Transcript
Graph Consensus: Autonomus and Controlled
Prepared by Abhijit Das
Many of the beautiful pictures are from a lecture by Ron Chen, City U. Hong KongPinning Control of Graphs
Natural and biological structures
Airline Route Systems
Distribution of galaxies in the universe
Motions of biological groups
Fishschool
Birdsflock
Locustsswarm
Firefliessynchronize
J.J. Finnigan, Complex science for a complex world
The internet
ecosystem ProfessionalCollaboration network
Barcelona rail network
Graph
Directed Graph or Diagraph
Un-directed Graph
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Two properties of diagraph nodes
• Out-degree: Number of connections going out from a node
• In-degree: Number of connections going in to a node
• Edge: Connection between any two nodes
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Important types of Diagraphs
Balanced
Strongly Connected
Tree
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What is Consensus among nodesConsensus in the English language is defined firstly as unanimous or general agreement
1h
2h
3h
4h
h h h h
Before Consensus After Consensus04/12/23 10ARRI, UTA
Graph Dynamics (Diagraph)
1
2
3
4
5
Adjacency Matrix
1 0 0 0 1 0
2 1 0 1 0 0
3 1 0 0 0 0
4 0 0 1 0 1
5 1 1 0 0 0
A
14
21 23
31
43 45
51 52
0 0 0 01
0 0 02
0 0 0 03
0 0 04
0 0 05
w
w w
A w
w w
w w
or
1 2 3 4 5 1 2 3 4 5
Diagonal Matrix
1 0 0 0 0
0 2 0 0 0
0 0 1 0 0
0 0 0 2 0
0 0 0 0 2
D
Laplacian matrix
L D A
1 0 0 1 0
1 2 1 0 0
1 0 1 0 0
0 0 1 2 1
1 1 0 0 2
L
21w
31w
51w
14w
43w
45w
23w
Note that is row stochastic I L04/12/23 11ARRI, UTA
Continuous Time System
• Each node if assumed to have simple integrator dynamics, for -th node,
• Input
• Resultant Dynamics of the graph with all node
i ix ui
i
i ij j ij
u a x x
x A D x Lx
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CommentAs is row stochastic
The first eigenvalue of will be 0
The right eigenvector corresponding to 0 eigenvalue will be
At steady state all state values will be equal
I L
L
1 1 1 1T
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State solution
Eigen decomposition and Left and right eigenvector
Right eigenvector Left eigenvector
R RLX X
0( ) Lt
x Lx
x t e x
L LX L XRX LX
1 1 1
L R L R
L L R R L L
X LX X X
L X X X X X X
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State solution (Contd..)
11 1
0 0 0! ! !
n n nL L L
L L L Ln n n
L X Xe X X X e X
n n n
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State solution (Contd..)
1
0
0
10
L L
Lt
X X t
tL L
x e x
x e x
x X e X x
At Steady state 0
1
1
1
tL c LX x e X x
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State solution (Contd..)1020
1 2 3 1 2 3 30
1 1
0
1
1
1
n ntc
L L
n
x
x
x e xX X
x
1
0 0 0
0 0
0 0 n
with
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Finding consensus value for SC graph
Considering only the first line of the equation
0
0
ii
i ic i i c
i i ii
xx x x
For balanced graph0i
ic
xx
n
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Simulation results (SC graph)
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
With
nor
mal
pro
toco
l
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What if there is one leader in the graph
Assuming rest of the graph is connected
The Laplacian matrix of a graph with a leader
1
0 0 0L
L
with 1L may be anything
Left eigenvector 1
1 0 0 0L
L
XX
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Consensus value for one leader graph
102030
1 1
0
1
1 0 0 0 1 0 0 01
1
tc
L L
n
x
x
x e xX X
x
10cx x
Note that if there is more than one leaders then no single solution is possible
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Simulation result (one leader case)
0 1 2 3 4 5 6 7 8 9 100.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
With
nor
mal
pro
toco
l
For tree network the result will be equivalent04/12/23 22ARRI, UTA
Graph contains a spanning tree
0 1 2 3 4 5 6 7 8 9 100.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
With
nor
mal
pro
toco
l
How the value of cx can be determined ?04/12/23 23ARRI, UTA
Eigenvalue properties
• For stability all the eigenvalues should be in the left half of the plane
• The second largest eigenvalue is of a standard laplacian matrix is known as Fiedler eigenvalue
• Fiedler eigenvalue determines the speed of the whole network, thus it is important to maximize its value
• Note that Fiedler eigenvalue in general can not be determined from the dominant eigenvalue of the inverse of laplacian matrix
s
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Gershgorin disk of a network
1
j
1 0 0 1
1 1 0 0
0 1 1 0
0 0 1 1
balL
0 0 0 0 0
1 1 0 0 0
1 0 1 0 0
0 1 0 1 0
0 1 0 0 1
treeL
1
j
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More comments
• Fiedler eigenvalue is also known as algebraic connectivity or spectral gap of a graph
• Algebraic connectivity is different from connectivity or vertex-connectivity
• Network synchronization speed does NOT depend on vertex-connectivity
• Number of zero eigenvalues in a laplacian matrix reveals, number of connected components in a graph
k
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Reducibility
Consider a matrix with . If is reducible, there exist an integer anda Permutation matrix such that
r rA 2r A
1n T
11
21 22
31 32 33
1 2 3
0 0 0
0 0
0T
n n n nn
B
B B
B B BT AT
B B B B
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Irreducibility
Consider a matrix . Then, is irreducible if and only if For any scalar .
r rA A
10
r
r rcI A
0c
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Comment on reducibility
• A connected graph (strongly/balanced) is generally have irreducible adjacency and laplacian matrix
• A tree network generally posses a reducible adjacency and laplacian matrix
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Discrete time system Murray-Saber, 2004
x Lx Continuous time system
max
( 1) ( ) ( ) ( )
( 1) ( )
0,1/
i
i i ij j ij N
x k x k a x k x k
x k P x k
P I L
d
maxd Max out-degree
Discretized
P Perron matrix
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Definition
1
Stochastic matrix: row sum =
Primitive matrix: If the matrix has one eigenvalue with maximum modulus
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Perron-Frobenius Theorem
Let be a primitive non-negetive matrix
with left and right eigenvectors and
Assumptions:
1. and
2. 1
Then, lim
T
k Tk
P
w v
Pv v wP w
vw
P vw
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Comment
max
When a perron matrix become
non-negetive, stochastic and primitive?
Hint:
1. Graph is a diagraph non-negetive
and row-stochastic
2. Graph is a SC diagraph with 0 1/
Primitive
P
G
G d
P
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State Solution- DT system
( ) (0) lim
lim ( ) (0) 1
(0)
(0)
k kk
k d
d i ii
ii
d
x k P x P
x k x v wx v
x w x
xx
n
with exist
with
For balanced graph
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Comparison
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Courtesy: Fax-Murray-Saber, 2006
Performance – Murray-Saber 2007
1
( ) ( )
( 1) ( )
0 ( ) ( 1)
c dx x x
t L t
k P k
L P k k
Error vector: where, = or
CT:
DT:
Note that, and
2
2 21
Algebric connectivity:
CT Graph:
DT Graph:
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Theorems
2 2
2 2
T
T
L
P
For balanced graph:
CT:
for all
DT:
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Alternative Laplacian-Structure: Fax-Murray 2004
1
1
1
1
(1 )
1, ( 1) ( )
i
i
i j ij Ni
i ij ij N
x x xN
N a d
x Qx
Q I D A L
P I L I D A
x k D Ax k
with
For does not converge
for every diagraphs (For example bipartite graph)04/12/23 38ARRI, UTA
Based on Vicsek model: Jadbabaie-Lin-Morse
1
1( 1) ( ) ( )
1
( 1) ( )
i
i i jj Ni
P
x k x k x kN
x k I D I A x k
Perron matrix
This Perron matrix is stable! How?
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Example: Bipartite graph
11
1
1
2
2
0 0 1 1
0 0 1 1
1 1 0 0
1 1 0 0
1 1.
A
P D A
P
P I D I A
P
Fax-Murray Formulation: contains
two eigenvalues at and So, is not Primitive
Jadbabaie-Lin-Morse:
is Primitive04/12/23 40ARRI, UTA
Trust Consensus: Ballal-Lewis-2008
1 2..
n
i
i
Tni i ii ii ii
i i
i ij j ij
ij ij
i ij j ij
t t t
u
u w
w c i
j
u
and
Baras-Jiang-2006
the confidence that node has
it its trust openion of node
Ballal-Lewis Bilinear Trust04/12/23 41ARRI, UTA
Bilinear trust Dynamics
1
( )
( )
1( 1) ( )
1
( 1) ( ) ( )
( ) ( ) ( )
i i
i
i ij j ij i ij j
n
i i ij j iji
n
u L t
L t I
k kn
k F k I k
F k I I D k L k
CT system:
For DT system (based on Vicsek model):
where, 04/12/23 42ARRI, UTA
Simulations
04/12/23 ARRI, UTA 43
1
2
4
3
Comment
1
( 1) ( ) ( )n
n n n
k F k I k
I D L I L I
CT and DT system described by Ballal-Lewis,
are not equivalent.
So, they have different consensus value.
For example, the equivalent CT system for
is
not
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Zhihua Qu’s formulation
1 1
1 1
( ) ( )
( ) ( )
( ) ( )
i i
n nij ij ij ij
i j i i jn nj j
il il il ill l
n
ij
x u
s t w s t wu x x x x
s t w s t w
x I D t x L t x
S s I A
D
where,
Note that, is a stochastic matrix.
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Comment
ij
D
w
If is irreducible (strongly connected/balanced)
then the algebric connectivity of the graph depends
on
Although, graph consensus can be achieved
successfully with the proposed control law
for irreduc Dible as well as reducible
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Passive system: Definition
1
0 0
( ) ( )
( )
( ), ( ) (0) 0
. . 0,
( ( )) ( (0)) ( ) ( ) ( ( ))
( ) ( ) ( ) ( )
t tT
Tf g
x f x g x u
y h x
V x S x C V
s t t
V x t V x u s y s ds S x s ds
L V x S x L V x h x
Consider a nonlinear system
is passive iff
and positive with
also, and
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Mark Spong’s Lyapunov formulation
1
1 1
2
2 2 ( )
0
i i
i
N
ii
N NT
f i g i i i i ii i
i ij j ij N
N
V V
VV x L V L Vu S x y u
x
u K y y V
Number of agent:
If then can be proved
for only balanced graph.
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Can we change for which iu
1
1 1
2 2
1 1
2 2
1 1
2 2
i ij j ij i ji ij j j
c r
T Ti ij j i i i i j j
j
u K y K y K y
u D D A
u K y y y y y y y
Some example:
Another one:
0?V
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Zhihua Qu’s Lyapunov formulation
2
1 1
1
( 1)
2
( ) ( ) ,
1
i i i
i
i
n n
c i j j ii j
nT
c i c c ci
T Tc i i
T n nc i i
n
x I D x Lx
V x x
V e Q e
Q G P I D I D P G
e x x G i
I
If then it can be shown that
where,
and eleminating th
column from 04/12/23 50ARRI, UTA
Comments: Zhihua Qu
D
D D
This Lyapunov formulation can successfully
be done for irreducible and reducible matrix
For reducible matrix, should be
lower-triangular complete
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Lihua Xie’s Lyapunov formulation
04/12/23 ARRI, UTA 52
01
n
i ij i j i ij
u
e a x x b x x
1. Considering a one leader network
2. Define a input based on terminal sliding mode
control surface (see addendum)
3. Define error as
Lihua Xie’s formulation contd…
04/12/23 ARRI, UTA 53
1 2 0
0
, ,......, 1
2. ( )
T
nx x x x
T x
If the conditions of the previous slide exist
Then,
1. The network will achieve consensus and
The consensus will achieve in finite time
(see addendum)
Scale free network
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Courtesy Wikipedia
Ron Chen’s pinning control
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Ron Chen’s Lyapunov formulation
04/12/23 ARRI, UTA 56
1
1
( ) ( )
1,2,3,.......,
( ) ( )
1,.....,
k k k k k k
k
k k k k k
k
N
i i i i j j i ijj i
N
i i i i j j ijj i
x f x c a x x u
k l
x f x c a x x
k l N
Consider a scale-free undi-rected network
Pinned
with
NOT pinned
with
Ron Chen’s formulation contd…
04/12/23 ARRI, UTA 57
( )
( )
k k k k k
k k k
i i i i i
i i i
T
u c d x x
c d
E x X
g x E U V E
U
Define a input
with some condition imposed on and
Error is defined as =
Then, if a lyapunov candidate is defined as
with, some symmetric and atleast semi
V definite
some positive definite matrix
Ron Chen’s formulation contd…
04/12/23 ARRI, UTA 58
If is symmetric then the whole network
can be stablilized ( ) 0 following some
conditions such as
0
where is a matrix such that ( ) is
uniformly decrasing
g x
U V G D I T
T f x Tx
V
Ron Chen’s formulation contd…
04/12/23 ARRI, UTA 59
min
min
( )
( ) 0
( , ( ))c
f x
G D
f L
Moreover, if is Lipschitz continuous
then, it can be shown that for the combination
network
with
Controlled consensus
If, and 1 then
algebric connectivity is increased by
i.e. one leader is connected to every node
with a weight
For all other case,
if the Graph is SC, then adding a leader to few node
Tc
x Lx Bu
u c B
c
c
s
decrease the algebric connectiMAY vity
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Some case studies
1
42
3
Consensus time approx 7.5 sec
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
Time
Sta
tes
with
diff
. In
i. co
nd.
04/12/23 61ARRI, UTA
Some case studies contd…
1
42
3
L Consensus time approx 8 sec
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
Time
Sta
tes
with
diff
. In
i. co
nd.
04/12/23 62ARRI, UTA
Some case studies contd…
1
42
3
LConsensus time approx: 3 sec
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
14
Time
Sta
tes
with
diff
. In
i. co
nd.
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A special case
2 1.3472
L
L
2 2
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A special case contd…
L
2 1.2451
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Mathematical formulation: Lewis, 09
1
( ) 0,
n
L D A
D D
L
L L
x Lx
Define new laplacian matrix
Note that the new laplacian has diagonal dominance
property over irreducibility.
So, is nonsingular and
i.e. is a AS system.04/12/23 66ARRI, UTA
Controlled consensus: Lewis-’09
1
0 0
( )
When there are more than one leader or a
leader network is present
ss
G l n
n l L
x L B x
u u
x L Bu
L CL
C L
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Leader-Graph network
Leader network
Graph network
Connection may be from both way
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One case study: based on Z. Qu’s Laplacian
11
21 22
31 32 33
1 2 3
0 0 0
0 0
0
n n n nn
d
d d
d d dD
d d d d
Consider a reducible graph (Ex: Tree)
N1
N3
N2
Lower triangularly complete
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One case study
1 10 0
0 0 0 0
0 0
n nd d
D
Now we add a leader/ leader
It is now possible to show that the new graph has
better algebric connectivity from Lyapun
virtual cl
ov anal
one
ysis
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Case study: contd…
1
1
0
Tn n
T T T Tn n n n n n n n
V
T T Tn n n n n n
V
V e Pe
V e PW I D G G I D W P e
e PW DG G DW P e
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Jadbabaie-Lin-Morse’s leader network
04/12/23 ARRI, UTA 72
0( )
0
1( 1) ( ) ( ) ( )
1 ( )
it can be shown that
lim ( ) 1
i
i i j ij N ki i
t
x k x k x k b k xN b k
x k x
Noisy information exchange: Ren-Beard-Kingston-2005
04/12/23 ARRI, UTA 73
*
*
* *
Noise on the edge: ,
with ~ 0,
Unknown consensus value:
Process noise: , with ~ 0,
Error Covariance:
( )( )
j i
ij j ij ij ij
Ti i i
v v
z x v v R
x
x w w Q
P E x x x x
Estimator dynamics
04/12/23 ARRI, UTA 74
1
( ) ( ) ( ) ( )
with Kalman gain:
and
( )
i
i
i ij ij ij ij
T
ij i j ij
i i ij j ij ij N
x t w K t z t x t
K P P R
P P w t P R P Q
Das-Lewis contribution
04/12/23 ARRI, UTA 75
( ) ( )i i i i ix f x w t u ˆ ( ) ( , )i i i iu f x v x t
ˆ ˆ ( )Ti i i i if x W x
ˆ ˆ( )Ti i i i i i i i iW F e p d b FW
0r
Select from Lyapunov
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
2
Synch. Motion
Control Node
04/12/23 ARRI, UTA 76
Thank you
Addendum: Zhihong Man
04/12/23 ARRI, UTA 77
1 2
2
11
1 2
2 1
( ) ( ) ( )
( ) ( ) ( )sgn( )
,0 ( ) , ( ) 0, 0, 0
q
p
q
p
x x
x f x g x b x u
qu b x f x x x l s
p
s x x l g x b x p q
Define a system as
Then TSM control law generally have the form
Addendum: Lihua Xie
04/12/23 ARRI, UTA 78
1 2
0 0 0
0
, ,......,
( ) ( ) ( ) ( )2 2
10
2( ) ( , , , )
T
n
t t
T
E e e e
S E t E t E t dt E t dt
V S S V V V
T x f V
Define, as error vector and
as sliding surface
If then
Addendum: Courtesy Fang-Antsaklis
04/12/23 ARRI, UTA 79
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