A spatiotemporal estimation framework for real-world LIDAR wind speed measurements

Despite significant advances in the remote sensing of fluid flows, 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. With this motivation in mind, this brief shows that it is possible to estimate the wind velocity, wind direction, and absolute pressure over the entire spatial region of interest. A key challenge is that most established estimation techniques 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 partial differential-algebraic equations (PDAEs). We show how a basis function decomposition method in conjunction with a pressure Poisson equation (PPE) formulation yields a spatially continuous, strangeness-free, reduced-order dynamic model for which a modified DAE form of the unscented Kalman filter (UKF) algorithm is used to estimate unmeasured velocities and pressure using sparse measurements from wind turbine-mounted LIDAR instruments. The approach 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 and the preview control of wind turbines

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

Latent state estimation in a class of nonlinear systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The problem of estimating latent or unobserved states of a dynamical system from observed data is studied in this thesis. Approximate filtering methods for discrete time series for a class of nonlinear systems are considered, which, in turn, require sampling from a partially specified discrete distribution. A new algorithm is proposed to sample from partially specified discrete distribution, where the specification is in terms of the first few moments of the distribution. This algorithm generates deterministic sigma points and corresponding probability weights, which match exactly a specified mean vector, a specified covariance matrix, the average of specified marginal skewness and the average of specified marginal kurtosis. Both the deterministic particles and the probability weights are given in closed form and no numerical optimization is required. This algorithm is then used in approximate Bayesian filtering for generation of particles and the associated probability weights which propagate higher order moment information about latent states. This method is extended to generate random sigma points (or particles) and corresponding probability weights that match the same moments. The algorithm is also shown to be useful in scenario generation for financial optimization. For a variety of important distributions, the proposed moment-matching algorithm for generating particles is shown to lead to approximation which is very close to maximum entropy approximation. In a separate, but related contribution to the field of nonlinear state estimation, a closed-form linear minimum variance filter is derived for the systems with stochastic parameter uncertainties. The expressions for eigenvalues of the perturbed filter are derived for comparison with eigenvalues of the unperturbed Kalman filter. Moment-matching approximation is proposed for the nonlinear systems with multiplicative stochastic noise

Implementation of Kalman Filtering for Differential-Algebraic Equations

This thesis describes two Kalman filters which are usable on semi-explicit index-1 differential-algebraic equations, prior to which a discussion of linear and nonlinear Kalman filters is presented. Performance between differential-algebraic equation-compatible Kalman filters and their ordinary differential equation counterparts is compared in two examples. Basic existence and uniqueness theory of linear differential-algebraic equations is discussed along with the process of numerically approximating the solution. Desire to estimate the state of charge of a lithium ion cell is used as motivation. The electrochemical processes of a lithium ion cell are discussed. When discretized, the model of a lithium ion cell results in a differential-algebraic equation

Optimal experimental design for parameter identification and model selection

Magdeburg, Univ., Fak. für Elektrotechnik und Informationstechnik, Diss., 2014René Schenkendor