f.l. lewis moncrief-o’donnell endowed chair head, … ctrl graph for...heading consensus using...

UTA Research Institute (UTARI)The University of Texas at Arlington

F.L. LewisMoncrief-O’Donnell Endowed Chair

Head, Controls & Sensors Group

Cooperative Control forTeams on Communication Networks

Supported by NSF, ARO, AFOSR

It is man’s obligation to explore the most difficult questions inthe clearest possible way and use reason and intellect to arriveat the best answer.

Man’s task is to understand patterns in nature and society.

The first task is to understand the individual problem, then toanalyze symptoms and causes, and only then to designtreatment and controls.

Ibn Sina 1002-1042(Avicenna)

Patterns in Nature and Society

Many of the beautiful pictures are from a lecture by Ron Chen, City U. Hong KongPinning Control of Graphs

1. Natural and biological structures

Distribution of galaxies in the universe

The Egyptian Twitter NetworkArab users in Red, English users in red

J.J. Finnigan, Complex science for a complex world

The internet

ecosystem ProfessionalCollaboration network

Barcelona rail network

Airline Route Systems

2. Motions of biological groups

Fishschool

Birdsflock

Locustsswarm

Firefliessynchronize

Herd and Panic Behavior During Emergency Building Egress

Helbring, Farkas, Vicsek, Nature 2000

Types of Graphs

1. Random Graphs – Erdos and Renyi

N nodesTwo nodes are connected with probability p independent of other edges

m= number of edges

There is a critical threshold m0(n) = N/2 above which a large connected component appears – giant clusters

Phase Transition

J.J. Finnigan, Complex science for a complex world

Poisson degree distributionmost nodes have about the same degreeave(k) depends on number of nodes

kave k

Connectivity- degree distribution is PoissonHomogeneity- all nodes have about the same degree

Watts & Strogatz, Nature 1998

2. Small World Networks- Watts and Strogatz

Start with a regular latticeWith probability p, rewire an edge to a random node.

Connectivity- degree distribution is PoissonHomogeneity – all nodes have about the same degree

Small diameter (longest path length)Large clustering coeff.- i.e. neighbors are connected

Phase TransitionDiameter and Clustering Coeficient

Clustering coefficient

Nr of neighbors of i = 4Max nr of nbr interconnections= 4x3/2= 6Actual nr of nbr interconnects= 2Clustering coeff= 2/6= 1/3

i

12

34

3. Scale-Free Networks– Barabasi and Albert

Nonhomogeneous- some nodes have large degree, most have small degreeScale-Free- degree has power law degree distribution

Start with m0 nodesAdd one node at a time:

connect to m other nodes

with probability

i.e. with highest probability to biggest nodes(rich get richer)

1( )( 1)

i

jj

dP id

2

3

2( ) mP kk

0 50 100 150 200 250 300 350 400 4500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4. Proximity Graphs

Randomly select N points in the planeDraw an edge (i,j) if distance between nodes i and j is within d

2d

x

y

When is the graph connected?for what values of (N,d)

What is the degree distribution?

Graphs and Dynamic Graphs

Communication Graph1

2

3

4

56

N nodes

[ ]ijA a

0 ( , )ij j i

i

a if v v E

if j N

oN1

Noi ji

jd a

Out-neighbors of node iCol sum= out-degree

42a

Adjacency matrix

0 0 1 0 0 01 0 0 0 0 11 1 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 1 0

A

iN1

N

i ijj

d a

In-neighbors of node iRow sum= in-degreei

(V,E)

i

1

2

3

4

56

Diameter= length of longest path between two nodes

Volume = sum of in-degrees1

N

ii

Vol d

Spanning treeRoot node

Strongly connected if for all nodes i and j there is a path from i to j.

Tree- every node has in-degree=1Leader or root node

Followers

Dynamic Graph- the Graphical Structure of ControlEach node has an associated state i ix u

Standard local voting protocol ( )i

i ij j ij N

u a x x

1

1i i

i i ij ij j i i i iNj N j N

N

xu x a a x d x a a

x

( )u Dx Ax D A x Lx L=D-A = graph Laplacian matrix

x Lx

If x is an n-vector then ( )nx L I x

x

1

N

uu

u

1

N

dD

d

Closed-loop dynamics

i

j

[ ]ijA a

The Power of Synchronization Coupled OscillatorsDiurnal Rhythm

1

2 3

4 5 6

12

3

4 5

6

Graph Eigenvalues for Different Communication Topologies

Directed Tree-Chain of command

Directed Ring-Gossip networkOSCILLATIONS

Graph Eigenvalues for Different Communication Topologies

Directed graph-Better conditioned

Undirected graph-More ill-conditioned

65

34

2

1

4

5

6

2

3

1

FlockingReynolds, Computer Graphics 1987

Reynolds’ Rules:Alignment : align headings

Cohesion : steer towards average position of neighbors- towards c.g.Separation : steer to maintain separation from neighbors

( )i

i ij j ij N

a

Distributed Adaptive Control for Multi‐Agent Systems

Nodes converge to consensus heading

( )ci

i ij j ij N

a

Consensus Control for Swarm Motions

heading angle

Convergence of headings

1

2

3

4

56

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

-350 -300 -250 -200 -150 -100 -50 0 50-300

-250

-200

-150

-100

-50

0

50

cossin

i i

i i

x Vy V

time x

y

Leader

Followers

0 5 10 15 20 25 30-120

-100

-80

-60

-40

-20

0

20

40

60

Time

Hea

ding

Heading Consensus using Equations (21) and (22)

Nodes converge to heading of leader

Consensus Control for Formations

heading angle

Formation- a Tree network

Convergence of headings

10 20 30 40 50 60 70 80 90 100-200

-150

-100

-50

0

50

100

150

200

Heading Update using Spanning Tree Trust Update

x

y

Leader

y

x

( )ci

i ij j ij N

a

cossin

i i

i i

x Vy V

time

leader

Herd and Panic Behavior During Emergency Building Egress

Helbring, Farkas, Vicsek, Nature 2000

Modeling Crowd Behavior in Stress SituationsHelbring, Farkas, Vicsek, Nature 2000

Radial compressionterm

Tangential frictionterm

Consensus term Interaction pot. fieldWall pot. field

Repulsive force

Modeling Crowd Behavior in Stress Situations

35

Balancing HVAC Ventilation Systems

SIMTech 5th floor temperature distribution

SIMTech

Automated VAV control system

AHUFan

C 1 C 2

CWRCWS

Air Flow

Diffuser outlets

VSD

Control Panel

Control stationVAV box

Room thermostat

Air diffuser

LEGENDS

Extra WSN temp. sensors

SIMTech

( 1) ( ) ( ) ( )i i i ix k x k f x u k

1( ) ( ) ( ( ) ( ))1

i

i i ij j ij Ni

u k k a x k x kn

1 12 4( ) 1, , ,...i k

Control damper position based on local voting protocol

Temperature dynamics

Unknown fi(x)

Under certain conditions this converges to steady-state desired temp. distribution

Adjust Dampers for desired Temperature distributionSIMTech

Open Research Topic - HVAC Flow and Pressure control

Synchronization Spong and Chopra

( ) ( )( )

i i i i i

i i

x f x g x uy h x

passive

0

( ) ( (0)) ( ) ( )t

Ti i i i i iV x V x u s y s ds

Synchronize if ( ) ( ), ,i jy t y t all i j

( )i

i j ij N

u K y y

Result -Let the communication graph be balanced. Then the agents synchronize.

Storage function

Local voting protocol with OUTPUT FEEDBACK

Circadian rhythm

Fireflies synchronize

Synchronization of Chaotic node dynamics – Ron Chen

Chen’s attractornode dynamics

Pinning control of largest node(for increasing coupling strengths)

c=0 c=10

c=15

c=20

The cloud-capped towers, the gorgeous palaces,The solemn temples, the great globe itself,Yea, all which it inherit, shall dissolve,And, like this insubstantial pageant faded,Leave not a rack behind.

We are such stuff as dreams are made on, and our little life is rounded with a sleep.

Our revels now are ended. These our actors, As I foretold you, were all spirits, and Are melted into air, into thin air.

Prospero, in The Tempest, act 4, sc. 1, l. 152-6, Shakespeare

Define as the trust that node i has for node jij

[ 1,1]ij -1 ………..………. 0 ………..…………. 1Distrust no opinion complete trust

Foundation work by John Baras

Define trust vector of node i as 1

2

3

4

5

6

i

i

ii

i

i

i

Trust node i has for node 3

N vector

( )i

i i ij j ij N

u a

Standard local voting protocol

Difference of opinion with neighbors

Inspired by social behavior in flocks, herds, teams

Trust Propagation and Consensus – Network Security

( )NL I

2

1

2 N

N

R

Closed-loop trust dynamics

Trust Propagation & Consensus1

2

3

4

56

Nodes 1, 2, 4 initially distrust node 5initial trusts are negative

Convergence of trust

0 5 10 15 20 25 30 35 40-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Other nodes agree that node 5 has negative trust

0

Nodes converge to consensus heading

Trust-Based Control: Swarms/Formations

Convergence of trust Convergence of headings

1

2

3

4

56

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

-350 -300 -250 -200 -150 -100 -50 0 50-300

-250

-200

-150

-100

-50

0

50

( )i

i ij j ij N

a

( )ci

i ij ij j ij N

a

heading angle

cossin

i i

i i

x Vy V

Trust dynamics

Motion dynamics

time time x

y

Causes Unstable Formation

( )ci

i ij ij j ij N

a

Trust-Based Control: Swarms/FormationsMalicious Node

Divergence of trust Divergence of headings

1

2

3

4

56

Node 5 injects negative trust values

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-20

-15

-10

-5

0

5

10

15

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

-30 -25 -20 -15 -10 -5 0 5 10 15 20-25

-20

-15

-10

-5

0

5

10

15

20

25

Internal attackMalicious node puts out bad trust values

i.e. false informationc.f. virus propagation

Trust-Based Control: Swarms/FormationsCUT OUT Malicious Node

heading angle

1

2

3

4

56

Node 5 injects negative trust values

If node 3 distrusts node 5,Cut out node 5

Convergence of trust0 5 10 15 20 25 30 35 40

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Other nodes agree that node 5 has negative trust

Restabilizes FormationConvergence of headings

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

Node 5

-400 -350 -300 -250 -200 -150 -100 -50 0 50 100-400

-350

-300

-250

-200

-150

-100

-50

0

50

Node 5

( )ci

i ij ij j ij N

a