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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
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 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
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 ith
v 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
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 of
neighbors 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 ?
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(
1
tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
finite-level symmetric
uniform quantizer
channel(j,i)
Distributed protocol
encoder φj
Z-1 )1(
1
tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
innovation
channel(j,i)
Distributed protocol
encoder φj
Z-1 )1(
1
tg q
)1( tg
)(tx j
)(tj
)1( tj )(tj
scaling
function
channel(j,i)
Distributed protocol
Symmetric uniform quantizer
0, -1/ 2 1/ 2
, (2 -1) / 2 (2 1) / 2( )
, (2 1) / 2
( ( )), 1/ 2
y
i i y iq y
K y K
q y y
(2K+1) levels
bits2log (2 )K
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
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
Main assumptions
A1) G is a connected graph
A2)
Denote
1 1max | (0) | , max | (0) |i x i
i N i Nx 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
0
2( )( )max ,
1/ 2x h x N
N
C C hCg
K 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
0
2( )( )max ,
1/ 2x h x N
N
C C hCg
K 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
0
2( )( )max ,
1/ 2x h x N
N
C C hCg
K 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
0
2( )( )max ,
1/ 2x h x N
N
C C hCg
K h
Convergence rate
for perfect
communication
• To save communication bandwidth
Minimize the number of quantization levels 2K+1
2 2 *
0 1
1 2 1lim lim
2 ( ) 2 2N
hh
N h 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 hN
h h M h K
0( ) tg t g
K
( , ) Kh
1K
1
1lim ( ) (0), 1, 2,...,
N
i jtj
x t x i NN
For a connected network, average-
consensus can be achieved
with exponential convergence rate
base on a single-bit exchange
between each pair of neighbors
at each time step
Parameter selection algorithm
Selecting a constant
Selecting the control gain
Choose
1
2
2 1( , ) | 0< , 1, ( , )
2KN
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
Kh
N 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
NNKQ
N
2N
N
Q
1K
Asymptotic convergence rate
Theorem. Suppose G is connected, then
for any given , we have
where
( , )
2
inflim 1
exp2
Kh
NNKQ
N
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
KQ
N
selection of edges:
initial states:
{( , ) } 0.2P i j E
(0) , 1, 2,...,30ix i i
a network with 30 nodes
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 JXr
X 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 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
A
A
means channel is active while
means channel
is the adjacency matri
is ina
x o
ctive
f
Distributed protocol
encoder φji
Z-1 )1(
1
tg
)1( tg
)(tx j
( )ji t
( 1)ji t ( )ji t
channel(j,i)
( )ija t
jitq
Distributed protocol
encoder φji
Z-1 )1(
1
tg
)1( tg
)(tx j
( )ji t
( 1)ji t ( )ji t
channel(j,i)
( )ija t
jitq
( )ijK 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 i
i N i Nx C C
1 ( ), 1, 2,...,i t iN N t i N
* 0d *
1 0max sup ( )i
i N td t d
0
( 1)
20
1
1inf ( )
m T
mk mT
L kT
Theorem. Suppose A3)-A6) hold. If
where ,
and the numbers of quantization levels satisfy
1/2*
1 10, ,
21
1
Thd h
T
0
sup ( ) / ( 1)t
g t g t
1(1) ,
(0) 2x
ij
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
hda t
K tt t
t
a
*( 1) ( 1) / ( ) 1( ) ( ( )) ,
( ) 2ij ij
g t g t g tt hd K t
g t
where
then
* 2
1/2
max | ( ) ( ) | 2limsup .
( )1 1
1
ij i j
Tt
x t x t N hd
g t hT
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 2
1 1 ( ) 1}
2 1
R
R
R
K TT
h
TT
h
h hhd
T
hdK
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
ij
ijij
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 exponential
convergence rate.
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