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