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 …