kalman and kalman bucy @ 50: distributed and intermittency
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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 PresentationTRANSCRIPT
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Kalman and Kalman Bucy @ 50: Distributed and Intermittency
José M. F. MouraJoint Work with Soummya Kar
Advanced Network ColloquiumUniversity of Maryland
College Park, MDNovember 04, 2011
Acknowledgements: NSF under grants CCF-1011903 and CCF-1018509, and AFOSR grant FA95501010291
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Outline Brief Historical Comments: From Kolmogorov to Kalman-Bucy Filtering Then … Filtering Today Consensus: Distributed Averaging in Random Environments Distributed Filtering: Consensus + innovations
Random field (parameter) estimation: Large scale Intermittency: Infrastructure failures, Sensor failures Random protocols: Gossip Limited Resources: Quantization
Linear Parameter Estimator: Mixed time scale Linear filtering: Intermittency – Random Riccati Eqn.
Stochastic boundedness Invariant distribution Moderate deviation
Conclusion
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Outline Brief Historical Comments: From Kolmogorov to Kalman-Bucy Filtering Then … Filtering Today Consensus: Distributed Averaging in Random Environments Distributed Filtering: Consensus + innovations
Random field (parameter) estimation: Large scale Intermittency: Infrastructure failures, Sensor failures Random protocols: Gossip Limited Resources: Quantization
Linear Parameter Estimator: Mixed time scale Linear filtering: Intermittency – Random Riccati Eqn.
Stochastic boundedness Invariant distribution Moderate deviation
Conclusion
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In the 40’s
Wiener Model
Wiener filter
Wiener-Hopf equation (1931; 1942)
1939-41: A. N. Kolmogorov, "Interpolation und Extrapolation von Stationaren Zufalligen Folgen,“ Bull. Acad. Sci. USSR, 1941
Dec 1940: anti-aircraft control pr.–extract signal from noise: N. Wiener "Extrap., Interp., and Smoothing of Stat. time Series with Eng. Applications," 1942; declassified, published Wiley, NY, 1949.
Carnegie MellonNorbert WIENER. The extrapolation, interpolation and smoothing of stationary time series with engineering applications. [Washington, D.C.: National Defense Research Council,] 1942.
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Kalman Filter @ 51
Trans. of the ASME-J. of Basic Eng., 82 (Series D): 35-45, March 1960
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Kalman-Bucy Filter @ 50
Transactions of the ASME-Journal of Basic Eng., 83 (Series D): 95-108, March 1961
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Outline Brief Historical Comments: From Kolmogorov to Kalman-Bucy Filtering Then … Filtering Today Consensus: Distributed Averaging in Random Environments Distributed Filtering: Consensus + innovations
Random field (parameter) estimation: Large scale Intermittency: Infrastructure failures, Sensor failures Random protocols: Gossip Limited Resources: Quantization
Linear Parameter Estimator: Mixed time scale Linear filtering: Intermittency – Random Riccati Eqn.
Stochastic boundedness Invariant distribution Moderate deviation
Conclusion
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Filtering Then …
Centralized
Measurements always available (not lost) Optimality: structural conditions – observability/controllability Applications: Guidance, chemical plants, noisy images, …
“Kalman Gain”
“Innovations”
“Prediction”
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Filtering Today: Distributed Solution Local communications
Agents communicate with neighbors No central collection of data
Cooperative solution In isolation: myopic view and knowledge Cooperation: better understanding/global knowledge
Iterative solution Realistic Problem: Intermittency
Sensors fail Local communication channels fail
Limited resources: Noisy sensors Noisy communications Limited bandwidth (quantized communications)
Optimality: Asymptotically Convergence rate
Structural Random Failures
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Outline Brief Historical Comments: From Kolmogorov to Kalman-Bucy Filtering Then … Filtering Today Consensus: Distributed Averaging
Standard consensus Consensus in random environments
Distributed Filtering: Consensus + innovations Random field (parameter) estimation Realistic large scale problem:
Intermittency: Infrastructure failures, Sensor failures Random protocols: Gossip Limited Resources: Quantization
Two Linear Estimators: LU: Stochastic Approximation GLU: Mixed time scale estimator
Performance Analysis: Asymptotics Conclusion
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Consensus: Distributed Averaging Network of (cooperating) agents updating their beliefs:
(Distributed) Consensus:
Asymptotic agreement: λ2 (L) > 0
DeGroot, JASA 74; Tsitsiklis, 74, Tsitsiklis, Bertsekas, Athans, IEEE T-AC 1986Jadbabaie, Lin, Morse, IEEE T-AC 2003
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Consensus: random links, comm. or quant. noise
Consensus (reinterpreted): a.s. convergence to unbiased rv θ:
Consensus in Random Environments
Xiao, Boyd, Sys Ct L., 04, Olfati-Saber, ACC 05, Kar, Moura, Allerton 06, T-SP 10, Jakovetic, Xavier, Moura, T-SP, 10, Boyd, Ghosh, Prabhakar, Shah, T-IT, 06
Var µ · 2M ¾2(1¡ p)N 2
Pi ¸ 0 ®(i)2
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Outline Brief Historical Comments: From Kolmogorov to Kalman-Bucy Filtering Then … Filtering Today Consensus: Distributed Averaging in Random Environments Distributed Filtering: Consensus + innovations
Random field (parameter) estimation: Large scale Intermittency: Infrastructure failures, Sensor failures Random protocols: Gossip Limited Resources: Quantization
Linear Parameter Estimator: Mixed time scale Linear filtering: Intermittency – Random Riccati Eqn.
Stochastic boundedness Invariant distribution Moderate deviation
Conclusion
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In/Out Network Time Scale Interactions
Consensus : In network dominated interactions 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
ζcomm ζsensing
ζcomm « ζsensing
time scale
ζcomm ~ ζsensing
time scale
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Filtering: Random Field
Random field: Network of agents: each agent observes:
Intermittency: sensors fail at random times
Structural failures (random links)/ random protocol (gossip):
Quantization/communication noise
spatially correlated, temporally iid,
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Consensus+Innovations: Generalized Lin. Unbiased
Distributed inference: Generalized linear unbiased (GLU)
Consensus: local avg “Innovations”“Prediction”
“Kalman Gain”
GainInnovations
WeightsConsensus
Weights
¯(i) > 0;P
i ¸ 0 ¯(i) = 1 ;P
i ¸ 0 ¯2(i) < 1
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Consensus+Innovations: Asymptotic Properties
Properties Asymptotic unbiasedness, consistency, MS convergence, As. Normality Compare distributed to centralized performance
Distributed observability condition: Matrix G is full rank
Distributed connectivity: Network connected in the mean
Structural conditions
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Consensus+Innovations: GLU Observation:
Assumptions: iid, spatially correlated, L(i) iid, independent Distributed observable + connected on average
Estimator:
A6. assumption: Weight sequences
Soummya Kar, José M. F. Moura, IEEE J. Selected Topics in Sig. Pr., Aug2011.
f»n(i)g Eµjj»(i)jj2+²1 < 1
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Consensus+Innovations: GLU Properties
A1-A6 hold, , generic noise distribution (finite 2nd moment)0· °0 < :5
Pµ¤ (limi ! 1 xn(i) = µ¤) = 1; 8n
Consistency: sensor n is consistent
Asymptotically normality:
Asymptotic variance matches that of centralized estimator
Efficiency: Further, if noise is Gauss, GLU estimator is asymptotically efficient
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Consensus+Innovations: Remarks on Proofs
Define Let Find dynamic equation for Show is nonnegative supermartingale, converges a.s.,
hence pathwise bounded (this would show consistency) 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
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Outline Intermittency: networked systems, packet loss Random Riccati Equation: stochastic Boundedness Random Riccati Equation: Invariant distribution Random Riccati Equation: Moderate deviation principle
Rate of decay of probability of rare events Scalar numerical example Conclusions
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Kalman Filtering with Intermittent Observations Model:
Intermittent observations:
Optimal Linear Filter (conditioned on path of observations) – Kalman filter with Random Riccati Equation
xt+1 = Axt +wt
yt = Cxt +vt
Pt = Eh¡
xt ¡ bxtjt¡ 1¢¡
xt ¡ bxtjt¡ 1¢T
j f ey(s)g0· s<t
i
Pt+1 = APtAT +Q ¡ °tAPtCT ¡CPtCT +R
¢¡ 1CPtAT
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Outline Intermittency: networked systems, packet loss Random Riccati Equation: stochastic Boundedness Random Riccati Equation: Invariant distribution Random Riccati Equation: Moderate deviation principle
Rate of decay of probability of rare events Scalar numerical example Conclusions
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Random Riccati Equation (RRE) Sequence is random
Define operators f0(X), f1(X) and reexpress Pt:
f Ptgt2T+
[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.
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Outline Intermittency: networked systems, packet loss Random Riccati Equation: stochastic Boundedness Random Riccati Equation: Invariant distribution Random Riccati Equation: Moderate deviation principle
Rate of decay of probability of rare events Scalar numerical example Conclusions
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Random Riccati Equation: Invariant Distribution
Stochastic Boundedness:°sb = inf
n° 2 [0;1] : fPtgt2Z+
is s.b.; 8P0 2 SN+
o
supp¡¹ °¢= cl(S)
¹ ° ¡©Y 2 SN
+ j Y º P ¤ª¢= 1
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Moderate Deviation Principle (MDP) 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:
Counting numbers of
String (f 0; f 1; f 1; f 1; f 0; f 0;P0) written concisely¡f 0; f 3
1 ; f20 ;P0
¢
Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” IEEE Tr. Automatic Control;
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MDP for Random Riccati Equation
P*
Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” IEEE Tr. Automatic Control
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Outline Intermittency: networked systems, packet loss Random Riccati Equation: stochastic Boundedness Random Riccati Equation: Invariant distribution Random Riccati Equation: Moderate deviation principle
Rate of decay of probability of rare events Scalar numerical example Conclusions
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Support of the Measure Example: scalar
Lyapunov/Riccati operators:
Support is independent of
0< ° < 1
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Self-Similarity of Support of Invariant Measure ‘Fractal like’:
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Class A Systems: MDP
Scalar system
Define
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MDP: Scalar Example Scalar system:
Soummya Kar and José M. F. Moura, “Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations,” accepted EEE Tr. Automatic Control
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Outline Intermittency: networked systems, packet loss Random Riccati Equation: stochastic Boundedness Random Riccati Equation: Invariant distribution Random Riccati Equation: Moderate deviation principle
Rate of decay of probability of rare events Scalar numerical example Conclusions
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Conclusion Filtering 50 years after Kalman and Kalman-Bucy: Consensus+innovations: Large scale distributed networked agents
Intermittency: sensors fail; comm links fail Gossip: random protocol Limited power: quantization Observ. Noise
Linear estimators: Interleave consensus and innovations Single scale: stochastic approximation 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
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Conclusion Intermittency: packet loss 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
P*
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Thanks
Questions?