Extended Kalman Filter Using Orthogonal Polynomials

Kundan Kumar,Paresh Date,Shovan Bhaumik

doi:10.1109/access.2021.3073289

Abstract

This paper reports a new extended Kalman filter where the underlying nonlinear functions are linearized using a Gaussian orthogonal basis of a weighted $\mathcal {L}_{2}$ space. As we are interested in computing the states’ mean and covariance with respect to Gaussian measure, it would be better to use a linearization, that is optimal with respect to the same measure. The resulting first-order polynomial coefficients are approximately calculated by evaluating the integrals using (i) third-order Taylor series expansion (ii) cubature rule of integration. Compared to direct integration-based filters, the proposed filter is far less susceptible to the accumulation of round-off errors leading to loss of positive definiteness. The proposed algorithms are applied to four nonlinear state estimation problems. We show that our proposed filter consistently outperforms the traditional extended Kalman filter and achieves a competitive accuracy to an integration-based square root filter, at a significantly reduced computing cost.

Highlights

IntroductionA. BAYESIAN FILTERING We consider the following state space model of a dynamic system in discrete time: Xk+1 = φ(Xk ) + ηk , (1) and
The coefficients of the first order polynomial are approximately calculated by evaluating the integrals using (i) Taylor series approximation of the function (ii) numerical evaluation of the Gaussian integral using a weighted sum of an appropriate set of deterministic sample points
This paper proposes a new approach that is based on linearization of the process and measurement equation with first-order orthogonal Hermite polynomial, to solve a nonlinear state estimation problem

Summary

Introduction

A. BAYESIAN FILTERING We consider the following state space model of a dynamic system in discrete time: Xk+1 = φ(Xk ) + ηk , (1) and. Where Xk ∈ Rnx is the state of the dynamic system at a time instant k, Yk ∈ Rny is the measurement, φ : Rnx → Rnx and γ : Rnx → Rny are known nonlinear functions. The process noise, ηk , and the measurement noise, νk+1 are white, Gaussian and uncorrelated to each other with zero mean and covariance Qk and Rk+1, respectively. In Bayesian filtering, the hidden state vector Xk+1 needs to be estimated by using measurement up to time k + 1. The pdf p(Xk+1|Y1:k+1) is constructed recursively in two steps: (i) prediction step (ii) update step. From the knowledge of p(Xk |Y1:k ), we construct the

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Conclusion