UCLA

In this dissertation, a real-time, analytic and recursive multivariate state-estimation algorithm is developed for time-invariant, time-varying and nonlinear dynamical systems. Unlike Gaussian based state-estimation algorithms, the proposed state-estimation algorithm uses Cauchy random variables to model the uncertainties in the process and measurement functions. For this reason, it is referred to as the multivariate Cauchy estimator (MCE). The MCE uses a characteristic function representation of the conditional probability density function of the system state vector, given the measurement history, which generates the conditional mean and covariance estimates of the system state vector at each estimation step. The characteristic function of the MCE is enhanced in this dissertation from its previous form by an innovative, computationally tractable, and reduced structure. In particular, the backward recursive, or tree-like, evaluation procedure of the previously-used characteristic function is replaced by a linear parameterization. This linear parameterization compresses the backward recursive characteristic function at each estimation step and allows similar terms of the characteristic function to now be combined together, which was previously not possible. Compressing the characteristic function is shown to lead to the elimination of over 99% of terms that previously comprised it after several estimation steps, although the number of terms after a measurement update still grows. Therefore, a method is developed to run the MCE for arbitrary simulation lengths and for the multivariate setting, despite the growing size of the characteristic function. Furthermore, the estimation structure of the MCE 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. It is then shown that the MCE algorithm can achieve real-time computational performance by exploiting the parallel structure of the compressed characteristic function, which is done by distributing the computation onto general-purpose graphical processing units. Through several linear and nonlinear dynamic simulations, the MCE and the extended MCE are shown to outperform the Kalman filter and the extended Kalman filter, respectively, for dynamical systems within heavy-tailed noise environments. Monte Carlo experiments illustrate the exciting robustness properties of the proposed estimator over the class of symmetric alpha-stable probability density functions.