Robust Estimation of Dynamic Systems with Additive Heavy-Tailed Uncertainties
For linear dynamic systems with additive Gaussian noises, the Kalman filter (KF) has been the main estimation paradigm in the fields of economics, finance and engineering. Though the Gaussian conditional probability density function generated in the Kalman filter is well-defined, tractable and computationally cheap, many real-world phenomena cannot be captured appropriately by its light probability tails. Recently, a multivariate state-estimator for linear dynamical systems was developed that explicitly models the associated initial state, process and measurement uncertainties using heavy-tailed Cauchy probability density functions (PDFs). Although few physical processes are explicitly Cauchy distributed since their tails over-bound other realistic densities, estimators that are based on the Cauchy PDFs are hypothesized to be robust to unknown physical densities. For this multivariate Cauchy estimator (MCE), rather than propagating the conditional probability density function (CPDF) given the measurement history, the analytic and recursive characteristic function of the unnormalized CPDF needs to be propagated. Two essential difficulties are the geometric increase in the number of terms at each measurement update that represent the characteristic function and the computation of each term. With the discovery of a basis representation for each term, a linear operation on this basis eliminates its computational difficulty and allows term addition, leading to the elimination of over 99% of terms. Although the number of terms after a measurement update still grows, a method based on a sliding measurement window is developed to run the MCE algorithm for arbitrary simulation lengths. Furthermore, the MCE structure is extended to handle nonlinearities in both the system dynamics and the measurement model in a fashion similar to that of the extended Kalman filter (EKF). Lastly, the MCE is implemented on a general purposed graphical processing unit. The massively parallel structure inherent within the MCE is exploited to achieve real-time computational performance. Through simulation, the MCE outperforms the KF/EKF in arbitrary heavy-tailed and nonlinear dynamic environments.
Dr. Jason L. Speyer received his B.S. in aeronautics and astronautics from the Massachusetts Institute of Technology in 1960 and his Ph.D. in applied mathematics from Harvard University in 1968. He was awarded an honorary doctorate from the Technion-Israel Institute of Technology in 2013. He is the Ronald and Valerie Sugar Distinguished Professor in the Mechanical and Aerospace Engineering Department and the Electrical Engineering Department at UCLA. He coauthored, with W.H. Chung, Stochastic Processes, Estimation, and Control (Society for Industrial and Applied Mathematics (SIAM), 2008) and coauthored, with D.H. Jacobson, Primer on Optimal Control Theory (SIAM, 2010). He served as an associate editor for Technical Notes and Correspondence (1975-76) and Stochastic Control (1978-79), the Institute of Electrical and Electronics Engineers (IEEE) Transactions on Automatic Control, for AIAA Journal of Guidance and Control (1977-78) and for the Journal of Optimization Theory and Applications (1981-present). He is a fellow of IEEE and an honorary fellow of the American Institute of Aeronautics and Astronautics (AIAA). He was awarded the AIAA Mechanics and Control of Flight Award, the AIAA Dryden Lectureship in Research, the Air Force Exceptional Civilian Decoration (1991 and 2001), the IEEE Third Millennium Medal, the AIAA Guidance, Navigation, and Control Award, the Richard E. Bellman Control Heritage Award and membership in the National Academy of Engineering.