F.L. Lewis, NAI. Cooperative Control of Multi Agent Systems on Communication Graphs

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2 F.L. Lews, NAI Moncref-O Donnell Char, UTA Research Insttute (UTARI) The Unversty of Texas at Arlngton, USA and Qan Ren Consultng Professor, State Key Laboratory of Synthetcal Automaton for Process Industres, Northeastern Unversty, Shenyang, Chna Cooperatve Control of Mult Agent Systems on Communcaton Graphs Supported by : NSF ONR ARO/TARDEC Chna Qan Ren Program, NEU Chna Educaton Mnstry Project 111 (No.B08015) Talk avalable onlne at

3 It s man s oblgaton to explore the most dffcult questons n the clearest possble way and use reason and ntellect to arrve at the best answer. Man s task s to understand patterns n nature and socety. The frst task s to understand the ndvdual problem, then to analyze symptoms and causes, and only then to desgn treatment and controls. Ibn Sna (Avcenna)

4 Patterns n Nature and Socety

5 1. Natural and bologcal structures Many of the beautful pctures are from a lecture by Ron Chen, Cty U. Hong Kong Pnnng Control of Graphs

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7 The nternet ecosystem Professonal Collaboraton network Barcelona ral network J.J. Fnngan, Complex scence for a complex world

8 Arlne Route Systems

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

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11 Herd and Panc Behavor Durng Emergency Buldng Egress Helbrng, Farkas, Vcsek, Nature 2000

12 1. Random Graphs Erdos and Reny N nodes Two nodes are connected wth probablty p ndependent of other edges Phase Transton m= number of edges There s a crtcal threshold m 0 (n) = N/2 above whch a large connected component appears gant clusters

13 Connectvty- degree dstrbuton s Posson Homogenety- all nodes have about the same degree J.J. Fnngan, Complex scence for a complex world P(k) Posson degree dstrbuton most nodes have about the same degree ave(k) depends on number of nodes ave k k

14 2. Small World Networks- Watts and Strogatz Start wth a regular lattce Wth probablty p, rewre an edge to a random node. Connectvty- degree dstrbuton s Posson Homogenety all nodes have about the same degree Small dameter (longest path length) Large clusterng coeff.-.e. neghbors are connected Watts & Strogatz, Nature 1998

15 Phase Transton Dameter and Clusterng Coefcent Clusterng coeffcent Nr of neghbors of = 4 Max nr of nbr nterconnectons= 4x3/2= 6 Actual nr of nbr nterconnects= 2 Clusterng coeff= 2/6= 1/3

16 3. Scale-Free Networks Barabas and Albert Start wth m 0 nodes Add one node at a tme: connect to m other nodes wth probablty P () d 1 ( d 1).e. wth hghest probablty to bggest nodes (rch get rcher) j j Nonhomogeneous- some nodes have large degree, most have small degree Scale-Free- degree has power law degree dstrbuton Pk ( ) 2m 3 k

17 4. Proxmty Graphs y 2d x Randomly select N ponts n the plane Draw an edge (,j) f dstance between nodes and j s wthn d When s the graph connected? for what values of (N,d) What s the degree dstrbuton? Moble Sensor Networks Project wth Sngapore A-Star

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19 The Power of Synchronzaton Coupled Oscllators Durnal Rhythm

20 Communcaton Graph Strongly connected f for all nodes and j there s a path from to j. Dameter= length of longest path between two nodes N Volume = sum of n-degrees Vol d 1 Tree- every node has n-degree=1 Leader or root node 1 Followers Spannng tree Root node

21 Communcaton Graph (V,E) N nodes 1 2 a Adjacency matrx A [ a j ] a 0 f ( v, v ) E j j f j N A d N a j1 j Row sum= n-degree N In-neghbors of node d o N a j1 j Col sum= out-degree N o Out-neghbors of node

22 Dynamc Graph- the Graphcal Structure of Control x u Each node has an assocated state j Standard local votng protocol u aj( xj x) jn x u x aj ajxj dx a1 a N jn jn x u1 u u N x d D 1 d N 1 N A [ a j ] u DxAx( DA) xlx L=D-A = graph Laplacan matrx x Lx Closed-loop dynamcs If x s an n-vector then x ( LI ) x n

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 stable 1 0 Graph strongly connected mples exsts a spannng tree

24 Consensus Value and Convergence Rate Closed-loop system wth local votng protocol x Lx Modal decomposton Let 1 0 be smple. Then for large t N N Lt t T T t j1 j1 xt () e x(0) ve wx(0) wx(0) e v N j1 xt ve wx ve wx ve wx x 2t 1 2 ( ) T t t (0) T (0) T (0) 1 (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 and the Fedler e-val Let graph have a spannng tree. Then all nodes reach consensus.

25 Consensus: Fnal Consensus Value Closed-loop system wth local votng protocol x Lx Modal decomposton Let 1 0 be smple. Then at steady-state 1 x1 (0) 1 N 1t T xt () ve 1 wx 1 (0) 1 N jxj(0) j1 1 x N (0) 1 T w1 1 N the normalzed left e-vector of 1 0 wth N N Lt t T T t j1 j1 xt () e x(0) ve wx(0) wx(0) e v Therefore N x () t x (0) for all nodes CONSENSUS j j j1 Consensus value depends on communcaton graph structure Importance of left e-vector of 1 0

26 Consensus value s N x () t x (0) j j j1 = 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 N a j1 N 1 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 w1 T 1 0 Then Consensus value s N 1 x() t xj(0) N = average of ntal condtons of nodes j 1 Independent of graph structure

27 Undrected Graphs e j s an edge f e j s an edge, and e j = e j A 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

28 Graph Egenvalues for Dfferent Communcaton Topologes Drected Tree- Chan of command Drected Rng- Gossp network OSCILLATIONS

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

30 Synchronzaton on Good Graphs Chrs Ellott fast vdeo

31 Synchronzaton on Gossp Rngs Chrs Ellott werd vdeo

32 Dynamc Graph- the Graphcal Structure of Control x u Each node has an assocated state j Standard local votng protocol u aj( xj x) jn x u x aj ajxj dx a1 a N jn jn x u1 u u N x d D 1 d N 1 N A [ a j ] u DxAx( DA) xlx L=D-A = graph Laplacan matrx x Lx Closed-loop dynamcs If x s an n-vector then x ( LI ) x n

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34 Flockng Reynolds, Computer Graphcs 1987 Reynolds Rules: Algnment : algn headngs a ( ) j j jn Coheson : steer towards average poston of neghbors- towards c.g. Separaton : steer to mantan separaton from neghbors

35 Dstrbuted Adaptve Control for Mult Agent Systems

36 500 BC 孙 Sun Tz bn fa

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

38 Chrs McMurrough Smulaton Seres for Swarm Moton

39 Trust-Based Control: Swarms/Formatons Malcous Node a ( ) j j j jn c Dvergence of trust Dvergence of headngs Node 5 njects negatve trust values Internal attack Malcous node puts out bad trust values.e. false nformaton c.f. vrus propagaton Causes Unstable Formaton

40 Trust-Based Control: Swarms/Formatons CUT OUT Malcous Node a ( ) headng angle j j j jn c Work by Sajal Das Other nodes agree that node 5 has negatve trust Convergence of trust Node 5 njects negatve trust values 6 5 Node If node 3 dstrusts node 5, Cut out node Node Convergence of headngs Restablzes Formaton

41 Consensus Control for Formatons Leader Followers Formaton- a Tree network a ( ) x j j jn y c V cos Vsn headng angle Headng Update usng Spannng Tree Trust Update Headng Consensus usng Equatons (21) and (22) leader y Leader 0 50 Headng Tme tme Convergence of headngs y x Nodes converge to headng of leader x

42 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 1 j j jn jn Closed-loop system x 0 I 0L 1L x x, x 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

43 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 1 j j jn jn where node 0 s a leader node. j s a desred separaton vector x 0 Leader Good for formaton offset poston control j Followers

44 b Controlled Consensus: Cooperatve Tracker x u Node state Local votng protocol wth control node v 0 u a ( x x ) b( vx ) j j jn Get rd of dependence on ntal condtons If control v s n the neghborhood of node u b aj x ajxj bv jn jn Control node s n some neghborhoods x ( LB) xb1v b 0 N B dag{ b } Ron Chen control node v Strongly connected graph L Theorem. Let at least one. Then L+B s nonsngular wth all e-vals postve and -(L+B) s asymptotcally stable 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 = [ ; ; ; ]; Lamda s = States wth dff. In. cond Tme Consensus tme approx 7.5 sec Average of ICS Controlled network 1 L States wth dff. In. cond Lamda s = Leader s IC Tme Consensus tme approx 8 sec

46 Balancng HVAC Ventlaton Systems Work wth SIMTech Sngapore Inst. Manufacturng Technology SIMTech 5 th floor temperature dstrbuton 74

47 Automated VAV control system SIMTech CWR CWS C 1 C 2 AHU VSD Fan Ar Flow Control Panel Dffuser outlets Control staton LEGENDS VAV box Room thermostat Ar dffuser Extra WSN temp. sensors

48 SIMTech Adjust Dampers for desred Temperature dstrbuton Temperature dynamcs x ( k1) x ( k) f ( x) u ( k) Unknown f (x) Control damper poston based on local votng protocol 1 u( k) ( k) aj( xj( k) x( k)) n 1 jn ( k) 1,,, Under certan condtons ths converges to steady-state desred temp. dstrbuton Open Research Topc - HVAC Flow and Pressure control

49 Herd and Panc Behavor Durng Emergency Buldng Egress Work wth Bar Polytechnc Unversty, Italy (Davd Naso) comes Aprl 24 Helbrng, Farkas, Vcsek, Nature 2000

50 Crowd Panc Behavor

51 Modelng Crowd Behavor n Stress Stuatons Helbrng, Farkas, Vcsek, Nature 2000 Consensus term Interacton pot. feld Wall pot. feld Repulsve force Radal compresson term Tangental frcton term

52 Smulated Crowd Panc Behavor Chrs McMurrough

53 Synchronzaton : Ron Chen Pnnng Control Connected undrected graphs Leader node dynamcs x f( x ) 0 0 x f( x ) u, y Cx x 0 (t) d a a j j j j a In-degree = out-degree d Dffusvty condton Pnnng Control nputs to some nodes x f( x ) c a C( x x ) cbc( x x ) j j 0 j ( ) ( ) 1 0 x f x c LB Cx cbc x L B DB A has e-vals b 1 f node s pnned B dag( b ) Results f( x) Node motons synchronze f s stable x cc Pn to the bggest node = hghest degree node= hghest socal standng- c.f. Baras Must have control gan c bg enough

54 Synchronzaton of Chaotc node dynamcs Ron Chen Pnnng control of largest node (for ncreasng couplng strengths) c=0 c=10 Chen s attractor node dynamcs c=20 c=15

55 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. 1, l , Shakespeare

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57 Swarm Stablty Analyss Gaz & Passno 2003, IEEE TAC x g( x x ) j j g( y) y( abexp y c 2 c.g. moton s nvarant All agents converge to c.g. form a hyperball of constant radus and ncreasng densty Locust Swarm

58 Results of Gaz and Passno 1. Center of gravty of swarm s statonary d x dt d 1 N dt N 1 x () t 0 g( y) y( abexp y c 2 2. All states converge n fnte tme to the regon and the fnal densty s 3 3a 4 b 3 N x bn ( 1) x an b a 3. Let nodes have fnte body sze of sphere wth radus Then all states converge to the regon and the fnal densty s x x N 1/3

59 Synchronzaton : Ron Chen Pnnng Control Connected undrected graphs Leader node dynamcs x f( x ) 0 0 x f( x ) u, y Cx x 0 (t) d a a j j j j a In-degree = out-degree d Dffusvty condton Pnnng Control nputs to some nodes x f( x ) c a C( x x ) cbc( x x ) j j 0 j ( ) ( ) 1 0 x f x c LB Cx cbc x L B DB A has e-vals b 1 f node s pnned B dag( b ) Results f( x) Node motons synchronze f s stable x cc Pn to the bggest node = hghest degree node= hghest socal standng- c.f. Baras Must have control gan c bg enough

60 Synchronzaton of Chaotc node dynamcs Ron Chen Pnnng control of largest node (for ncreasng couplng strengths) c=0 c=10 Chen s attractor node dynamcs c=20 c=15

61 Synchronzaton on Gossp Rngs wth Leader Chrs Ellott vdeo #3

62 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. 1, l , Shakespeare

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64 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 Nonsngular 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 =1 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 k1 k 1 1 E ( I D) ( I A) I ( I D) L x Ex

65 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 F21 F22 0 T F TET. Fp 1 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

66 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 1. 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, 1n 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.

67 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, 1. E has rank n 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 1 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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69 Le Guo Soft control Do not change the local protocols of the nodes can only add addtonal neghbors to nfluence exstng nodes u 1 u 2 u 3 External control nodes added How to pck Injecton nodes? Orgnal network to be controlled Extenson of vrtual leader approach

70 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 : 1, N m} { x, x,, x, u,, u } 1 2 N 1 m New connectvty matrx a11 a12 a1 N b11 b12 b1 m a b b A A B [ a ] R an1 an2 ann bn1 bnm Row sum = m N( Nm) j d Laplacan d1a11 a12 a b 1N 11 b12 b 1m a b b2 m D A B a b N1 an2 dn a NN N1 bnm L DA

71 Modfed local votng protocol x a ( x x ) j j jn 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 j1, Nm where Modfed Laplacan L D AR x LxBu N N New closed-loop system Row sum of [ ] j SOFT CONTROL, 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

72 1 L11 L12 L1N L L L L L NN L DAL Lj R N1 N2 N NN [ ] = -th row sum of control matrx B. Lemma Let L have row sum zero and be rreducble. At least one 0 Then 1 L11 L12 L1N L L L L L NN L DAL [ Lj ] R N1 N2 N NN s rreducbly dagonally domnant and hence nonsngular. Lemma. Then L s asymp. stable.

73 1 L11 L12 L1N L L L L L NN L DAL Lj R N1 N2 N NN [ ] = -th row sum of control matrx B. RTP 1. 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

74 My Theorem Defne dag{ } Let 1 L11 L12 L1N L L [ ] LN1 LN2 N LNN NN L D A L Lj R Then the determnant of k L 1 2 N N R s1 1j j j k s j 1 j 2 j s j j j s j j j L L L L s gven by s Example- one dagonal entry postve L Mnor wth rows and columns struck out L L L L L L L 0 L L 1 L L 1 L L 1 L L L L L L L L 0 L L L N N 11 1N N1 N NN N1 N NN N1 NN 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

75 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=1 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

76 Control outputs y 2 y1 u 2 u 1 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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