distributed consensus with limited communication data rate
DESCRIPTION
ANU RSISE System and Control Group Seminar. Distributed Consensus with Limited Communication Data Rate. Tao Li. Key Laboratory of Systems & Control, AMSS, CAS, China. October, 15 th , 2010. Co-authors. Xie Lihua , School of EEE, Nanyang Technological University, Singapore. - PowerPoint PPT PresentationTRANSCRIPT
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Distributed Consensus with Limited Communication Data
Rate
Tao Li
Key Laboratory of Systems & Control, AMSS, CAS, China
ANU RSISE System and Control Group Seminar
October, 15th, 2010
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Co-authors
Xie Lihua, School of EEE, Nanyang Technological University, Singapore.
Fu Minyue, School of EE&CS, University of Newcastle, Australia.
Zhang Ji-feng, AMSS, Chinese Academy of Sciences, China
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Outline Background and motivation
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Outline Background and motivation Case with time-invariant topology
protocol design and closed-loop analysis parameter selection algorithm asymptotic convergence rate
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Outline Background and motivation Case with time-invariant topology
protocol design and closed-loop analysis parameter selection algorithm asymptotic convergence rate
Communication energy minimization
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Outline Background and motivation Case with time-invariant topology
protocol design and closed-loop analysis parameter selection algorithm asymptotic convergence rate
Communication energy minimization Case with time-varying topologies
protocol design and closed-loop analysis finite duration of link failures
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Outline Background and motivation Case with time-invariant topology
protocol design and closed-loop analysis parameter selection algorithm asymptotic convergence rate
Communication energy minimization Case with time-varying topologies
protocol design and closed-loop analysis finite duration of link failures
Concluding remarks
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Background and Motivation
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Distributed estimation over sensor networks
(0) , 1, 2,..., ---- : the parameter to be estimated ---- (0) : the meauremnt by the sensor ---- : the measurement noise of the sensor
i i
i
i
x v i N
x ithv ith
Maximum likelihood estimate:1
1 (0)N
ii
xN
Xiao, Boyd & Lall, ISIPSN, 2005
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Central strategy : remote fusion centerdrawbacks : high communication cost low robustness
1
1 (0)N
ii
xN
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Distributed consensus: to achieve agreement by distributed information exchange
1
1( ) (0), N
i jj
x t x tN
average
consensus
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Literature: precise information exchange Olfati-Saber & Murray TAC 2004 Beard, Boyd, Kingston, Ren, Xiao, et al.Features exact information exchange exponentially fast convergence zero steady-state error
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The communication channels between
agents have limited capacity. Quantization
plays an important role.
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Literature: quantized information exchange Kashyap et al. Automatica 2007 Carli et al. NeCST 2007 Frasca et al. IJNRC 2009 Kar & Moura arXiv:0712 2009 Features infinite levels nonzero steady-state error
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Theoretic motivation: Exponentially fast
exact average-consensus with finite data-rate
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Power limitation is a critical constraint
Energy totransmit 1 bit
Energy for1000-3000operations
≈ Shnayder et al. 2004
How to select the number of quantization levels (data rate) and
convergence rate to minimize the communication energy
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Channel uncertainties :Link failuresPacket dropouts Channel noises
How to design robust encoding-decoding schemes and
control protocols against channel uncertainties
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Problem 1:
How many bits does each pair ofneighbors need to exchange at each time step to achieve
consensus of the whole network ?
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Problem 2:
What is the relationship between the consensus
convergence rate and the number of quantization
levels (data rate)?
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Problem 3:
What is the relationship of the communication energy cost,
the convergence rate and the data rate ?
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Problem 4:
How to design encoders and decoders when the
communication topology is time-varying or the transmission is
unreliable ?
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Case with time-invariant topology
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i encoding transmission decoding j
States are real-valued , but only finite bits of information are
transmitted at each time-step
( )ijx t( )ix t
{ , , }G V E A
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i encoding transmission decoding j
States are real-valued , but only finite bits of information are
transmitted at each time-step
( )ijx t( )ix t
( 1) ( ) ( ), 0,1,..., 1, 2...,i i ix t x t u t t i N
{ , , }G V E A
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Difficulties quantization error
divergence of the states finite-level quantizer
nonlinearity, unbounded quantization errorKey points design proper encoder, decoder and protocol
stabilize the whole network eliminate the effect of quantization error on
achieving exact consensus
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Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
channel(j,i)
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Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
channel(j,i)
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Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
finite-level symmetric
uniform quantizer
channel(j,i)
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Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
innovation
channel(j,i)
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Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
scalingfunction
channel(j,i)
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Distributed protocol Symmetric uniform quantizer
0, -1/ 2 1/ 2, (2 -1) / 2 (2 1) / 2
( ), (2 1) / 2( ( )), 1/ 2
yi i y i
q yK y K
q y y
(2K+1) levels bits2log (2 )K
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Distributed protocol
decoder ψji
)1( tg
Z-1
)(tj )(ˆ tx jichannel(j,i)
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Distributed protocol
decoder ψji
)1( tg
Z-1
)(tj )(ˆ tx ji
Encoder
φj
channel
(j,i)
Decoder
ψji
)(tj )(tj )(ˆ tx ji)(tx j
channel(j,i)
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Distributed protocol
protocol( ) ( ( ) ( )), 0,1,...,
1, 2,...,i
jii ij ij N
u t h a x t t t
i N
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Distributed protocol
protocol( ) ( ( ) ( )), 0,1,...,
1, 2,...,i
jii ij ij N
u t h a x t t t
i N
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Distributed protocol protocol
( ) ( ( ) ( )), 0,1,...,
1,2,...,i
jii ij ij N
u t h a x t t t
i N
error compensation
( ) ( ( ) ( ))
- ( ( ) ( ))
( ( ) ( ))
i
i
i
i ij j ij N
jiij jj N
ijji ij N
u t h a x t x t
h a x t x t
h a x t x t
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Distributed protocol
protocol( ) ( ( ) ( )), 0,1,...,
1, 2,...,i
jii ij ij N
u t h a x t t t
i N
average preserved
( ) ( ( ) ( ))
- ( ( ) ( ))
( ( ) ( ))
i
i
i
i ij j ij N
jiij jj N
ijji ij N
u t h a x t x t
h a x t x t
h a x t x t
1
( ) 0N
ii
u t
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control gain: h
quantization levels: 2K+1
scaling fucntion: g(t)
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Main assumptions
A1) G is a connected graph
A2)
Denote
1 1max | (0) | , max | (0) |i x ii N i N
x C C
2
10 0
max |1 |
1 1( ) ( ) (0), , ( ) (0)
h ii N
TN N
j N jj j
h
t x t x x t xN N
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( 1) ( ( ), ( ))( 1) ( ( ), ( ))t f t e t
e t g t e t
Nonlinear coupling between consensus error and quantization
error
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( 1) ( ( ), ( ))( 1) ( ( ), ( ))t f t e t
e t g t e t
Nonlinear coupling between consensus error and quantization
error
( 1) ( ( ), ( ))
( 1) ( ( ), ( ))
t f t t
t g t t
Adaptive control
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( 1) ( ( ), ( ))t f t e t 0
( 1) , , ( ) sup ( )k t
t h K g t e k
( 1) ( ( ), ( ))e t g t e t
Selecting proper h, K, g(t)
0sup ( )t
e t
( ) (1), t o t
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Lemma. Suppose A1)-A2) hold. For any given
and
by taking with
(0,2 / )Nh ( ,1)h
0( ) tg t g
1If 2 1 2 ( , ) 1K K h
1
1 then lim ( ) (0), 1, 2,...,N
i jt j
x t x i NN
02( )( )max ,
1/ 2x h x N
N
C C hCgK h
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Lemma. Suppose A1)-A2) hold. For any given
and
by taking with
(0,2 / )Nh ( ,1)h
0( ) tg t g
1If 2 1 2 ( , ) 1K K h
1
1 then lim ( ) (0), 1, 2,...,N
i jt j
x t x i NN
02( )( )max ,
1/ 2x h x N
N
C C hCgK h
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where
convergence rate:
|| ( ) || ( ), tt O t
2 2 *
11 2 1( , ) - 1
2 ( ) 2 2N
h
N h hdK h
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Lemma. Suppose A1)-A2) hold. For any given
and
by taking with
(0,2 / )Nh ( ,1)h
0( ) tg t g
1If 2 1 2 ( , ) 1K K h
1
1 then lim ( ) (0), 1, 2,...,N
i jt j
x t x i NN
02( )( )max ,
1/ 2x h x N
N
C C hCgK h
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Lemma. Suppose A1)-A2) hold. For any given
and
by taking with
(0,2 / )Nh ( ,1)h
0( ) tg t g
1If 2 1 2 ( , ) 1K K h
1
1 then lim ( ) (0), 1, 2,...,N
i jt j
x t x i NN
02( )( )max ,
1/ 2x h x N
N
C C hCgK h
Convergence rate for perfect
communication
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2 2 *
11 2 1( , ) - 1
2 ( ) 2 2N
h
N h hdK h
),(1
hKh
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• To save communication bandwidth
Minimize the number of quantization levels 2K+1
2 2 *
0 1
1 2 1lim lim2 ( ) 2 2
N
hh
Nh hd
2 10 1
lim lim log 2 ( , ) 1h
K h
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Theorem. Suppose A1)-A2) hold. For any given
let
Then (i) is not empty
(ii) for any given , let
12
2 1( , ) | 0 , 1, ( , )2K h
N
h h M h K
0( ) tg t g
K
( , ) Kh
1K
1
1lim ( ) (0), 1, 2,...,N
i jt j
x t x i NN
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For a connected network, average-consensus can be achieved
with exponential convergence ratebase on a single-bit exchange
between each pair of neighborsat each time step
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Parameter selection algorithm
Selecting a constant Selecting the control gain
Choose
1
2
2 1( , ) | 0< , 1, ( , )2K
N
M K
0 (0,1) *
0(0, ( ))Kh h
* 0 20 2 * 2
2 2 0 2 0 0
22( ) min ,2 (2 1) (1 )K
N N
KhN d K
0 21 (1 )h Nonlinear coupled inequalities
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Asymptotic convergence rate
Theorem. Suppose G is connected, then
for any given , we have
where
( , )2
inflim 1
exp2
Kh
NNKQN
2N
N
Q
1K
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Asymptotic convergence rate
Theorem. Suppose G is connected, then
for any given , we have
where
( , )2
inflim 1
exp2
Kh
NNKQN
2N
N
Q
1K
Synchronizability
Barahona & Pecora PRL 2002
Donetti et al. PRL 2005
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The highest asymptotic convergence rate increases as the number of
quantization levels and the synchronizability increase
and decreases as the network expands
2
( , )inf exp
2K
N
h
KQN
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selection of edges: initial states:
{( , ) } 0.2P i j E
(0) , 1,2,...,30ix i i
a network with 30 nodes
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1 bit quantizer convergence factor : 0.95
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1 bit quantizerconvergence factor : 0.975
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Communication energy minimization
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convergence time constant
Xiao & Boyd 2004
1(ln(1/ ))asym asymr
1/
(0) (0)
|| ( ) (0) ||sup lim|| (0) (0) ||
t
asym tX JX
X t JXrX JX
Consensus error decreasing by 1/e
Model of communication energy
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Model of communication energy
2 1 2( ) log (2 ) ( | | 2 | |) asymK K c c V E
Energy for transmitting a single bit
Energy for receiving
a single bit
The faster, the larger
The faster, the smaller
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Fast convergence does not imply power saving. The optimization of the total
communication cost leads to tradeoff between number of
quantization levels and convergence rate.
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2 1 2( ) log (2 ) ( | | 2 | |) asymK K c c V E
Energy for transmitting a single bit
Energy for receiving
a single bit
11 2 2 1( ) ( | | 2 | |) log (2 ( , )) [ln(1/ )]K c V c E K h
Optimization w.r.t.
1( , ), asymK K h r
Suboptimal solution
( )hB
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Network with 30 nodes and 0-1 weights
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Case with time-varying topologies
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Network model
1
( ) { , ( )}, 1,2...,
( ) 1 (j, i)
( ) [ ( )] ( )
,
( ) 0 (j, i)
,
( ) { | ( ) 1} (
.
, )ij
i ij i t k
ij
ij
t i
t a t G t
N t
G t
j V a
V t t
t N
a t
k
a t
N
AA
means channel is active while
means channel
is the adjacency matri
is ina
x o
ctive
f
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Distributed protocol
encoder φji
Z-1 )1(1tg
)1( tg
)(tx j
( )ji t
( 1)ji t ( )ji t
channel(j,i)
( )ija t
jitq
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Distributed protocol
encoder φji
Z-1 )1(1tg
)1( tg
)(tx j
( )ji t
( 1)ji t ( )ji t
channel(j,i)
( )ija t
jitq
( )ijK t
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Distributed protocol
decoder ψji
)1( tg( )ji t
Z-1
)(ˆ tx jichannel
(j,i)( )ija t
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Distributed protocol
protocol( ) ( ( ) ( )), 0,1,...,
1, 2,...,i
jii ij ijj N
u t h a x t t t
i N
1
( ) 0N
ii
u t
1 1
( ) (0)N N
i ii i
x t x
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Main assumptions
A3)
A4)
A5) There is a constant , such that
A6) There exist an integer T>0 and a real constant
such that
1 1max | (0) | , max | (0) |i x ii N i N
x C C
1 ( ), 1, 2,...,i t iN N t i N
* 0d *
1 0max sup ( )ii N t
d t d
0
( 1)
20 1
1inf ( )m T
m k mT
L kT
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Theorem. Suppose A3)-A6) hold. If
where ,
and the numbers of quantization levels satisfy
1/2*
1 10, , 2
11
Thd h
T
0
sup ( ) / ( 1)t
g t g t
1(1) ,(0) 2
xij
CK
g
1
2
( , (0), (1)), (1) 1,(2)
( , (0), (1)), (1) 0,ij
ijij
h g g aK
h g g a
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*
,
,
(2 1) 1 , ( ) 1,( 1) 2 2
( ), , ( ) 0
2,3,
,
...
h ijij
h ij ij
hd a tK t
t t
t
a
*( 1) ( 1) / ( ) 1( ) ( ( )) ,( ) 2ij ij
g t g t g tt hd K tg t
where
then* 2
1/2
max | ( ) ( ) | 2limsup .( )
1 11
ij i jT
t
x t x t N hdg t h
T
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Finite duration of link failures
NCS with finite dropoutsXiao, et al., IJNRC, 2009Xiong & Lam, Automatica, 2007Yu, et al., CDC, 2004
A7) There is an integer , such that 0RT
( 1) ( ) , 1,2,..., , .ij ij R it k s k T i N j N
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Theorem. Suppose A3)-A7) hold. For any given
let
Then
is nonempty, and if
, *1/2
*
,
*,
1 1{( , ) | (0, ), (1, ),2 (1 )
1(2 1) 1 ,
2 21 1 ( ) 1}
2 1
R
RR
K TT
h
TT
h
h h hdT
hd K
K hd K
1K
, RK T
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selecting h and g(t), such that , and
Note that
(1) (2) ,
, ( ) 1,( 1) 2,3,...,
1, ( ) 0,
ij ij
ijij
ij
K K K
K a tK t t
K a t
,( , )EK Th
1
1lim ( ) (0), 1, 2,...,N
i jtj
x t x i NN
2log 5 3
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If the duration of link failures in the network is bounded, then control gain and scaling function can be selected properly , s. t. 3-bit quantizers suffice for asymptotic average-consensus with an exponentialconvergence rate.
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Concluding remarks
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Fixed topology: protocol based on
difference encoder and decoder
Asymptotic average-consensus
with a single-bit exchange
Relationship between asymptotic
convergence rateand system parameters
Algorithm for selectingthe control
parameters and energyminimization
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Time-varying topology: different quantizersfor different channels
Adaptive algorithm forselecting the numbersof quantization levels
Special case with jointly-connected graph
flow and finiteduration of link failures:
3-bit quantizer
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Future topics Noisy channels Random link failures Model uncertainty …
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