Distributionally Robust State Estimation for Jump Linear Systems

In practice, the designed nominal model set for a jump (Markov) linear system might be uncertain: 1) Every candidate model might be inexact due to, e.g., mismatched modeling assumptions or model's identification errors; 2) The nominal model set might be incomplete (e.g., the true system has three operating modes but the designed model set includes only two of them). Moreover, the designed model transition probability matrix (TPM) might be uncertain: There is a discrepancy between the designed and true TPMs. Neglecting such model uncertainties and employing nominally optimal multi-model state estimators may cause significant performance losses. Therefore, this article proposes a robust state estimation framework for jump linear systems that is insensitive to these three types of model uncertainty, technically by leveraging the distributionally robust optimization theory. Specifically, the model uncertainties are quantified by a collection of mixture distributions lying in a distributional ball centered at the nominal mixture distribution, where the nominal mixture distribution represents the nominal state-measurement distribution defined by nominal candidate models and their nominal model weights. Then, the robust state estimation is taken over the least-favorable distribution in the employed distributional ball. We show that the distributionally robust state estimation problem for jump linear systems can be reformulated into a tractable optimization equivalent such as a quadratic program or a positive semi-definite program, and in a special case, it can be analytically solved. Simulated and real-world experiments suggest that the proposed method is particularly useful when large model uncertainties exist.