About the true type of smoothers

TL;DR

This paper analyzes the steady-state error of linear non-causal smoothers using variational methods, showing they outperform filters in robustness and error reduction, especially for polynomial inputs.

Contribution

It provides a complete description of steady-state errors in smoothers for polynomial inputs and compares their performance to filters, highlighting their robustness advantages.

Findings

Steady-state error in smoothers is similar to that in a filter of double the type.

Optimal smoothers have significantly smaller steady-state error than Kalman filters.

Smoothing offers greater robustness to model uncertainty than filtering.

Abstract

We employ the variational formulation and the Euler-Lagrange equations to study the steady-state error in linear non-causal estimators (smoothers). We give a complete description of the steady-state error for inputs that are polynomial in time. We show that the steady-state error regime in a smoother is similar to that in a filter of double the type. This means that the steady-state error in the optimal smoother is significantly smaller than that in the Kalman filter. The results reveal a significant advantage of smoothing over filtering with respect to robustness to model uncertainty.