A Polynomial Least Squares Multiple-Model Estimator: Simple, Optimal, Adaptive, and Practical

Abstract

Original polynomial least squares (LS) fits data by constructing coefficients that minimize the sum of squared deviations of deterministic samples from assumed polynomials. Contemporary LS estimates existing polynomial coefficients by filtering out corrupting statistical errors. Kalman Filter (KF) state estimation from noisy data is analogous to polynomial LS estimation. A major problem in both LS and KF target tracking is matching estimator order to target dynamics. 2nd order estimators match constant velocity targets; 3rd order estimators match accelerating targets. Filtered error variances from 3rd order estimators are larger than from 2nd order estimators. 2nd order estimators applied to accelerating targets produce increasing biases as more data are filtered, causing their MSEs (variance plus bias-squared) to rapidly exceed 3rd order variances (MSEs). This becomes troublesome when recurrently maneuvering targets make acceleration jumps. A trade-off between 2nd and 3rd order MSEs is needed. The interacting multiple-model (IMM) addresses this problem adaptively by making adjustments between 2nd and 3rd order KFs with model probabilities as functions of likelihoods from KF residuals and transition probabilities of assumed acceleration jumps. The IMM does not address biases, perform variance/bias-squared trade-offs, or minimize MSEs. In this paper linear interpolation is established between the 2nd and 3rd order polynomial estimators creating the LS multiple-model (LSMM), the 2nd order acceleration bias is defined, and the LSMM MSE is minimized in a variance/bias-squared trade-off. A sequence of optimal LSMMs matched to accelerations covering the spectrum of acceleration between zero and assumed maximum are derived and an adaptive algorithm is designed.