of 38/38

Carnegie Mellon Kalman and Kalman Bucy @ 50: Distributed and Intermittency José M. F. Moura Joint Work with Soummya Kar Advanced Network Colloquium University of Maryland College Park, MD November 04, 2011 Acknowledgements: NSF under grants CCF-1011903 and CCF-1018509, and AFOSR grant FA95501010291

View

56Download

1

Embed Size (px)

DESCRIPTION

Kalman and Kalman Bucy @ 50: Distributed and Intermittency. José M. F. Moura Joint Work with Soummya Kar Advanced Network Colloquium University of Maryland College Park, MD November 04, 2011. Acknowledgements: NSF under grants CCF-1011903 and CCF-1018509, and AFOSR grant FA95501010291. - PowerPoint PPT Presentation

Higher dimensional consensus algorithms in sensor networksTexPoint
fonts used in EMF.

Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAAA

José M. F. Moura

Advanced Network Colloquium

University of Maryland

College Park, MD

November 04, 2011

*

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

1939-41: A. N. Kolmogorov, "Interpolation und Extrapolation von Stationaren Zufalligen Folgen,“ Bull. Acad. Sci. USSR, 1941

*

Carnegie Mellon

Kalman Filter @ 51

*

Kalman-Bucy Filter @ 50

*

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

Optimality: structural conditions – observability/controllability

“Kalman Gain”

Cooperative solution

Cooperation: better understanding/global knowledge

Filtering Then … Filtering Today

Random protocols: Gossip

Limited Resources: Quantization

Two Linear Estimators:

LU: Stochastic Approximation

Performance Analysis: Asymptotics

(Distributed) Consensus:

Asymptotic agreement: λ2 (L) > 0

DeGroot, JASA 74; Tsitsiklis, 74, Tsitsiklis, Bertsekas, Athans, IEEE T-AC 1986

Jadbabaie, Lin, Morse, IEEE T-AC 2003

Distributed architecture, no fusion center, nor parallel architecture

*

Consensus (reinterpreted): a.s. convergence to unbiased rv θ:

Consensus in Random Environments

*

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

fast comm. (cooperation) vs slow sensing (exogenous, local)

Consensus + innovations: In and Out balanced interactions

communications and sensing at every time step

Distributed filtering: Consensus +Innovations

Structural failures (random links)/ random protocol (gossip):

Quantization/communication noise

Distributed inference: Generalized linear unbiased (GLU)

Consensus: local avg

Compare distributed to centralized performance

Structural conditions

Distributed connectivity: Network connected in the mean

*

Estimator:

A6. assumption: Weight sequences

*

Consistency: sensor n is consistent

Asymptotically normality:

*

Define

Let

Strong convergence rates: study sample paths more critically

Characterize information flow (consensus): study convergence to averaged estimate

Study limiting properties of averaged estimate:

Rate at which convergence of averaged estimate to centralized estimate

Properties of centralized estimator used to show convergence to

*

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Model:

Intermittent observations:

Optimal Linear Filter (conditioned on path of observations) – Kalman filter with Random Riccati Equation

Carnegie Mellon

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Define operators f0(X), f1(X) and reexpress Pt:

[2] S. Kar, Bruno Sinopoli and J.M.F. Moura, “Kalman filtering with intermittent observations: weak convergence to a stationary distribution,” IEEE Tr. Aut Cr, Jan 2012.

Carnegie Mellon

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Stochastic Boundedness:

Carnegie Mellon

Interested in probability of rare events:

As ϒ 1: rare event: steady state cov. stays away from P* (det. Riccati)

RRE satisfies an MDP at a given scale:

Pr(rare event) decays exponentially fast with good rate function

String:

Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” IEEE Tr. Automatic Control;

Carnegie Mellon

Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” IEEE Tr. Automatic Control

P*

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

‘Fractal like’:

Carnegie Mellon

Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” accepted EEE Tr. Automatic Control

Carnegie Mellon

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Intermittency: sensors fail; comm links fail

Gossip: random protocol

Limited power: quantization

Mixed scale: can optimize rate of convergence and limiting covariance

Structural conditions: distributed observability+ mean connectivitiy

Asymptotic properties: Distributed as Good as Centralized

unbiased, consistent, normal, mixed scale converges to optimal centralized

*

Stochastically bounded as long as rate of measurements strictly positive

Random Riccati Equation: Probability measure of random covariance is invariant to initial condition

Support of invariant measure is ‘fractal like’

Moderate Deviation Principle: rate of decay of probability of ‘bad’ (rare) events as rate of measurements grows to 1

All is computable

Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAAA

José M. F. Moura

Advanced Network Colloquium

University of Maryland

College Park, MD

November 04, 2011

*

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

1939-41: A. N. Kolmogorov, "Interpolation und Extrapolation von Stationaren Zufalligen Folgen,“ Bull. Acad. Sci. USSR, 1941

*

Carnegie Mellon

Kalman Filter @ 51

*

Kalman-Bucy Filter @ 50

*

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

Optimality: structural conditions – observability/controllability

“Kalman Gain”

Cooperative solution

Cooperation: better understanding/global knowledge

Filtering Then … Filtering Today

Random protocols: Gossip

Limited Resources: Quantization

Two Linear Estimators:

LU: Stochastic Approximation

Performance Analysis: Asymptotics

(Distributed) Consensus:

Asymptotic agreement: λ2 (L) > 0

DeGroot, JASA 74; Tsitsiklis, 74, Tsitsiklis, Bertsekas, Athans, IEEE T-AC 1986

Jadbabaie, Lin, Morse, IEEE T-AC 2003

Distributed architecture, no fusion center, nor parallel architecture

*

Consensus (reinterpreted): a.s. convergence to unbiased rv θ:

Consensus in Random Environments

*

Filtering Then … Filtering Today

Distributed Filtering: Consensus + innovations

Intermittency: Infrastructure failures, Sensor failures

Random protocols: Gossip

Limited Resources: Quantization

Stochastic boundedness

Invariant distribution

Moderate deviation

fast comm. (cooperation) vs slow sensing (exogenous, local)

Consensus + innovations: In and Out balanced interactions

communications and sensing at every time step

Distributed filtering: Consensus +Innovations

Structural failures (random links)/ random protocol (gossip):

Quantization/communication noise

Distributed inference: Generalized linear unbiased (GLU)

Consensus: local avg

Compare distributed to centralized performance

Structural conditions

Distributed connectivity: Network connected in the mean

*

Estimator:

A6. assumption: Weight sequences

*

Consistency: sensor n is consistent

Asymptotically normality:

*

Define

Let

Strong convergence rates: study sample paths more critically

Characterize information flow (consensus): study convergence to averaged estimate

Study limiting properties of averaged estimate:

Rate at which convergence of averaged estimate to centralized estimate

Properties of centralized estimator used to show convergence to

*

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Model:

Intermittent observations:

Optimal Linear Filter (conditioned on path of observations) – Kalman filter with Random Riccati Equation

Carnegie Mellon

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Define operators f0(X), f1(X) and reexpress Pt:

[2] S. Kar, Bruno Sinopoli and J.M.F. Moura, “Kalman filtering with intermittent observations: weak convergence to a stationary distribution,” IEEE Tr. Aut Cr, Jan 2012.

Carnegie Mellon

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Stochastic Boundedness:

Carnegie Mellon

Interested in probability of rare events:

As ϒ 1: rare event: steady state cov. stays away from P* (det. Riccati)

RRE satisfies an MDP at a given scale:

Pr(rare event) decays exponentially fast with good rate function

String:

Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” IEEE Tr. Automatic Control;

Carnegie Mellon

Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” IEEE Tr. Automatic Control

P*

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

‘Fractal like’:

Carnegie Mellon

Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” accepted EEE Tr. Automatic Control

Carnegie Mellon

Random Riccati Equation: Moderate deviation principle

Rate of decay of probability of rare events

Scalar numerical example

Intermittency: sensors fail; comm links fail

Gossip: random protocol

Limited power: quantization

Mixed scale: can optimize rate of convergence and limiting covariance

Structural conditions: distributed observability+ mean connectivitiy

Asymptotic properties: Distributed as Good as Centralized

unbiased, consistent, normal, mixed scale converges to optimal centralized

*

Stochastically bounded as long as rate of measurements strictly positive

Random Riccati Equation: Probability measure of random covariance is invariant to initial condition

Support of invariant measure is ‘fractal like’

Moderate Deviation Principle: rate of decay of probability of ‘bad’ (rare) events as rate of measurements grows to 1

All is computable