About the true type of smoothers

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.