Unified
Forms for Kalman and Finite Impulse Response Filtering and Smoothing

Dan

Cleveland, Ohio

Kalman filters are commonly used to estimate the states of a dynamic system. However, Kalman filters have an infinite impulse response, which can lead to instability in the presence of unmodeled dynamics. Finite impulse reponse (FIR) filters thus provide a robust alternative to Kalman filters. This paper derives a unified form for the Kalman filter and Kalman smoother, derives a unified form for an FIR filter and smoother, and compares the filters and smoothers via simulation results. The examples in the paper below can be replicated with the below m-files, which can be run in the MATLAB environment.

RTSEq1.m illustrates the equivalence
of the unified Kalman filter/smoother and the RTS smoother

RTSEq4.m was used to derive the results
of Example 1 in the paper.

FIRNopt.m was used to derive the
results of Example 2 in the paper.

RTSFIRMonte.m was used to derive
the results of Example 3 in the paper.

*References*

- D. Simon and Y. Shmaliy, “Unified Forms for Kalman and Finite
Impulse Response Filtering and Smoothing,”
*Automatica*, vol. 49, no. 6, pp. 1892-1899, June 2013

Department of Electrical and Computer Engineering

Last Revised: December 14, 2013