A graph is a pair G= (V,E) with V a nonempty finite set of nodes or vertices V { v1 a set of edges or arcs E V V. We assume ( vi, vi) E,

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1 Prepared by FL Lews Updated: hursday, October 6, 008 Graphs A graph s a par G= (V,E) wth V a nonempty fnte set of nodes or vertces V { v,, v N } and a set of edges or arcs E V V We assume ( v, v) E,, e no self loops Elements of E are denoted as e ( v, v ) whch s termed an edge or arc from v to v, and represented as an arrow wth tal at v and head at v Edge e ( v, v ) s sad to be outgong wrt node v and ncomng wrt v ; node v s termed the parent and v the chld If every possble arc exsts, the graph s sad to be complete If ( v, v) E( v, v) E,, the graph s sad to be undrected, otherwse t s drected, termed a dgraph he edges can be represented by an adacency or connectvty matrx A [ a ] wth a f ( v, v ) E and a =0 otherwse Note that a =0 For an undrected graph, A s symmetrc he n-degree of v s the number of edges havng v as a head, e -th row sum a he out-degree of a node v s the number of edges havng v as a tal, e -th column sum If the n-degree equals the out-degree for all nodes v V the graph s sad to be balanced a Lemma An undrected graph s balanced Connectvty he set of neghbors of a node v s N { v : ( v, v) E}, e the set of nodes wth arcs ncomng to v he number of neghbors of a node s equal to ts n-degree, e N a If subset U V s nonempty, ts neghbors are N N { v : ( v, v ) E for somev U} U v U [Saber and Murray 004] Some defne NU N \ U [Moreau 005] vu Defne the degree matrx as a dagonal matrx D= [d ] wth dagonal element d N equal to the number of neghbors of node v, e ts n-degree Defne the graph Laplacan matrx as L= D-A Note that the row sums of L are all zero For an undrected graph, L 0 [Saber and Murray 004]

2 A drected path s a sequence of nodes v0, v,, vr such that ( v, v ) E, {0,,, r} Node v s sad to be connected to node v f there s a drected path from v to v Graph G s sad to be strongly connected f v, v are connected for all dstnct nodes v, v G A weak path s a sequence of nodes v 0, v,, vr such that ether v, v E or ( v, v) E, {0,,, r} Nodes v, v are sad to be weakly connected f there s a weak path from v to v Graph G s sad to be weakly connected f v, v are weakly connected for all dstnct nodes v, v G A graph n whch there are dsont sets of vertces that are not strongly connected s termed dsconnected he communcaton cost C(G) of dgraph G=(V,E) s defned as the total number of drected edges C E he graph algebrac connectvty s defned as ( L) s the Fedler egenvalue of L Let U be a nonempty subset of V hen U s sad to be closed f (, vu) E, uu, vv U, e set U has no ncomng arcs, e N U U (or N U by Moreau 005) If there s a drected path from v to v, then, node v s sad to be connected to node v and node v s sad to be reachable from node v A node that s reachable from all nodes n the graph s sad to be globally reachable If there s a node from whch all other nodes are reachable, t s called a base node A base node s connected to all nodes n the graph Queston?- Is observablty defned? If there s a drected path from v to v, then, node v s sad to be connected to node v and node v s sad to be observable from node v A node that s observable from all nodes n the graph s sad to be globally observable e, a base node s globally observable A drected tree s a dgraph where every node except one, called the root, has n-valence equal to one, e exactly one parent A spannng tree of a dgraph s a drected tree formed by graph edges that connects all the nodes of the graph A graph s sad to have a spannng tree f a subset of the edges forms a spannng tree Lemma Let G be an undrected graph hen G s strongly connected f and only f t s weakly connected Lemma Let G be a dgraph that s balanced hen G s strongly connected f and only f t s weakly connected Lemma [check t] An undrected graph s strongly connected ff t has a spannng tree Lemma A strongly connected dgraph has a spannng tree

3 Lemma A graph havng a spannng tree has a base node, eg the root node Lemma [Ren, Beard, Atkns 005] A graph has a spannng tree ff t has a node connected to all other nodes, e a base node Lemma [Ln, Francs, Maggore 005] A dgraph wth V has no base node f and only f t has at least two dsont closed subsets NOE- Defne neghborhood backwards, e N { v : ( v, v) E}, e the set of nodes wth arcs outgong from v he number of neghbors of a node s equal to ts out-degree If subset U V s nonempty, ts neghbors are NU { v : ( v, v) E for somev U} hen U s sad to be closed f ( uv, ) E, uu, vv U, e set U has no outgong arcs, e NU U Defne the adacency or connectvty matrx A [ a ] wth a f ( v, v ) E and a =0 otherwse Defne the degree matrx as a dagonal matrx D= [d ] wth dagonal element d N equal to the number of neghbors of node v, e ts out-degree Defne the graph Laplacan matrx as L= D-A hen we have: Lemma [Ln, Francs, Maggore 005] A dgraph wth V has no globally reachable node f and only f t has at least two dsont closed subsets Lemma A dgraph has a globally reachable node f and only f 0 s a smple egenvalue of the Laplacan L Weghted Graphs A weghted graph s a trple G= (V,E,W) wth nodes V, edges E, and a map W : E R 0 from E to the postve real numbers R 0 assocatng wth each edge v, v E a strctly postve weght w he map W can be represented as a weghted adacency matrx W wth elements w For a weghted graph, defne the degree matrx as a dagonal matrx D= [d ] wth dagonal element d w, the weghted n-degree Defne the Laplacan matrx of a weghted graph as L= D-W Defne the Perron matrx P I L [Saber and Murray 004] Let d max max d be the largest n-degree hen P s a nonnegatve stochastc matrx for all (0,/ d ) max 3

4 Matrces Gershgorn (sp Ger s gorn) heorem Let A [ a ] M and defne the deleted absolute row sums R ( A) a,n hen all egenvalues of A are located n the unon of the n ' n n ' Gershgorn dscs n zc: za R( A) G( A), wth G(A) known as the Gershgorn Regon Furthermore, f a unon of k of these dscs forms a connected regon that s dsont from all other dscs, there are precsely k egenvalues of A n ths regon Can also apply ths to A and get Gershgorn dscs n terms of column sums domnant f A matrx s sad to be dagonally domnant f a ' R( A), n n a R( A) a,n ' and strctly Lemma L s dagonally domnant (wth row sums equal to zero), dagonal elements d > 0, and d 0, herefore, all ts egenvalues are located n N dscs around d, all of whch are contaned n the open rght-half plane Lemma [Saber and Murray 004] Denote the maxmum n-degree of a graph G by d max DA ( ) zc: zd d hen all egenvalues of the Laplacan L are located n the dsk max max A nonnegatve matrx wth all ts row sums equal to s a (row) stochastc matrx A n stochastc matrx P s ndecomposable and aperodc (SIA) f lm P y wth a column n vector wth all entres equal to one and y a vector See Luenberger s book [979], sectons 7, 73 about stochastc matrces Lemma [Saber and Murray 004] he row sums of L are all zero herefore L has a zero egenvalue correspondng to the rght egenvector w r = herefore, rank( L) N Lemma [Chopra and Spong 005] he algebrac multplcty of =0 s equal to the number of connected components n the graph 4

5 Lemma [Saber and Murray 004] An undrected graph G s connected f and only f rank( L) N, e 0 s a smple egenvalue of L Lemma [Saber and Murray 004] Let dgraph G be strongly connected hen rank( L) N, e 0 s a smple egenvalue of L (Does NO go the other way) Lemma [Fax and Murray 004] If the graph s undrected, then all egenvalues are real Lemma [Saber and Murray 004] A dgraph G s balanced ff s a left egenvector of =0, e 0 L Lemma [Ln, Francs, Maggore 005], [Moreau 005] A dgraph has a node connected to all other nodes (e a base node) f and only f 0 s a smple egenvalue of the Laplacan L xlx Lemma For an undrected graph mn ( L), where x0 x ( L) s the Fedler egenvalue of x0 L, or the graph algebrac connectvty (Specal case of Courant-Fscher heorem) A matrx s nonnegatve f all ts elements are greater than or equal to zero A nonnegatve matrx G s rreducble f there does not exst a permutaton matrx P such that PGP s block trangular, otherwse called reducble [Fax and Murray 004] A dgraph s rreducble ff strongly connected [Ren, Beard, Atkns 005] Perron-Frobenus heorem [Fax and Murray 004] Let A be a nonnegatve, rreducble matrx hen: ) spectral radus ( A) 0 ) ( A) s a smple egenvalue of A, and any egenvalue of A of the same modulus s also smple 3) A has a postve egenvector correspondng to ( A) Lemma (cf Perron-Frobenus heorem for nonnegatve matrces) Let G be a connected undrected graph Let the rght and left egenvectors of =0 be Lw 0, w L 0, w w hen R lm exp( Lt) w w t r l r l l r Lemma [Saber and Murray 004] Let G be a strongly connected dgraph Let the rght and left egenvectors of =0 be Lw 0, w L 0, w w hen R lm exp( Lt) w w r l l r t r l 5

6 Lemma [Ren and Beard 005] If A B, wth 0 and the graph of B has a spannng tree, then the graph of A has a spannng tree Lemma [Ren and Beard 005] Gven a matrx M= [m ] (eg the negatve Laplacan) wth m 0, m 0,, and all row sums equal to zero, then M has at least one zero egenvalue and all of the nonzero egenvalues are n the open left-half plane Furthermore, M has exactly one zero egenvalue f and only f ts dgraph has a spannng tree Lemma [Ren and Beard 005] If a nonnegatve matrx A=[a ] has all row sums equal to the same postve constant 0, then s an egenvalue of A wth assocated egenvector of and the spectral radus of A s ( A) In addton, the egenvalue has algebrac multplcty equal to one f and only f the assocated dgraph has a spannng tree Moreover, f the assocated graph has a spannng tree and a > 0, then s the unque egenvalue of maxmum modulus Note- row sum of A equals the n-degree equals the number of neghbors Lemma [Ren and Beard 005] A stochastc matrx has algebrac multplcty equal to one for ts egenvalue = ff the assocated graph has a spannng tree Furthermore, a stochastc matrx wth postve dagonal elements has the property that for every egenvalue not equal to one Lemma [Ren and Beard 005] Let A=[a ] be a stochastc matrx If A has an egenvalue = wth algebrac multplcty equal to one, and all other egenvalues satsfy, then A s n SIA lm P y n Moreover y satsfes A y =, and y It follows that y s nonnegatve y, e y s a left egenvector correspondng to Incdence Matrx [Chopra and Spong 005 from Jadbabae, Motee, and Barahona 004] Let the number of edges n a graph by e hen the graph can also be represented by an N e ncdence matrx B [ b ] wth b f edge s ncomng wrt, b f edge s outgong wrt and b =0 otherwse he Laplacan s L BB Assocate to each edge a postve weght w and defne the weght matrx W= dag{w } hen the weghted Laplacan L BWB has the same propertes as the Laplacan Networks and Dynamc Graphs A network s a set of nodes that collaborates to acheve what each cannot acheve alone A network, or algebrac graph, s G x =(G,x) wth N nodes, wth vector x [ x x ] R N and x R a value or state assocated to node v of graph G N 6

7 Defne the Laplacan potental of an undrected graph G as VG w( x x) x Lx wth L the Laplacan, A dynamc graph, or dynamc network, s a network whose state evolves accordng to some dynamcs x f ( x, u ) Lemma [Saber and Murray 004] For an undrected graph, L= L and n fact L 0 (Proof- Gershgorn) 004] For a drected graph, L s nonsymmetrc and may even be ndefnte [Saber and Murray Let G= (V,E,W) be a weghted dgraph Defne the reversed graph G ( V, E, W ) wth the set of reverse edges E r defned as ( v, v ) E f ( v, v ) E, e reverse all the edge arrows r Defne the mrror graph as Gm ( V, Em, Wm) wth E m E E r and symmetrc weghted adacency matrx W m W W Snce W m s symmetrc, G m s an undrected graph r r L L Lemma [Saber and Murray 004] he symmetrc part Ls of the Laplacan L for dgraph G s a vald Laplacan for mrror graph G m ff G s balanced hen, Ls 0 Control graphs [Saber and Murray 004] Gven a graph G=(V,E), we nterpret v, v E to mean that node v can obtan nformaton from node v for feedback control purposes Let G x =(G,x) be a dynamc graph wth N nodes and wth x f( x, u) he control be gven by u k ( x, x,, x ) s sad to be a protocol wth topology G f m v v N, {, m} he protocol s called dstrbuted f m N, he communcaton cost C(G) of dgraph G=(V,E) s defned as the total number of drected edges C E For a weghted graph one has C sgn( w ) N, A network desgn problem- for fxed C, select the best weghts w gven some performance measure 7

8 Agreement Nodes v, v n a network agree f x =x Synchronzaton Problem Nodes are sad to synchronze f lm x ( t) x ( t) 0,, t Consensus Problem Nodes have reached consensus f x x,, Usually refers to constant steady-state value Asymptotc Consensus Problem Nodes acheve consensus asymptotcally f, for all ntal condtons x (0), lm x ( t) x ( t) 0,, t N -Consensus problem Let ( x, x,, xn ): R R he -Consensus problem s to fnd a way to compute the global quantty ( x(0)) usng a dstrbuted (local) protocol Asymptotc -Consensus A protocol asymptotcally solves the -Consensus problem f there * exsts an asymptotcally stable equlbrum x* of the closed-loop system satsfyng x ( x(0)) for all Average consensus problem Fnd a dstrbuted protocol to compute ( x(0)) Ave( x(0)) x (0) N, N Also, max-consensus problem, mn-consensus, etc Lnear ntegrator dynamcs and lnear couplng Let G be a dynamc graph wth dynamcs x u For an undrected graph wth Laplacan potental VG x Lx a gradent-based control law s gven by u VG( x) w( x x) hs s equvalent to u w( x x) and so s a protocol he closed-loop system s N x wx wx dx wx so that N N N x ( DW) xlx wth L the graph Laplacan Note also that u= -Lx For an undrected graph, W and hence L s symmetrc, and n fact L 0 For a dynamc graph wth dscrete-tme state x ( k) x ( k) u ( k), the gradent-based control gves the closed-loop system x( k) Px( k) wth P the Perron matrx 8

9 Lemma [Saber and Murray 004] Let G be a dgraph hen G s balanced ff w ( x N x ) 0 hat s, u 0, xr wth u w( x x) the gradent-based, control (Proof- Balanced ff 0 L Lx u u N Lemma [Saber and Murray 004] Let G be a strongly connected dgraph Let the left egenvector of L for =0 be w [ ] hen x Lx, wth L the Laplacan, solves the consensus problem gven by ( x(0)) l x(0) Lemma [Saber and Murray 004] A strongly connected dgraph G solves the average consensus problem ff G s balanced (Proof- then w l = for = 0) Lemma [cf Saber and Murray 004] Consder a strongly connected dynamc graph G wth ntegrator dynamcs and control u k w ( x x ), wth k a control gan Let the left N egenvector of L for =0 be wl [ ] hen the control solves the consensus problem gven by x(0) ( x(0)) wth weghtng / k hus, a state usng smaller control gans wll end up more heavly weghted n the consensus value Lemma [Ren and Beard 005] (rough statement)- he gradent-based control law acheves consensus asymptotcally ff the graph has a spannng tree Changng opology he graph connectvty or topology may change over tme Let G { G, G,, G M } wth M fnte be the set of all dgraphs defned for a gven set of nodes V Let the Laplacan for graph G be denoted L he unon of any group of dgraphs contaned n G s a dgraph wth nodes V and edge set gven by the unon of the edge sets of that group A group of graphs n s sad to be ontly (strongly) connected f ther unon s (strongly) connected [Jadbabae, Ln, Morse 003] Consder a sequence of dgraphs (V,E(t)) wth t N A node v k s connected to a dstnct node vl V \{ vk} across an nterval I N f v k s connected to v l n the dgraph V, E( t) ti [Moreau 005] 9

10 Wth swtchng, the gradent-based control s ut () Ltxt () () and the closed-loop system s x () t L() t x() t wth L the graph Laplacan More specfcally ut () Lt ( k) xt (), t [ tk, tk ) and x () t L( tk) x(), t t[ tk, tk ) wth t k the swtchng tmes Lemma [Saber and Murray 004] Let the swtched control law be ut ( ) Lt ( k ) xt ( ) where the correspondng graphs Gt ( k ) are each strongly connected and balanced hen the closed-loop system converges asymptotcally to the average consensus soluton x Moreover, V x x L L s a common Lyapunov functon (Proof- V ( xx) ( xx) ( Ls ) xx ) CHECK HIS- Lemma [anon] A swtched gradent-based control law ut ( ) Lt ( k ) xt ( ) acheves consensus asymptotcally f there exsts an nfnte sequence of bounded nonoverlappng tme ntervals wth the property that the unon of dgraphs over each nterval s strongly connected hen, f each unon s also balanced, average consensus s solved Lemma [Ren and Beard 005] (rough statement)- A swtched gradent-based control law ut () Lt ( k ) xt () acheves consensus asymptotcally f there exsts an nfnte sequence of bounded nonoverlappng tme ntervals wth the property that the unon of dgraphs over each nterval has a spannng tree Furthermore, f the unon of graphs after some tme does not have a spannng tree, consensus cannot be acheved asymptotcally Remark [Jadbabae, Ln, Morse 003] For swtchng topologes, there may not exst a common Lyapunov functon even f consensus s reached Vcsek Models From [Jadbabae, Ln, Morse 003] hey consder UNDIRECED graphs Let x( t) x( t) u( t) and ut () ( I Dt ()) Ltxt () (), wth the average headng error gven by et () Lt () xt () (e u(t) s the gradent-based control) hen the closed-loop system s the Vcsek [995] system x( t) x( t) x( t) n ( t), where n() t N() t Note that N () t x( t) ( I D()) t ( I A()) t x() t hs s a stochastc matrx Lemma [Jadbabae, Ln, Morse 003] Let every graph n the swtchng set be connected hen lm x( t) x, where x ss s a number dependng only on x(0) and the swtchng sgnal t ss 0

11 Lemma [Jadbabae, Ln, Morse 003] Let the swtched system have an nfnte sequence of contguous, nonempty, bounded tme ntervals wth the property that, across each tme nterval the graphs are ontly connected hen lm x( t) x t ss here may NO EXIS a common Lyapunov functon n such cases Jont Lyapunov functon s too strong a requrement Defne ut () Gt () Ltxt () () wth Gt ( ) gi and g>n a const hen closed-loop x( t) I L( t) x( t) system s g NOW HERE DOES EXIS a common Lyapunov Functon If swtched graphs are ontly connected over ntervals hey can handle the case of a leader- namely one node connected to all other nodes, but not connected from any node Moreau [005] consders DIRECED GRAPHS He studes the case that ncludes weghts, x( t) x( t) wx( t) n ( t), wth n( t) w the weghted neghbors N () t Note that x( t) ( I D( t)) ( I A( t)) x( t) hs s a stochastc matrx In general there does not exst a tme-nvarant quadratc Lyapunov functon for ths system [Moreau 005], [Jadbabae, Ln, Morse 003] Lemma [Moreau 005] Drected Graphs Let the nonzero weghts be bounded above and below If there s a 0 such that for all t 0 N there s a node connected to all other nodes across [ t0, t0 ], then the N values x (t) converge to a common value as t Lemma [Moreau 005] Undrected Graphs wth dfferent weghts w kl w lk Let the nonzero weghts be bounded above and below If for all t 0 N there s a node connected to all other nodes across [ t0, ), then the N values x (t) converge to a common value Communcaton me Delays Consdered n [Saber and Murray 004] for nonswtchng topology Nyqust encrclements of s Le

12 Control of Interconnected Lnear Systems wth Dynamcs [Fax and Murray 004] If the nodes have ther own dynamcs, they must be taken nto account along wth the control graph hen, dependng on the dynamcs, addng lnks can destablze the system An r-cycle s a path for whch v 0 =v r A graph wthout cycles s called acyclc A graph wth the set of all cycle lengths havng a common dvsor k> s called k-perodc, otherwse aperodc If graph G s aperodc, then the adacency matrx W s called prmtve [Fax and Murray m 004] W s prmtve ff W 0 for some m [Ren, Beard, Atkns 005] A matrx s nonnegatve f all ts elements are greater than or equal to zero A nonnegatve matrx G s rreducble f there does not exst a permutaton matrx P such that PGP s block trangular, otherwse called reducble Lemma [Fax and Murray 004] An rreducble graph s strongly connected Lemma [Fax and Murray 004] If a graph s aperodc (prmtve A matrx), all nonzero egenvalues le n the nteror of the Perron dsk, e there exsts unque e-value of max modulus If a graph s k-perodc, L has k evenly spaced egenvalues on the boundary of the Perron dsk Consder dentcal node (vehcle) dynamcs x Ax Bu, x u K( x x) and n the node n-degree (Actually they consder also controller n N n R wth feedback dynamcs) hen u D Lx Lx wth L I D A the normalzed Laplacan he closed- x I A I B LI (Kronecker product) loop system s ( )( ) N N n It s assumed that each vehcle can sense at least one other vehcle Now the nterconnectons are more crucal In fact, addng more lnks can destablze the system Lemma he formaton (overall system) s stable ff the local systems are all stable wth controls u K( x x), wth the egenvalues of the normalzed Laplacan L n N he zero egenvalue of L corresponds to the unobservablty of the absolute moton of the formaton he relatve formaton dynamcs s sad to be stable f the ndvdual systems n the Lamme are stable for all other than the zero egenvalue

13 Lemma Let the graph be strongly connected, or have only one leader (e one base node) hen the only equlbrum pont s the desred relatve poston of the vehcles Lemma If the systems are SISO, hen the formaton s relatvely stable ff the net encrclement of by the Nyqust plot of P(s) s zero for all nonzero I, wth P(s) the vehcle transfer functons Addng lnks can destablze the formaton f the new lnks make the graph more nearly k- s perodc he example presented s the dynamcs wth delay x e u Passvty-Based Control of Swtched Nonlnear Systems [Spong work] If the node dynamcs are passve, the good propertes are reganed and the stablty only depends on the control graph Consder a dynamc graph wth node dynamcs x f( x) g( x) u, y h( x) whch are passve he outputs synchronze f lm y( t) y( t) 0,, t Consder control u K( y y) N Lemma Let control graph be connected and balanced (e strongly connected) hen the system s globally stable and the outputs synchronze Consder swtchng topology u K()( t y y ) wth the dwell tme constrant, N () t swtchng tmes are greater than Lemma Let each control graph be balanced Let there exst an nfnte sequence of bounded, non-overlappng tme ntervals across whch the graphs are ontly connected hen the system s globally stable and the outputs synchronze Consder constant bounded tme delays he outputs synchronze f lm y( t) y( t) 0,,, where s the sum of tme delays along the path from v to v t Consder tme-delayed control u() t K( y( t) y) N Lemma Let control graph be connected and balanced (e strongly connected) Let there be a unque path between any two dstnct nodes hen the system s globally stable and the outputs synchronze 3

14 Consder the nonlnear couplng control u ( y y) wth () a nonlnear functon that s contnuous, locally Lpschtz, and antsymmetrc Lemma Let control graph be undrected and connected hen the system s globally stable and the traectores converge to the largest set where the Laypunov dervatve s equal to zero Can steer the dfference of the outputs to a desred set by selectng () [cf Kevn Passno] N Also consdered the case of nonlnear couplng wth tme delay, usng the scatterng transformaton 4