Cooperative Control for Teams on Communication Networks

Transcription

1 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

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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

6 Dstrbuton of galaxes n the unverse

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

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

10 Arlne Route Systems

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

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

13 Types of Graphs

14 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

15 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

16 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

17 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

18 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

19 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?

20 Graphs and Dynamc Graphs

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 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

23 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

24 The Power of Synchronzaton Coupled Oscllators Durnal Rhythm

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

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

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28 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

29 Dstrbuted Adaptve Control for Mult Agent Systems

30 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

31 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

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

33 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

34 Modelng Crowd Behavor n Stress Stuatons

35 Balancng HVAC Ventlaton Systems SIMTech SIMTech 5 th floor temperature dstrbuton 35

36 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

37 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

38 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

39 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

40 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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42 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

43 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

44 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

45 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

46 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