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Kalman and Kalman Bucy @ 50: Distributed and Intermittency

This talk discusses the historical background of filtering, consensus in distributed averaging, distributed filtering with innovations, random field estimation, intermittency, random protocols, limited resources, and linear parameter estimators.

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Kalman and Kalman Bucy @ 50: Distributed and Intermittency

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  1. 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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAAA

  2. 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

  3. 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

  4. In the 40’s • 1939-41: A. N. Kolmogorov, "Interpolation und Extrapolation von Stationaren Zufalligen Folgen,“ Bull. Acad. Sci. USSR, 1941 • Wiener Model • Wiener filter • Wiener-Hopf equation (1931; 1942) • 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.

  5. Norbert WIENER. The extrapolation, interpolation and smoothing of stationary time series with engineering applications. [Washington, D.C.: National Defense Research Council,] 1942.

  6. Kalman Filter @ 51 Trans. of the ASME-J. of Basic Eng., 82 (Series D): 35-45, March 1960

  7. Kalman-Bucy Filter @ 50 Transactions of the ASME-Journal of Basic Eng., 83 (Series D): 95-108, March 1961

  8. 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

  9. Filtering Then … • Centralized • Measurements always available (not lost) • Optimality: structural conditions – observability/controllability • Applications: Guidance, chemical plants, noisy images, … “Kalman Gain” “Prediction” “Innovations”

  10. 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

  11. 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

  12. 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 1986 Jadbabaie, Lin, Morse, IEEE T-AC 2003

  13. Consensus in Random Environments • Consensus: random links, comm. or quant. noise • Consensus (reinterpreted): a.s. convergence to unbiased rv θ: 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

  14. 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

  15. 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 time scale ζsensing ζcomm ζcomm«ζsensing time scale ζcomm ~ ζsensing

  16. 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,

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

  18. Consensus+Innovations: Asymptotic Properties • Properties • Asymptotic unbiasedness, consistency, MS convergence, As. Normality • Compare distributed to centralized performance • Structural conditions • Distributed observability condition: Matrix G is full rank • Distributed connectivity: Network connected in the mean

  19. 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.

  20. Consensus+Innovations: GLU Properties • A1-A6 hold, , generic noise distribution (finite 2nd moment) • Consistency: sensor n is consistent • Asymptotic variance matches that of centralized estimator • Asymptotically normality: • Efficiency: Further, if noise is Gauss, GLU estimator is asymptotically efficient

  21. 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

  22. 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

  23. Kalman Filtering with Intermittent Observations • Model: • Intermittent observations: • Optimal Linear Filter (conditioned on path of observations) – Kalman filter with Random Riccati Equation

  24. 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

  25. Random Riccati Equation (RRE) • Sequence is random • 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.

  26. 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

  27. Random Riccati Equation: Invariant Distribution • Stochastic Boundedness:

  28. 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;

  29. 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

  30. 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

  31. Support of the Measure • Example: scalar • Lyapunov/Riccati operators: • Support is independent of

  32. Self-Similarity of Support of Invariant Measure • ‘Fractal like’:

  33. Class A Systems: MDP • Define • Scalar system

  34. 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

  35. 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

  36. 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

  37. 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*

  38. Thanks Questions?

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