On the accuracy and convergence of the minimax filtering algorithm for chaotic signals

The article is focused on the filtering problem for chaotic signals. The original discrete signal is generated by a one-dimensional chaotic system, and the measured signal is corrupted by additive errors. The goal is to estimate the unknown system states from measurements. The minimax filtering algorithm is developed in the context of guaranteed state estimation that is based on a set-membership description of uncertainty. It assumes that the unknown variables (system states and measurement errors) are bounded by intervals (sets of possible values). The proposed algorithm is a recursive procedure based on interval analysis. It computes interval estimates that are guaranteed to contain the true states of the system (true values of the original signal). The computation of the interval estimate consist of three steps (prediction, measurement, and correction) that are similar to the computation of the information set for linear dynamical systems. The point estimates are obtained by an algorithm that is similar to the Kalman filter. This paper studies the accuracy and convergence properties of the minimax filter. The aims of this study are the following: to confirm the effectiveness of the proposed algorithm for computation of the point estimates, to compare the results of the minimax filter and the unscented Kalman filter, and to derive the sufficient conditions for obtaining the exact value of the state. The computational scheme of the minimax filter and numerical simulations are given for the logistic map.