Distributed Consensus over Wireless Sensor Networks Affected by Multipath Fading

Distribu ted Consensu s o v er Wireless Sen sor Net w orks Affected b y Mul ti path F adin g Gesualdo Scutari and Sergio Barbarossa Dpt. INF OCOM, Univ. of Rome “La Sapienza ”, Via Eudossiana 18, 00 184 Rome, Italy E-mail: { scutari, sergio } @info com.uniroma1.it . ∗ Paper submitted to IEEE T r ansactions on Signal Pr o c essing , August 2007 . Revised Nov ember 30, 2007 . Accepted J anuary 14, 2008 . Abstract The design of sensor net w or k s capable of reaching a consensus o n a glo bally optimal decis ion tes t, without the need for a fusio n cent er, is a problem that ha s received considerable attention in the last years. Many consensus algor ithms have b een prop osed, with convergence conditions dep ending on the g raph describing the interaction among the nodes . In most works, the graph is undir ected and there are no pr opagation delays. Only r ecently , the analysis ha s b een extended to co nsensus algorithms incor po rating propag ation delays. In this w ork , w e propo se a consensus algorithm able to conv erge to a glob al ly optimal decision sta tis tic, using a wideb and wireless netw ork , gov erned by a fairly s imple MAC mechanism, where each link is a multipath, frequency-s e le ctive, c hannel. The main contribution of the pap er is to derive necessar y and sufficien t conditions on the netw ork top ology a nd sufficient conditions on the channel transfer functions gua ranteeing the exp onential conv ergence of the consens us algor ithm to a globally optimal decision v alue, for any b ounded dela y condition. 1 In tro duction Distributed algorithms for ac hieving c onsensu s in wireless sensor n et w orks, without the need for a fusion cen ter, ha v e b een the sub ject of many recent works. Tw o excellen t tutorials on the sub ject are [1, 2] (see also references ther ein). The conditions for ac hieving a consensus o ver a globally optimal decision test ultimately d ep end on the prop erties of the graph mo deling the interac tion among the no des. Most w orks consider und irected graphs and neglect propagation dela ys. There are only a few w orks that s tu dy the impact of delays in consensus-ac hieving algorithms, namely [3 ] − [6], fo cusing on time-con tin uous systems, and [7] − [9 ], dealing w ith d iscrete-time sy s tems. Among these works, it is useful to distinguish b et w een consensu s algorithms, [1] − [3], wher e the states of all the sensors con verge to a prescrib ed function (t y p ically the a v erage) of the s ensors’ initial v alues, and agreemen t algorithms, [4] − [9], t ypically used for coord inating the motion of sets of vehicles, where the states of the no d es con v erge to a common v alue, bu t this v alue is not a sp ecified function of the initial v alues. A recen t ∗ This work has b een partially funded by the WINSO C pro ject, a Sp ecific T argeted Researc h Pro ject (Contract Nu m b er 0033914 ) co-fun d ed by the INFSO DG of the Europ ean Commission within the R TD activities of the Thematic Priorit y Information So ciety T echnolo gies, and by ARL/ERO Con tract N 62558-05 -P-0458. 1 w ork prop osed a randomized gossip algorithm [10 ] to ac hieve distribu ted consensus, with a simple in teraction mec hanism, w here eac h no de in teracts with one no de at the time, in a rand omized fashion. In this w ork, we are interested in distributed consensus algorithms wher e the consensus coincides with a globally optimal decision statistic. Our goal is to deriv e the conditions on the c hannels b et we en eac h p air of no des, guarantee ing that eac h sensor will ev entuall y conv erge to the globally optimal decision statistic, in a totally distributed manner , i.e. without requirin g the pr esence of a fu sion cen ter. In [1 , 3], the authors pro vided n ecessary and sufficient conditions for th e conv ergence of a linear consensus proto col, in the case of a common time-in v arian t d elay v alue for all the links, i.e., τ ij = τ ∀ i 6 = j , an d assu m ing symmetric c hannels among the no des (mo deled as a und irected graph ). Under these assum ptions, the av erage consensus in [1, 3] is reac hed if and only if the common dela y τ is smaller than a top ology-dep end en t v alue. Ho w ev er, the assump tions of h omogeneous dela ys and n onrecipro cal channels are not appropr iate for describing the propagation in a common n et w ork deplo ying scenario, where the dela ys dep end on trav eled distances an d the comm unication c hannels ma y b e asymmetric . In [11], w e generalize d th e consens us algorithms to net w orks with inhomogeneous dela ys and asymmetric fl at-fading c hann els. In this corresp ond en ce, we extend our p revious wo rk to the more general case wh ere eac h link is m o deled as a multi path c hann el. W e assume baseband comm unications, m otiv ated b y the u se of imp ulse radio tec hnologies. The main contributions of this pap er are the follo wing: i) W e p ro vide necessary and su fficien t conditions on the netw ork top ology and sufficient conditions on the transf er fun ction of eac h c hann el ensurin g global con verge nce to the optimal d ecision test, for any set of finite propagation dela ys; ii) W e prov e that the conv ergence is exp onent ial, with con v ergence rate d ep ending, in general, on the c hann el parameters and propagation dela ys; iii) W e sho w h o w to reac h a distributed consensu s coinciding with the globally optimal decision statistics, ac hiev able by a central ized system ha ving error-fr ee access to all the n o des measurement s and observ atio n parameters, without the need of estimating neither the c hann el co efficien ts nor the dela ys. 2 Ho w to Ac hiev e Consensus on a G lobally Optimal Decision T est in a Decen tralized W a y Let us consider a set of N sen sors, eac h measur in g a s calar p arameter y i , i = 1 , . . . , N . The goal of the net w ork is to compute a sufficien t statistic of the measured d ata exp ressible as f ( y 1 , y 2 , . . . , y N ) = h   X N i =1 c i g i ( y i ) X N i =1 c i   , (1) where { c i } are p ositiv e co efficien ts and { g i } and h are arbitrary (p ossibly n onlinear) r eal functions on R , i.e., g i , h : R 7→ R . Eve n th ough the class of fu nctions expressible as in (1) is not the most general one, it do es include many cases of practical interest, lik e, e.g., b est lin ear unbiased estimati on or ML estimation under linear signal mo dels, m ultiple hyp othesis testing, detection of Gaussian pro cesses in Gaussian noise, computation of maximum, minimum, geometric mean or the histograms of the 2 gathered data [11, 13]. In this pap er , we consider only the scalar obs er v ation case, b ut the extension of (1) to the ve ctor case is straigh tforward, along the same guidelines of [11, 12]. T o compute f u nctions in the form (1) in a d istributed w a y , we consid er a linear inte raction mo del among the no des, and w e generalize the ap p roac h of [11 , 12] to a n et w ork wher e the c hannel b e- t w een eac h pair of no des is a multip ath c hannel, with, in general, asymmetric c hannel coefficient s and geometry-dep enden t dela ys. In eac h n o de there is a dyn amical system whose state x i ( t ; y ) ev olv es according to th e follo wing linear differen tial equation ˙ x i ( t ; y ) = g i ( y i ) + K c i X j ∈N i L X l =1 a ( l ) ij  x j ( t − τ ( l ) ij ; y ) − x i ( t ; y )  , t > 0 , x i ( ϑ ; y ) = e φ i ( ϑ ) , ϑ ∈ [ − τ , 0] , i = 1 , . . . , N , (2) where y = { y i } N i =1 is the set of measur emen ts; g i ( y i ) is a function of the lo cal measurement , wh ose form dep ends on the sp ecific decision test; c i is a p ositiv e coefficient that is chosen in ord er to ac hiev e the desired consensus, as in (1); K is a p ositive co efficient con trolling the con v ergence rate; a ( l ) ij and τ ( l ) ij are the amplitude and the dela y asso ciated to the l -th p ath of the c hann el b et w een no des i and j ; N i = { j = 1 , . . . , N : ∃ a ( l ) ij 6 = 0 , l = 1 , . . . , L } denotes the set of neigh b ors of no d e i , i.e., the no des that s en d signals to n o de i . It is w orth noticing that the state function of, let us sa y , no de i dep ends, directly , only on the measurement y i tak en by the no de itself and only indirectly on th e measuremen ts gathered by the other no des. In other w ords, ev en though the state x i ( t ; y ) gets to dep end, even tually , on all the measurements, through the interac tion with the other n o des, eac h n o de needs to know only its o w n measurement. The c hannel th r ough w hic h no d e r receiv es the signal from no d e q is a multipath c hannel with transfer fu nction H r q ( j ω ) = P L l =1 a ( l ) r q e − j ω τ ( l ) r q , for all r 6 = q . W e assume that the c hannel co efficien ts are sufficiently slo wly v aryin g to b e considered constan t for the time int erv al n ecessary for the netw ork to conv erge, w ithin a prescrib ed accuracy . In S ection 3, we will sho w that the conv er gence of (2) is exp onent ial and we will d eriv e a b ound for the con ve rgence rate. Knowing this rate, our metho d is applicable for those c hannels w hose coherence time is sufficien tly greater than the con v ergence time. W e are in terested in baseband comm unications, m otiv ated from a p ossible implemen tation of the radio in terface allo wing for the interacti on describ ed by (2) w ith an impulse radio using pulse-p osition mo dulation (IR-PPM), where the p osition of th e pu lse trans m itted b y no d e i is p r op ortional to the state of no de x i ( t ). In general, w e allo w the c hannels to b e asymmetric, i.e., a ( l ) r q ma y b e different fr om a ( l ) q r (and th us H r q ( j ω ) 6 = H q r ( j ω )) . W e also assume, r ealistica lly , that the maxim um dela y is b ounded, with maximum v alue τ = max r,q,l τ ( l ) r q . Because of the d ela ys, the state evo lution (2) for, let u s sa y , t > 0, is u niquely defin ed pr o vided that the initial state v ariables x i ( t ; y ) are sp ecified in the interv al from − τ to 0 , i.e., x i ( ϑ ; y ) = e φ i ( ϑ ) , for all i = 1 , . . . , N , and ϑ ∈ [ − τ , 0] . Some imp ortant commen ts about the interac tion mec hanism (2) are appropriate. D istribu ted consensus algorithms ha v e a clear adv an tage w ith resp ect to cen tralized systems, as they are less prone to congestion eve nts or failures of some of the no d es. T hey are also inherently scalable. Ho wev er, as opp osed to cen tralized systems, they t ypically require an iterativ e mec hanism to con verge to the 3 desired decision test. In most a v ailable works on distribu ted consensus, it is tacitly assumed that eac h n o de is ab le to receiv e the signals sen t b y its neighbors separately . This, of cour se, r equires a prop er medium access control (MA C) mec hanism to av oid collisions. But, when com bined with the iterativ e nature of distrib u ted consensus algorithms, a collision av oidance MA C pr otocol ma y b ecome rather complicated. Even the simple randomized gossip algorithm of [10] requires some form of MAC to a vo id collisions. Unfortun ately , enforcing a MA C control go es against the requ iremen t of simplicit y and scalabilit y , which are some of the ma jor motiv ations u nderlying the u se of distribu ted consensus algorithms. C on v ersely , we are intereste d in distr ib uted consensus mec hanisms where all no des tran s mit o v er a common shared p h ysical c hann el and there are no collision a v oidance or resolution mec hanisms wh atsoever, so th at eac h n o de receiv es a linear com bin ation of the signals transmitted b y the other no des, p ossibly thr ough a multipat h pr op agation c hannel. This motiv ates the interact ion mo del expressed b y (2), from wh ic h it turns out that eac h n o de do es not need to resolv e the receiv ed signals to b e able to up date its o wn state function. I n this corresp ondence, w e do not study the radio in terface allo wing for the no de interac tion giv en by (2). Neverthele ss, some preliminary stud ies, see e.g., [13, 20, 21] s uggest that im p ulse radios with pulse p osition mo dulation or d istributed phase-lo c k circuits are p ossible candidates for implemen ting (2), where the state v alues are exc hanged through pulse p osition mo dulation or ph ase mo dulation, resp ectiv ely . Ho w ev er, the adv an tages of distrib u ted consensus based algorithms as describ ed ab o v e come at the price of a p enalt y: the final consensus is reac hed thr ough an iterativ e pro cedu re th at consumes time and energy . The o ve rall energy necessary to ac hieve the final d ecision is the sum of the p o wers transmitted by eac h sensor multiplie d by the conv ergence time (in the next section, we will giv e an upp er b ound of such a v alue). On one hand , to sa ve en ergy , w e w ould lik e to use the minimum transmit p o wer that ens u res netw ork connectivit y . But a small transmit p ow er has an effect on the net w ork top ology , as it leads to a redu ced n umb er of links and, as a consequence, to a small algebraic connectivit y . Hence, a small individual transmit p o w er imp lies a long con v ergence time. C on v ersely , to reduce the conv ergence time, the n et w ork should hav e a high connectivit y , but this requires a large transmit p o w er. It is then intuitiv e to exp ect an optimal trade-off. Th is tr ade-off has b een studied in [22], wh er e we r emand to the in terested reader. T he fo cus of this corresp ondence is on find ing the conditions on the c hann el parameters that guarant ee the conv ergence of (2) to the desired consensus v alue. Consensus on t he state deriv ativ e . Differen tly from most p ap ers dealing w ith a ve rage consensu s problems [1] − [3], [4] − [9], we adopt here the alternativ e d efinition of consensus already introd uced in our pr evious wo rks [11] − [13]: W e define the consensu s (or net wo rk sync hronization) with resp ect to the state derivative , r ather than to the state. Definition 1 Given the dynamic al system in (2), we say that a solution { x ⋆ i ( t ; y ) } of (2) is a synchro- nized state of the system, if ˙ x ⋆ i ( t ; y ) = α ⋆ ( y ) , ∀ i = 1 , 2 , . . . , N . The system (2) is said to globally syn- chr onize if ther e e xi sts a synchr onize d state α ⋆ ( y ) , and all the state derivatives asympt otic al ly c onver ge to this c ommo n v alue, for any given set of i nitial c onditions { e φ i } , i.e., lim t 7→∞ | ˙ x i ( t ; y ) − α ⋆ ( y ) | = 0 , 4 ∀ i = 1 , 2 , . . . , N , wher e { x i ( t ; y ) } is a solution to (2). The synchr onize d state is said to b e globally asymptotically stable if the system glob al ly synchr onizes, in the se nse sp e cifie d ab ove. Observe that, according to Definition 1, if there exists a globally asymptotically stable synchronized state, then it m ust necessarily b e unique (in the deriv ativ e). O ne of the reasons to in tro du ce this definition of consens us, as opp osed to the consensus on the state [1 ] − [3], [4] − [9], is th at, as will b e sho wn in the next section, the con verge nce on the s tate deriv ativ e is not affected by th e presence of propagation dela ys. O ne more reason is that, in the presence of coupling noise, state-con v ergen t algorithms giv e r ise to a noise with div erging v ariance , wh er eas the algorithm conv erging on the state deriv ative exhibits a fin ite v ariance [12, 13]. 3 Necessary and Su ffic ien t Conditions for Ac hieving Consensus T o d eriv e our main results, we rely on some basic notions of directed graph (d igraph) theory , as briefly recalled next. More details are giv en in [11, App endix A]. A d igraph G is defined as G = { V , E } , where V is the set of vertic es and E ⊆ V × V is the set of edges, with th e con ve ntio n that e ij = ( v i , v j ) ∈ E if there exists an edge fr om v j to v i , i.e., the information flows f r om v j to v i . A d igraph is weig hted if a p ositive wei ght, denoted by a ij , is asso ciated with eac h edge e ij . The in-degree of a vertex is defined as the sum of the weigh ts of all its incoming ed ges. The out-degree is similarly defined. The Laplacian matrix L = L ( G ) of the digraph asso ciated to s ystem (2) is L = D − A , wher e D is the diagonal matrix of verte x in-degrees and A is the adjacency matrix. F or reasons that will b e clarified within the pro of of next theorem, the ab o ve matrices are b uilt as follo w s: [ D ] ii = P j ∈N i P L l =1 a ( l ) ij and [ A ] ij = P L l =1 a ( l ) ij . A digraph is a d irected tree if it has N v ertices and N − 1 edges and there exists a ro ot verte x (i.e., a zero in-degree vertex) with directed paths to all other v ertices. A d ir ected tree is a sp anning dir ected tree of a digraph G if it has the same vertic es of G . A digraph is Str ongly Connected (SC) if, for ev ery pair of no d es v i and v j , there exists a directed path fr om v i to v j and vicev ersa. A digraph is Quasi- Str ongly Connected (QSC) if, for ev ery pair of no des v i and v j , there exists a no de r that can reac h b oth v i and v j b y a directed path. T he fundamenta l result of this p ap er is stated in the follo win g. Theorem 1 L et L b e the L aplacian matrix asso ciate d to the digr aph G = { V , E } of system (2), and let γ = [ γ 1 , . . . , γ N ] T b e the left eigenve ctor of L c orr esp onding to the zer o eigenvalue, i.e., γ T L = 0 T N . Given system (2), assume that the f ol lowing c onditions ar e satisfie d: a1) The c oupling gain K and the c o efficients { c i } ar e p ositive; a2) The pr op agation delays { τ ( l ) ij } ar e finite, the c o efficients { a ( l ) ij } ar e r e al and the channel tr ansfer functions { H r q ( j ω ) } ar e su ch that H r q (0) > 0 , ∀ q , r 6 = q , and P q ∈N r | H r q ( j ω ) | P q ∈N r H r q (0) ≤ 1 , ∀ ω ∈ R , ∀ r 6 = q ; (3) a3) The initial c onditions ar e taken in the set of c ontinuously differ e ntiable and b ounde d functions mapping the interval [ − τ , 0] to R N . 5 Then, system (2) glob al ly synchr onizes, for any set of pr op agation delays, if and only if the digr aph G is QSC. The synchr onize d state is α ⋆ ( y ) = X N i =1 γ i c i g i ( y i ) X N i =1 γ i c i + K X N i =1 γ i X j ∈N i X L l =1 a ( l ) ij τ ( l ) ij , (4) wher e γ i > 0 if and only if no de i c an r e ach al l the other no des of the digr aph by a dir e cte d p ath, otherwise γ i = 0 . The c onver genc e is exp onential, with asympto tic c onver genc e r ate arbitr arily close to r , − min i {| Re { s i }| : p ( s i ) = 0 and s i 6 = 0 } , wher e p ( s ) is the char acteristic f u nction asso ciate d to system (2) (se e (12) in the App endix). Pro of. See the App endix. Remark 1 - Robustness against multipath c hannels: Theorem 1 sho ws that, d ifferen tly from classical linear consensus proto cols [1, 2 ], th e prop osed algorithm is robust aga inst propagation dela ys, since its con v ergence condition is not affe cte d by the delays . Moreo v er, the prop osed approac h is v alid for fr e quency-sele ctive and asymmetric channels. Th e only significant constraint is that the c hannel co efficien ts are real and th at, acco rd in g to a2) , their summation, o ver eac h channel, has to b e a p ositiv e quan tit y . This implies a sort of implicit coheren t com bination, conceptually similar to the t yp e-based approac h of [19], eve n though the wo rk of [19] was aimed at study in g the multiple access for sensor net w orks with a f u sion cen ter, wh ereas our sc heme do es not need a fu sion cen ter. The additional constrain ts on the channel transfer fun ctions, i.e., P q ∈N r | H r q ( j ω ) | ≤ P q ∈N r H r q (0), f or all ω ∈ R , is certainly v alid if the channels are lo w-pass filters with maxim um gain in ω = 0. But this is only a sufficien t condition and we w ill later rep ort some numerical r esults showing that if this condition is not satisfied, the metho d can still conv erge. Finally , giv en the conv ergence rate r , we kno w for whic h class of channels the metho d is applicable: the channels whose coherence time is sufficiently greater than 1 /r . Remark 2 - Effect of netw ork top ology: According to Theorem 1, a global consensus is p ossible if and only if there exists at least one no d e (the ro ot no d e of the span n ing dir ected tree of the digraph) that can reac h all the other no des by a directed path. If no suc h a no d e exists, the information gathered b y eac h sensor has no w a y to propagate through the whole n et w ork an d th us a global consensus cannot b e reac hed. Moreo v er, the only no d es con tributing to the final consensus v alue are the ones havi ng a directed path linking them to all th e other no des [see (4)]. As a consequence, the final decision dep ends on the measurements gathered b y al l the no des if and only if the net work is strongly connected. When the digraph is not QSC, system (2) may still con v erge, b u t it forms separated clusters of consensus , as pro ve d in [11, 13]. Remark 3 - Un biased decisions wit hout estimating the channel parameters: The closed form expr ession of the s ync hronized state giv en in (4) sho ws a d ep endence of the final consensus on the n et w ork top ology and propagation p arameters. This implies that the final consensus v alue (4), in general, do es not coincide with the desired decision statistics as giv en in (1), except for the trivial case of flat-fading channels w ith zero d ela ys and balanced (and th us strongly connected) digraph. 6 Nev ertheless, in the follo wing we pro vide a metho d to get an unbiase d estimate, without ha ving to get an y preliminary estimation of the c hannel parameters, i.e. a ( l ) i,j , τ ( l ) i,j , incorp orating, only in the case of unbala nced net wo rks, a d ecen tralized estimation of the topology d ep endent coefficients γ i . The bias due to the propagation dela ys and path amplitudes can b e remo ved using the f ollo w ing t w o-step algorithm. W e let system (2 ) to ev olv e t wice: The first time, the system ev olv es acco rd ing to (2) and w e denote by α ⋆ ( y ) the synchronized state, as in (4); the second time, we set g i ( y i ) = 1 in (2), for all i , and the system is let to ev olv e again, denoting the new sync hron ized state by α ⋆ ( 1 ). T aking the ratio α ⋆ ( y ) /α ⋆ ( 1 ) = ( P N i =1 γ i c i g i ( y i )) / ( P N i =1 γ i c i ) , we obtain the same consensus v alue that would ha v e b een ac hiev ed in the absence of multipath propagation. If the netw ork is s trongly connected and balanced, γ i = 1 , ∀ i and then the comp ensated consen- sus coincides w ith the desired v alue (1). If the n et w ork is unbalanced, th e comp ensated consensus α ⋆ ( y ) /α ⋆ ( 1 ) do es not dep en d on the multipath co efficient s, but it is still b iased, w ith a bias dep endent on γ , i.e., on the net w ork top ology . This residual dep end ence can b e eliminated in a d ecen tralized w a y if eac h no de is able to estimate its o wn γ i . In fact, in su c h a case, α ⋆ ( y ) /α ⋆ ( 1 ) can b e made to coincide with the desired expression in (1) b y simply replacing eac h c i in (2) with c i /γ i , for all i su c h that γ i 6 = 0 (sup p ose that there are N r of such no d es, w.l.o.g.). Interestingly , th e estimate of eac h γ i can also b e obtained in a decen tralized wa y , using the follo wing pro cedu re. At the b eginning, every no de sets g i ( y i ) = 1 and the net work is let to evo lv e. Th e final consens u s v alue will b e, in this case α ⋆ ( 1 ). Then, the net w ork is let to ev olv e N r times, according to the follo wing proto col. A t step i , with i = 1 , . . . N r , n o de i sets g i ( y i ) = 1, while all the other no des set g k ( y k ) = 0 for all k 6 = i ; all no des are then let to ev olv e according to (2); let us denote b y α ⋆ ( e i ) the final consensus v alue, where e i is the canonical v ector ha ving all zeros, except the i -th comp onen t, equal to one. Eac h no de is no w able to take th e r atio α ⋆ ( e i ) /α ⋆ ( 1 ), whic h coincides with the ratio ˜ γ i := γ i / P k γ k . Th us, after N r + 1 steps, ev ery no de kno ws its o wn (normalized) ˜ γ i and it may then u se it in the s u bsequent run of the consensus algorithm, setting c i = c / ˜ γ i , to ac hiev e a top ology ind ep endent estimate. Obs er ve that, since the eigen v ector γ do es not d ep end on the observ ations { y i } , the prop osed algorithm to estimate γ is required to b e p erformed only once ev ery c hann el coherence p erio d. In s u mmary , th e effects of b oth dela ys and c hann el co efficien ts can b e eliminated from the fin al consensus v alue, eve n if at the price of a slight increase of complexit y and the need f or some co ord ination among the no des. 4 Numerical Results and Conc lusion As a numerical example, in the top r o w of Fig. 1, we r ep ort t w o examples of top ologies: the left graph is SC, wh ereas th e r igh t graph is Q SC. F or the QSC digraph in the fi gure, w e sk etc h its decomp osition in Strongly Connected Comp onents (SCC ), whose r o ot is denoted b y RSCC. 1 The b eh avior of th e state deriv ativ es v ersus the iteration index, in the t wo cases, is illustrated on th e b ottom ro w of the 1 A SCC of a digraph is a max imal subgraph whic h is also SC, meaning that there is no larger SC subgraph contai ning the nodes of the considered comp onent. A RSCC is a SCC containing all no des that can reac h all the other no d es in t he digraph by a directed path [11, A pp endix A]. 7 figure 2 . The ed ges sho wn in b oth graph s show the activ e link. Eac h lin k is mo deled as an FIR fi lter, mo deling the m ultipath fading. Eac h filter has maximum length L = 5 and the co efficien ts ha v e b een generated as a ( l ) ij = ( A + w ( i, j, l )) e − lT /τ 0 , l = 0 , . . . , L − 1, w here th e constan t A = 1 r epresen ts a deterministic comp onent, whereas w ( i, j, l ) are i.i.d. random Gaussian v ariables w ith zero mean an d standard deviation σ n = 0 . 5, m o deling th e f ad in g. Obs erv e that, usin g this setting, some c hannel co efficien ts are also negativ e. The exp onential mo d els th e attenuati on as a fu nction of distance and τ 0 represent s the dela y spread; T is the sampling time. The dela ys τ ( l ) ij on eac h link ha ve b een mo d eled as τ ( l ) ij = d ij /c + ( l − 1) T , where d ij is the distance b etw een nod es i and j and c is the sp eed of ligh t. The dimension of the net wo rk has b een computed in order to mak e th e maximum d ela y τ max = d max /c m uch larger than the sampling time T . In particular, w e chose the parameters so that τ max = 30 T , in order to test the algo rithm under a seve re propagation dela y . The constant lines with arrows rep orted in the b ottom ro w of Fig. 1 represent the theoretical v alue, as giv en by (4). W e can verify that the sim ulation curv es tend to approac h the theoretical v alues for b oth S C and QS C top ologies, as predicted b y the theory . It is worth men tioning that, in b oth cases, w e u s ed c hannels that resp ect the condition H r q (0) > 0 , ∀ r , q , but d o not necessarily resp ect the condition P q ∈N r | H r q ( j ω ) | ≤ P q ∈N r H r q (0). Nonetheless, the sim ulation results are still in goo d agreemen t with our theoretical find ings. There is no con trast with the theory b ecause the condition P q ∈N r | H r q ( j ω ) | ≤ P q ∈N r H r q (0), ∀ r 6 = q , is only a sufficien t condition. The estimates rep orted in Fig. 1 sho w a go o d agreemen t b et w een theory and sim ulation, b u t the fi nal r esult do es not coincide with the th eoretica l optimal v alue, b ecause of the b ias induced by the multipath co efficien ts and dela ys. Ho wev er, as suggested at the en d of the previous section, it is p ossible to r emo v e the bias, w ithout h a ving to estimate n either th e channel amplitudes a ( k ) i,j nor the dela ys τ ( k ) i,j . As an example of this comp ensation tec hnique, in Fig. 2 , we rep ort the runnin g state deriv ative ˙ x i ( t ; y ) (solid lines) and the comp ensated estimate ˙ x i ( t ; y ) / ˙ x i ( t ; 1 ) (dotted line), together with the theoretic al limits (constan t lines) ac h iev able without comp ensation (triangle marks) and with comp ensation (circle marks). The last v alue coincides with the globally optimal estimat e. The results sho wn in Fig. 2 ha ve b een ac hiev ed with m ultipath fading c hannels of length L = 11, u n der the same fad in g mo del used in the previous example. Fig. 2 sh o ws that, as pr edicted by the theory , the consensus algorithm with comp ensation is able to reac h the globally optimal estimate, without the need of estimating the channel co efficien ts. In summary , in this wo rk w e hav e derived the conditions allo wing a distribu ted consensus mec ha- nism to r eac h globally optimal decision statistics, in the presence of multipat h propagation in the link b et wee n eac h pair of no d es. The metho d is v alid for real (baseband) c hannels and it requires that the summation of the c hannel co efficien ts ov er eac h link is str ictly p ositiv e. Thanks to th e closed f orm expression d eriv ed in this pap er, w e hav e also sho wn ho w to get unbiased, globally optimal estimates, without the need to resolv e the signals receiv ed from different no d es (thus allo wing for a v ery simple MA C mec hanism), n or to estimate th e c hannel co efficient s. A crucial in v estigation, motiv ated fr om 2 Clearly , the sim ulations ha ve b een p erformed on th e discretized versio n of ( 2 ); in such a case, given the sampling time T , there is a maximum v alue of K guaranteei ng th e conv ergence of (2): K T m ust b e sufficiently smalle r than 1 / d eg max in , where deg max in is the maximum in-d egree of the graph Laplacian (cf. [11, Ap p endix A]). 8 (a) (b) 1 2 4 5 6 8 11 9 10 12 13 3 7 14 1 2 4 5 6 8 11 9 10 12 13 3 7 14 0 50 100 150 200 250 300 -50 0 50 100 150 State derivative Iteration number 0 50 100 150 200 250 300 -50 0 50 100 150 200 state derivative Iteration number RSCC 2 SCC 1 SCC Figure 1: Example of global consensus in a SC and QSC n etw ork. this w ork, is the design of the most ap p ropriate r adio int erface allo wing for the inte rn o de interac tion enabling the distributed consensus. 5 App endix P art of the pro of of th e theorem is based on the same appr oac h we follo w ed in [11, App endix C]. Th us, in the follo wing w e mak e use of the general resu lts of [11, App en dix C] and w e fo cus only on the sp ecific asp ects of the mo del used in this p ap er. W e start studying the existence of a s ync hronized s tate in the form (4). Then, we pro ve th at s uc h a state is also globally asymp toticall y stable (cf. Definition 1). Th roughout the pro of, we assume th at cond itions a1) - a4) are satisfied and that th e digraph G asso ciated to (2) is QSC. In the follo wing, for the sak e of n otatio n simplicit y , we drop the dep en d ence of the state fun ction f r om the observ ation, as this dep endence do es not p la y an y role in our pro of. Existence of a sync hronized stat e : The set of d ela y ed differential equations (2) admits a solution in the form x ⋆ i ( t ) = αt + x ⋆ i, 0 , i = 1 , . . . , N , (5) where α ∈ R and { x ⋆ i, 0 } are a set of co efficients that dep end in general on the system parameters and on the initial conditions, if and only if { x ⋆ i ( t ) } satisfies (2), i.e., if and only if there exist α and { x ⋆ i, 0 } 9 0 100 200 300 400 500 600 700 800 900 1000 −100 −50 0 50 100 150 200 250 300 iteration index state derivatives compensated state derivative state derivative Figure 2: U ncomp ensated ru nning estimate ( solid lines), i.e. ˙ x i ( t ; y ) and compensated run ning estimate (dotted lines), i.e. ˙ x i ( t ; y ) / ˙ x i ( t ; 1 ). suc h that the follo w in g system of linear equations is feasible: c i ∆ i ( α ) K + X j ∈N i  X L l =1 a ( l ) ij   x ⋆ j, 0 − x ⋆ i, 0  = 0 , (6) ∀ i = 1 , . . . , N , where ∆ i ( α ) , g i ( y i ) − α  1 + K c i X j ∈N i X L l =1 a ( l ) ij τ ( l ) ij  . (7) In tro du cing the w eigh ted Laplacian L = L ( G ) asso ciated to G , system (6) can b e equiv alen tly r ewritten in v ector form as K Lx ⋆ 0 = D c ∆ ( α ) , (8) where x ⋆ 0 , [ x ⋆ 1 , 0 , . . . , x ⋆ N , 0 ] T , D c , diag( c 1 , . . . , c N ) , and ∆ ( α ) , [∆ 1 ( α ) , . . . , ∆ N ( α )] T , with ∆ i ( α ) defined in (7). O b serv e that, b ecause of a2) and the quasi-strong conn ectivit y of G , the graph Laplacian L h as the follo wing prop erties: i) rank( L ) = N − 1; ii) N ( L ) = R ( 1 N ) ; and iii) N ( L T ) = R ( γ ) , where N ( · ) an d R ( · ) denote the (righ t) n ull-space and th e range space op erators, r esp ectiv ely , and γ is a left eigen v ector of L corresp onding to the (simple) zero eigen v alue of L , i.e., γ T L = 0 T . It follo ws from i)-iii) that, for any given α, (8) ad m its a solution if and on ly if D c ∆ ( α ) ∈ R ( L ). Usin g again p rop erties i)-iii), w e ha ve: D c ∆ ( α ) ∈ R ( L ) ⇔ γ T D c ∆ ( α ) = 0 . It is easy to chec k th at the v alue of α that s atisfies the latter condition is α = α ⋆ , with α ⋆ defined in (4). Hence, if α = α ⋆ , th e sync hronized state in the desired form (5) is a solution to (2 ), for an y giv en set of { τ ( l ) ij } , { g i } , { c i } , { a ( l ) ij } and K > 0. The structure of the left eigen vec tor γ asso ciated to the zero eigen v alue of L as giv en in the th eorem follo ws from [11, Lemma 4]. Setting α = α ⋆ , system (8) admits ∞ 1 solutions, giv en b y x ⋆ 0 = 1 K L ♯ D c ∆ ( α ⋆ ) + R ( 1 N ) , x 0 + R ( 1 N ) , where x 0 , L ♯ D c ∆ ( α ⋆ ) /K, (9) ∆ i ( α ⋆ ) is obtained b y (7) setting α = α ⋆ and L ♯ is the generaliz ed in verse of the Laplacian L . Global asymptotic stability : W e prov e now that th e sync hr onized state of sys tem (2) is globally asymptotically stable. T o this end, we use the follo w ing inte rmed iate r esults. 10 Let C + = { s ∈ C : Re { s } > 0 } , C − = { s ∈ C : Re { s } < 0 } , and C + b e the closure of C + , i.e., C + = { s ∈ C : Re { s } ≥ 0 } . Denoting by H n × m the set of n × m matrices whose en tries are analytic 3 and b ounded fun ctions in C + , let us introd uce the degree matrix ∆ ≥ 0 (where “ ≥ ”has to b e int ended comp onen t-wise) and the complex matrix H ( s ) ∈ C N × N , defined resp ectiv ely as ∆ , d iag ( k 1 deg in ( v 1 ) , ..., k N deg in ( v N )) , [ H ( s )] ij ,    0 , if i = j, k i P L l =1 a ( l ) ij e − sτ ( l ) ij , if i 6 = j, (10) where d eg in ( v i ) = P j ∈N i P L l =1 a ( l ) ij ≥ 0 is the in-degree of no d e v i and k i , K /c i > 0 . Ob serv e that H ( s ) ∈ H N × N . Lemma 1 Consid er the fol lowing line ar fu nc tional differ ential e quation: ˙ x i ( t ) = k i X j ∈N i X L l =1 a ( l ) ij  x j ( t − τ ( l ) ij ) − x i ( t )  , t > 0 , x i ( ϑ ) = φ i ( ϑ ) , ϑ ∈ [ − τ , 0] . i = 1 , . . . , N , (11) and assume that the f ol lowing c onditions ar e satisfie d: b1. The initial value f u nctions φ ar e taken in the set C 1 of c ontinuously differ entiable functions that ar e b ounde d in the norm 4 | φ | s = sup − τ ≤ ϑ ≤ 0 k φ ( ϑ ) k ∞ , and the solutions x ( t ) with initial functions φ ar e b ounde d; b2. The c har acteristic e quation asso ciate d to (11) p ( s ) , det ( s I + ∆ − H ( s )) = 0 , (12) with ∆ and H ( s ) define d i n (10), has al l r o ots { s r } r ∈ C − , with at most one simple r o ot at s = 0 . 5 Then, system (11) is mar ginal ly stable, i.e., ∀ φ ∈ C 1 and Re { s 1 } < c < 0 , ther e e xi st t 1 and α , with t 0 < t 1 < + ∞ and 0 < α < + ∞ , indep e ndent of φ , and a ve ctor x ∞ , with k x ∞ k < + ∞ , such that k x ( t ) − x ∞ k ≤ α | φ | s e ct , ∀ t > t 1 . (13) Pro of. Because of sp ace limitation, w e omit the pro of that can b e obtained follo wing the same steps of the pr o of in [11, Lemma 5], after observing that system (11) can b e rewritten in the canonical form of [16, C h. 6, Eq. (6.3.2 )], [17, Ch. 3, Eq. (3.1)]. Lemma 2 ([15, Theorem 2.2]) L et H ( s ) ∈ H N × N and ρ ( H ( s )) denote the sp e ctr al r adius of H ( s ) . Then, ρ ( H ( s )) is a subharmonic 6 b ounde d (ab ove) fu nction on C + .  3 A complex function is said to b e analytic (or holomorphic) on a region D ⊆ C if it is complex d ifferentia ble at ever y p oin t in D , i.e., for any z 0 ∈ D th e function satisfies the Cauch y-Riemann eq u ations and has contin uous first partial deriv ative s in the neighborho o d of z 0 (see, e.g., [14, Theorem 11.2]). 4 W e u sed, without loss of generalit y , as v ector norm in R N the infinity norm k·k ∞ , defined as k x k ∞ , max i | x i | . Of course, the same conclusions can b e obtained using any other norm. 5 W e assume, w.l.o.g., that the ro ots { s r } are arranged in nonincreasing order with respect to th e real part, i.e., 0 = Re { s 0 } > Re { s 1 } ≥ Re { s 2 } ≥ ... . 6 See, e.g., [15], for the definition of subharmonic function. 11 W e are ready to pro v e the global asymptotic stabilit y of the sync hronized state of (2). Ap plying the follo wing c hange of v ariables: Ψ i ( t ) , x i ( t ) − ( α ⋆ t + x i, 0 ), for all i = 1 , . . . , N , where α ⋆ and { x i, 0 } are defined in (4) and (9), resp ectiv ely , and using (9), the original system (2) can b e equiv alentl y rewritten in terms of { Ψ i ( t ) } i as ˙ Ψ i ( t ) = k i X j ∈N i X L l =1 a ( l ) ij  Ψ j ( t − τ ( l ) ij ) − Ψ i ( t )  , t ≥ 0 , (14) with Ψ i ( ϑ ) = φ i ( ϑ ) , e φ i ( ϑ ) − α ⋆ ϑ − x i, 0 for ϑ ∈ [ − τ , 0] , where { e φ i } are the initial v alue functions of the original system (2). It follo ws from (14) that the s y n c hronized state of system (2), as giv en in (5) (with α = α ⋆ ), is globally asymp totica lly stable (according to Definition 1) if system (14) is marginally stable. According to Lemma 1, th e marginal stabilit y of system (14) is guaran teed if: b1 ) the tra jectories { Ψ i ( t ) } are b ound ed for all t > 0 , giv en φ ∈ C 1 ; b2 ) the c haracteristic equ ation (12 ) asso ciated to (14), h as all ro ots in C − , with at most one simple ro ot at s = 0 . F ollo wing the same steps as in [11, App endix C], one can prov e that, under a1 )- a3 ), all the solutions { Ψ i ( t ) } to (14), with initial conditions in C 1 , are uniformly b ound ed , as required by assumption b1 ) in Lemma 1. Because of sp ace limitation we omit the details. W e study instead the charact eristic equation (12), and p ro v e th at, under a1 )- a3 ) and the quasi-strong connectivit y of the digraph, assumption b2 ) of Lemma 1 is satisfied. First of all, observe th at, since ∆ − H (0) = K D c L , w e ha ve p (0) = det ( ∆ − H (0)) = d et ( K D c ) det ( L ) = 0 , (15) where th e last equalit y in (15) is due to rank( L ) = N − 1. It follo w s from (15) that p ( s ) has a ro ot in s = 0 , corresp onding to the zero eigen v alue of the Laplacian L (recall that det ( K D c ) 6 = 0). Since the digraph is assu med to b e QSC, according to [11, Corollary 2], such a ro ot is simp le. Thus, to complete the pro of, w e need to sh o w that p ( s ) do es not h av e an y solution in C + \{ 0 } , i.e., det ( s I + ∆ − H ( s )) 6 = 0 , ∀ s ∈ C + \{ 0 } . (16) Since s I + ∆ is nonsin gular in C + \{ 0 } [recall that, u nder a1)-a2) , ∆ ≥ 0 , with at least one p ositiv e diagonal en try], (16) is equiv alen t to det  I − ( s I + ∆ ) − 1 H ( s )  6 = 0 , ∀ s ∈ C + \{ 0 } , (17) whic h leads to the follo win g sufficient condition for (16): ρ ( s ) , ρ  ( s I + ∆ ) − 1 H ( s )  < 1 , ∀ s ∈ C + \{ 0 } . (18) Since ( s I + ∆ ) − 1 ∈ H N × N and H ( s ) ∈ H N × N , it follo ws fr om Lemma 2 th at the sp ectral radius ρ ( s ) in (18) is a su bharmonic function on C + . As a direct consequen ce, w e h a v e, among all, that ρ ( s ) is a contin uous b oun d ed fu nction on C + and satisfies the maximum mo dulus principle (see, e.g., [15] and r eferences th er ein): ρ ( s ) ac hiev es its glob al maxim um only on the b oundary of C + . Since 12 ρ ( s ) is strictly prop er in C + , i.e., ρ ( s ) → 0 as | s | → + ∞ wh ile k eeping s ∈ C + , it follo ws that sup s ∈ C + ρ ( s ) < s up ω ∈ R ρ ( j ω ) . According to the latter inequalit y , condition (18) is satisfied if ρ ( j ω ) = ρ  ( j ω I + ∆ ) − 1 H ( j ω )  < 1 , ∀ ω ∈ R \{ 0 } . (19) Denoting by k A k ∞ , max r P q | [ A ] r q | the maximum r ow sum matrix norm and using ρ ( A ) ≤ k A k ∞ ∀ A ∈ C N × M [18], w e ha ve ρ ( j ω ) ≤    ( j ω I + ∆ ) − 1 H ( j ω )    ∞ = max r P q ∈N r | k r H r q ( j ω ) |    j ω + k r P q ∈N r H r q (0)    ≤ max r P q ∈N r | H r q ( j ω ) | P q ∈N r H r q (0) ≤ 1 , where in the last inequalit y we used (3) [see assumption a2) ]. Since in the second inequalit y , the equalit y is r eac hed if and only if ω = 0 , w e ha ve ρ ( j ω ) < 1 for all ω 6 = 0 , which guarant ees that condition (18) is satisfied. Hence, according to Lemma 1, giv en any set of initial conditions { φ i } satisfying a 4) , the tra j ectories Ψ ( t ) → Ψ ∞ as t → + ∞ , with exp onent ial r ate arbitrarily close to r , −{ min i | Re { s i }| : p ( s i ) = 0 and s i 6 = 0 } , w here p ( s ) is defined in (12) and Ψ ∞ ∈ R ( 1 N ) (b ecause of LΨ ∞ = 0 and rank( L ) = N − 1 ) . In other wo rd s, system (14) exp onen tially reac hes the consensus on the state. Necessit y : The necessit y of quasi-strong connectivit y of the digraph for the n et w ork to reac h a glob al consensus can b e pr o v ed as in [11 , App end ix C.2] by showing that, if the digraph asso ciated to (2 ) is not QS C, different clusters of n o des synchronize on differen t v alues [11, Corollary 1]. 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