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1 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

2 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

3 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

4 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.

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

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

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

8 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

9 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”

10 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

11 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

12 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

13 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

14 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

15 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

16 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,

17 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

18 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

19 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.

20 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

21 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

22 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

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

24 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

25 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.

26 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

27 Carnegie Mellon Random Riccati Equation: Invariant Distribution  Stochastic Boundedness: 

28 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;

29 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

30 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

31 Carnegie Mellon Support of the Measure  Example: scalar  Lyapunov/Riccati operators:  Support is independent of

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

33 Carnegie Mellon Class A Systems: MDP    Scalar system  Define

34 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

35 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

36 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

37 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*

38 Carnegie Mellon Thanks Questions?


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