A Novel Kalman Filter Formulation for Improving Tracking Performance of the Extended Kernel RLS
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A Novel Kalman Filter Formulation for Improving Tracking Performance of the Extended Kernel RLS Tiago S. Façanha1 · Guilherme A. Barreto1
· José T. Costa Filho1
Received: 28 November 2019 / Revised: 18 August 2020 / Accepted: 23 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, two contributions are presented. Firstly, a novel variant of the Kalman filter is developed and used to improve the tracking performance of the extended kernel recursive least squares algorithm (Ex-KRLS). Without resorting to the Riccati equation, the proposed formulation of the Kalman filter relies on principles of optimization and convex duality to obtain a stable quadratic form. It allows for performance and stability efficiency by minimizing the estimation error with the manipulation of a single scalar variable. Secondly, the proposed Kalman filter formulation is embedded into the Ex-KRLS-KF algorithm, an extension of the Ex-KRLS, to further improve its tracking performance. For this purpose, the state model of the proposed Ex-KRLS-KF variant is constructed in the original state space, while the hidden state is then estimated using the proposed Kalman filter formulation. The standard KRLS algorithm learns the measurement model used in hidden state estimation. We show through a comprehensive set of computer experiments that the proposed hybrid algorithm has more flexible state and noise models than competing alternative algorithms. We evaluate the proposed algorithm in two benchmarking tasks (nonlinear Rayleigh multipath channel tracking and Lorenz system modeling) and compare its performance with those provided by the state-of-the-art algorithms. Keywords Kalman filter · Kernel RLS · Extended KRLS · Nonlinear dynamical systems · Tracking performance
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Guilherme A. Barreto [email protected] Tiago S. Façanha [email protected] José T. Costa Filho [email protected]
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Graduate Program in Teleinformatics Engineering, Center of Technology, Federal University of Ceará, Campus of Pici, Fortaleza, Ceará, Brazil
Circuits, Systems, and Signal Processing
1 Introduction The filtering problem is formulated for dynamical systems on the premise of linearity and Gaussian conditions [20] for disturbance and noise characteristics. In this regard, the widely known Kalman filter (KF) algorithm [12] is an estimator for the linear quadratic Gaussian problem, which is based on Riccati’s differential equations. However, the application of the KF to nonlinear and non-Gaussian systems usually results in poor tracking performances, and its extension to these challenging scenarios is not straightforward. As a result, several variants of the KF algorithm which are applied to nonlinear dynamic systems have been developed, such as the extended Kalman filter (EKF) [6], the unscented Kalman filter (UKF) [11], the cubature Kalman filter (CKF) [1], and the kernel Kalman filter (KKF) [17]. Another essential aspect to consider is that it is not possible for some applications to correctly obtain the state-tran
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