Distributed Consensus with Limited Communication Data Rate Tao Li Key Laboratory of Systems &...

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



Communication energy minimization



Communication energy minimization Case with time-varying topologies

protocol design and closed-loop analysis finite duration of link failures



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 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

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 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 ?

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(

1

tg q

)1( tg

)(tx j

)(tj

)1( tj )(tj

channel(j,i)


encoder φj

Z-1 )1(

1

tg q

)1( tg

)(tx j

)(tj

)1( tj )(tj

finite-level symmetric

uniform quantizer

channel(j,i)


encoder φj

Z-1 )1(

1

tg q

)1( tg

)(tx j

)(tj

)1( tj )(tj

innovation

channel(j,i)


encoder φj

Z-1 )1(

1

tg q

)1( tg

)(tx j

)(tj

)1( tj )(tj

scaling

function

channel(j,i)


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


decoder ψji

)1( tg

Z-1

)(tj )(ˆ tx jichannel

(j,i)


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)


protocol

( ) ( ( ) ( )), 0,1,...,

1, 2,...,i

jii ij ij N

u t h a x t t t

i N


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


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 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

where

convergence rate:

|| ( ) || ( ), tt O t

2 2 *

1

1 2 1( , ) - 1

2 ( ) 2 2N

h

N h hdK h


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


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

2 2 *

1

1 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 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

1 bit quantizer

convergence factor ： 0.95

1 bit quantizer

convergence 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 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 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

A

A

means channel is active while

means channel

is the adjacency matri

is ina

x o

ctive

f


encoder φji

Z-1 )1(

1

tg

)1( tg

)(tx j

( )ji t

( 1)ji t ( )ji t

channel(j,i)

( )ija t

jitq


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


decoder ψji

)1( tg( )ji t

Z-1

)(ˆ tx jichannel

(j,i)( )ija t


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.

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 …

Distributed Consensus with Limited Communication Data Rate Tao Li Key Laboratory of Systems &...

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