Distributionally Robust Kalman Filter Under Likelihood Model Mismatch From an Optimization Perspective—Part I: Theoretical Framework

Over the past few decades, numerous robust Kalman filters have been proposed. However, existing methods rely heavily on a relatively accurate likelihood model, and their performance deteriorates significantly when the likelihood model exhibits severe mismatch. In this paper, through reformulating the Bayesian posterior inference to be an optimization problem and examining the sensitivity mechanism to likelihood model mismatch, a robust cost function which is less sensitive to the likelihood model mismatch is designed to replace the traditional Bayesian cost function. Then a distributionally robust posterior inference framework is proposed, which is less sensitive to the likelihood model mismatch. Building upon this robust posterior inference framework, a robust Kalman filter is derived by specifying the prior distribution and likelihood distribution as Gaussian distributions while an improved natural gradient method is introduced for solving the coupled Gaussian parameters. The convexity of the cost function is proven to ensure that the optimization algorithm converges to the global optimum. A stochastic stability analysis is conducted, proving that the estimation errors of the proposed algorithm are exponentially bounded in the mean square sense. Three different cases of numerical simulations are performed to verify the validity and superiority of the proposed algorithms.