Changing Topology and Communication Delays

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1 Prepared by F.L. Lews Updated: Saturday, February 3, 00 Changng Topology and Communcaton Delays Changng Topology The 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. We tae the case where the graph topology swtches between graphs n ths set. The frst result taes the case where all the graphs G are connected. Lemma [Saber and Murray 004]. Let the swtched control law be ut ( ) Lt ( ) xt ( ) where the correspondng graphs Gt ( ) are each strongly connected and balanced. Then the closed-loop system converges asymptotcally to the average consensus soluton x. Moreover, V x x T T L L T s a common Lyapunov functon. (Proof- V ( xx) ( xx) ( Ls ) xx.) Normally, however, each graph G s not connected. Then, we must tal about how the graphs are related to each other over tme ntervals. The 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 jontly (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 s connected to a dstnct node vl V \{ v} acro an nterval I N f v s connected to v l n the dgraph V, E( t) ti [Moreau 005]. A row stochastc matrx M s stochastcally ndecomposable and aperodc (SIA) (or T ergodc) f lm M w wth w a vector. That s, the lmt has ran of one. Lemma. Let M be a non-negatve row stochastc matrx wth a smple e-val at z=. Then all T other e-vals are strctly nsde the unt crcle. Then M s SIA. Furthermore, lm M w where w s the normalzed left e-vector of,.e. wm T w T, w T. Fnally every element of w s nonnegatve.

2 Recall that M s a non-negatve row stochastc matrx wth a smple e-val at z= ff ts graph contans a spannng tree, and f the graph s strongly connected. The next case consders products of matrces, each one of whch may not be SIA. Lemma (Wolfowtz). Let { M, M,, M m } be a fnte set of SIA matrces wth the property that every fnte product of the M s also SIA. Then for each nfnte sequence of matrces M one has lm T M M M w Note that n ths theorem the set of matrces must be fnte. Lemma (Ren and Beard). Let { M, M,, M m } be a set of row stochastc matrces. If the unon of the drected graphs of these matrces contans a spannng tree, then the product M M M s SIA. m Dscrete-Tme Swtchng Models From [Jadbabae, Ln, Morse 003]. They consder UNDIRECTED 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.) Then the closed-loop system s the Vcse [995] system x( t) x( t) xj( t) n ( t) jn () t where n () t N () t. Then the global state dynamcs s x t I D t I A t x t F t x t ( ) ( ()) ( ()) () () () where each F(t) s a stochastc matrx. Lemma. The system acheves consensus f and only f lm FtFt ( ) ( ) F() F(0) w T for some vector w. Frst the easy case. t Lemma. [Jadbabae, Ln, Morse 003]. Let every graph n the swtchng set be connected. Then lm x( t) x, where x s a number dependng only on x(0) and the swtchng sgnal. t

3 Note t s not obvous n the swtched topology case how x depends on the vector w mentoned above. Now the case where every graph s not connected. Lemma. [Jadbabae, Ln, Morse 003]. Let the swtched system have an nfnte sequence of contguous, nonempty, bounded tme ntervals wth the property that, acro each tme nterval the graphs are jontly connected. Then lm x( t) x. t Lemma. [Ren and Beard]. Let the swtched system have an nfnte sequence of contguous, nonempty, bounded tme ntervals wth the property that, acro each tme nterval the graphs contan a spannng tree. Then lm x( t) x. Moreover, f the unon of graphs after some fnte t tme does not have a spannng tree, consensus s not acheved n the lmt. About Lyapunov Functons for swtched graphs: There may NOT EXIST 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. Then closed-loop x( t) I L( t) x( t). system s g NOW THERE DOES EXIST a common Lyapunov Functon. If swtched graphs are jontly connected over ntervals. Remar. [Jadbabae, Ln, Morse 003]. For swtchng topologes, there may not exst a common Lyapunov functon even f consensus s reached. Moreau [005] consders DIRECTED GRAPHS. He studes the case that ncludes weghts, x( t) x( t) wjxj( t) n ( t), wth n() t wj the weghted neghbors. jn () t j Note that x( t) ( I D( t)) ( I A( t)) x( t). Ths 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 T 0 such that for all t 0 N there s a node connected to all other nodes acro [ t0, t0 T], then the N values x (t) converge to a common value as t. 3

4 Lemma [Moreau 005]. Undrected Graphs wth dfferent weghts w l w l. Let the nonzero weghts be bounded above and below. If for all t 0 N there s a node connected to all other nodes acro [ t0, ), then the N values x (t) converge to a common value. Contnuous-Tme Swtchng Models Wth swtchng, the local votng protocol for contnuous-tme systems s ut () Ltxt () () and the closed-loop system s x () t L() t x() t wth L the graph Laplacan. More specfcally ut () Lt ( ) xt (), t [ t, t ) and x () t L( t) x(), t t[ t, t ) wth t the swtchng tmes. One way to prove consensus for swtched contnuous-tme systems s to loo for jont Lyapunov functons. However, such Lyapunov functons may not exst. Another way s to consder the matrx exponental. Lemma. Let L be the Laplacan of a strongly connected dgraph. Then L has row sum of zero Lt and exactly one e-val at s=0. Moreover, s row stochastc and has exactly one e-val at z=. e Lemma. Let the swtchngs occur at tmes t such that the graph topology s constant over fnte dwell tmes t t bounded below by a postve constant. Then consensus s acheved f L( t ) L( t ) L( t ) T 0 0 lm For some vector w. e e e w Lemma. [anonymous]. A swtched gradent-based control law ut ( ) Lt ( ) 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. Then, f each unon s also balanced, average consensus s solved. Lemma. [Ren and Beard 005]. The swtched CT protocol ut ( ) Lt ( ) xt ( ) acheves consensus asymptotcally f there exsts an nfnte sequence of contguous bounded 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.. Communcaton Tme Delays If there are tme delays along the communcatons channels, the contnuous-tme protocol gves x () t a ( x ( t ) x ()) t j j jn 4

5 It s shown by Moreau (004) that consensus s stll acheved. However, ths protocol does not preserve the average consensus. An alternatve s to also add a desgn delay to the node s own state so that x () t a ( x ( t ) x ( t )) j j jn Then the global state dynamcs s x Lx( t ) Ths protocol does preserve the average consensus. For undrected graphs, t reaches consensus ff 0 max ( L) s See [Saber and Murray 004]. The proof reles on checng the Nyqust encrclements of Le. Note that the convergence speed of consensus depends on the smallest nonzero e-val of L, namely the Fedler e-val ( L), whle the robustne to tme delays depends on the largest e- val of L ( L). max 5