Moncrief-O’Donnell Chair, UTA Research Institute (UTARI ...

Moncrief-O’Donnell Chair, UTA Research Institute (UTARI)The University of Texas at Arlington, USA

Talk available online at http:s://lewisgroup.uta.edu

Cooperative Control of Multi‐Agent Systemson Communication Graphs

Supported by :NSFONRARO/TARDEC

F.L. Lewis National Academy of Inventors

Invited by Pei Hailong

Li Huiliang

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

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 purple

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

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

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

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

kave k

P(k)

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?

Mobile Sensor Networks Project with Singapore A-Star

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

The Interaction Between Communication and Control

The way we communicate decides the way we evolve dynamically

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

The Power of Synchronization Coupled OscillatorsDiurnal Rhythm

Synchronization on Good Graphs

Chris Elliott fast video

65

34

2

1

1

2 3

4 5 6

Regular mesh

Synchronization on Gossip Rings

Chris Elliott weird video

12

3

4 5

6

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

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

Work by Sajal Das

39

Balancing HVAC Ventilation Systems

SIMTech 5th floor temperature distribution

Work with SIMTech – Singapore Inst. Manufacturing Technology

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

Herd and Panic Behavior During Emergency Building EgressWork with Bari Polytechnic University, Italy (David Naso) – comes April 24

Helbring, Farkas, Vicsek, Nature 2000

Crowd Panic Behavior

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

Radial compressionterm

Tangential frictionterm

Consensus term Interaction pot. fieldWall pot. field

Repulsive force

Controlled Consensus: Cooperative Tracker

Node state i ix uLocal voting protocol with control node v

( ) ( )i

i ij j i i ij N

u a x x b v x

( ) 1x L B x B v

i i

i i ij i ij j ij N j N

u b a x a x b v

0ib If control v is in the neighborhood of node i

Control node is in some neighborhoods iN

{ }iB diag b

Theorem. Let at least one . Then L+B is nonsingular with all e-vals positiveand -(L+B) is asymptotically stable

0ib

So initial conditions of nodes in graph A go away.Consensus value depends only on vIn fact, v is now the only spanning node

Get rid of dependence on initial conditions control node vRon Chen

Strongly connected graph L

Pinning Control

ibPinninggains

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

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