Abstract
In this paper, we develop in part and review various iterative unbiased finite impulse response (UFIR) algorithms (both direct and two‐stage) for the filtering, smoothing, and prediction of time‐varying and time‐invariant discrete state‐space models in white Gaussian noise environments. The distinctive property of UFIR algorithms is that noise statistics are completely ignored. Instead, an optimal window size is required for optimal performance. We show that the optimal window size can be determined via measurements with no reference. UFIR algorithms are computationally more demanding than Kalman filters, but this extra computational effort can be alleviated with parallel computing, and the extra memory that is required is not a problem for modern computers. Under real‐world operating conditions with uncertainties, non‐Gaussian noise, and unknown noise statistics, the UFIR estimator generally demonstrates better robustness than the Kalman filter, even with suboptimal window size. In applications requiring large window size, the UFIR estimator is also superior to the best previously known optimal FIR estimators.
Highlights
In optimal estimation theory, unbiasedness is a key condition that is used to derive linear and nonlinear estimators
We develop in part the results achieved in the field of unbiased finite impulse response (UFIR) filtering and review a family of iterative UFIR algorithms for filtering, smoothing, and prediction of time-varying (TV) and time-invariant (TI) discrete state-space models in white Gaussian noise environments
1.10.1 Optimal UFIR (OUFIR) vs. optimal FIR (OFIR) Beginning with the early limited memory filter of Jazwinski [5], OFIR filtering has been under development for several decades
Summary
1.1 Introduction In optimal estimation theory, unbiasedness is a key condition that is used to derive linear and nonlinear estimators. An extremely useful property of the BLUE and UFIR is that noise statistics are not required Another example is the maximum likelihood estimator (MLE), which obtains the estimate at an extremum of the density function of the state conditioned on the measurements [5]. In view of the fact that noise statistics and initial errors are commonly not well known, especially for time-variant models, theoretically optimal estimators end up being suboptimal in practical applications. In this regard, engineering experience says the following [9]: Practical implementation of the Kalman filter is often difficult due to the inability in getting a good estimate of the noise covariance matrices. We wish to estimate the estimation errors and generalize the properties to facilitate a comparison with the OFIR and Kalman algorithms
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