A fast convergence speed of an observer helps improve the capability to track the states of a system for an arbitrary divergence between a real and an estimated initial conditions. This property of the observers is significantly useful if a system has fast dynamics and its states change rapidly. Thus, the convergence time is one of the main performance criteria of linear and non-linear state observers.

This article presents a comparative analysis of observers for both linear and nonlinear systems in terms of the time of convergence of the observers. The following observers was chosen for this study: the Kalman filter (KF), extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), Luenberger observer (LO), and fuzzy-based Luenberger observer (Fuzzy-LO). The listed observers were studied using a non-linear mathematical model of an open-link locomotion module, which movements were studied in stochastic terrain conditions. The mathematical model was then simplified and simulated as a linear model with the purpose to estimate the efficiency of the linear observers. The Fuzzy-LO with an adaptive gain to the estimation error gives better results than the LO, especially in steady states. The PF with a simple Gaussian distribution provides a lower convergence speed than the KF, EKF, and UKF. To faster the convergence of the PF, a novel approach, PF*, that utilizes mixture probability density function of the distribution of initial particles was introduced in the article.

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