State-space models are used in a wide range of time series analysis applications. Kalman filtering and smoothing are work-horse algorithms in these settings. While classic algorithms assume Gaussian errors to simplify estimation, recent advances use a broad range of optimization formulations to allow outlier-robust estimation, as well as constraints to capture prior information.Here we develop methods on state-space models where either transition or error covariances may be singular. These models frequently arise in navigation (e.g. for 'colored noise' models or deterministic integrals) and are ubiquitous in auto-correlated time series models such as ARMA. We reformulate all state-space models (singular as well as nonsingluar) as constrained convex optimization problems, and develop an efficient algorithm for this reformulation. The convergence rate is locally linear, with constants that do not depend on the conditioning of the problem.Numerical comparisons show that the new approach outperforms competing approaches for nonsingular models, including state of the art interior point (IP) methods. IP methods converge at superlinear rates; we expect them to dominate. However, the steep rate of the proposed approach (independent of problem conditioning) combined with cheap iterations wins against IP in a run-time comparison. This suggests that the proposed approach can be a default choice for estimating state space models outside of the Gaussian context for singular and nonsingular models. To highlight the capabilities of the new framework, we focus on navigation applications that use singular process covariance models, and analyze data from a drifting mooring as a proxy for an autonomous underwater vehicle.

We develop data-driven algorithms to fully automate sensor fault detection in systems governed by underlying physics. The proposed machine learning method uses a time series of typical behavior to approximate the evolution of measurements of interest by a linear timeinvariant system. Given additional data from related sensors, a Kalman observer is used to maintain a separate real-time estimate of the measurement of interest. Sustained deviation between the measurements and the estimate is used to detect anomalous behavior. A decision tree, informed by integrating other sensor measurement values, is used to determine the amount of deviation required to identify a sensor fault. We validate the method by applying it to three test systems exhibiting various types of sensor faults: commercial flight test data, an unsteady aerodynamics model with dynamic stall, and a model for longitudinal flight dynamics forced by atmospheric turbulence. In the latter two cases we test fault detection for several prototypical failure modes. The combination of a learned dynamical model with the automated decision tree accurately detects sensor faults in each case.

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Tracking underwater autonomous platforms is often difficult because of noisy, biased, and discretized input data. Classic filters and smoothers based on standard assumptions of Gaussian white noise break down when presented with any of these challenges. Robust models (such as the Huber loss) and constraints (e.g. maximum velocity) are used to attenuate these issues. Here, we consider robust smoothing with singular covariance, which covers bias and correlated noise, as well as many specific model types, such as those used in navigation. In particular, we show how to combine singular covariance models with robust losses and state-space constraints in a unified framework that can handle very low-fidelity data. A noisy, biased, and discretized navigation dataset from a submerged, low-cost inertial measurement unit (IMU) package, with ultra short baseline (USBL) data for ground truth, provides an opportunity to stress-test the proposed framework with promising results. We show how robust modeling elements improve our ability to analyze the data, and present batch processing results for 10 minutes of data with three different frequencies of available USBL position fixes (gaps of 30 seconds, 1 minute, and 2 minutes). The results suggest that the framework can be extended to real-time tracking using robust windowed estimation.