Expository & survey paper: A unified approach to smoothing formulas

Authors:
Lennart Ljung;Thomas Kailath
Affiliations:
Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305, U.S.A.;Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305, U.S.A.
Venue:
Automatica (Journal of IFAC)
Year:
1976

Citing 6
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Supplement to “a survey of data smoothing”

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A solution of the smoothing problem for linear dynamic systems

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A survey of data smoothing for linear and nonlinear dynamic systems

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A view of three decades of linear filtering theory

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Smoothed state estimates under abrupt changes using sum-of-norms regularization

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Abstract

Based on various approaches, several different solutions to the smoothing problem have been given. The relationships between these solutions are not immediate, although they solve the same problem. Making use of a certain framework from scattering theory, we derive two families of solutions, with equations evolving forwards and backwards in time, respectively. Within these families three major previous approaches are obtained as special cases, and their relationships are clarified. The set of solutions also contains as a fourth special case a (new) backwards analog of the innovations solution. The Mayne-Fraser two-filter formula belongs to the set of backwards solutions, and within this framework certain difficulties with its interpretation can be resolved.