Distributed Kalman Filtering : Weak Consensus Under Weak Detectability

The paper presents the gossip interactive Kalman filter (GIKF) for distributed Kalman filtering for networked systems and sensor networks, where intersensor communication and observations occur at the same time-scale. The communication among sensors is random; each sensor occasionally exchanges its filtering state information with a neighbor depending on the availability of the appropriate network link. We show that under a weak distributed detectability condition: 1) the GIKF error process remains stochastically bounded, irrespective of the instability of the random process dynamics; and 2) the network achieves weak consensus, i.e., the conditional estimation error covariance at a (uniformly) randomly selected sensor converges in distribution to a unique invariant measure on the space of positive semidefinite matrices (independent of the initial state). To prove these results, we interpret the filtering states (estimates and error covariances) at each node in the GIKF as stochastic particles with local interactions. We analyze the asymptotic properties of the error process by studying as a random dynamical system the associated switched (random) Riccati equation, the switching being dictated by a nonstationary Markov chain on the network graph.

[1]  Andrey V. Savkin,et al.  The problem of state estimation via asynchronous communication channels with irregular transmission times , 2003, IEEE Trans. Autom. Control..

[2]  Soummya Kar,et al.  Kalman Filtering with Intermittent Observations: Weak Convergence and Moderate Deviations , 2009, ArXiv.

[3]  M. Rosenzweig,et al.  Learning by Doing and Learning from Others: Human Capital and Technical Change in Agriculture , 1995, Journal of Political Economy.

[4]  Lei Guo,et al.  On stability of random Riccati equations , 1999 .

[5]  I. Chueshov.  Monotone Random Systems Theory and Applications , 2002 .

[6]  A. Goldsmith,et al.  Kalman filtering with partial observation losses , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[7]  V. Araújo.  Random Dynamical Systems , 2006, math/0608162.

[8]  Subhrakanti Dey,et al.  Stability of Kalman filtering with Markovian packet losses , 2007, Autom..

[9]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[10]  Soummya Kar,et al.  Moderate Deviations of a Random Riccati Equation , 2012, IEEE Transactions on Automatic Control.

[11]  José M. F. Moura,et al.  Distributing the Kalman Filter for Large-Scale Systems , 2007, IEEE Transactions on Signal Processing.

[12]  P. R. KumarDepartment.  ON KALMAN FILTERING FOR CONDITIONALLY GAUSSIAN SYSTEMS WITH RANDOM MATRICES , 1989 .

[13]  Bruno Sinopoli,et al.  Kalman filtering with intermittent observations , 2004, IEEE Transactions on Automatic Control.

[14]  Lihua Xie,et al.  Stability of a random Riccati equation with Markovian binary switching , 2007, 2007 46th IEEE Conference on Decision and Control.

[15]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[16]  W. Marsden.  I and J , 2012 .

[17]  Babak Hassibi,et al.  A Stieltjes transform approach for studying the steady-state behavior of random Lyapunov and Riccati recursions , 2008, 2008 47th IEEE Conference on Decision and Control.

[18]  Richard M. Murray,et al.  On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage , 2006, Autom..

[19]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[20]  Yannis M. Ioannides,et al.  Job Information Networks, Neighborhood Effects, and Inequality , 2004 .

[21]  Kaivan Munshi.  Social learning in a heterogeneous population: technology diffusion in the Indian Green Revolution , 2004 .

[22]  V. Solo,et al.  Stability of the Kalman filter with stochastic time-varying parameters , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[23]  Francesco Bullo,et al.  On Kalman Filtering for Detectable Systems With Intermittent Observations , 2009, IEEE Transactions on Automatic Control.

[24]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[25]  P. Bougerol.  Kalman filtering with random coefficients and contractions , 1993 .

[26]  Ruggero Carli,et al.  Distributed Kalman filtering using consensus strategies , 2007, 2007 46th IEEE Conference on Decision and Control.

[27]  Soummya Kar,et al.  Distributed Parameter Estimation in Sensor Networks: Nonlinear Observation Models and Imperfect Communication , 2008, IEEE Transactions on Information Theory.

[28]  Ueli Maurer.  Random Systems: Theory and Applications , 2007, ICITS.

[29]  H. F. Durrant-Whyte,et al.  Fully decentralised algorithm for multisensor Kalman filtering , 1991 .

[30]  Soummya Kar,et al.  Kalman Filtering With Intermittent Observations: Weak Convergence to a Stationary Distribution , 2009, IEEE Transactions on Automatic Control.

[31]  Persi Diaconis,et al.  Iterated Random Functions , 1999, SIAM Rev..

[32]  Panganamala Ramana Kumar,et al.  Sending the most recent observation is not optimal in networked control: Linear temporal coding and towards the design of a control specific transport protocol , 2007, 2007 46th IEEE Conference on Decision and Control.

[33]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[34]  Ruggero Carli,et al.  Distributed Kalman filtering based on consensus strategies , 2008, IEEE Journal on Selected Areas in Communications.

[35]  Andrea Censi,et al.  On the performance of Kalman filtering with intermittent observations: A geometric approach with fractals , 2009, 2009 American Control Conference.

[36]  R. Olfati-Saber,et al.  Distributed Kalman Filter with Embedded Consensus Filters , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[37]  Andrey V. Savkin,et al.  The problem of state estimation via asynchronous communication channels with irregular transmission times , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[38]  Asuman E. Ozdaglar,et al.  Convergence of rule-of-thumb learning rules in social networks , 2008, 2008 47th IEEE Conference on Decision and Control.

[39]  L. Arnold,et al.  Order-preserving random dynamical systems: equilibria, attractors, applications , 1998 .

[40]  Jean C. Walrand,et al.  Achieving 100% throughput in an input-queued switch , 1996, Proceedings of IEEE INFOCOM '96. Conference on Computer Communications.

[41]  J.P. Hespanha,et al.  Estimation under uncontrolled and controlled communications in Networked Control Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.