Multi-agent Consensus and Algebraic Graph Theory

Transcription

1 F.L. Lews, AI Moncref-O Donnell Endowed Char Head, Controls & Sensors Group UTA Research Insttute (UTARI) The Unversty of Texas at Arlngton Mult-agent Consensus and Algebrac Graph Theory Supported by SF, ARO, AFOSR

2 Motons of bologcal groups Fsh school Brds flock Locusts swarm Frefles synchronze

3 Graphs and Dynamc Graphs

4 Communcaton Graph Algebrac Graph Theory G= (V,E) nodes 2 a Adjacency matrx A [ a j ] a 0 f ( v, v ) E j j f j A d a j j Row sum= n-degree In-neghbors of node d o a j j Col sum= out-degree o Out-neghbors of node

5 Strongly connected f for all nodes and j there s a path from to j. Dameter= length of longest path between two nodes Volume = sum of n-degrees Vol d Tree- every node has n-degree= Leader or root node Followers Spannng tree Root node

6 Exsts a spannng tree ff there s a node havng a path to all other nodes Graph strongly connected mples exsts a spannng tree Strongly connected mples Every ode s a root node Quas-strongly connected f for all nodes and j there exsts a node k wth a path to and a path to j Exsts a spannng tree ff quas-strongly connected k j

7 Dynamc Graph- the Graphcal Structure of Control Frst-order Consensus of Mult-agent systems Each node has an assocated state Standard local votng protocol u aj( xj x) Dynamcs x j u x a a x d x a a j j j j j Global Vectors x u u u x a ( x x ) j j j d D d u x x A [ a j ] d 0 a2 a u x x a2 0 u Dx Ax u x x d a 0 j

8 Dynamc Graph- the Graphcal Structure of Control Frst-order Consensus of Mult-agent systems d 0 a2 a u x x a2 0 u Dx Ax u x x d a 0 u DxAx( DA) xlx L=D-A = graph Laplacan matrx x a ( x x ) j j j x Lx Global Closed-loop dynamcs If x s an n-vector then x ( LI ) x n

9 Closed-loop system wth local votng protocol x Lx Lt xt () e x(0) L= graph Laplacan matrx Global Closed-loop dynamcs depends on L d 0 a2 a a2 0 L D A d a 0, d a j j = row sum L has row sum of L L Lc For any constant c

10 Mult-agent Cooperatve Regulator Dynamcs x u Cooperatve Control Closed-loop system u a ( x x ) j j j Local eghborhood Consensus Error x a ( x x ) j j j e a ( x x ) j j j Mult-agent Cooperatve Regulator In the sngle-agent regulator problem, the state goes to zero. In the mult-agent cooperatve regulator problem, the states go to a nonzero consensus value

11 The Mult-agent Consensus Problem Cooperatve Control Agent dynamcs x u Fnd control u so that all agents reach the same steady-state value x x 0,, j j

12 Flockng Reynolds, Computer Graphcs 987 Reynolds Rules: Algnment : algn headngs a ( ) j j j Coheson : steer towards average poston of neghbors- towards c.g. Separaton : steer to mantan separaton from neghbors

13 Dstrbuted Adaptve Control for Mult Agent Systems

14 Consensus Control for Swarm Motons a ( ) j j j x y c V cos Vsn headng angle y tme x Convergence of headngs odes converge to consensus headng

15 Consensus Control for Formatons Leader Followers Formaton- a Tree network a ( ) x j j j y c V cos Vsn headng angle Headng Update usng Spannng Tree Trust Update Headng Consensus usng Equatons (2) and (22) leader y Leader 0 50 Headng Tme tme Convergence of headngs y x odes converge to headng of leader x

16 Egenstructure of Graphs 6

17 Closed-loop system wth local votng protocol x a ( x x ) j j j x Lx L= D-A= graph Laplacan matrx Lt xt () e x(0) Closed-loop dynamcs depends on L Egenvalues of L determne tme response x Lx At steady-state 0 s s L =D-A has row sum of 0 0 L c Lc Therefore at steady state and consensus s reached xs c c For any constant c

18 Egenstructure of Graph Laplacan Matrx State Space Transformaton In general the Jordan form matrx s Rght Egenvectors L MJM J block dag We consder the case of smple L whch has dagonal Jordan form Then So that and J 2 LM MJ L v v v v v v Lv 2 2 v Defne transformaton matrx as the modal matrx M v v v 2 2 or v ( I L) v 0 s sad to be a rght egenvector for the egenvalue

19 Left Egenvectors L MJM LM MJ So can wrte ether as before Or M L JM Defne nverse modal matrx as M w T T w2 w T T T Then w w T T w 2 2 w2 M L JM L w w T T and wl w T T or w T w ( I L) 0 s sad to be a left egenvector for the egenvalue

20 Connectvty for Undrected Graphs a j a j A [ a ] [ a ] A j T T L DA DA L j T Connected f for all nodes and j there s a path from to j. If there s a path from to j, there s a path from j to

21 Any undrected graph has Hence, all ts egenvalues are real and can be ordered as

22 Snce L has all row sums zero : where and c s any constant 0L c( I L) v f 0 and v c Is an egenvalue and v c so 0 s the rght e-vector

23 Theorem. Graph contans a spannng tree ff e-val at s smple. Then 2 0 Then L has one e-val at zero and all the rest n open rght-half plane Then -L has one e-val at zero and all the rest stable Graph strongly connected mples exsts a spannng tree 0 Theorem : L has rank -,.e 0 s non-repeated, f and only f graph G has a spannng tree.

24 Modal decomposton w L MJM v v v v w x Lx Lt x() t e x(0) T T 2 w2 T 2 j T w Modal decomposton j Lt t T x() t e x(0) ve w x(0)

25 Closed-loop system wth local votng protocol x Lx Modal decomposton j Lt t T xt () e x(0) ve wx(0) Closed-loop dynamcs depends on egenstructure of L d 0 a2 a a2 0 L D A d a 0 L has row sum of 0 Jordan ormal Form L MJM 0 L L or 0 L ( I L) v, d a j v = rght e-vectors w = left e-vectors j = row sum so 0 and v s the rght e-vector

26 Consensus: Fnal Consensus Value Closed-loop system wth local votng protocol x Lx Modal decomposton Let 0 be smple. Then at steady-state x (0) xt ve wx x (0) t T () (0) (0) w T wth Lt t T T t j j xt () e x(0) ve wx(0) wx(0) e v j j j x the normalzed left e-vector of 0 Therefore x () t x (0) for all nodes COSESUS j j j Consensus value depends on communcaton graph structure Importance of left e-vector of 0

27 Consensus Leaders Spannng tree Root node COSESUS x () t x (0) j j j for all nodes the normalzed left e-vector of 0 w T wth Theorem. 0 ff node s a Root node So the consensus value s a weghted average of the ntal condtons of all the Root nodes Corollary. Let the graph be a tree. Then all nodes converge to IC of the root node

28 Convergence Rate Closed-loop system wth local votng protocol x Lx Modal decomposton Let 0 be smple. Then for large t Lt t T T t j j xt () e x(0) ve wx(0) wx(0) e v j xt ve wx ve wx ve wx x 2t 2 ( ) T t t (0) T (0) T (0) (0) j j 2 determnes the rate of convergence and s called the FIEDLER e-value There s a bg push to fnd expressons for the left e-vector for 0 and the Fedler e-val 2

29 Fedler e-val depends on left e-vect for 0 Olshevsky and Tstskls T w For a certan type of graph mn a ( x x ) 2 2 xs j j j 2

30 Bounds for Fedler e-val: Convergence Rate Undrected Graphs 2 Dvol( G) Row sum = col sum, n-degree = out-degree, balanced D= dameter= length of longest path between 2 nodes vol( G) d Strogatz Small World etworks are faster 2, where s the mnmum n-degree In-degree = out-degree Drected Graphs- less s known Star network has largest Fedler e-val of any graph wth the same number of nodes and edges n a ( L) mnmax out degree, max n degree n OUT degree s mportant Book by Cha Wah Wu Work of Fan Chung

31 Left e-vector for 0 For a connected undrected graph T w d 2L Hovareshth & Baras where d s the out-degree= n-degree, the number of nodes, L the number of edges

32 Consensus value s x () t x (0) j j j = weghted average of ntal condtons of nodes Consensus value depends on communcaton graph structure A graph s balanced f n-degree=out-degree A [ a j ] d d o a j j a j Row sum= n-degree Column sum= out-degree Balanced means that row sum= column sum Then L has row sum=0 and column sum=0 wl means that left e-vector s also w T 0 Then Consensus value s x () t x (0) j = average of ntal condtons of nodes j Independent of graph structure

33 Undrected Graphs e j s an edge f e j s an edge, and e j = e j A A T A s symmetrc Row sum= column sum so that n-degree= out-degree L=D-A s symmetrc and postve semdefnte A symmetrc graph s balanced Connected undrected graph has average consensus value Then Consensus value s x () t x (0) j = average of ntal condtons of nodes j Independent of graph structure

34 Bdrectonal graph a 0a 0 j j Olshevsky and Tstskls But maybe a j a j Equal neghbor a j, a n n n umber of neghbors + (.e. nclude node tself) Then left evector for 0 T w n where Vol( G) n Vol( G) Consensus value s n x () t x (0) x (0) Vol( G)

35 Graph Laplacan Egenstructure Laplacan L DA Frst order consensus system x Lx Frst egenvalue s and v 0 c s the rght e-vector All other egenvalues have postve real parts Iff there exsts a spannng tree A suffcent condton for ths s strongly connected. Then the Fedler egenvalue has postve real part And s a measure of the convergence rate 2 What about the other egenvalues? Where are they?

36 Thm. Gerschgorn Crcle Crteron. All egenvalues of matrx E [ e ] R j n n are located wthn the followng unon of n dscs n zc: ze e j d 0 a2 a a2 0 L D A d a 0 j All e-vals are n the crcles Cd (, d) 2 0 d max 2d max L has row sum = 0 mples there s an egenvalue at s=0 0 2 E-vals of -L -2d max -dmax 0

37 Thm. Gerschgorn Crcle Crteron. All egenvalues of matrx E [ e ] R j n n are located wthn the followng unon of n dscs n zc: ze e j j ormalzed Laplacan D L D ( D A) I D A has dagonal entres = All e-vals are n the crcle L has row sum = 0 mples there s an egenvalue at s=0 0 2 E-vals of -L -2-0

38 4 5 E.G. f graph s k-cyclc (all crcular paths have a GCD of k) then e-vals of D L are unformly spaced around ths crcle e.g. k=4 0 2 Then consensus tme plot oscllates untl steady-state Captures the structure of socal networks and rumor passng

39 Graph Egenvalues for Dfferent Communcaton Topologes Drected Tree- Chan of command Drected Rng- Gossp network OSCILLATIOS 3 6 2

40 Graph Egenvalues for Dfferent Communcaton Topologes Drected graph- Better condtoned Undrected graph- More llcondtoned

41 Synchronzaton on Good Graphs Regular 2D mesh Chrs Ellott fast vdeo

42 Synchronzaton on Gossp Rngs Chrs Ellott werd vdeo 2 cycle

43 Control of a etwork wth a Leader ode

44 b Controlled Consensus: Cooperatve Tracker x u ode state Local votng protocol wth control node v 0 u a ( x x ) b( vx ) j j j Get rd of dependence on ntal condtons If control v s n the neghborhood of node u b aj x ajxj bv j j Control node s n some neghborhoods x ( LB) xbv b 0 B dag{ b } Ron Chen control node v Pnnng gans Strongly connected graph L Pnnng Control Theorem. Let at least one. Then L+B s nonsngular wth all e-vals postve and -(L+B) s asymptotcally stable b So ntal condtons of nodes n graph A go away. Consensus value depends only on v In fact, v s now the only spannng node

45 Controlled Consensus Orgnal network A = [ ; 0 0; ; 0 0 0]; Lamdas = 0 2 States wth dff. In. cond Tme Consensus tme approx 7.5 sec Average of ICS Controlled network L States wth dff. In. cond Lamdas = Leader s IC Tme Consensus tme approx 8 sec

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47 Open Research Topc Tme-varyng edge weghts a ( ) j j j j c j Is the control nput j???? An extenson of Adaptve Control Methods Lyapunov Desgn? S. Boyd showed that f some edge weghts are negatve, convergence speed s faster

48 Second Order Consensus Kevn Moore and We Ren Consensus works because the closed-loop system s Type I. x Lx has an ntegrator- smple e-val at 0. Let each node have an assocated state x u Second-order local votng protocol u a ( x x ) a ( x x ) 0 j j j j j j Closed-loop system x 0 I 0L L x PD cooperatve control Reaches consensus n both ff graph has a spannng tree and gans are chosen for stablty Has 2 ntegrators- Can follow a ramp consensus nput x, x

49 Second Order Controlled Consensus for Poston Offset Control Kevn Moore and We Ren node state x u v Second-order controlled protocol u a ( x x ) a ( x x ) b ( x x ) ( x x ) 0 j j j j j 0 0 j j where node 0 s a leader node. j s a desred separaton vector x 0 Good for formaton offset poston control j Leader Followers

50 Our revels now are ended. These our actors, As I foretold you, were all sprts, and Are melted nto ar, nto thn ar. The cloud-capped towers, the gorgeous palaces, The solemn temples, the great globe tself, Yea, all whch t nhert, shall dssolve, And, lke ths nsubstantal pageant faded, Leave not a rack behnd. We are such stuff as dreams are made on, and our lttle lfe s rounded wth a sleep. Prospero, n The Tempest, act 4, sc., l. 52-6, Shakespeare

51 Graph Matrx Theory 52

52 A 0 L DA wth row sum postve= d s an M matrx Zhhua Qu 2009 book M 0 0 Off-dagonal entres 0 Prncpal mnors nonnegatve sngular M matrx onsngular M matrx f all prncpal mnors postve L also has all row sums = 0 Do not confuse wth stochastc matrx E 0 s row stochastc f all row sums = E e Lt s row stochastc wth postve dagonal elements Let E 0 be row stochastc. Then A=I-E s an M matrx wth row sums zero. Dscrete-tme votng protocol gves stochastc c.l. matrx k k E ( I D) ( I A) I ( I D) L x Ex

53 Irreducblty Matrx E s reducble f t can be brought by row/column permutatons to the form * 0 * * Two matrces that are smlar usng permutaton matrces are sad to be cogredent. A graph G(A) s strongly connected ff ts adjacency matrx A s rreducble. A reducble matrx E can be brought by a permutaton matrx T to the lower block trangular (LBT) Frobenus canoncal form F 0 0 F2 F22 0 T F TET. Fp Fp2 Fpp where F s square and rreducble. (note- f F s a scalar, t s equal to 0.) F s sad to be lower trangularly complete f n every row there exsts at least one such that Fj 0 (.e. t has least one nonzero entry). F s sad to be lower trangularly postve f F 0, j F s lower trang. Complete ff the assocated graph has a spannng tree j j

54 n n Matrx E [ e ] R s dagonally domnant f, for all, e ej j It s strctly dagonally domnant f these nequaltes are all strct. E s strongly dagonally domnant f at least one of the nequaltes s strct [Serre 2000] E s rreducbly dagonally domnant f t s rreducble and at least one of the nequaltes s strct. Let E be a dagonally domnant M matrx (.e. nonpostve elements off the dagonal, nonnegatve elements on the dagonal). 0 Dagonal Domnance Then s an egenvalue of E ff all row sums are equal to 0. Moreover, let E be rreducble wth all row sums equal to zero. Then j 0 has multplcty of. Thm. Let E be strctly dagonally domnant or rreducbly dagonally domnant. Then E s nonsngular. If n addton, the dagonal elements of E are all postve real numbers, then Re ( E) 0, n SO. Let graph be rreducble. Then the Laplacan L has smple e-value at 0. Add a postve number to any dagonal entry of L to get L. Then L s nonsngular and L s stable.

55 Zhhua Qu Book 2009 Thm. Propertes of Irreducble M matrces. Let E=sI-A be an rreducble M matrx, that s, Interpret E as the Laplacan matrx L A 0 and s rreducble. Then,. E has rank n-. 2. there exsts a vector v>0 such that Ev=0. 3. Ex 0 mples Ex=0. 4. Each prncpal submatrx of order less that n s a nonsngular M matrx 5. ( D E) exsts and s nonnegatve for all nonnegatve dagonal matrces D wth at least one postve element. 6. Matrx E has Property c. 7. There exsts a postve dagonal matrx P such that PE T E P s postve semdefnte, That s, matrx E s pseudo-dagonally domnant, That s, matrx -E s Lyapunov stable. Used for Lyapunov Proofs

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57 Le Guo Soft control Do not change the local protocols of the nodes can only add addtonal neghbors to nfluence exstng nodes u u 2 u 3 External control nodes added How to pck Injecton nodes? Orgnal network to be controlled Extenson of vrtual leader approach

58 Add m addtonal control nodes B [ b j ] 0 where bj 0 s the weght from control node u j to exstng network node v. Augmented state { x :, m} { x, x,, x, u,, u } 2 m ew connectvty matrx a a2 a b b2 b m a b b A A B [ a ] R a a2 a b bm Row sum = m ( m) j d Laplacan da a2 a b b2 b m a b 2 2 b2 m D A B a b a2 d a bm L DA

59 Modfed local votng protocol x a ( x x ) j j j modfed dagonal matrx of n-degrees D dag{ d } wth d a the -th row sum of, whch ncludes the new control nodes. d d A j, m where Modfed Laplacan L D AR x LxBu ew closed-loop system Row sum of [ ] j SOFT COTROL, ncludes new control nodes n some nbhds L B D A B = -th row sum of control matrx B. s zero Le Guo But -th row sum of L D A has been ncreased by

60 L L2 L L L L L L L DAL Lj R 2 [ ] = -th row sum of control matrx B. Lemma Let L have row sum zero and be rreducble. At least one 0 Then L L2 L L L L L L L DAL [ Lj ] R 2 s rreducbly dagonally domnant and hence nonsngular. Lemma. Then L s asymp. stable.

61 L L2 L L L L L L L DAL Lj R 2 [ ] = -th row sum of control matrx B. RTP. Let L dag{ } L, 0 and L be rreducble Then relate egenvalues of L to those of L. They are shfted rght by some amount. Specal case. c, Then all e-vals are shfted rght by c { } Defne dag Conjecture. Let L be rreducble. Then L

62 My Theorem Defne dag{ } Let L L2 L L L [ ] L L2 L L D A L Lj R Then the determnant of k L 2 R s j j j k s j j 2 j s j j j s j j j L L L L s gven by 2 2 s Example- one dagonal entry postve L Mnor wth rows and columns struck out L L L L L L L 0 L L L L L L L L L L L L L L 0 L L L To ncrease the determnant as much as possble- Add to the node wth the largest OUT-degree,.e. largest column sum We call the out-degree of node ts nfluence or socal standng. - John Baras

63 Overall Dynamcs x L Bx u 0 0 u aug L B D 0 A B L has m e-vals at 0 Does not reach consensus unless matrx s rreducble. aug L rreducble ff m= Add control graph L aug L B D 0 A B G G 0 L 0 D 0 G x u L 0 B L G x u Control graph wth desred structure Orgnal network to be controlled wth fxed structure Induced Strogatz Small World Structure Reduced dameter= longest path length, larger Fedler e-val, so faster

64 Control outputs y 2 y u 2 u External control nodes added Orgnal network to be controlled Results. To make the controlled network as fast as possble, Tap nto the node wth the LARGEST out-degree (hghest socal standng) And take measured outputs from the nodes wth the SMALLEST out-degree - Zhhua Qu

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