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Carnegie Mellon Kalman and Kalman 50: Distributed and Intermittency TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAAA 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 and CCF , and AFOSR grant FA

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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon In the 40’s Wiener Model Wiener filter Wiener-Hopf equation (1931; 1942) : 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.

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Carnegie Mellon Norbert 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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Carnegie Mellon Kalman 51 Trans. of the ASME-J. of Basic Eng., 82 (Series D): 35-45, March 1960

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Carnegie Mellon Kalman-Bucy 50 Transactions of the ASME-Journal of Basic Eng., 83 (Series D): , March 1961

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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon Consensus: Distributed Averaging Network of (cooperating) agents updating their beliefs: (Distributed) Consensus: Asymptotic agreement: λ 2 (L) > 0 DeGroot, JASA 74; Tsitsiklis, 74, Tsitsiklis, Bertsekas, Athans, IEEE T-AC 1986 Jadbabaie, Lin, Morse, IEEE T-AC 2003

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Carnegie Mellon 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

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Carnegie Mellon 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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Carnegie Mellon 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 ζ comm ζ sensing ζ comm « ζ sensing time scale ζ comm ~ ζ sensing time scale

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Carnegie Mellon 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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Carnegie Mellon Consensus+Innovations: Generalized Lin. Unbiased Distributed inference: Generalized linear unbiased (GLU) Consensus: local avg “Innovations” “Prediction” “Kalman Gain” Gain Innovations Weights Consensus Weights

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Carnegie Mellon 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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Carnegie Mellon 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.

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Carnegie Mellon Consensus+Innovations: GLU Properties A1-A6 hold,, generic noise distribution (finite 2 nd moment) 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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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon Kalman Filtering with Intermittent Observations Model: Intermittent observations: Optimal Linear Filter (conditioned on path of observations) – Kalman filter with Random Riccati Equation

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Carnegie Mellon 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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Carnegie Mellon Random Riccati Equation (RRE) Sequence is random Define operators f 0 (X), f 1 (X) and reexpress P t : [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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Carnegie Mellon 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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Carnegie Mellon Random Riccati Equation: Invariant Distribution Stochastic Boundedness:

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Carnegie Mellon 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 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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Carnegie Mellon MDP for Random Riccati Equation P*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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Carnegie Mellon 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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Carnegie Mellon Support of the Measure Example: scalar Lyapunov/Riccati operators: Support is independent of

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Carnegie Mellon Self-Similarity of Support of Invariant Measure ‘Fractal like’:

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Carnegie Mellon Class A Systems: MDP Scalar system Define

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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon 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*P*

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Carnegie Mellon Thanks Questions?

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