Linear recursive discrete-time estimators using covariance information under uncertain observations

This paper, using the covariance information, proposes recursive least-squares (RLS) filtering and fixed-point smoothing algorithms with uncertain observations in linear discrete-time stochastic systems. The observation equation is given by y(k) = γ(k)Hx(k) + v(k), where {γ(k)} is a binary switching sequence with conditional probability distribution verifying Eq. (3). This observation equation is suitable for modeling the transmission of data in multichannels as in remote sensing situations. The estimators require the information of the system matrix Φ concerning the state variable which generates the signal, the observation vector H, the crossvariance function Kxz(k,k) of the state variable with the signal, the variance R(k) of the white observation noise, the observed values, the probability p(k): P{γ(k)= 1} that the signal exists in the uncertain observation equation and the (2,2) element [P(k|j)]2,2 of the conditional probability matrix of γ(k), given γ(j).