Moncrief-O’Donnell Chair, UTA Research Institute (UTARI ... coop... · 3. Scale-Free Networks–...
Transcript of Moncrief-O’Donnell Chair, UTA Research Institute (UTARI ... coop... · 3. Scale-Free Networks–...
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Moncrief-O’Donnell Chair, UTA Research Institute (UTARI)The University of Texas at Arlington, USA
F.L. Lewis, NAI
Talk available online at http://www.UTA.edu/UTARI/acs
Cooperative Control of Multi‐Agent Systemson Communication Graphs
Supported by :NSFONRARO/TARDEC
China Qian Ren Program, NEUChina Education Ministry Project 111 (No.B08015)
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He who exerts his mind to the utmost knows nature’s pattern.
The way of learning is none other than finding the lost mind.
Man’s task is to understand patterns in nature and society.
Meng Tze
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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)
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Patterns in Nature and Society
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Many of the beautiful pictures are from a lecture by Ron Chen, City U. Hong KongPinning Control of Graphs
1. Natural and biological structures
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Distribution of galaxies in the universe
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The Egyptian Twitter NetworkArab users in Red, English users in purple
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J.J. Finnigan, Complex science for a complex world
The internet
ecosystem ProfessionalCollaboration network
Barcelona rail network
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Airline Route Systems
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2. Motions of biological groups
Fishschool
Birdsflock
Locustsswarm
Firefliessynchronize
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Herd and Panic Behavior During Emergency Building Egress
Helbring, Farkas, Vicsek, Nature 2000
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Communication Graph
N nodes (agents) interconnected by communication links.Each agent can only get information from its neighbors.
iN In-neighbors of node ii
Each agent has dynamics i i ix Ax Bu
Study the interaction of control and communication
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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
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J.J. Finnigan, Complex science for a complex world
Poisson degree distributionmost nodes have about the same degreeave(k) depends on number of nodes
Connectivity- degree distribution is PoissonHomogeneity- all nodes have about the same degree
kave k
P(k)
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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
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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
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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
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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?
Mobile Sensor Networks Project with Singapore A-Star
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Communication Graph
N nodes (agents) interconnected by communication links.Each agent can only get information from its neighbors.
iN In-neighbors of node ii
Each agent has dynamics i i ix Ax Bu
Study the interaction of control and communication
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The Interaction Between Communication and Control
The way we communicate decides the way we evolve dynamically
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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
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The Power of Synchronization Coupled OscillatorsDiurnal Rhythm
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Synchronization on Good Graphs
Chris Elliott fast video
65
34
2
1
1
2 3
4 5 6
Regular mesh
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Synchronization on Gossip Rings
Chris Elliott weird video
12
3
4 5
6
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Distributed Adaptive Control for Multi‐Agent Systems
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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
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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
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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
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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
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1
Other nodes agree that node 5 has negative trust
0
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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
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0.8
0.9
1
0 5 10 15 20 25 30 35 400
1
2
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-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
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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
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15
0 0.5 1 1.5 2 2.5 30
5
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15
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25
30
35
40
-30 -25 -20 -15 -10 -5 0 5 10 15 20-25
-20
-15
-10
-5
0
5
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15
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Internal attackMalicious node puts out bad trust values
i.e. false informationc.f. virus propagation
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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
Work by Sajal Das
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38
Balancing HVAC Ventilation Systems
SIMTech 5th floor temperature distribution
Work with SIMTech – Singapore Inst. Manufacturing Technology
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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
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( 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
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Herd and Panic Behavior During Emergency Building EgressWork with Bari Polytechnic University, Italy (David Naso) – comes April 24
Helbring, Farkas, Vicsek, Nature 2000
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Crowd Panic Behavior
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Modeling Crowd Behavior in Stress SituationsHelbring, Farkas, Vicsek, Nature 2000
Radial compressionterm
Tangential frictionterm
Consensus term Interaction pot. fieldWall pot. field
Repulsive force
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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
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Synchronization : Ron Chen – Pinning Control
Leader node dynamics
Connected undirected graphs
ii ia d
i ij jij i j i
d a a
In-degree = out-degree
Diffusivity condition
0( ) ( ) ( )i i ij j i i ij
x f x c a C x x cbC x x
Pinning Control – inputs to some nodes
1ib if node i is pinned
Results –Node motions synchronize if is stable
Pin to the biggest node = highest degree node= highest social standing- c.f. Baras
0( ) ( ) 1x f x c L B Cx cB C x
( )i
f x c Cx
L B D B A has e-vals i
Must have control gain c big enough
x0(t)
( )iB diag b
0 0( )x f x
( ) ,i i i i ix f x u y Cx
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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
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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
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