-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Asymptotic Mean Ergodicity of Average ConsensusEstimators
Bryan Van Scoy, Randy A. Freeman, Kevin M. Lynch
Northwestern University
June 6, 2014
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
What is average consensus?
Group of n agents
Each agent has a local inputui
Communication withneighbors represented bydirected graph
Want all agents to be ableto calculate the average of
all the inputs,1
n
n∑i=1
ui
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
What is average consensus?
Group of n agents
Each agent has a local inputui
Communication withneighbors represented bydirected graph
Want all agents to be ableto calculate the average of
all the inputs,1
n
n∑i=1
ui
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
What is average consensus?
Group of n agents
Each agent has a local inputui
Communication withneighbors represented bydirected graph
Want all agents to be ableto calculate the average of
all the inputs,1
n
n∑i=1
ui
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
What is average consensus?
Group of n agents
Each agent has a local inputui
Communication withneighbors represented bydirected graph
Want all agents to be ableto calculate the average of
all the inputs,1
n
n∑i=1
ui
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Why average consensus?
Average consensus is a key building block in many
distributedalgorithms such as the following:
Formation control
Distributed Kalman filtering
Distributed sensor fusion
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Why random switching graphs?
[fragile]
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Why random switching graphs?
[fragile]
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Assumptions
The graph Laplacian at time k is Lk ≡ Dk − Ak where Dk is
thedegree matrix and Ak is the adjacency matrix of the graph.
Assumptions
E[Lk ] balanced and connected
Lk i.i.d.
Lk independent of the estimator initial state for all k
Note
We do not require Lk to be balanced or connected at every
timestep.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Assumptions
The graph Laplacian at time k is Lk ≡ Dk − Ak where Dk is
thedegree matrix and Ak is the adjacency matrix of the graph.
Assumptions
E[Lk ] balanced and connected
Lk i.i.d.
Lk independent of the estimator initial state for all k
Note
We do not require Lk to be balanced or connected at every
timestep.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Outline
1 Introduction
2 Average Consensus EstimatorsInitial condition estimatorP
estimatorPI estimator
3 Polynomial Linear ProtocolDefinition and ExamplesSeparated
System
4 Asymptotic Mean Ergodicity and Main Theorem
5 Conclusions
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Initial Condition Estimator
Consider the well-known distributed algorithm
x ik+1 = xik −
∑j∈Ni
aij(xik − x
jk) (agent i)
xk+1 = xk − Lkxk (vectorized)
where xk is the state and Lk is the graph Laplacian at time k,
andx0 = u is the input.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Simulation
Consensus is achieved.
Estimate converges to a random variable whose mean is thecorrect
average (Li and Zhang, 2010).
Average could be approximated by averaging multiple trials.
This is inefficient...
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Simulation
Consensus is achieved.
Estimate converges to a random variable whose mean is thecorrect
average (Li and Zhang, 2010).
Average could be approximated by averaging multiple trials.
This is inefficient...
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Simulation
Consensus is achieved.
Estimate converges to a random variable whose mean is thecorrect
average (Li and Zhang, 2010).
Average could be approximated by averaging multiple trials.
This is inefficient...
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Simulation
Consensus is achieved.
Estimate converges to a random variable whose mean is thecorrect
average (Li and Zhang, 2010).
Average could be approximated by averaging multiple trials.
This is inefficient...
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
P Estimator
The P estimator equations are
xk+1 = (1− γ)xk − kpLkykyk = xk + u
where xk is the internal state and yk is the output at time k,
and γand kp are system parameters.
Special case
For γ = 0 and kp = 1, we have
yk+1 = yk − Lkyk
where y0 = x0 + u.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
P Estimator
The P estimator equations are
xk+1 = (1− γ)xk − kpLkykyk = xk + u
where xk is the internal state and yk is the output at time k,
and γand kp are system parameters.
Special case
For γ = 0 and kp = 1, we have
yk+1 = yk − Lkyk
where y0 = x0 + u.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
P Estimator (γ 6= 0)
Consensus is not achieved.
The time average of the output converges to the
statisticalaverage.
But the statistical average is not the average of the
inputs...
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
P Estimator (γ 6= 0)
Consensus is not achieved.
The time average of the output converges to the
statisticalaverage.
But the statistical average is not the average of the
inputs...
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
P Estimator (γ 6= 0)
Consensus is not achieved.
The time average of the output converges to the
statisticalaverage.
But the statistical average is not the average of the
inputs...
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
PI Estimator
The PI estimator equations are
νk+1 = (1− γ)νk + γu − kpLkνk + kILkηkηk+1 = ηk − kILkνk
yk = νk
where νk and ηk are the internal states at time k and γ, kp, and
kIare system parameters.
Convex combination of input and previous state.
Proportional error term.
Integral error term.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
PI Estimator
The PI estimator equations are
νk+1 = (1− γ)νk + γu − kpLkνk + kILkηkηk+1 = ηk − kILkνk
yk = νk
where νk and ηk are the internal states at time k and γ, kp, and
kIare system parameters.
Convex combination of input and previous state.
Proportional error term.
Integral error term.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
PI Estimator
The PI estimator equations are
νk+1 = (1− γ)νk + γu − kpLkνk + kILkηkηk+1 = ηk − kILkνk
yk = νk
where νk and ηk are the internal states at time k and γ, kp, and
kIare system parameters.
Convex combination of input and previous state.
Proportional error term.
Integral error term.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
PI Estimator
The PI estimator equations are
νk+1 = (1− γ)νk + γu − kpLkνk + kILkηkηk+1 = ηk − kILkνk
yk = νk
where νk and ηk are the internal states at time k and γ, kp, and
kIare system parameters.
Convex combination of input and previous state.
Proportional error term.
Integral error term.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
PI Estimator
Consensus is achieved for the time average process.
The time average of the output converges to the
statisticalaverage.
The statistical average is the average of the inputs, so
averageconsensus is achieved!
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
PI Estimator
Consensus is achieved for the time average process.
The time average of the output converges to the
statisticalaverage.
The statistical average is the average of the inputs, so
averageconsensus is achieved!
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
PI Estimator
Consensus is achieved for the time average process.
The time average of the output converges to the
statisticalaverage.
The statistical average is the average of the inputs, so
averageconsensus is achieved!
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
For average consensus, we need
Time average = Statistical average (ergodicity)
Statistical average = Average of inputs (correctness).
Then we can low-pass filter the output process to obtain
theaverage of the inputs.
AverageConsensusEstimator
Low−passFilter
Neighboring Agents
ui y i ỹ i
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
For average consensus, we need
Time average = Statistical average (ergodicity)
Statistical average = Average of inputs (correctness).
Then we can low-pass filter the output process to obtain
theaverage of the inputs.
AverageConsensusEstimator
Low−passFilter
Neighboring Agents
ui y i ỹ i
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Estimator Properties
Estimator Ergodic Correct
P, γ = 0 No Yes1
P, γ 6= 0 Yes No
PI Yes Yes
1 If the expectation of the initial state is zero.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Contribution: Confirm simulations with analysis
Strategy: Do analysis for a general estimator and applyresults
to the P and PI estimators
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Initial condition estimatorP estimatorPI estimator
Contribution: Confirm simulations with analysis
Strategy: Do analysis for a general estimator and applyresults
to the P and PI estimators
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Outline
1 Introduction
2 Average Consensus EstimatorsInitial condition estimatorP
estimatorPI estimator
3 Polynomial Linear ProtocolDefinition and ExamplesSeparated
System
4 Asymptotic Mean Ergodicity and Main Theorem
5 Conclusions
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Polynomial Linear Protocol
A polynomial linear protocol (Freeman, Nelson, and Lynch,
2010)of degree ` is the collection Σ(L) = [A(L),B(L),C (L),D(L)]
where
A(L) ≡∑̀i=0
Li ⊗ Ai B(L) ≡∑̀i=0
Li ⊗ Bi
C (L) ≡∑̀i=0
Li ⊗ Ci D(L) ≡∑̀i=0
Li ⊗ Di
are polynomials in L which describe the linear system
xk+1 = A(L)xk + B(L)uk
yk = C (L)xk + D(L)uk .
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Examples
Example 1 (P Estimator)
The P estimator is a polynomial linear protocol of degree one
withparameters γ and kp where[
A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[−kp −kp
0 0
]
Example 2 (PI Estimator)
The PI estimator is a polynomial linear protocol of degree one
withparameters γ, kp, and kI where[
A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ −kp kI 0−kI 0 0
0 0 0
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Examples
Example 1 (P Estimator)
The P estimator is a polynomial linear protocol of degree one
withparameters γ and kp where[
A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[−kp −kp
0 0
]
Example 2 (PI Estimator)
The PI estimator is a polynomial linear protocol of degree one
withparameters γ, kp, and kI where[
A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ −kp kI 0−kI 0 0
0 0 0
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Objective
Objective
Want conditions under which the output process yk of apolynomial
linear protocol Σ(Lk) is
1 Asymptotically mean ergodic
2 Correct (i.e., the expectation converges to the average of
theinputs)
Then the low-pass filtered output converges to the average of
theinputs.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Correctness
A polynomial linear protocol Σ(Lk) of degree one is correct
ifand only if Σ(E[Lk ]) converges to the average of the inputs.
This has been characterized (Freeman, Nelson, and
Lynch,2010).
A necessary condition is A0 must have an eigenvalue at one.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Correctness
A polynomial linear protocol Σ(Lk) of degree one is correct
ifand only if Σ(E[Lk ]) converges to the average of the inputs.
This has been characterized (Freeman, Nelson, and
Lynch,2010).
A necessary condition is A0 must have an eigenvalue at one.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Correctness
A polynomial linear protocol Σ(Lk) of degree one is correct
ifand only if Σ(E[Lk ]) converges to the average of the inputs.
This has been characterized (Freeman, Nelson, and
Lynch,2010).
A necessary condition is A0 must have an eigenvalue at one.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Note
A0 must have an eigenvalue at one for the system to
becorrect.
The Laplacian always has an eigenvalue at zero.
Therefore, correct systems have an eigenvalue at one.
Problem
The steady-state variance of the state could be infinite!
Solution
The state corresponding to the eigenvalue at one must
beunobservable.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Note
A0 must have an eigenvalue at one for the system to
becorrect.
The Laplacian always has an eigenvalue at zero.
Therefore, correct systems have an eigenvalue at one.
Problem
The steady-state variance of the state could be infinite!
Solution
The state corresponding to the eigenvalue at one must
beunobservable.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Note
A0 must have an eigenvalue at one for the system to
becorrect.
The Laplacian always has an eigenvalue at zero.
Therefore, correct systems have an eigenvalue at one.
Problem
The steady-state variance of the state could be infinite!
Solution
The state corresponding to the eigenvalue at one must
beunobservable.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition and ExamplesSeparated System
Separated System
Definition 3 (Reduced Laplacian)
The reduced Laplacian L̂ is defined as L̂ := STLS whereQ =
[v S
]∈ Rn×n is orthogonal and v = 1n/
√n.
Performing the change of variable x̃k = (Q ⊗ I )T xk , the
separatedsystem Σ̃(L) is
Ã(L) =
[A0 (v ⊗ I )TA(L)(S ⊗ I )0 A(L̂)
]B̃(L) =
[(v ⊗ I )TB(L)(S ⊗ I )TB(L)
]C̃ (L) =
[v ⊗ C0 C (L)(S ⊗ I )
]D̃(L) = D(L).
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
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Conclusions
Definition and ExamplesSeparated System
Separated System
Definition 3 (Reduced Laplacian)
The reduced Laplacian L̂ is defined as L̂ := STLS whereQ =
[v S
]∈ Rn×n is orthogonal and v = 1n/
√n.
Performing the change of variable x̃k = (Q ⊗ I )T xk , the
separatedsystem Σ̃(L) is
Ã(L) =
[A0 (v ⊗ I )TA(L)(S ⊗ I )0 A(L̂)
]B̃(L) =
[(v ⊗ I )TB(L)(S ⊗ I )TB(L)
]C̃ (L) =
[v ⊗ C0 C (L)(S ⊗ I )
]D̃(L) = D(L).
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Outline
1 Introduction
2 Average Consensus EstimatorsInitial condition estimatorP
estimatorPI estimator
3 Polynomial Linear ProtocolDefinition and ExamplesSeparated
System
4 Asymptotic Mean Ergodicity and Main Theorem
5 Conclusions
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Definition 4 (Asymptotically Wide-Sense Stationary)
The process Xk is asymptotically wide-sense stationary if and
onlyif the mean and covariance of the steady-state process do
notchange with time; that is, the limits
mX ≡ limn→∞
E [Xn] and CX (k) ≡ limn→∞
COV[Xk+n,Xn]
exist and are finite where mX is the mean and CX (k) is
thecovariance of the steady-state process.
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IntroductionAverage Consensus Estimators
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Conclusions
Theorem 5 (Asymptotic Mean Ergodicity)
Let {Xk}∞k=1 be a single-sided asymptotically wide-sense
stationarydiscrete-time random process with limiting mean mX and
limitingcovariance CX (k). The process is asymptotically mean
ergodic,that is,
limT→∞
limn→∞
1
T
T−1∑k=0
Xk+n = mX
in the mean square sense if and only if
limT→∞
1
T
T−1∑k=−(T−1)
(1− |k |
T
)CX (k) = 0
(similar to a result in (Leon-Garcia, 2008)).
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
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IntroductionAverage Consensus Estimators
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Corollary 6
An asymptotically wide-sense stationary random process Xk
withsteady-state covariance given by
CX (k) = λ|k|
is asymptotically mean ergodic if and only if |λ| ≤ 1 and λ 6=
1.
Corollary 7
An asymptotically wide-sense stationary random process Xk
withsteady-state covariance given by
CX (k) = CA|k|B
where A ∈ Rn×n is convergent, B ∈ Rn×1, C ∈ R1×n, and
anyeigenvalue of A at one is either uncontrollable through B
orunobservable through C, is asymptotically mean ergodic.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
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IntroductionAverage Consensus Estimators
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Conclusions
Corollary 6
An asymptotically wide-sense stationary random process Xk
withsteady-state covariance given by
CX (k) = λ|k|
is asymptotically mean ergodic if and only if |λ| ≤ 1 and λ 6=
1.
Corollary 7
An asymptotically wide-sense stationary random process Xk
withsteady-state covariance given by
CX (k) = CA|k|B
where A ∈ Rn×n is convergent, B ∈ Rn×1, C ∈ R1×n, and
anyeigenvalue of A at one is either uncontrollable through B
orunobservable through C, is asymptotically mean ergodic.
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IntroductionAverage Consensus Estimators
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Conclusions
Main Theorem
Theorem 8 (Asymptotically Mean Ergodic)
Consider the time-varying polynomial linear protocol Σ(Lk)
ofdegree ` based on the time-varying Laplacian Lk where E[Lk ]
isbalanced and connected, and Lk are i.i.d. and independent of
theinitial state for all k. The output process due to a constant
inputis asymptotically mean ergodic if the following hold:
1 A0 is convergent,
2 any eigenvalues of A0 at one are unobservable through C0,
3 ρ(E[A(L̂k)
])< 1, and
4 Ci = Di = 0 for 1 ≤ i ≤ `.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
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IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
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Conclusions
Outline
1 Introduction
2 Average Consensus EstimatorsInitial condition estimatorP
estimatorPI estimator
3 Polynomial Linear ProtocolDefinition and ExamplesSeparated
System
4 Asymptotic Mean Ergodicity and Main Theorem
5 Conclusions
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1−��7
0γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 1: γ = 0
! A0 is convergent
% any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct (if the expectation of the
initial state is zero)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1−��7
0γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 1: γ = 0
! A0 is convergent
% any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct (if the expectation of the
initial state is zero)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1−��7
0γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 1: γ = 0
! A0 is convergent
% any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct (if the expectation of the
initial state is zero)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1−��7
0γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 1: γ = 0
! A0 is convergent
% any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct (if the expectation of the
initial state is zero)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1−��7
0γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 1: γ = 0
! A0 is convergent
% any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct (if the expectation of the
initial state is zero)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1−��7
0γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 1: γ = 0
! A0 is convergent
% any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct (if the expectation of the
initial state is zero)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 2: γ 6= 0! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, γ)
! Ci = Di = 0 for 1 ≤ i ≤ `% Correct (need A0 to have an
eigenvalue at one)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 2: γ 6= 0! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, γ)
! Ci = Di = 0 for 1 ≤ i ≤ `% Correct (need A0 to have an
eigenvalue at one)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 2: γ 6= 0! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, γ)
! Ci = Di = 0 for 1 ≤ i ≤ `% Correct (need A0 to have an
eigenvalue at one)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 2: γ 6= 0! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, γ)
! Ci = Di = 0 for 1 ≤ i ≤ `% Correct (need A0 to have an
eigenvalue at one)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 2: γ 6= 0! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, γ)
! Ci = Di = 0 for 1 ≤ i ≤ `% Correct (need A0 to have an
eigenvalue at one)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
P Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
[1− γ 0
1 0
]+ L⊗
[− kp −kp
0 0
]Case 2: γ 6= 0! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, γ)
! Ci = Di = 0 for 1 ≤ i ≤ `% Correct (need A0 to have an
eigenvalue at one)
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
PI Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ − kp kI 0− kI 0 0
0 0 0
! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, kI , γ)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
PI Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ − kp kI 0− kI 0 0
0 0 0
! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, kI , γ)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
PI Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ − kp kI 0− kI 0 0
0 0 0
! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, kI , γ)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
PI Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ − kp kI 0− kI 0 0
0 0 0
! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, kI , γ)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
PI Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ − kp kI 0− kI 0 0
0 0 0
! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, kI , γ)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
PI Estimator
[A(L) B(L)
C (L) D(L)
]= I ⊗
1− γ 0 γ0 1 01 0 0
+ L⊗ − kp kI 0− kI 0 0
0 0 0
! A0 is convergent
! any eigenvalues of A0 at one are unobservable through C0
! ρ(E[A(L̂k)
])< 1 (for appropriate kp, kI , γ)
! Ci = Di = 0 for 1 ≤ i ≤ `! Correct
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Estimator Properties
Estimator Ergodic Correct
P, γ = 0 No Yes1
P, γ 6= 0 Yes No
PI Yes Yes
1 If the expectation of the initial state is zero.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Summary
Defined asymptotic mean ergodicity and gave an
ergodictheorem.
Characterized the asymptotic mean ergodicity property
forpolynomial linear protocols.
Applied results to the P and PI estimators to explain
behaviorover i.i.d. random graphs.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Summary
Defined asymptotic mean ergodicity and gave an
ergodictheorem.
Characterized the asymptotic mean ergodicity property
forpolynomial linear protocols.
Applied results to the P and PI estimators to explain
behaviorover i.i.d. random graphs.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
Summary
Defined asymptotic mean ergodicity and gave an
ergodictheorem.
Characterized the asymptotic mean ergodicity property
forpolynomial linear protocols.
Applied results to the P and PI estimators to explain
behaviorover i.i.d. random graphs.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main
Theorem
Conclusions
ReferencesCai, Kai and H. Ishii (2012). “Average Consensus on
Arbitrary Strongly Connected Digraphs with Dynamic
Topologies”. In: Proceedings of the 2012 American Control
Conference, pp. 14–19.Chen, Yin et al. (2010). “Corrective
Consensus: Converging to the Exact Average”. In: Proceedings of the
49th
IEEE Conference on Decision and Control, pp. 1221–1228. doi:
10.1109/CDC.2010.5717925.Cortes, J. (2009). “Distributed Kriged
Kalman Filter for Spatial Estimation”. In: IEEE Transactions on
Automatic
Control 54.12, pp. 2816–2827. issn: 0018-9286. doi:
10.1109/TAC.2009.2034192.Freeman, R.A., T.R. Nelson, and K.M. Lynch
(2010). “A Complete Characterization of a Class of Robust
Linear
Average Consensus Protocols”. In: Proceedings of the 2010
American Control Conference, pp. 3198–3203.Freeman, R.A., Peng
Yang, and K.M. Lynch (2006). “Stability and Convergence Properties
of Dynamic Average
Consensus Estimators”. In: Proceedings of the 45th IEEE
Conference on Decision and Control, pp. 338–343.doi:
10.1109/CDC.2006.377078.
Leon-Garcia, A. (2008). Probability, Statistics, and Random
Processes for Electrical Engineering. Pearson/PrenticeHall.
Li, Tao and Ji-Feng Zhang (2010). “Consensus Conditions of
Multi-Agent Systems With Time-Varying Topologiesand Stochastic
Communication Noises”. In: IEEE Transactions on Automatic Control
55.9, pp. 2043–2057.issn: 0018-9286. doi:
10.1109/TAC.2010.2042982.
Peterson, Cameron K. and Derek A. Paley (2013). “Distributed
Estimation for Motion Coordination in an UnknownSpatially Varying
Flowfield”. In: Journal of Guidance, Control, and Dynamics 36.3,
pp. 894–898. issn:0731-5090. doi: 10.2514/1.59453.
Vaidya, N.H., C.N. Hadjicostis, and A.D. Dominguez-Garcia
(2012). “Robust Average Consensus over PacketDropping Links:
Analysis via Coefficients of Ergodicity”. In: Proceedings of the
51st IEEE Conference onDecision and Control, pp. 2761–2766. doi:
10.1109/CDC.2012.6426252.
Yang, Peng, R.A. Freeman, and K.M. Lynch (2008). “Multi-Agent
Coordination by Decentralized Estimation andControl”. In: IEEE
Transactions on Automatic Control 53.11, pp. 2480–2496. issn:
0018-9286. doi:10.1109/TAC.2008.2006925.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average
Consensus Estimators
http://dx.doi.org/10.1109/CDC.2010.5717925http://dx.doi.org/10.1109/TAC.2009.2034192http://dx.doi.org/10.1109/CDC.2006.377078http://dx.doi.org/10.1109/TAC.2010.2042982http://dx.doi.org/10.2514/1.59453http://dx.doi.org/10.1109/CDC.2012.6426252http://dx.doi.org/10.1109/TAC.2008.2006925
IntroductionAverage Consensus EstimatorsInitial condition
estimatorP estimatorPI estimator
Polynomial Linear ProtocolDefinition and ExamplesSeparated
System
Asymptotic Mean Ergodicity and Main TheoremConclusions