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
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
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
Outline Background and motivation
Outline Background and motivation Case with time-invariant topology
protocol design and closed-loop analysis parameter selection algorithm asymptotic convergence rate
Outline Background and motivation Case with time-invariant topology
protocol design and closed-loop analysis parameter selection algorithm asymptotic convergence rate
Communication energy minimization
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
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
Background and Motivation
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
Central strategy : remote fusion centerdrawbacks : high communication cost low robustness
1
1 (0)N
ii
xN
Distributed consensus: to achieve agreement by distributed information exchange
1
1( ) (0), N
i jj
x t x tN
average
consensus
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
The communication channels between
agents have limited capacity. Quantization
plays an important role.
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
Theoretic motivation: Exponentially fast
exact average-consensus with finite data-rate
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
Channel uncertainties :Link failuresPacket dropouts Channel noises
How to design robust encoding-decoding schemes and
control protocols against channel uncertainties
Problem 1:
How many bits does each pair ofneighbors need to exchange at each time step to achieve
consensus of the whole network ?
Problem 2:
What is the relationship between the consensus
convergence rate and the number of quantization
levels (data rate)?
Problem 3:
What is the relationship of the communication energy cost,
the convergence rate and the data rate ?
Problem 4:
How to design encoders and decoders when the
communication topology is time-varying or the transmission is
unreliable ?
Case with time-invariant topology
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
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
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
Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
channel(j,i)
Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
channel(j,i)
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)
Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
innovation
channel(j,i)
Distributed protocol
encoder φj
Z-1 )1(1tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
scalingfunction
channel(j,i)
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
Distributed protocol
decoder ψji
)1( tg
Z-1
)(tj )(ˆ tx jichannel(j,i)
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)
Distributed protocol
protocol( ) ( ( ) ( )), 0,1,...,
1, 2,...,i
jii ij ij N
u t h a x t t t
i N
Distributed protocol
protocol( ) ( ( ) ( )), 0,1,...,
1, 2,...,i
jii ij ij N
u t h a x t t t
i N
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
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
control gain: h
quantization levels: 2K+1
scaling fucntion: g(t)
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
( 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 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
( 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
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
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
where
convergence rate:
|| ( ) || ( ), tt O t
2 2 *
11 2 1( , ) - 1
2 ( ) 2 2N
h
N h hdK h
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
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
2 2 *
11 2 1( , ) - 1
2 ( ) 2 2N
h
N h hdK h
),(1
hKh
• 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
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
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
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
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
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
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
selection of edges: initial states:
{( , ) } 0.2P i j E
(0) , 1,2,...,30ix i i
a network with 30 nodes
1 bit quantizer convergence factor : 0.95
1 bit quantizerconvergence factor : 0.975
Communication energy minimization
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
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
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.
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
Network with 30 nodes and 0-1 weights
Case with time-varying topologies
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
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
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
Distributed protocol
decoder ψji
)1( tg( )ji t
Z-1
)(ˆ tx jichannel
(j,i)( )ija t
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
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
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
*
,
,
(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
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
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
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
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.
Concluding remarks
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
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
Future topics Noisy channels Random link failures Model uncertainty …