Comparative Analysis of the Performances of a Nonlinear Observer and Nonlinear Kalman Filters in the Presence of Non-Gaussian Disturbances

Abstract

This paper focuses on state estimation for a fairly general class of systems, involving nonlinear functions and disturbances in both the process dynamics and output equations. A nonlinear observer that satisfies a $H_{\infty }$ disturbance attenuation constraint in addition to providing asymptotic stability in the absence of disturbances is developed using Lyapunov analysis. A weighted form of this observer is able to adjust the estimation performance for systems that have states with considerably different levels of magnitude. The observer is shown analytically to provide a guaranteed upper bound on the vector norm of the estimation error, and this upper bound is utilized to guarantee the stability of observers in disturbed systems that are designed to be stable over a finite domain. The performance of the nonlinear observer is compared with the performance of the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). Three different applications are utilized for the comparison, consisting of a magnetic position estimation problem, a state-of-charge battery application, and a vehicle tracking application. In the case of the disturbances being Gaussian noise, the UKF and the nonlinear observer provide approximately the same level of performance, and they both surpass the performance of the EKF. However, in the case of 2-norm-bounded non-Gaussian noise, such as spikes/pulses, the nonlinear observer is shown to significantly outperform both the UKF and the EKF. Extensive experimental results and comparisons using a range of covariance choices demonstrate the superiority of the nonlinear observer, confirming that it is not just an artifact of specific tests.