Computationally Efficient Implementation of the Weighted Kalman Filter for Quadratic Systems

The weighted Kalman filter (WKF) can be per- ceived as a variant of the extended Kalman filter (EKF), which incorporates the so-called weighted linearisation. The price behind such an extension is expressed by an increased computational complexity, which aims at improving the overall convergence of the filter. Owing to the increased complexity, the WKF is in general not suitable for high-order nonlinear systems. To overcome this difficulty, the objective of this paper is to show that a closed-form expression of the WKF can be obtained for a specific class of nonlinear systems, i.e., those characterised by a quadratic state equation. This closed-form of the WKF scales well with the order of the system, enabling the application of the WKF to high-order nonlinear quadratic systems, as exemplified in the final part of the paper using a comprehensive Monte Carlo analysis applied to a large number of random systems.