Estimation methods such as the Kalman filter identify best state estimates based on certain optimality criteria using a model of the system and the observations. A common assumption underlying the estimation is that the noise is Gaussian. In practical systems though, one quite frequently encounters thick-tailed, non-Gaussian noise. Statistically, contamination by this type of noise can be seen as inducing outliers among the data and leads to significant degradation in the KF. While many nonlinear methods to cope with non-Gaussian noise exist, a filter that is robust in the presence of outliers and maintains high statistical efficiency is desired. To solve this problem, a new robust Kalman filter framework is proposed that bounds the influence of observation, innovation, and structural outliers in a discrete linear system. This filter is designed to process the observations and predictions together, making it very effective in suppressing multiple outliers. In addition, it consists of a new prewhitening method that incorporates a robust multivariate estimator of location and covariance. Furthermore, the filter provides state estimates that are robust to outliers while maintaining a high statistical efficiency at the Gaussian distribution by applying a generalized maximum likelihood-type (GM) estimator. Finally, the filter incorporates the correct error covariance matrix that is derived using the GM-estimator's influence function.

This dissertation also addresses robust state estimation for systems that follow a broad class of nonlinear models that possess two or more equilibrium points. Tracking state transitions from one equilibrium point to another rapidly and accurately in such models can be a difficult task, and a computationally simple solution is desirable. To that effect, a new robust extended Kalman filter is developed that exploits observational redundancy and the nonlinear weights of the GM-estimator to track the state transitions rapidly and accurately.

Through simulations, the performances of the new filters are analyzed in terms of robustness to multiple outliers and estimation capabilities for the following applications: tracking autonomous systems, enhancing actual speech from cellular phones, and tracking climate transitions. Furthermore, the filters are compared with the state-of-the-art, i.e. the H_â -filter for tracking an autonomous vehicle and the extended Kalman filter for sensing climate transitions.