Prepared by F.L. Lews Updated: Saturday, February 06, 200 Dynamc Systems on Graphs Control Graphs and Consensus 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 nodes, wth vector x [ x x ] T R and x R a value or state assocated to node v of graph G. A dynamc graph, or dynamc network, s a network whose state evolves accordng to some dynamcs x f ( x, u ).. Control Graphs [Saber and Murray 2004] 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 nodes and wth x f( x, u). The control be gven by u k ( x, x,, x ) s sad to be a protocol wth topology G f v v, {, m} 2 m. The protocol s called dstrbuted f m,. The 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 )., A network desgn problem- for fxed C, select the best weghts w gven some performance measure. Agreement. odes v, v n a network agree f x =x. Synchronzaton Problem. odes are sad to synchronze f lm x ( t) x ( t) 0,,. t Consensus Problem. odes have reached consensus f x x,,. Usually refers to constant steady-state value. Asymptotc Consensus Problem. odes acheve consensus asymptotcally f, for all ntal
condtons x (0), lm x ( t) x ( t) 0,,. t -Consensus problem. Let ( x, x2,, x ): R R. The -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)., Also, max-consensus problem, mn-consensus, etc. 2 Frst Order Consensus 2. Contnuous-Tme Systems Let G be a dynamc graph wth dynamcs x u. Consder the local votng control protocol u a ( x x ). Ths s equvalent to u a ( x x ). The closed-loop system s x ax ax dx ax so that x ( DA) xlx wth L the graph Laplacan and D dag{ d } the dagonal n-degree matrx. ote also that u= -Lx. Lemma. [Saber and Murray 2004]. Let G be a strongly connected dgraph. Let the left egenvector of L for =0 be w [ ]. Then x Lx, wth L the Laplacan, solves the consensus problem gven by ( x(0)) l x(0). Lemma. [c.f. Saber and Murray 2004]. Consder a strongly connected dynamc graph G wth ntegrator dynamcs and control u k a ( x x ), wth k a control gan. Let the left egenvector of L for =0 be w [ ]. Then the control solves the consensus problem gven by l 2
( x(0)) x (0) wth weghtng / k. Thus, a state usng smaller control gans wll end up more heavly weghted n the consensus value. Lemma. [Ren and Beard 2005]. (rough statement)- The gradent-based control law acheves consensus asymptotcally ff the graph has a spannng tree. Lemma. [Saber and Murray 2004]. Let G be a dgraph. Then G s balanced ff a ( x x ) 0. That s, u 0, xr wth u a( x x) the gradent-based, T T T control. (Proof- Balanced ff 0 L Lx u u. Lemma. [Saber and Murray 2004]. A strongly connected dgraph G solves the average consensus problem ff G s balanced. (Proof- then w l = for = 0.) ormalzed votng protocol x u a( x x) d wth d the n-degree of node. Then x D ( DA) xd Lx uses the normalzed Laplacan. u D ( DA) x D Lx 2.2 Dscrete-Tme Systems: Vcsek Type Standard local votng protocol, normalzed x ( k) x ( k) u ( k) u( k) a( x( k) x( k)) d x( k) x( k) a( x( k) x( k)) d ( d ) x( k) a( x( k) x( k)) d x( k) ax( k) d 3
So the global dynamcs can be wrtten x( k) x( k) u( k) uk ( ) ( I D) L( ID) ( D A) x( k) x( k) ( I D) Lx( k) x( k) ( I D) ( I A) x( k) x( k) Fx( k) where the DT system matrx s F I ( I D) L( I D) ( I DL) ( I D) ( I A) F 0 s a stochastc matrx wth postve dagonal elements. 2.3 Dscrete-Tme Systems: Perron Type For a dynamc graph wth dscrete-tme state x( k) x( k) u( k), the local votng control protocol above gves the closed-loop system x ( k) x ( k) w x ( k) x ( k) or x( k) Px( k) wth P I L the Perron matrx, whch s a row stochastc matrx. Lemma. Let the graph be strongly connected so that L has one e-val at s=0 and the rest n the open-rght half s-plane. Then P has one e-val at z= and the rest strctly nsde the unt crcle f 2 Wth the largest e-val of L. Ths s guaranteed n terms of the maxmum n-degree f max d 2.4 Egenvalues Laplacan L has at least one e-val at 0 and the rest n the rght-half s-plane. It has exactly one e- val at 0 ff the graph has a spannng tree, and f the graph s strongly connected. Then L has all e-vals n the open left half s-plane except for one e-val at s=0. Let E have egenvalues at wth egenvectors v. Then:. E I has egenvalues at wth egenvectors v. 2. I E has egenvalues at wth egenvectors v. 4
0 2 ( E) ' 0 0 ' 2 ' 2 2 2 Egenvalue regon of stochastc matrx. Example: E D A Egenvalue regon of normalzed M matrx w/ row sum= and dag. entres =. Example: L I D A Let L be an M matrx wth row sum zero. Then: G I L s a Metzler matrx wth row sum of one. G s an M matrx wth row sum of. Laplacan L=D-A s an M matrx wth row sum zero. Then:. G I L I D A s a Metzler matrx wth row sum of one. G s an M matrx wth row sum of. 2. GI D LD A 0 s a row stochastc matrx wth dagonal elements equal to 0. 3. F I ( I D) L( I D) ( I DL) ( I D) ( I A) s a row stochastc matrx wth dagonal elements equal to. A matrx wth off-dagonal elements nonnegatve s called a Metzler matrx. * 0 Metzler 0 * Let Z denote the set of matrces whose off-dagonal elements are nonpostve. * 0 Z 0 * Metzler matrx = -(Z matrx) A Z matrx s called an (sngular) M matrx f all ts prncpal mnors are nonnegatve. It s a nonsngular M matrx f all the prncpal mnors are postve. 0 M 0 A matrx E 0 s row stochastc f all ts row sums equal to..e. f E M () A matrx E 0 s doubly stochastc f all ts row sums and column sums equal to. 5
The maxmum egenvalue of a stochastc matrx s. The DT system matrx F has at least one e-val at n the z-plane and all the rest nsde the unt crcle. It has exactly one e-val equal to ff the graph has a spannng tree, and f the graph s strongly connected. Then F has all e-vals strctly nsde the unt crcle except for one e-val at z=. 6