In this paper, we present an ε-contamination observation model for a time-series to multidimensional systems. From this model, expression of the least favorable contaminating density is obtained. Also, the optimal robust estimator and its associated conditional covariance matrix are obtained in recursive form for linear systems. This robust estimator and its covariance are identical to the estimator and the covariance given by the Kalman filter in the absence of data outliers (when data is ‘clean.’) In the presence of observation data outliers, we then evaluate numerically performance of the optimal robust filter comparing to performance of the Kalman filter. From the numerical example result, we observe that the optimal robust estimates have less mean square error than the estimates given by the Kalman filter at the stages where outliers occur.