Consensus Problems in Networks

1
Consensus Problems in Networks

• Aman Agarwal
• EYES 2007 intern
• Advisor Prof. Mostofi
• ECE, University of New Mexico
• July 5, 2007

2
Background

• Cooperative control for multi agent systems have
  a lot of applications.
• Formation control
• Non formation cooperative control
• Key issue is shared information

3
Consensus Protocols

• xi be the information state of agent i
• Continuous time
• xi(t) S aij (xj - xi )
• ? x(t) - L x(t)
• ? x(t) e-Lt x(0) Lt t?8 e-Lt?1vT vT1
  1 vTL 0
• ? x(t) 1vTx(0) where v is the eigen vector
  corr to eigen value 0
• Discrete time
• xi k1 S aijk xik
• ? xk1 Dk xk
• ? xk1 Dk x0 Lt k?8 Dk?1 vT vT1 1
  vTD vT
• ? x(t) 1vTx(0) where v is the eigen
  vector corr to eigen value 1
• L has an eigen value 0 corresponding to the
  solution and D has an eigen value 1 corresponding
  to the solution

4
Convergence of Consensus Protocols

• Equilibrium value function of the initial state.
• Agents that can pass info to all the other
  vehicles have a say in the final value.
• Second smallest eigen value (L) or second largest
  eigen value(-L) or Fiedler eigen value
• Determines the speed of convergence
• Dense graphs ? ?2 is relatively large.
• Sparse graphs ? ?2 is relatively small.
• The third smallest eigen value (of L) should be
  far away from ?2 for faster convergence.
• Multiple values of ?2 also affect the speed of
  convergence.
• Ideally we would like to have ?2 as a simple
  eigen value for fast convergence.

5
Binary Consensus Problems

• In most consensus applications, the agents will
  communicate their status wirelessly.
• On the bit level there is receiver noise.
• Noise is not bounded ?no transition point beyond
  which consensus is guaranteed ?a probabilistic
  approach to characterize and understand the
  behavior of the network.
• To examine this effect we look at binary
  consensus problems.
• Assume that the network is fully connected.
• A majority poll to assert if the majority of the
  nodes are in consensus and ? node updates its own
  information

6
Binary Consensus Problems

• Model 1
• noise
  decision
• bj(k1) Dec( S bj,i(k)/ M ) Dec(x) 1 x ?
  0.5      0 x lt 0.5
• Dec( S bj(k)/ M S
  nj,i(k)/ M )
• Dec( S S(k)/ M wi(k) )
• S(k) state of the system at time k S bj(k)
• ?i(k) probability state S(k) i
• ?(k) ?0(k) ?1(k) ?n(k) probability
  vector
• Pij probability S(k) j S(k) i
• MCj kij (1-ki)M-j where ki prob
  i / M wi(k) gt 0.5 S(k)i
• ?(k1) PT ?(k) P Pij
• ?(k) ( PT )k ?(0) asymptotic behavior of
  probabilities

bj(k)
bji(k)
bj(k1)
7
Model 1 M4 X(0)0 1 1 1
Probability plot for Model 1 with sigma 0.5
Probability plot for Model 1 with sigma 0.75
Probability plot for Model 1 with sigma 1
Probability plot for Model 1 with sigma 2
8
Binary Consensus Problems

• Model 2(a) 2(b)
• Noise
  noise filtering decision
• bj,i D(k) Dec( bj,i(k) ) Dec(x) 1 x
  ? th (normally 0.5)
  
  0         x lt th
• bj(k1) Dec( S bj,i D(k)/ M ) Dec(x) 1
  x ? 0.5
  
  0        x lt 0.5
• Dec( S Dec( bj,i(k) )/ M )
• Pij probability S(k) j S(k) i
• MCj kij (1-ki)m-j where ki
  prob S Dec( bj,i(k) )/ M gt 0.5 s(k)i
• 
  M
• S MCl P bj,i D(k) 1
  j P bj,i D(k) 0 m-j
• 
  L ?m/2?
• Pbj,i d(k) 1 Pbj,i d(k)1
  bj(k)1Pbj(k) 1Pbj,i d(k)1
  bj(k)0Pbj(k) 0
• i / m Q(0.5/?)(1 - 2i / M)
• Pbj,i d(k) 0 Pbj,i d(k)0 bj(k)
  1Pbj(k) 1Pbj,i d(k)0 bj(k)0Pbj(k)
  0
• 1- i / M -
  Q(0.5/?)(1 - 2i / M)

bj(k)
bji(k)
bj(k1)
bji D(k)
9
Model 2(a) The noise is filtered first by
thresholding the received values at threshold
level of 0.5 to ensure that the majority decision
is made on correct data only. M4 X(0)0 1 1
1
Probability plot for Model 2(a) with sigma 0.5
Probability plot for Model 2(a) with sigma 0.75
Probability plot for Model 2(a) with sigma 1
Probability plot for Model 2(a) with sigma 2
10
Model 2(b) In this case the threshold for the
comm. noise is dynamically chosen by monitoring
the values that the nodes are sending and then
updating the threshold based on the differential
probabilities of sending a 1 or a 0.
11
Model 2(b) M4 X(0)0 1 1 1
Probability plot for Model 2(b) with sigma 0.5
Probability plot for Model 2(b) with sigma 0.75
Probability plot for Model 2(b) with sigma 1
12
Binary Consensus Problems

• Model 3
• Noise noise
  filtering soft info
  decision
• bj,i D(k) Dec( bj,i(k) ) Dec(x) 1
  x ? th (normally 0.5)
  
  
  0         x lt th
• bj(k1) Dec( S E bj(k) bj,i(k) / M )
  Dec(x) 1 x ? 0.5
• 0 x lt 0.5
• Where E bj(k) bj,i(k)
  f( bj,i(k) - 1 ) P bj(k) 1
  
• 
  f( bj,i(k) - 1 ) P bj(k) 1 f( bj,i(k) )
  P bj(k) 0
• And f(x) pdf of N ( 0 , ?2 )
• Pij probability s(k) j s(k) i
  finding the probability of transition becomes
  very tedious and complex in this case so we
  simulate the case and calculate the probability
  statistically by taking a lot of samples ( min
  1000 )

bj(k)
bji(k)
bj(k1)
bji D(k)
Ebj(k)bji D(k)
13
Model 3
Probability plot for Model 2 with sigma 0.5
Probability plot for Model 3 with sigma 0.75
Probability plot for Model 3 with sigma 1
Probability plot for Model 3 with sigma 2
14
Comparison of models

• Model 1 ? performance sharply degrades for larger
  noise variances (sigma gt 0.5).
• Model 2(a) ? Better than model 1 but cant handle
  large noise variances (sigma gt 1).
• Model 2(b) ? better than model 2(a). The dynamic
  threshold works but only if the noise variance is
  lt 1, because for larger noises a threshold
  between 0,1 will not work.
• Model 3 ? is very robust and can perform with
  large noises also (sigma gt1) but we trade off
  speed of convergence for handling larger noises.

15
Detection Estimation

• A group of nodes where each node has limited
  sensing capabilities ? rely on the group for
  improving its estimation/detection quality.
• Estimation ? each agent has an estimate of the
  parameter of interest which can take values over
  an infinite set or a known finite set.
• Detection ? parameter of interest takes values
  from a finite known set

16
Sensing noise noise
filtering decision
comm. noise noise filtering
decision
Binary Detection
Oj(k1)
S
Sj(k)
Sj(k)
Oj(k)
Oji(k)
Oj D(k)

• For k 1
• Sj(k) event sensed at time k
• Oj(k) opinion formed at time k
• Oji(k) Oj(k) sn ji
• Oi D(k) Dec ( Oji(k) ) Dec(x) 1 x
  0.5
• 0 x lt 0.5
• Oj(k1) Dec( ( ?Oi D(k) Oj(k) Sj(k) ) /
  M1 )

sn 0.5 , ss 1 S1
17
Binary Detection

• Every node has M 1 different values to weigh
  every time
• Weigh nodes with better communication or better
  sensing differently
• Define trust factor
• Trust factors ? either time invariant or time
  variant
• Should update themselves over time

18
Binary Detection

• Trust factors one way of implementing this is as
  follows

Average consensus
Different weights
How nodes with good sensing and good
communication affect the consensus X(0) 0 0 0
0 0 1 1 1