Invited by Karl Mathia

Transcription

1

2

3 Invted by Karl Matha

4 F.L. Lews Moncref-O Donnell Endowed Char Head, Controls & Sensors Group UTA Research Insttute (UTARI) The Unversty of Texas at Arlngton Cooperatve Control for Teams on Communcaton Networks Supported by NSF, ARO, AFOSR

5

6 Lao Tze The way that can be told s not the Constant Way The name that can be named s not the Constant Name For nameless s the true way Beyond the myrad experences of the world To experence wthout ntenton s to sense the world All experence s an arch wherethrough gleams that untravelled land whose margns fade forever as we move Dao ke dao fechang dao Mng ke mng fechang mng

7 Meng Tze He who exerts hs mnd to the utmost knows nature s pattern. The way of learnng s none other than fndng the lost mnd. Man s task s to understand patterns n nature and socety.

8 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)

9 Patterns n Nature and Socety

10 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

11 Dstrbuton of galaxes n the unverse

12

13

14

15 The Egyptan Twtter Network Arab users n Red, Englsh users n red

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

17 Arlne Route Systems

18

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

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

21 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

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

23 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

24 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

25 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

26 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?

27

28 The Power of Synchronzaton Coupled Oscllators Durnal Rhythm

29 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

30 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

31 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

32 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 Node 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

33 Closed-loop system wth local votng protocol x Lx Modal decomposton Closed-loop dynamcs depends on egenstructure of L N N Lt T t T t j1 j1 xt () e x(0) we vx(0) wx(0) e v d1 0 a12 a1n a21 0 L D A d a 0 L has row sum of 0 N N1 1 0 L L1 or 0 L1 ( 1I L) v1 1, d N a j1 v = rght e-vectors w = left e-vectors j = row sum so 1 0 and v1 1 s the rght e-vector

34 Thm. Gerschgorn Crcle Crteron. All egenvalues of matrx E [ e ] R j n n are located wthn the followng unon of n dscs n 1 N N1 zc: ze e j d1 0 a12 a1n a21 0 L D A d a 0 j All e-vals are n the crcles Cd (, d) 1 2 N 0 d max 2d max L has row sum = 0 mples there s an egenvalue at s= N E-vals of -L -2d max -dmax 0

35 Thm. Gerschgorn Crcle Crteron. All egenvalues of matrx E [ e ] R j n n are located wthn the followng unon of n dscs n 1 zc: ze e j j Normalzed Laplacan D L D ( D A) I D A has dagonal entres =1 All e-vals are n the crcle 1 2 N L has row sum = 0 mples there s an egenvalue at s= N E-vals of -L

36 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

37 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

38 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 x (0) j = average of ntal condtons of nodes N j 1 Independent of graph structure

39 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

40 Left e-vector for 1 0 For a connected undrected graph T w1 1 N d 1 2L N Hovareshth & Baras where d s the out-degree= n-degree, N the number of nodes, L the number of edges

41 Olshevsky and Tstskls Bdrectonal graph a 0a 0 j j Equal neghbor a j 1 1, a n n n N 1 Number of neghbors +1 (.e. nclude node tself) Then left evector for 1 0 T w1 1 N n where Vol( G) n Vol( G) Consensus value s n x () t x (0) x (0) N N 1 1Vol( G)

42 Run Two Consensus Algorthms at each node Olshevsky & Tstskls State 1 Get rd of left-egenvector dependence y, (0) 1 u, y, n u a ( y y ) j j jn n 1 N ( ) Vol( G) N N () y(0) 1 1Vol G n y t Learns global graph propertes State 2 x (0) z w, z(0), n n x (0) 1 z () t z (0) x (0) N N N 1 1Vol( G) n Vol( G) 1 Set x () t z ()/ t y () t w a ( z z ) j j jn then N 1 x () t x (0) Independent of graph structure N 1 Average consensus!

43 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.

44 Bounds for Fedler e-val: Convergence Rate Undrected Graphs 1 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 Networks are faster n 2, where s the mnmum n-degree n 1 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 a1 ( L) mnmax out degree, max n degree n 1 OUT degree s mportant Book by Cha Wah Wu Work of Fan R.K. Chung

45 Fedler e-val depends on left e-vect for 1 0 Olshevsky and Tstskls T w1 1 N For a certan type of graph 1 mn a ( x x ) xs N N 1 j1 j j 2

46 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= Then consensus tme plot oscllates untl steady-state Captures the structure of socal networks and rumor passng

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

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

49 Synchronzaton on Gossp Rngs Chrs Ellott vdeo

50

51 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

52 Dstrbuted Adaptve Control for Mult Agent Systems

53 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

54 Trust Propagaton and Consensus Network Securty Inspred by socal behavor n flocks, herds, teams Foundaton work by John Baras Defne j as the trust that node has for node j [ 1,1] j Dstrust no opnon complete trust 1 Defne trust vector of node as N 6 R N Standard local votng protocol Closed-loop trust dynamcs ( LI N ) u a ( ) j j jn Trust node has for node 3 N vector Dfference of opnon wth neghbors

55 Trust Propagaton & Consensus Nodes 1, 2, 4 ntally dstrust node 5 ntal trusts are negatve Other nodes agree that node 5 has negatve trust Convergence of trust

56 Trust-Based Control: Swarms/Formatons 1 2 Trust dynamcs a ( ) j j jn Moton dynamcs a ( ) j j j jn x y c V cos Vsn headng angle y tme Convergence of trust tme Convergence of headngs x Nodes converge to consensus headng

57 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

58 Trust-Based Control: Swarms/Formatons CUT OUT Malcous Node a ( ) headng angle j j j jn c 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

59 Consensus Leaders Spannng tree Root node CONSENSUS N x () t x (0) j j j1 for all nodes the normalzed left e-vector of 1 0 w1 1 N 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

60 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

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

62 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

63 Open Research Topc Tme-varyng edge weghts a ( ) j j j jn 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

64 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 a x a x bv j j j 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

65 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

66 Balancng HVAC Ventlaton Systems SIMTech SIMTech 5 th floor temperature dstrbuton 66

67 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

68 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

69

70 How about consensus control for Buldng Egress?

71 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

72 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 0 0 jn jn where node 0 s a leader node. j s a desred separaton vector x 0 Good for formaton offset poston control j Leader Followers

73 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

74 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

75 Synchronzaton Spong and Chopra x f ( x ) g( x ) u y h( x ) passve t T 0 V( x ) V( x (0)) u ( s) y ( s) ds Storage functon Synchronze f y () t y (), t all, j Local votng protocol wth OUTPUT FEEDBACK u K( y y ) j jn j Frefles synchronze Result - Let the communcaton graph be balanced. Then the agents synchronze. Crcadan rhythm

76 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

77 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

78 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

79

80 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

81 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

82 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.

83 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

84

85 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

86 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

87 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

88 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.

89 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

90 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

91 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

92 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

93

94

95 F.L. Lews Moncref-O Donnell Endowed Char Head, Controls & Sensors Group Krstan Hengster-Movrc Stablty vs. Optmalty of Mult Agent Cooperatve Control UTA Research Insttute (UTARI) The Unversty of Texas at Arlngton Supported by AFOSR, NSF, ARO

96 Invted by Andrea Gasparr

97 Invted by Davd Naso

98 Structure of Natural and Manmade Systems The Internet Local nature of Physcal Laws Peer-to-Peer Relatonshps n networked systems Clusters of galaxes J.J. Fnngan, Complex scence for a complex world ecosystem Professonal Collaboraton network Barcelona ral network

99 Motons of Bologcal Groups Local / Peer-to-Peer Relatonshps n soco-bologcal systems Fsh school Brds flock Locusts swarm Frefles synchronze

100 Stablty vs. Optmalty of Cooperatve Control Outlne A. Stable Desgn for Synchronzaton of Cooperatve Systems B. Optmal Desgn for Synchronzaton of Cooperatve Systems Issues: For cooperatve control on graphs - Local stablty of each agent s NOT the same as stable synchronzaton of the team Local optmalty of each agent s NOT the same a global optmalty of the team

101 Stablty vs. Optmalty of Cooperatve Control Outlne A. Stable Desgn for Synchronzaton of Cooperatve Systems A.1 Contnuous-tme desgn A.2 Dscrete-tme desgn B. Optmal Desgn for Synchronzaton of Cooperatve Systems Issues: For cooperatve control on graphs - Local stablty of each agent s NOT the same as stable synchronzaton of the team Local optmalty of each agent s NOT the same a global optmalty of the team

102 Bascs on Graphs, Cooperatve Control, and Consensus Communcaton Graph G=(V,E) N nodes State at node s x () t Consensus or Synchronzaton problem x () t x () t 0,, j j

103 Communcaton Graph 1 G=(V,E) 2 a 42 4 N nodes 3 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 Socal Standng

104 Standard Dstrbuted Control Protocol wth Lnear Integrator System x u Each node has an assocated state Standard local votng protocol u a ( x x ) j j jn 1 u x a a x d x a a j j j N jn jn u1 u u N x d D 1 d N u DxAx( DA) xlx L=D A = graph Laplacan matrx x x 1 N j x Lx Closed loop dynamcs L has row sum zero mples 1 0 and 1 st rght egenvector s v T T

105 Convergence Value and Rate Closed loop system wth local votng protocol x Lx Modal decomposton N N Lt T t T t j1 j1 xt () e x(0) we vx(0) wx(0) e v Let be smple. Then and for large t 1 0 N j1 xt ve wx ve wx ve wx x 2t 1 2 ( ) T t t (0) T (0) T (0) 1 (0) Left e vector L has an e val at zero w T determnes the consensus value n terms of the ntal condtons 2 determnes the rate of convergence Fedler e value s smple f the graph s strongly connected Depends on Communcaton Graph Topology No freedom to determne the consensus value We call ths the Cooperatve Regulator Problem j j

106

107 A. STABLE DESIGN FOR COOPERATIVE CONTROL ON GRAPHS We want Desgn Freedom that overcomes graph topology constrants Decouple Control Desgn from Graph Topology constrants Guaranteed synchronzaton for general Drected graphs Guaranteed stablty for contnuous-tme mult-agent systems on graphs - Hongwe Zhang, F.L. Lews, and Abhjt Das Optmal desgn for synchronzaton of cooperatve systems: state feedback, observer and output feedback IEEE Trans. Automatc Control, vol. 56, no. 8, pp , August 2011.

108 A.1 State Feedback Desgn for Cooperatve Systems on Graphs Cooperatve Regulator vs. Cooperatve Tracker problem N nodes wth dynamcs x Ax Bu n, x R, u R m x 0 (t) Control node or Command generator x Ax 0 0 (Exosystem) Synchronzaton Tracker desgn problem 0 x () t x (), t Local neghborhood trackng error e ( x x ) g ( x x ) j j 0 jn L=D-A Ron Chen- pnnng control Lhua Xe- local nbhd error j Overall error vector n 0 e LG I xx LG I n = Local quantty where T T T T nn nn e 1 2 N R, x0 Ix0 R, 1 I I R n nn n Consensus or synchronzaton error x x R 0 nn = Global quantty

109 Local Neghborhood Trackng Error e ( x x ) g ( x x ) j j 0 jn n 0 e LG I xx LG I n Local quantty Global quantty Local control objectves mply global performance

110 Coop. nbhd SVFB u ck ck ej( xj x) g( x0 x) jn Closed loop system x Ax Bu Ax cbk ej( x j x) g( x0 x) jn Overall state Dstrbuted form of control Overall c.l. dynamcs Global synch. error dynamcs T T T T nn 1 2 N, x x x x R ( N ) ( ) ( ) 0 x I A c LG BK xc LG BK x ( I A) c( LG) BK N Graph structure Control structure x Ix 0 u c ( LG) K Fax and Murray 2004 MIXES UP CONTROL DESIGN AND GRAPH STRUCTURE

111 u ck ck ej( xj x) g( x0 x) jn OPTIMAL Desgn at Each node Lews and Syrmos 1995 Lews and Zhang IEEE TAC 2011 DECOUPLES CONTROL DESIGN FROM COMMUNICATION GRAPH STRUCTURE LOCAL OPTIMAL DESIGN Guarantees Global Synchronzaton mnmzes 1 T T 2 0 J ( Q u Ru ) dt

112 OPTIMAL Desgn at Each node Optmal Control 3 rd ed Lews, Vrabe, Syrmos 2012 L, Duan, Chen- Fnsler s Lemma OPTIMAL Desgn at each node gves global guaranteed performance on any strongly connected communcaton graph Emre Tuna 2008 paper onlne

113 Example: Unbounded Regon of Consensus for Optmal Feedback Gans. A cbk E-vals of (L+G) Im{ } a. Bounded Consensus Regon for Arbtrarly Chosen Stablzng SVFB Gan A, B Re{ } K Example from [L, Duan, Chen 2009] Im{ } b. Unbounded Consensus Regon for Optmal SVFB Gan Q=I, R=1 K Re{ }

114 Results: Local Rccat Desgn yelds guaranteed stable synchronzaton Decouples Controls Desgn from Graph Propertes

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

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

117

118 Dstrbuted Systems

119

120 Stablty vs. Optmalty of Cooperatve Control Outlne A. Stable Desgn for Synchronzaton of Cooperatve Systems B. Optmal Desgn for Synchronzaton of Cooperatve Systems Issues: For cooperatve control on graphs - Local stablty of each agent s NOT the same as stable synchronzaton of the team Local optmalty of each agent s NOT the same a global optmalty of the team Have seen that LOCAL OPTIMAL DESIGN Guarantees Global Synchronzaton

121 A.1. Coop. nbhd SVFB u ck ck ej( xj x) g( x0 x) jn Closed loop system x Ax Bu Ax cbk ej( x j x) g( x0 x) jn Overall state Dstrbuted form of control Overall c.l. dynamcs Global synch. error dynamcs T T T T nn 1 2 N, x x x x R ( N ) ( ) ( ) 0 x I A c LG BK xc LG BK x ( I A) c( LG) BK N Graph structure Control structure x Ix 0 u c ( LG) K Fax and Murray 2004 MIXES UP CONTROL DESIGN AND GRAPH STRUCTURE

122 u c ( LG) K OPTIMAL Desgn at Each node Lews and Syrmos 1995 Lews and Syrmos 1995 DECOUPLES CONTROL DESIGN FROM COMMUNICATION GRAPH STRUCTURE LOCAL OPTIMAL DESIGN Guarantees Global Synchronzaton mnmzes 1 T T 2 0 J ( Q u Ru ) dt

123 B. OPTIMAL DESIGN FOR COOPERATIVE CONTROL ON GRAPHS The method just shown guarantees synchronzaton on arbtrary graphs It s a LOCAL OPTIMAL DESIGN at each agent What about Global Optmalty of cooperatve control on graphs? Problem- the global optmal control s not dstrbuted The global optmal control s generally dstrbuted only on a complete graph We Ren Agent dynamcs Global dynamcs LQR ARE x Ax Bu n x ( IA) x( IB) u AxBu ( I A) ( IB) u A Bu ( T T J Q u Ru) dt T 1 T A P PA Q PBR B P x Ix 0 Control u 1 T R B P s dstrbuted only on a complete graph- We Ren BUT- a dstrbuted control must have the form ( ) u c L G K So Q and R must depend on the graph topology

124 Inverse Optmalty Krstan Hengster-Movrc LQR case- ARE T 1 T APPAQPBR BP Gven A, B, and the dstrbuted control form, fnd Q and R u c ( LG) K

125 n System x Ax Bu Leader x 0 Ax0 Global dsagreement error Local nbhd trackng error Dstrbuted Control Closed loop system x ( IA) x( IB) u ( I A) ( IB) u e ( x x ) g ( x x ) j j 0 jn u ck2 ck2 ej( xj x) g( x0 x) jn u c( LG) K e LG I x Ax Bu Ax cbk2 ej( xj x) g( x0 x) jn ( ) ( ) ( ) x I A c LG BK xc LG BK x N 2 n x Ix Global synch. error dynamcs ( I A) c( LG) BK N 2 Graph structure Control structure

126 Krstan Hengster-Movrc B.1 Optmal Cooperatve Tracker for Sngle-Integrator Dynamcs System Leader node x x u, x R x x x x T 1 N uu1 u N Local nbhd trackng error u e ( x x ) g ( x x ) j j 0 jn e ( LG) e 1 N T 0 0 T G dag{ g } Global dsagreement error control Closed loop System u x Ix 0 u ( LG) u ( LG) Graph structure Control structure No control structure here Focus on graph structure

127 Krstan Hengster-Movrc Q Use local nbhd trackng error In the cost functon! Condton on graph topology x u T 1 T A P PA Q PBR B P

128 B.2 Cooperatve Tracker for Identcal LTI Dynamcs System x Ax Bu Local nbhd trackng error Control Closed loop system x ( IA) x( IB) u n ( I A) ( IB) u u ck2 ck2 ej( xj x) g( x0 x) jn u c( LG) K2 x Ax Bu Ax cbk2 ej( xj x) g( x0 x) jn x ( I A) c( LG) BK xc ( LG) BK x N e ( x x ) g ( x x ) ( I A) c( LG) BK N j j 0 jn e LG I n leader Graph structure Control structure x x Ix 0 Ax 0 0 T e 1 N

129 Krstan Hengster-Movrc TWO CONDITIONS Q ( R( LG) ( Q K R K )) c (( ) ( ) ) T max T T mn LG R1 LG K2 R2 K2

130 Proof: ARE Select System ( I A) ( IB) u A Bu T 1 T A P PA Q PBR B P 1 ( ) T T I A PP( I A) QP( I B) R ( I B) P 0 PPP 1 2 R R R 1 2 Krstan Hengster-Movrc ARE Choose Q PA P PPAQ( PPB)( R R )( PB P) 0 T 1 1 T P( A P PA) QPR P( PBR B P) 0 T 1 1 T T T Q c LG K2 R1R2 LG K2 cr1 LG A P2 P2A (( ) ) ( )(( ) ) ( ) ( ) c ( LG) R ( LG) K R K cr ( LG) ( Q P BR B P ) 2 T T 1 T PR P PBR B P P ( Q PBR B P ) 1 1 T 1 T Q P BR B P P ( Q P BR B P ) 1 T 1 T ARE Q c ( LG) R ( LG) 2 T 1 1 P ( A P P A Q PBR B P ) T T 1 T ( Q PR P) ( PBR B P ) T condtons: AP PA Q PBR BP T T 1 P cr ( L G) 1 1 Control u R B PR ( I B) P ( R R )( I B )( P P) 1 T 1 T 1 1 T R P R B P c( LG) K 1 1 T Dstrbuted!!

131 Two Condtons for global optmal desgn on the graph 1. Condton on graph topology P cr ( LG) 1 1 T T For some P1 P1 0, R1 R Local agent control desgn condton Same as before-local optmal control T 1 T AP2 PA 2 Q2 PBR 2 2 BP2 0 T T T For some P2 P2 0, R2 R2 0, Q2 Q2 0 Always holds f (A,B) reachable Locally optmal desgn s also globally optmal on the graph f condton 1 holds

132 Condton on Graph Topology P cr ( L G) Undrected Graphs LG ( LG) T Equvalent to R ( LG) ( LG) T R Krstan Hengster-Movrc 1 1 P P 0, R R 0 T T The condton becomes a Commutatvty Requrement R ( L G) ( LG) R 1 1 Case 1. R1 For sngle-ntegrator dynamcs I J(, u) ( T ( LG) T ( LG) uudt T ) ( ee T uudt T ) Case 2. R ( LG) ( LG) R 1 1 Iff R,( L G) have the same egenvectors 1 Let L TT T Jordan form Select T RTT 0 For any 0 R depends on graph topology- ALL e-vectors dagonal

133 P cr ( L G) Equvalent to R 1 1 1( LG) ( LG) T R1 2. Detal Balanced Graphs e e j j j 1 N for T Then s a left egenvector for L for e-val= 0 N L DP, D dag{1/ } 0 wth a symmetrc graph Laplacan matrx 1 ( ) LG DP G D P D G DP P D 1 ( LG) R( LG) Detal balanced mples reversblty of an assocated Markov Process Detal balanced mples balanced P P P 0, R R 0 T T R depends on graph topology prncpal left e-vector

134 A new class of dgraphs P cr ( L G) Equvalent to R 1 1 1( LG) ( LG) T R1 3. Drected Graphs wth Smple Graph Laplacan L+G P P 0, R R 0 T T TL ( GT ) 1 Dagonal Jordan form TL GT T L G T 1 ( ) T T T T ( ) Select TTL T ( G) ( LG) T TT T T T R T T R 0 Denns Bernsten Matrx book 1 ( ) ( ) T T LG R LG R, RR 0 R depends on graph topology- ALL e-vectors

135

136

137 A.2 Dscrete Tme Optmal Desgn for Synchronzaton K. Hengster-Movrc, Keyou You, F.L. Lews, and Lhua Xe,, Synchronzaton of Dscrete-Tme Mult-agent Systems on Graphs Usng Rccat Desgn, Automatca, to appear. Dstrbuted systems x k Ax k Bu k 1 Command generator x k Ax k Local Nbhd Trackng Error e x x g x x 0 j j jn Local cooperatve SVFB weghted 1 1 u c d g K x Local closed loop dynamcs 1 k 1 Ax k c1 d g BK k Global dsagreement error dynamcs k A ki AcI DG 1 LGBK k 1 c N Weghted Graph Matrx Weghted graph egenvalues 1 I D G L G Decouple controls desgn from graph topology k, k 1, N k x k x ( k) 0

138 Synchronzaton error dynamcs k A ki AcI DG 1 LGBK k 1 c N Weghted Graph Matrx Weghted graph egenvalues 1 I D G L G k, k 1, N MIXES UP CONTROL DESIGN AND GRAPH STRUCTURE

139 Decouple Controls Desgn From Graph Topology Krstan Movrc Ctrl desgn r r0 Graph Props. 1 c0 Synchronzaton regon contans ths crcle Coverng crcle of graph egenvalues

140 Sngle Input case wth Real Graph Egenvalues r0 1/2 T T 1 T 1/2 condton r max ( Q A PB( B PB) B PAQ ) c 0 r If graph egenvalues are real c 1/2 0 max mn 0 max mn. For SI systems, for proper choce of Q Mahler measure r u 1 u A u log 2 mn max A ntrnsc entropy rate = mnmum data rate n a networked control system that enables stablzaton of an unstable system Guoxang Gu, Qu L, We Chen Balleul and others. / Egen rato = condton number of the communcaton graph Work on log quantzaton- Ela & Mtter, Lhua Xe

141 Sngle Input case wth Real Graph Egenvalues Mahler Measure M ( A) ( A) unstable Graph Condton Number ( LG) Lke to have mn max mn ( G) 1 Varshney large means fast convergence Synchronzaton guaranteed f Topologcal Entropy ha ( ) log( M( A)) 1 M( A) 1 Guoxang Gu, L. Maronovc, and F.L. Lews, Consensusablty of dscrete-tme dynamc mult-agent systems, IEEE Trans. Automatc Control, to appear, New defnton- Graph Channel Capacty 1 1 CL ( G) log 1 1

142 max mn ( A) 1 ( A) 1 Graph Condton Number ( G) max mn egenrato= mn max Lke to have ( G) 1 mn large means fast convergence L.R. Varshney, Dstrbuted nference wth costly wres

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

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

145 Sngle Input case wth Real Graph Egenvalues u ck ejxj xgx0 x 1 x k Ax k Bu k u ( A) u max mn A max mn jn Is equvalent to max mn ( A) 1 ( A) 1 Guoxang Gu & Lews IEEE TAC Add stable flter u cf( z) K e x x g x x jn 0 j j Fltered protocol gves synch. f select ( A) P 0 the stablzng soluton to mn 2 2 T 1 mn max mn T K ( I (1 ) B PB) B PA ( A) 1 ( A) T ( (1 ) T mn ), T 2 mn P A P I BB P A B PB Improvement 1 T( z) mnk( zi Amn BK) B Complementary senstvty F( z) (1 ) T( z)

146

147 1. Cooperatve State Feedback, Observers, Dualty, and Optmal Desgn for Synchronzaton Results: Dstrbuted dynamc regulator for synchronzaton of teams usng only output measurements Dualty structure theory extended to networked cooperatve feedback systems on graphs Optmal Desgn yelds synchronzaton on ANY strongly connected Communcaton d graph CONTROL Cooperatve system node dynamcs Dualty x Ax Bu DISTRIBUTED ESTIMATION & SENSOR FUSION Output measurements at each node y Cx State feedback wth local neghborhood trackng error u ck ck ej( xj x) g( x0 x) jn Pnnng control node Overall Cooperatve Team Dynamcs ( N ) ( ) ( ) 0 x I A c LG BK xc LG BK x Local nbhd. estmaton error o ( ) ( ˆ ej yj y g y0 y), y y y jn Use local coop. observer dynamcs at each node o xˆ Axˆ Bu cf Overall Team Observer/Sensor Fuson Dynamcs x ˆ ( I A) c( LG) FC xc ( LG) F y( I B) u ˆ N N Thm 1. Desgn of SVFB Gan for Coop. Trackng Stablty Use OPTIMAL feedback gan 0, T 1 T 1 T A P PA Q PBR B P K R B P Then synchronzaton s acheved for ANY strongly connected dgraph Thm 2. Desgn of Observer Gan for Coop. Estmaton Use OPTIMAL observer gan 0 AP PA Q PC R CP, F PC R T T 1 T 1 Then estmates converge for ANY strongly connected dgraph Unbounded Regon of Consensus for Optmal Gans f couplng gan s c 1 mn Re( ( L G))

148 B. Observer Desgn for Cooperatve Systems on Graphs N nodes wth dynamcs x Ax Bu, y Cx State and output estmates n xˆ () t R, yˆ () t Cxˆ () t R p State and output estmaton errors x x xˆ, y y yˆ Control node or Command generator dynamcs Cooperatve Observer desgn problem Local neghborhood estmaton error Overall estmaton error x Ax, 0 0 y0 Cx0 x () t 0, xˆ () t x (), t or e ( y y ) g ( y y ) o j j 0 jn Ren; Beard, Kngston 2008 n 0 n o e LG I yy LG I y LG C x j

149 Local node observers o xˆ Axˆ Bu cf Overall observer dynamcs xˆ ˆ Ax Bu cf ej( y j y ) g( y 0 y ) jn xˆ ˆ Ax Bu cfc ej( x j x ) g( x 0 x ) jn ˆ x ˆ ( I A) c( LG) FC xc ( LG) F y( I B) u N Gr Gr xˆ A xˆ F y( I B) u o o N N where T T T T nn 1 2 N xˆ x ˆ xˆ xˆ R Overall estmaton error dynamcs Gr x ( I A) c( LG) FC x A x N o L, Duan, Ron Chen 2009

150 OPTIMAL Desgn at Each node OPTIMAL Desgn at Each node gves guaranteed performance On any strongly connected communcaton d-graph topology c.f. Fnsler Lemma desgn n L, Duan, Ron Chen 2009 Emre Tuna 2008 paper onlne (wthout observer desgn)

151 C. Control/Observer Dualty on Graphs SVFB u ck ck e x x g x x j( j ) ( 0 ), x Ax Bu cfc ej x j x g x 0 x jn jn Must use local nbhd. Trackng error and local nbhd. Estmaton error or dualty does not work Converse, reverse, or transpose graph = reverse edge arrows Observer ˆ ˆ ( ) ( )

152 D. Dynamc Tracker for Synchronzaton of Cooperatve Systems Usng Output Feedback N nodes wth dynamcs y Cx x Ax Bu, Command generator dynamcs (exosystem) x Ax, 0 0 y0 Cx0 Observers at each node Estmated SVFB o xˆ Axˆ Bu cf xˆ ˆ Ax Bu cf ej( y j y ) g( y 0 y ) jn u ˆ ( ˆ ˆ ) ( ˆ ck ck ej xj x g x0 x) jn Closed loop systems x ( ˆ ˆ ) ( ˆ Ax cbk ej xj x g x0 x) jn Overall system/observer dynamcs 0 x ( I A) xc( LG) BKxˆ c ( LG) BK x N ( ˆ N ) ( ) ( ) cl ( G) BKxˆ c( L G) BKx 0 xˆ I A c LG FC xc LG F y Must use local nbhd. Trackng error and local nbhd. Estmaton error or t s not nce.

153 OPTIMAL Desgn at Each node gves guaranteed performance On any strongly connected communcaton d-graph topology

154 Three Regulator Desgns 1. Neghborhood Controller and Neghborhood Observer Nbhd Observers Nbhd Controls xˆ ˆ Ax Bu cf ej( y j y ) g( y 0 y ) jn u ˆ ( ˆ ˆ ) ( ˆ ck ck ej xj x g x0 x) jn j j 2. Neghborhood Controller and Local Observer Local Observers xˆ Axˆ Bu cf y. j Nbhd Controls u ˆ ( ˆ ˆ ) ( ˆ ck ck ej xj x g x0 x) jn j 3. Local Controller and Neghborhood Observer Nbhd Observers xˆ ˆ Ax Bu cf ej( y j y ) g( y 0 y ) jn j Local Controls u Kxˆ j

155

156 Motons of Bologcal Groups Local / Peer-to-Peer Relatonshps n soco-bologcal systems Fsh school Brds flock Locusts swarm Frefles synchronze

157

158

159