Developments in the theory of linear least-squares estimation in the last thirty years or so are outlined. Particular attention is paid to early mathematica[ work in the field and to more modern developments showing some of the many connections between least-squares filtering and other fields.

A view of three decades of linear filtering theory

This paper addresses a general p-shift linear optimal finite impulse response (FIR) estimator intended for solving universally the problems of filtering (p = 0), smoothing (p 0) of discrete time-invariant models in state space. An optimal solution is found in the batch form with the initial mean square state function self-determined by solving the discrete algebraic Riccati equation. An unbiased solution represented both in the batch and recursive forms does not involve any knowledge about noise and initial state. The mean square errors in both the optimal and unbiased estimates are found via the noise power gain (NPG) and a recursive algorithm for fast computation of the NPG is supplied. Applications are given for FIR filtering with fixed, receding, and full averaging horizons.

Linear Optimal FIR Estimation of Discrete Time-Invariant State-Space Models

SUMMARY

A general p-shift linear optimal finite impulse response (FIR) estimator is proposed for filtering (p  =  0), p-lag smoothing (p 0) of discrete time-varying state-space models An optimal solution is found in the batch form with the mean square initial state function self-determined by solving the discrete algebraic Riccati equation An unbiased batch solution is shown to be independent on noise and initial conditions The mean square errors in both the optimal and unbiased estimates have been determined along with the noise power gain and estimate error bound The following important inferences have been made on the basis of numerical simulation Unlike the time-invariant Kalman filter, the relevant optimal FIR one is very less sensitive to noise, especially when N  ≫ 1 Both time varying, the optimal FIR and Kalman estimates trace along almost the same trajectories with similar errors and sensitivities to noise Overall, the optimal FIR estimator demonstrates better robustness than the Kalman one against faults in the noise description Copyright © 2011 John Wiley & Sons, Ltd

Time‐variant linear optimal finite impulse response estimator for discrete state‐space models

CONCLUSION Because nonlinear problems are inherently more complex than their linear counterparts, there has been a tendency to overlook the fact that results from one nonlinear problem may be useful in obtaining the solution to another. The simple results presented in Baum’s earlier paper’ and Table I of this correspondence suggest that the picture may not be as bleak as first appears and that additional results may be possible for more general nonlinear problems.

Comments on "On Finite-Memory Recursive Filters

A Note on the Derivation of Finite-Memory Almost-Least-Squares Recursive Filters

It has not been noted in the literature that finite-memory least-squares, recursive filters are unstable, and hence unsuited for a large number of recursions. This correspondence modifies the finite-memory least-squares recursive filters to allow for stable solutions.

G. Epstein

Papers

On finite-memory recursive filters (Corresp.)