Estimation of temporal and spatio-temporal nonlinear descriptor systems

As advances in the remote sensing of fluid flows forge ahead at an impressive rate, we face an increasingly compelling question of how best to exploit this progress. Light detection and ranging (LIDAR) measurement equipment still presents the problems of having only radial (line-of-sight) wind speed measurements (Cyclops' dilemma). Substantial expanses of unmeasured flow still remain and range weighting errors have a considerable influence on LIDAR measurements. Clearly, more information needs to be extracted from LIDAR data and an estimation problem naturally arises. A key challenge is that most established estimation techniques, such as Kalman filters, cater for systems that are finite-dimensional and described by ordinary differential equations (ODEs). By contrast, many fluid flows are governed by the Navier-Stokes equations, which are nonlinear partial differential-algebraic equations (PDAEs). With this motivation in mind, this thesis proposes a novel statistical signal processing framework for the model-based estimation of a class of spatio-temporal nonlinear partial differential-algebraic equations (PDAEs). The method employs finite-dimensional reduction that converts this formulation to a nonlinear DAE form for which new unscented transform-based filtering and smoothing algorithms are proposed. Gaussian approximations are derived for differential state variables and more importantly, extended to algebraic state variables. A mean-square error lower bound for the nonlinear descriptor filtering problem is obtained based on the posterior Cramér-Rao inequality. The potential of adopting a descriptor systems approach to spatio-temporal estimation is shown for a wind field estimation problem. A basis function decomposition method is used in conjunction with a pressure Poisson equation (PPE) formulation to yield a spatially-continuous, strangeness-free, reduced-order descriptor flow model which is used to estimate unmeasured wind velocities and pressure over the entire spatial region of interest using sparse measurements from wind turbine-mounted LIDAR instruments. The methodology is validated for both synthetic data generated from large eddy simulations of the atmospheric boundary layer and real-world LIDAR measurement data. Results show that a reconstruction of the flow field is achievable, thus presenting a validated estimation framework for potential applications including wind gust prediction systems, the preview control of wind turbines and other spatio-temporal descriptor systems spanning several disciplines.