Kalman and Kalman Bucy @ 50: Distributed and Intermittency

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

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

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.

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

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

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

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

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

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

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

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

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

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,

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

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

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.

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

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

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

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

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.

Random Riccati Equation: Invariant Distribution • Stochastic Boundedness:

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;

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

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

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

Class A Systems: MDP • Define • Scalar system

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

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

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*

Thanks Questions?

Kalman and Kalman Bucy @ 50: Distributed and Intermittency