asymptotic mean ergodicity of average consensus estimators · 2020. 12. 15. · 5 conclusions van...
TRANSCRIPT
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main Theorem
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 Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main Theorem
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.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main Theorem
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.
Van Scoy, Freeman, Lynch Asymptotic Mean Ergodicity of Average Consensus Estimators
-
IntroductionAverage Consensus Estimators
Polynomial Linear ProtocolAsymptotic Mean Ergodicity and Main Theorem
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 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
-
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