asymptotic mean ergodicity of average consensus estimators · 2020. 12. 15. · 5 conclusions van...

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

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

Conclusions

Group of n agents

all the inputs,1

n

n∑i=1

ui

Conclusions

Group of n agents

all the inputs,1

n

n∑i=1

ui

Conclusions

Group of n agents

all the inputs,1

n

n∑i=1

ui

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

Conclusions

Why random switching graphs?

[fragile]

Conclusions

Why random switching graphs?

[fragile]

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.

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.

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

Conclusions

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.

Conclusions

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...

Conclusions

Simulation

Conclusions

Simulation

Conclusions

Simulation

Conclusions

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.

Conclusions

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.

Conclusions

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...

Conclusions

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.

Conclusions

PI Estimator

yk = νk

Conclusions

PI Estimator

yk = νk

Conclusions

PI Estimator

yk = νk

Conclusions

PI Estimator

Consensus is achieved for the time average process.

The statistical average is the average of the inputs, so averageconsensus is achieved!

Conclusions

PI Estimator

Conclusions

PI Estimator

Conclusions

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

Conclusions

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

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.

Conclusions

Contribution: Confirm simulations with analysis

Strategy: Do analysis for a general estimator and applyresults to the P and PI estimators

Conclusions

Contribution: Confirm simulations with analysis

Strategy: Do analysis for a general estimator and applyresults to the P and PI estimators

Conclusions

Definition and ExamplesSeparated System

Outline

1 Introduction

5 Conclusions

Conclusions

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 .

Conclusions

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

Conclusions

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

Conclusions

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.

Conclusions

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.

Conclusions

Correctness

Conclusions

Correctness

Conclusions

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.

Conclusions

Note

Problem

Solution

Conclusions

Note

Problem

Solution

Conclusions

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).

Conclusions

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).

Conclusions

Outline

1 Introduction

5 Conclusions

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.

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)).

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

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.

Conclusions

Corollary 6

CX (k) = λ|k|

is asymptotically mean ergodic if and only if |λ| ≤ 1 and λ 6= 1.

Corollary 7

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.

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 ≤ `.

Conclusions

Outline

1 Introduction

5 Conclusions

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)

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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)

Conclusions

P Estimator

[A(L) B(L)

C (L) D(L)

]= I ⊗

[1− γ 0

1 0

]+ L⊗

[− kp −kp

0 0

! ρ(E[A(L̂k)

Conclusions

P Estimator

[A(L) B(L)

C (L) D(L)

]= I ⊗

[1− γ 0

1 0

]+ L⊗

[− kp −kp

0 0

! ρ(E[A(L̂k)

Conclusions

P Estimator

[A(L) B(L)

C (L) D(L)

]= I ⊗

[1− γ 0

1 0

]+ L⊗

[− kp −kp

0 0

! ρ(E[A(L̂k)

Conclusions

P Estimator

[A(L) B(L)

C (L) D(L)

]= I ⊗

[1− γ 0

1 0

]+ L⊗

[− kp −kp

0 0

! ρ(E[A(L̂k)

Conclusions

P Estimator

[A(L) B(L)

C (L) D(L)

]= I ⊗

[1− γ 0

1 0

]+ L⊗

[− kp −kp

0 0

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

])< 1 (for appropriate kp, kI , γ)

! Ci = Di = 0 for 1 ≤ i ≤ `! Correct

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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

! ρ(E[A(L̂k)

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.

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.

Conclusions

Summary

Conclusions

Summary

Conclusions

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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

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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.

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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

asymptotic mean ergodicity of average consensus estimators · 2020. 12. 15. · 5 conclusions van...

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