Nonlinear Filters for Linear Models (A Robust Approach)

Robert Sh Liptser, Wolfgang J. Runggaldier, Wolfgang J. Runggaldier

Research output: Contribution to journalArticlepeer-review


We consider the filtering problem for linear models where the driving noises may be quite general, nonwhite and non-Gaussian, and where the observation noise may only be known to belong to a finite family of possible disturbances. Using diffusion approximation methods, we show that a certain nonlinear filter minimizes the asymptotic filter variance. This nonlinear filter is obtained by choosing at each moment, on the basis of the observations, one of a finite number of Kalman-type filters driven by a suitable nonlinear transformation of the “innovations.” As a byproduct we obtain also the asymptotic identification of the a priori unknown observation noise disturbance. By yielding an asymptotically efficient filter in face of an unknown observation noise, our approach may also be viewed as a robust approach to filtering for linear models.

Original languageEnglish
Pages (from-to)1001-1009
Number of pages9
JournalIEEE Transactions on Information Theory
Issue number4
StatePublished - Jul 1995


  • Linear and nonlinear filtering
  • diffusion approximations
  • non-Gaussian noises
  • robust filtering


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