Distributionally Robust State Estimation for Nonlinear Systems
Shixiong Wang
Abstract
Uncertainties unavoidably exist in modeling for nonlinear systems: state equation, measurement equation, and/or noises statistics might be uncertain. Such model mismatches render the performance of nominally optimal state estimators being deteriorated or even unsatisfactory. Therefore, robust filters that are insensitive to modeling uncertainties have to be designed. The challenge is to quantitatively describe the uncertainties and then design accordingly efficient robust filters. Since uncertainties in nominal models make prior state distributions and likelihood distributions uncertain as well, this article proposes a distributionally robust particle filtering framework for nonlinear systems subject to modeling uncertainties. Specifically, we use worst-case prior state distributions (near the nominal prior state distributions) to generate prior state particles and/or determine their weights. Likewise, worst-case likelihood distributions (near the nominal likelihood distributions) are used to evaluate the worst-case likelihoods of prior state particles at given measurements. The “worst-case” scenario is quantified by entropy of distributions, and maximum entropy distributions are found in balls centered at nominal distributions with radii defined by statistical similarity measures such as moments-based similarity, Wasserstein distance, and Kullback-Leibler divergence. We prove that Gaussian approximation filters (e.g., unscented/cubature/ensemble Kalman filter) are distributionally robust in the sense that they use maximum entropy prior state distributions and maximum entropy likelihood distributions. Moreover, we show that the distributionally robust particle filtering framework provides a likelihood evaluation method for general nonlinear measurement equation with non-additive and non-multiplicative measurement noises. At last, we discuss measurement outlier treatment strategies in the distributionally robust particle filtering framework.