Robust Estimators for Missing Observations in Linear Discrete-Time Stochastic Systems with Uncertainties

Abstract

As a first approach to estimating the signal and the state, Theorem 1 proposes recursive least-squares (RLS) Wiener fixed-point smoothing and filtering algorithms that are robust to missing measurements in linear discrete-time stochastic systems with uncertainties. The degraded quantity is given by multiplying the Bernoulli random variable by the degraded signal caused by the uncertainties in the system and observation matrices. The degraded quantity is observed with additional white observation noise. The probability that the degraded signal is present in the observation equation is assumed to be known. The design feature of the proposed robust estimators is the fitting of the degraded signal to a finite-order autoregressive (AR) model. Theorem 1 is transformed into Corollary 1, which expresses the covariance information in a semi-degenerate kernel form. The autocovariance function of the degraded state and the cross-covariance function between the nominal state and the degraded state is expressed in semi-degenerate kernel forms. Theorem 2 shows the robust RLS Wiener fixed-point and filtering algorithms for estimating the signal and state from degraded observations in the second method. The robust estimation algorithm of Theorem 2 has the advantage that, unlike Theorem 1 and the usual studies, it does not use information on the existence probability of the degraded signal. This is a unique feature of Theorem 2.