Distributionally Robust State Estimation for Linear Systems Subject to Uncertainty and Outlier

Parameter uncertainties and measurement outliers unavoidably exist in a real linear system. Such uncertainties and outliers make the true joint state-measurement distributions (induced by the true system model) deviate from the nominal ones (induced by the nominal system model) so that the performance of the optimal state estimator designed for the nominal model becomes unsatisfactory or even unacceptable in practice. The challenges are to quantitatively describe the uncertainties in the model and the outliers in the measurements, and then robustify the estimator in a right way. This article studies a distributionally robust state estimation framework for linear systems subject to parameter uncertainties and measurement outliers. It utilizes a family of distributions near the nominal one to implicitly describe the uncertainties and outliers, and the robust state estimation in the worst case is made over the least-favorable distribution. The advantages of the presented framework include: 1) it only uses a few scalars to parameterize the method and does not require the structural information of uncertainties; 2) it generalizes several classical filters (e.g., the fading Kalman filter, risk-sensitive Kalman filter, relative-entropy Kalman filter, outlier-insensitive Kalman filters) into a unified framework. We show that the distributionally robust state estimation problem can be reformulated into a linear semi-definite program and in some special cases it can be analytically solved. Comprehensive comparisons with existing state estimation frameworks that are insensitive to parameter uncertainties and measurement outliers are also conducted.