Extended Kalman Filter Using Orthogonal Polynomials

• Published in: IEEE access Vol. 9; pp. 59675 - 59691
• Main Authors: Kumar, Kundan, Bhaumik, Shovan, Date, Paresh
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Computational efficiency; Computing costs; Extended Kalman filter; filtering; functional approximation; Kalman filter; Kalman filters; Linearization; Mathematical analysis; Monte Carlo methods; nonlinear filter; numerical analysis; orthogonal polynomial; Polynomials; Roundoff error; Series expansion; State estimation; target tracking; Taylor series; Thermal stability
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2021.3073289

This paper reports a new extended Kalman filter where the underlying nonlinear functions are linearized using a Gaussian orthogonal basis of a weighted <inline-formula> <tex-math notation="LaTeX">\mathcal {L}_{2} </tex-math></inline-formula> 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.