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Lecture VI:Collective Behavior of Multi-
Agent Systems II: Intervention
Zhixin Liu
Complex Systems Research Center, Complex Systems Research Center, Academy of Mathematics and Systems Academy of Mathematics and Systems
Sciences, CASSciences, CAS
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In the last lecture, we talked about
Collective Behavior of Multi-Agent Systems I: Analysis
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Introduction Model: Vicsek model
In the last lecture, we talked about
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Multi-Agent System (MAS)
MAS Many agents Local interactions between agents Collective behavior in the population level
More is different.---Philp Anderson, 1972 e.g., small-world, swarm intelligence, panic, phase transition, coordination, s
ynchronization, consensus, clustering, aggregation, ……
Examples:Physical systemsBiological systemsSocial and economic systems Engineering systems… …
Autonomy: capable of autonomous action Interactions: capable of interacting with other agents
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Vicsek Model (T. Vicsek et al. , PRL, 1995)
http://angel.elte.hu/~vicsek/http://angel.elte.hu/~vicsek/
r
A bird’s neighborhoodAlignment: steer towards the average heading of neighbors
)(ti
v: the constant speed of birdsr: radius of neighborhood
: heading of agent ixi(t) : position of agent i in the plane
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Vicsek Model
r
A bird’s neighborhood
Alignment: steer towards the average heading of neighbors
http://angel.elte.hu/~vicsek/
})()(:{)( rtxtxjtN jii Neighbors:
))1(sin),1((cos)()1( ttvtxtx iiii Position:
Synchronization: There exists θ, such that
Heading:
)()(cos
)()(sin
arctan)1(
ti
Njt
j
ti
Njt
j
ti
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Introduction Model Theoretical analysis Concluding remarks
In the last lecture, we talked about
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.))1(sin),1((cos)()1(
,)()(
1)1(
)(
ttvtxtx
ttN
t
iiii
tNjj
ii
i
.))1(sin),1((cos)()1(
),()()1(
ttvtxtx
ttPt
otherwise
jiiftNtp
tptP
iij
ij
0
~|)(|
1)(
)},({)(
The Linearized Vicsek Model
A. Jadbabaie , J. Lin, and S. Morse, IEEE Trans. Auto. Control, 2003.
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Related result: J.N.Tsitsiklis, et al., IEEE TAC, 1984
Joint connectivity of the neighbor graphs on each time interval [th, (t+1)h] with h >0
Synchronization of the linearized Vicsek model
Theorem 2 (Jadbabaie et al. , 2003)
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Random Framework
Random initial states:
1) The initial positions of all agents are uniformly and independently distributed in the unit square;
2) The initial headings of all agents are uniformly and independently distributed in [-+ε, -ε] with ε∈ (0, ). The initial headings and positions are independent.
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For any given system parameters
and when the number of agnets n
is large, the Vicsek model will synchronize almost surely.
0v,0r
Theorem 7
High Density Implies Synchronization
This theorem is consistent with the simulation result.
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Let and the velocity
satisfy
Then for large population, the MAS will synchronize almost surely.
),(log
),1(61
nn ron
nor
.
log 2/3
6
n
nrOv n
n
Theorem 8High density with short distance interaction
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Three Categories of Research on Collective Behavior
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Three Categories of Research on Collective Behavior
J.Han, M.Li, L.guo, JSSC,2006
AnalysisGiven the local rules of the agents, what is the collective behavior of the overall system ? (Bottom Up)
DesignGiven the desired collective behavior, what are the local rules for agents ? (Top Down)
InterventionGiven the local rule of the agents, how we intervene the collective behavior?
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Example 1: Synchronization
r
A bird’s Neighborhood
Alignment: steer towards the average heading of neighbors
Simulation Result
Q: Under what conditions such a system can reach consensus?
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Example 2: Escape Panic
Normal, no panic Fire, panic
D. Helbing, et al., Nature, Vol. 407, 2000
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Three Categories of Research on Collective Behaviors
J.Han, M.Li, L.guo, JSSC,2006
AnalysisGiven the local rules of the agents, what is the collective behavior of the overall system ? (Bottom Up)
DesignGiven the desired collective behavior, what are the local rules for agents ? (Top Down)
InterventionGiven the local rule of the agents, how we intervene the collective behavior?
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How we design the control law of each plane to maintain the form ?
Example 1: Formation control
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Example 2: Swarm Intelligence (Marco Dorigo et al., 2001-2004)
www.answers.com/topic/s-bot-mobile-robot
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Example 3:Distributed Control in Boid Model
Each agent is described by a double integrator (Newton's second law of motion ):
i i
i i
x v
v u
R. Olfati-Saber, IEEE Trans. Auto. Control ,2006.
where xi, vi and ui represent the position, velocity and the control input of the agent i.
What information can be used to design the controller?The position and velocity of neighbors
Goal: 1) Avoid collision 2) Alignment 3) Cohension
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Algorithm
where A=[aij(q)] is the adjacency matrix,
Theorem 1:
If the neighbor graphs are connected at each time instant. Then
1) The group will form cohesion.
2) All agents asymptotically move with the same velocity.
3) No interagent collisions occur.
Neighbor graph
Controller design:
(·) is the action function, )(
isσ-norm, and
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Three Categories of Research on Collective Behaviors
J.Han, M.Li, L.guo, JSSC,2006
AnalysisGiven the local rules of the agents, what is the collective behavior of the overall system ? (Bottom Up)
DesignGiven the desired collective behavior, what are the local rules for agents ? (Top Down)
InterventionGiven the local rule of the agents, how we intervene the collective behavior?
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Example 1: Can we guide the birds’ flight if we know how they fly ?
Intervention
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Example 2: Leadership by Numbers
Couzin, et al., Nature, Vol. 433, 2005
The larger the group is, the smaller the leaders are needed.
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Example 3: Cockroach
J.Halloy, et al., Science, November 2007
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III. Intervention Given the local rule of the agents, how we intervene the collective behavior?
The current control theory can not be applied directly, because It is a many-body self-organized system. The purpose of control aims to collective behavior. Not allowed to change the local rules of the existing
agents; Distributed Control: special task of formation, Pinning Control: Networked system, imposed
controllers on selected nodes
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Intervention Via
Soft Control
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Soft Control
The multi-agent system: Many agents Each agent follows the local rules
Autonomous, distributed Agents are connected, the local effect will affect the whole.
From Jing Han’s PPT
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Soft Control
y(t)u(t)
The “Control”: No global parameter to adjust Not to change the local rule of the existing agents;
Put a few “shill” agents to guide (seduce) Shill: is controlled by us, not following the local rules, is treated as an ordinary agent by other ordinary agents The power of shill seems limited
The ‘control’ is soft and seems weak
From Jing Han’s PPT
an associate of a person selling goods or services or a political group, who pretends no association to the seller/group and assumes the air of an enthusiastic customer.
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Soft Control
y(t)U(t)
Key points:Key points: Different from distributed control approach.
Intervention to the distributed system Not to change the local rule of the existing agents Add one (or a few) special agent – called “shill” based on the syst
em state information, to intervene the collective behavior; The “ shill” is controlled by us, but is treated as an ordinary agent
by all other agents. Shill is not leader, not leader-follower type. Feedback intervention by shill(s).
From Jing Han’s PPTThis page is very important!
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There Are Lots of Questions …
What is the purpose/task of control here? Synchronization/consensus Group connected / Dissolve a group Turning (Minimal Circling) Lead to a destination (in a shortest time) Avoid hitting an object Tracking …
In what degree we can control the shill? (heading, position, speed, …)
How much information the shill can observe ? (positions, headings, …)
…
From Jing Han’s PPT
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A Case Study
Problem statement: System: A group of n agents with initial headings i(0)[0,
); Goal: all agents move to the direction of eventually. Soft control: Design one shill agent based on the agents’ state
information.
Assumptions: The local rule about the ordinary agents is known The position x0(t) and heading 0(t) of the spy can be controlled
at any time step t The state information (headings and positions) of all ordinary
agents are observable at any time step
From Jing Han’s PPT
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Vicsek Model
r
A bird’s Neighborhood
Alignment: steer towards the average heading of neighbors
http://angel.elte.hu/~vicsek/
})()(:{)( rtxtxjtN jii Neighbors:
))(sin),((cos)()1( ttvtxtx iiii Position:
Heading:
)()(cos
)()(sin
arctan)1(
ti
Njt
j
ti
Njt
j
ti
Synchronization: There exists θ, such that
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Problem statement: System: A group of n agents with initial headings i(0)[0, ); Goal: all agents move to the direction of eventually. Soft control: Design one shill agent based on the agents’ state information.
Assumptions: The local rule about the ordinary agents is known
The position x0(t) and heading 0(t) of the shill can be controlled at any time step t
The state information (headings and positions) of all ordinary agents are observable at any time step
From Jing Han’s PPT
A Case Study
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The Control Law u
Control the Shill agent
From Jing Han’s PPT
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Theorem 4: For any initial headings and positions
i(0)[0, ), xi(0)R2, 1 i n, the updat
e rule and the control law uβ will lead to th
e asymptotic synchronization of the group.
Control the Shill agent
It is possible to control the collective behavior of a group of agents by a shill.
J.Han, M.Li, L.guo, JSSC,2006
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Simulation
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(t),u
(t),uu
r
βt
,2,1, kkht
where
)),(())(),((: )(00 txttxu trr
)}({maxarg)( 1
)(:txtr i
ti i
An Alternative Control Law
Result: The control law ut will also lead to asymptotic synchronization of the group.
otherwise
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Simulations
Control Law u
Switching between u and ur
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Remarks on Soft Control It is not just for the above model Can be applied to other MAS ,e.g.,
Panic in Crowd Evolution of Language Multi-player Game ……
“Add the special agent(s)” is just one way Should be other ways for different systems: Remove agents Put obstacle … …
We need a theory for Soft Control !We need a theory for Soft Control !
From Jing Han’s PPT
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Intervention ViaLeader-Follower Model (LFM)
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Example 1: Leadership by Numbers Couzin, et al., Nature, Vol. 433, 2005
The larger the group is, the smaller the leaders are needed.
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Leader-Follower Model
Problem statement:
System: A group of n agents;
Goal: All agents move with the expected direction eventually.
Intervention by leaders: Add some information agents-called “leaders”, which move
with the expected direction.
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Leader-Follower ModelOrdinary agents
Information agents
Key points: Not to change the local rule of the existing agents. Add some (usually not very few) “information” agents –
called “leaders”, to control or intervene the MAS; But the existing agents treated them as ordinary agents.
The proportion of the leaders is controlled by us (If the number of leaders is small, then connectivity may not be guaranteed).
Open-loop intervention by leaders.
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Mathematical Model
})()(:{)( ''' rtxtxjtN
jii
})()(:{)( rtxtxjtN jii Neighbors:
Heading:
)(
)(
)(cos
)(sinarctan)1(
tNj
tNj
i
i
i
tj
tj
t
Position: ))1(sin),1((cos)()1( ttvtxtx iiii
Ordinary agents (labeled by 1,2,…,n):
0)1(' tiHeading:
Position: ))1(sin),1((cos)()1( '''' ttvtxtxiiii
Leader agents (labeled by ):)(,2,1 '''nn nM
)(#)(),(#)( '' tNtntNtn iiii
)()(
)()(
'
'
)(cos
)(sinarctan)1(
tNtNj
tNtNj
i
ii
ii
tj
tj
t
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Simulation Example
N=1000
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Q: How many leaders are required for consensus/synchronization?
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Assumption on the initial states
1) The initial positions of all agents are independently and uniformly distributed in the unit square.
2) The initial headings of the agents are uniformly and independently distributed in [-π, π), and the initial headings of the leaders are . The headings and the positions are mutually independent.
0
Random Framework
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))(,),(()( 1 tdtddiagtD n
)(min),(max)(,,,1),( minmax tddtdtdnitd ii
ii
i Degree:
Degree matrix:
Adjacency matrix:
0
,1)(taij
If i ~ j
Otherwise)},({)( tatA ij
Some Notations
otherwise
jiifttatatA
j
ijij
0
~)(cos)(~)],(~[)(
)()(
)(cos),(cos)(1 tNj
jtNj
j
n
ttdiagtG
Weighted adjacency matrix:
Weighted degree matrix:
));(,),(()( ''1 tntndiagtN nLeader degree matrix:
)())()(()( 1 tAtNtDtP
)(~
))()(()(~ 1 tAtNtGtP
Average matrix:
Weighted average matrix:
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“Normalized Laplacian” :2/12/1 ))0()0())(0()0(())0()0(()0( NDNLNDM
)0()0(0 1 n Spectrum :
Vj jjj
Vj jjji ji
fn nnf
fnff
))0()0((
)0()(sup)0(
'2
2'
~
2
0
)0(1,)0(1max)0( 1 n “Spectral gap”:
where
Laplacian : L(0)=D(0) – A(0)
Some Notations (cont.)
Vj jjj
Vj jjji ji
f nnf
fnff
))0()0((
)0()(inf)0(
'2
2'
~
2
01
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Key Steps in the Analysis of the LFM
Analysis of the system dynamics
Estimation of the rate of consensus
Dealing with the matrices with increasing dimension
Dealing with the inherent nonlinearity
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Analysis of the System Dynamics
Evolution of the distance
Lemma 1: For any two agents i and j, their distance satisfy the following inequality:
where
is important for the evolution of the distance!
![Page 53: Lecture VI: Collective Behavior of Multi- Agent Systems II: Intervention Zhixin Liu Complex Systems Research Center, Academy of Mathematics and Systems](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d2a5503460f949fede8/html5/thumbnails/53.jpg)
Analysis of the System Dynamics
Step 1: Projection
)(tan)(cos)(
)(cos)1(tan
)()(
't
ttn
tt j
tNjtNj
ji
ji
i
i
0)()( tt ii
Evolution of the headings
)(tan)(~
)(tan)(~
))()(()1(tan 1 ttPttAtNtDt
![Page 54: Lecture VI: Collective Behavior of Multi- Agent Systems II: Intervention Zhixin Liu Complex Systems Research Center, Academy of Mathematics and Systems](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d2a5503460f949fede8/html5/thumbnails/54.jpg)
Analysis of the System Dynamics
Step 2: Analyze the stability of
Step 3: Dealing with the changing neighbor graphs
)0()(sup)()(~
sup11
PkPkPkPtktk
)0()(~
sup1
PkPtk
)0()0(min
)0()0(max'
1
'
1
iini
iini
nn
nnk
where
'
n n
' 'n n
{ : (1 ) (1 ) }
{ : (1 ) (1 ) }
i j i
i ij
R j r x x r
R j r x x r
n(1 )r
r
(1 )n r
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Estimation of Consensus Rate
1) A key lemma: For any vector f=[f1,f2,…,fn]τ, we have
)0(The consensus rate depends on
Vj jjj
Vj jjji ji
fn nnf
fnff
))0()0((
)0()(sup)0(
'2
2'
~
2
0
Vj jjj
Vj jjji ji
f nnf
fnff
))0()0((
)0()(inf)0(
'2
2'
~
2
01
..)),1(1(1)0( saoM
n
n
2)
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Dealing with the Matrices with Increasing Dimension
Estimation of multi-array martingales
..,log34
3),(maxmax
11
11sanS
Cnkwf n
wm
jjj
nknm
.),(),(sup, 21
,11
2 nkFnkwECfS jjnjk
w
n
jjn
where
..log3),(maxmax1
111
sanSCnkwf nw
m
jjj
nknm
,log4 1 nCS wnMoreover, if then we have
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Dealing with the Matrices with Increasing Dimension
.1.,.,log1
1)(cosmax)4
..,log1
)1(tanmax)3
..,log)0(cosmax)2
..,log)0(sinmax)1
1
1
)0(1
)0(1
tsan
nOt
san
nO
sannO
sannO
nj
ni
nj
ni
Njj
ni
Njj
ni
i
i
Using the above corollary, we have for large n
where .1log6
nn
n
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The Degree of The Initial Graph
])log[()0(#max)4
..],)log[()0(#max)3
..],)log[()0(max)2
..],)log[()0(max)1
2/1
1))0((
1
2/1
1))0((
'
1
2/1
1))0((
1
2/1
1))0((
'
1
''
''
nnOEIR
sannOEIR
sannOEIn
sannOEIn
n
jRji
ni
n
M
jRji
ni
n
jNji
ni
n
M
jNji
ni
i
n
i
i
n
i
Lemma: For initial graph G0, we have for large n
![Page 59: Lecture VI: Collective Behavior of Multi- Agent Systems II: Intervention Zhixin Liu Complex Systems Research Center, Academy of Mathematics and Systems](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d2a5503460f949fede8/html5/thumbnails/59.jpg)
saornR
saornR
sanCnnn
sanCnnn
saon
n
nnini
nini
nini
nini
ini
ini
ni
i
.)),1(1(4)0(#max)5
.)),1(1(4)0(#max)4
.,)0(min,)0(max)3
..,)0(min,)0(max)2
..)),1(1()0(
)0()1
2'
1
2
1
2'
1
'
1
111
'
Corollary:
The Degree of The Initial Graph
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Dealing with the Inherent Nonlinearity
Proposition 1For any positive v and r, we have for large n
..,1)),1(1()0()( satordtd nijij
..,1)),1(1(3
11))(()0( 21 satotk n
where
.3
1,)()(
~sup)(
,24
161,4min
21
1
2
22
nts
nn
sPsPt
r
r
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Theorem 5 Let the velocity v > 0 and radius r > 0 be positive
constants. If the proportion of the leaders satisfies
where C is a constant depending on v and r, then the headings of all agents will converge to almost surely when the population size n is large enough.
6
log
n
nC
n
M n
0
Main Result
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Concluding Remarks
In this talk, we talked about intervention to the multi-agent systems:
Soft control
Design the control law of the “shill”
Leader-follower model
Control the number of the leaders
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Concluding RemarksThese two lectures mainly focus on the
collective behavior of the MAS.
In the next lecture, we will talk about game theory.
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Thank you!