A Survey of the Probability Density Function Control for Stochastic Dynamic Systems

Probability density function (PDF) control strategy investigates the controller design approaches in order to to realise a desirable distributions shape control of the random variables for the stochastic processes. Different from the existing stochastic optimisation and control methods, the most important problem of PDF control is to establish the evolution of the PDF expressions of the system variables. Once the relationship between the control input and the output PDF is formulated, the control objective can be described as obtaining the control input signals which would adjust the system output PDFs to follow the pre-specified target PDFs. This paper summarises the recent research results of the PDF control while the controller design approaches can be categorised into three groups: 1) system model-based direct evolution PDF control; 2) model-based distribution-transformation PDF control methods and 3) databased PDF control. In addition, minimum entropy control, PDF-based filter design, fault diagnosis and probabilistic decoupling design are also introduced briefly as extended applications in theory sense

A Direct Quadrature Based Nonlinear Filtering with Extended Kalman Filter Update for Orbit Determination

Abstract-An optimal estimation of the states of a nonlinear continuous system with discrete measurements can be achieved through the solution of the Fokker-Planck equation, along with the Bayes' formula. However, solving the Fokker-Planck equation is restrictive in most cases. Recently a nonlinear filtering algorithm using a direct quadrature method of moments and the extended Kalman filter update mechanism was proposed, in which the associated Fokker-Planck equation was solved efficiently and accurately via discrete quadrature and the measurement update was done through the extended Kalman filter update mechanism. In this paper this hybrid filter based on the DQMOM and the EKF update is applied to the orbit determination problem with appropriate modification to mitigate the filter smugness. Unlike the extended Kalman filter, the hybrid filter based on the DQMOM and the EKF update does not require the burdensome evaluation of the Jacobian matrix and Gaussian assumption for system noise, and can still provide more accurate estimation of the state than those of the extended Kalman filter especially when measurements are sparse. Simulation results indicate that the advantages of the hybrid filter based on the DQMOM and the EKF update make it a promising alternative to the extended Kalman filter for orbit estimation problems

Complete analytic solution to Brownian unicycle dynamics

This paper derives a complete analytical solution for the probability distribution of the configuration of a non-holonomic mobile robot that moves in two spatial dimensions by satisfying the unicycle kinematic constraints. The proposed solution differs from previous solutions since it is obtained by deriving the analytical expression of any-order moment of the probability distribution. To the best of our knowledge, an analytical expression for any-order moment that holds even in the case of arbitrary linear and angular speed, has never been derived before. To compute these moments, a direct integration of the Langevin equation is carried out and each moment is expressed as a multiple integral of the deterministic motion (i.e., the known motion that would result in absence of noise). For the special case when the ratio between the linear and angular speed is constant, the multiple integrals can be easily solved and expressed as the real or the imaginary part of suitable analytic functions. As an application of the derived analytical results, the paper investigates the diffusivity of the considered Brownian motion for constant and for arbitrary time-dependent linear and angular speed

Nonlinear Bayesian Estimation via Solution of The Fokker-Planck Equation

A general approach to optimal nonlinear filtering can be described by a recursive Bayesian approach. The key step in this approach is to determine the probability density function of the state vector conditioned on available measurements. However, an optimal solution to the Bayesian filtering problem can only be obtained exactly for a small class of problems such as linear and Gaussian cases. Therefore, in practice, approximate solutions, such as the extended Kalman filter, have been used.An optimal nonlinear filtering in a recursive Bayesian approach is a two-step process which consists of the prediction and the update process. In the update process, the priori conditional state probability density function (PDF) from the prediction process is updated through Bayes' rule using measurements from sensors. The prediction of conditional state PDF can be made by solving the Fokker-Planck equation (FPE) that governs the time-evolution the conditional state PDF. However, it is extremely difficult to obtain an analytical solution of the Fokker-Planck equation with the exception of a few special cases. So far this estimation method has not been employed much in practice because of the high computational cost needed in solving the FPE numerically. In this dissertation, methods to improve the efficiency of the numerical method in solving the FPE are investigated to enhance the efficiency of the nonlinear filtering.Two finite difference methods, namely i) the explicit forward method and ii) the alternating direction implicit (ADI) method, are used to solve the FPE numerically. Although the explicit forward method is much simpler to implement, the ADI method is preferred for its low computational cost. To reduce the computational cost further, as the first contribution of the dissertation, a moving domain scheme is developed to reduce the domain of integration required for solving the Fokker-Planck equation numerically. Simulation results show that the accuracy of the estimation is improved as compared with the Extended Kalman Filter, and at the same time the computational cost is significantly lower with the proposed moving grid scheme than the case without it.Recently a nonlinear filtering algorithm using a direct quadrature method of moments was proposed, where the associated Fokker-Planck equation is solved efficiently via discrete quadrature based on moment constraints. For some problems, however, this approach showed the phenomenon similar to the "degeneracy'' in a particle filter, which is the concentration of weight on particular particles. The possible cause of the phenomenon is that only the weights are updated through the modified Bayes' rule. Therefore, in this dissertation, as another contribution, a new hybrid filter is proposed where the measurement update equations in the extended or the unscented Kalman filter are used along with the direct quadrature method of moments to solve the FPE. In this way the "degeneracy'' problem can be mitigated.Then, new proposed filtering methods are applied to several challenging problems such as i) the bearing-only tracking problem, ii) the relative orbit position estimation problem, and iii) the orbit determination problem to demonstrate their advantages. Simulation results indicate that the performance of the proposed filters are better than existing nonlinear filtering methods, such as the Extended Kalman Filter especially with less measurement updates

Filtraggio e stima dello stato nei sistemi dinamici non lineari

In questa tesi viene mostrato un metodo per la stima dello stato nei sistemi dinamici non lineari e non gaussiani. La tecnica principale mostrata è il filtro a particelle. Viene mostrato un metodo di ricampionamento alternativo basato sul concetto di massima entropia e vengono presentate diverse simulazioni a supporto di tale idea

Time-lapse seismic modeling and production data assimilation for enhanced oil recovery and CO2 sequestration

Production from a hydrocarbon reservoir is typically supported by water or carbon dioxide (CO2) injection. CO2 injection into hydrocarbon reservoirs is also a promising solution for reducing environmental hazards from the release of green house gases into the earth’s atmosphere. Numerical simulators are used for designing and predicting the complex behavior of systems under such scenarios. Two key steps in such studies are forward modeling for performance prediction based on simulation studies using reservoir models and inverse modeling for updating reservoir models using the data collected from field. The viability of time-lapse seismic monitoring using an integrated modeling of fluid flow, including chemical reactions, and seismic response is examined. A comprehensive simulation of the gas injection process accounting for the phase behavior of CO2-reservoir fluids, the associated precipitation/dissolution reactions, and the accompanying changes in porosity and permeability is performed. The simulation results are then used to model the changes in seismic response with time. The general observation is that gas injection decreases bulk density and wave velocity of the host rock system. Another key topic covered in this work is the data assimilation study for hydrocarbon reservoirs using Ensemble Kalman Filter (EnKF). Some critical issues related to EnKF based history matching are explored, primarily for a large field with substantial production history. A novel and efficient approach based on spectral clustering to select ‘optimal’ initial ensemble members is proposed. Also, well-specific black-oil or compositional streamline trajectories are used for covariance localization. Approach is applied to the Weyburn field, a large carbonate reservoir in Canada. The approach for optimal member selection is found to be effective in reducing the ensemble size which was critical for this large-scale field application. Streamline-based covariance localization is shown to play a very important role by removing spurious covariances between any well and far-off cell permeabilities. Finally, time-lapse seismic study is done for the Weyburn field. Sensitivity of various bulk seismic parameters viz velocity and impedance is calculated with respect to different simulation parameters. Results show large correlation between porosity and seismic parameters. Bulk seismic parameters are sensitive to net overburden pressure at its low values. Time-lapse changes in pore-pressure lead to changes in bulk parameters like velocity and impedance

The Sparse-grid based Nonlinear Filter: Theory and Applications

Filtering or estimation is of great importance to virtually all disciplines of engineering and science that need inference, learning, information fusion, and knowledge discovery of dynamical systems. The filtering problem is to recursively determine the states and/or parameters of a dynamical system from a sequence of noisy measurements made on the system. The theory and practice of optimal estimation of linear Gaussian dynamical systems have been well established and successful, but optimal estimation of nonlinear and non-Gaussian dynamical systems is much more challenging and in general requires solving partial differential equations and intractable high-dimensional integrations. Hence, Gaussian approximation filters are widely used. In this dissertation, three innovative point-based Gaussian approximation filters including sparse Gauss-Hermite quadrature filter, sparse-grid quadrature filter, and the anisotropic sparse-grid quadrature filter are proposed. The relationship between the proposed filters and conventional Gaussian approximation filters is analyzed. In particular, it is proven that the popular unscented Kalman filter and the cubature Kalman filter are subset of the proposed sparse-grid filters. The sparse-grid filters are employed in three aerospace applications including spacecraft attitude estimation, orbit determination, and relative navigation. The results show that the proposed filters can achieve better estimation accuracy than the conventional Gaussian approximation filters, such as the extended Kalman filter, the cubature Kalman filter, the unscented Kalman filter, and is computationally more efficient than the Gauss-Hermite quadrature filter

Nonlinear estimation with sparse temporal measurements

Nonlinear estimators based on the Kalman filter, the extended Kalman filter (EKF) and unscented Kalman filter (UKF) are commonly used in practical application. The Kalman filter is an optimal estimator for linear systems; the EKF and UKF are sub-optimal approximations of the Kalman filter. The EKF uses a first-order Taylor series approximation to linearize nonlinear models; the UKF uses an approximation of the states' joint probability distribution. Long measurement intervals exacerbate approximation error in each approach, particularly in covariance estimation. EKF and UKF performance under varied measurement frequency is studied through two problems, a single dimension falling body and simple pendulum. The EKF is shown more sensitive to measurement frequency than the UKF in the falling body problem. However, both estimators display insensitivity to measurement frequency in the simple pendulum problem. The literature's lack of consensus as to whether the EKF or UKF is the superior nonlinear estimator may be explained through covariance approximation error. Tools are presented to analyze EKF and UKF measurement frequency sensitivity. Covariance is propagated forward using the approximations of the EKF and UKF. Each propagated covariance is compared for similarity with a Monte Carlo propagation. The similarity of the covariance matrices is shown to predict filter performance. Portions of the state trajectory susceptible to EKF divergence are found using the Frobenius norm of the Jacobian matrix, limiting the need to consider covariance propagation along the entire state trajectory. Long measurement intervals also reveal a commonly overlooked challenge in UKF application: sigma point selection methods may produce sigma point vec-tors that violate physical state constraints. Although the UKF can function under this condition over short measurement intervals, unexpected failure may occur without consideration of physical constraints. A novel constrained UKF, using the scaled unscented transform, is proposed to address this issue.http://archive.org/details/nonlinearestimat1094550547Commander, United States NavyApproved for public release; distribution is unlimited