evolution of the leftmost agents (1,2,. . . , ⌊ N / 2 ⌋ ), the evolution of the o thers b eing symmetrical. F or ear ly po sitive times, all age n ts a r e still except agent 1 which mov es to the r ight with cons ta n t sp eed ∆. Then, at time T ′ = 1 2 we hav e that x 1 ( T ′ ) = ∆ / 2, that is agent 1 r eaches the b order of the fir s t qua nt izatio n interv a l. Since K q ( x 1 ( T ′ )) ∋ ∆, there is one Kras ovskii solution such that for t ∈ ( T ′ , 2 T ′ ), x 1 ( t ) is constant while ag ent 2 moves to the r ight unt il it reaches x 2 (2 T ′ ) = 3∆ / 2, so that K q ( x 2 (2 T ′ )) ∋ 2∆ and K q ( x 1 (2 T ′ )) ∋ ∆. Then, for t ∈ (2 T ′ , 4 T ′ ) the only age nt on the mov e is ag ain ag ent 1, until x 1 (4 T ′ ) = 3 ∆ / 2. At time t = 4 T ′ , the tw o agents hav e the same sta te x 2 (4 T ′ ) = x 1 (4 T ′ ) . After this time, agents 3, 2 a nd 1 mov e to the right during three successive time interv als, so that at t = 9 T ′ they a re all collo ca ted as x 1 ( t ) = x 2 ( t ) = x 3 ( t ) = 5 ∆ / 2 . By rep eating this r easoning, we obser ve that the constructed solution x ( · ) rea ches the limit configura tion o f Cor ollary 3 at time T con = 1 2 ⌊ N 2 ⌋ X k =0 (1 + 2 k ) = 1 2  N 2   N 2  + 2  ≥ 1 8 N ( N − 1) . Since q ( x N (0)) − q ( x 1 (0)) = ( N − 1)∆ , then T con ≥ 1 8 N q ( x N (0)) − q ( x 1 (0)) ∆ . On the other hand, N is the num b er of age nts, and the bo und (7) allows for a convergence time which is exp onential in N . The following example provides a family of solutions such that T con ≥ C 2 N , (8) for a p ositive consta n t C . W e obser ve that in or de r to have an exp onential-in- N conv ergence time, the solution m ust stay on a disco nt inuit y of the right-hand side for a finite duration of time. Example 2 (Slow conv ergence: T con ∼ e N ) . W e let I = { 1 , . . . , N } and we assume that, given 0 < a ≤ b      ˙ x 1 = a  q ( x 2 ) − q ( x 1 )  ˙ x i = a  q ( x i +1 ) − q ( x i )  + b  q ( x 1 ) − q ( x i )  if 2 ≤ i ≤ N − 1 ˙ x N = 0 . W e also a s sume that the qua n tizer is uniform w ith ∆ = 1 a nd that the initial condition is      x 1 (0) = 0 x i (0) = 1 2 if 2 ≤ i ≤ N − 1 x N = 1 . Note that x i (0) is on a discontin uity po int of q fo r 2 ≤ i ≤ N − 1: then the Krasovskii conv exification is no n trivia l and we hav e ˙ x = − Lz , denoting the conv exified v alues as z i = (1 − α i ) × 0 + α i × 1 = α i . O ne c an immediately verify that there exists a K r asovskii solution x ( · ) having the following pro pe r ties: (a) for every t ≥ 0, it holds that x N ( t ) = 1 and x i ( t ) = 1 2 if 2 ≤ i ≤ N − 1; (b) α i =  a a + b  N − i for all 2 ≤ i ≤ N − 1 and for t ≤ T con ; 9 (c) ˙ x 1 ( t ) = a  a a + b  N − 1 almost always for t ≤ T con ; (d) at time T con = 1 2 a  a + b a  N − 1 ≥ 1 2 a 2 N − 1 the agents r each quantized consensus in the int erv al [1 / 2 , 1]. Then (8) follows cho osing C = 1 4 a . The q ua litative b ehavior of the c o nv ergence time of Krasovskii solutions, o utlined a b ove, should b e contrasted with that of nonquantized consensus dyna mics. Let T ε con be the time for conv ergence within a precisio n ε in a suitable no rm. Then, c o nsensus dyna mics without quantization typically yield a lo garithmic dep endence on ε , T ε con ≤ C log ε − 1 , where C is a constant which dep ends on the initial condition and o n the top ology of the int era ction gra ph, a nd entails a dep endence on N which is at mo st p olyno mial. W e conclude that our theoretica l results predict a qualita tive degr adation of conv erge nc e sp eed due to q uantization. Ho wev er , Prop ositio n 4 is intrinsically a worst-case result, and not every solution needs to achiev e the p erfor mance b ound. Indeed, it is arg ue d in [10, Remark 5] that, far from the eq uilibria, the quantized dynamics c onv erges exp one ntially fast and has the sa me rate of conv er g ence as the nonquantized linear co nsensus dynamics. This is confirmed by simulations rep orted in the same pa pe r , which show lo g arithmic conv erg ence times in b oth ca s es. These remarks e n tail no c ontradiction: far aw ay from the equilibria the quantized dynamics is well approximated by the nominal linea r dynamics, and the effect of quantization can b e s tudied as a bo unded disturbance (cf. [1 4, 3, 15]). On the other hand, in a neighbor ho o d of the equilibria the approximation is no long er go o d and the co nsequences of qua nt izatio n may fully come o ut, as we have shown a b ove. 5 Summary and future w ork This pap er has demonstrated that a mathematical framework combining graph theory and Krasovskii differential inclusio ns c a n b e use ful to solve pr oblems of distributed control with quantized communication. Complete Kras ovskii solutions of quantized consensus dynamics exist for any initial conditio n, a nd it is p ossible to s tudy their conv erg e to equilibria of “pr ac- tical consens us”. Under a mild connectivity a ssumption, which tr a nslates to the unbounded int era ctions graph the usua l co nnectivity condition for consensus on s tatic networks, solu- tions ar e shown to reach a neighborho o d of co nsensus after a finite time. The s ize of such neighborho o d only depe nds on the quantizer, a nd can thus b e made arbitr arily s mall by de- sign. O n the other hand, the c o nv ergence time can b e exp onentially increasing in the num b er of no des for so me solutions which s lide on a sur fa ce of disco nt inuit y of the dynamics. A few natura l generaliza tions o f the pres en t work would be of interest: we br ie fly men tion three of them. (i) In this pap er, the states of the ag ent s are communicated thro ugh a non-smo oth map which is a q ua nt izer , that is, w ho se ra nge is a discr e te spa ce. How ever, our pro o f tech- nique based on mono tonicity properties seems to b e promising for s tudying conv ergence of systems featuring more genera l non-smo o th interaction maps. (ii) Theorem 1 states sufficient conditions for consensus : is it then natural to a s k whether these assumptions are necessary . While it is clear that the connectedness o f G ∞ is 10 necessary for consensus, we believe that the a rgument of Theo rem 1 ca n b e extended in such a wa y to rela x the non-degener acy a ssumption a min > 0. A s ufficient co nnectivit y condition would then b e: there e xist T > 0, δ > 0 and a gr aph G = ( I , E ) which has a globally reachable no de and is such that if ( i, j ) ∈ E , then R t 0 + T t 0 a ij ( t ) dt > δ for every t 0 > 0. W e leav e the pro of of this extension to future res earch. On the o ther hand, when G ∞ is not connected but is cut -b alanc e d in the se ns e of [1 7, Assumption 1], we exp ect r e sults of par tial consens us and cluster ization [4, 9]. (iii) In this w ork , connectivity is a function of time determined by an exog enous signal. How- ever, there are applications in which connectivity b e t ween ag ent s is sta te-dep endent . Whic h would b e the conv erg ence prop erties of quantized contin uous-time dynamics o n a state-dep endent netw or k descr ibed by in tera ction functions of type a ij ( t, x )? 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