graph consensus: a review
TRANSCRIPT
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Graph Consensus: Autonomus and Controlled
Prepared by Abhijit Das
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Many of the beautiful pictures are from a lecture by Ron Chen, City U. Hong KongPinning Control of Graphs
Natural and biological structures
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Airline Route Systems
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Distribution of galaxies in the universe
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Motions of biological groups
Fishschool
Birdsflock
Locustsswarm
Firefliessynchronize
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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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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
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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
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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
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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
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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
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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
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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
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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
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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
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Simulations
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1
2
4
3
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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
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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
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Lihua Xie’s formulation contd…
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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)
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Scale free network
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Courtesy Wikipedia
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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
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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
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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
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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
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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
04/12/23 60ARRI, UTA
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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
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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
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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.
04/12/23 63ARRI, UTA
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A special case
2 1.3472
L
L
2 2
04/12/23 64ARRI, UTA
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A special case contd…
L
2 1.2451
04/12/23 65ARRI, UTA
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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
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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
04/12/23 67ARRI, UTA
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Leader-Graph network
Leader network
Graph network
Connection may be from both way
04/12/23 68ARRI, UTA
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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
04/12/23 69ARRI, UTA
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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
04/12/23 70ARRI, UTA
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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
04/12/23 71ARRI, UTA
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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
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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
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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
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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
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04/12/23 ARRI, UTA 76
Thank you
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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
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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
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Addendum: Courtesy Fang-Antsaklis
04/12/23 ARRI, UTA 79