Novel Methodologies in State Estimation for Constrained Nonlinear Systems under Non-Gaussian Measurement Noise & Process Uncertainty

Abstract

Chemical processes often involve scheduled/unscheduled changes in the operating conditions that may lead to non-zero mean non-Gaussian (e.g., uniform, multimodal) process uncertainties and measurement noises. Moreover, the distribution of the variables of a system subjected to process constraints may not often follow Gaussian distributions. It is essential that the state estimation schemes can properly capture the non-Gaussianity in the system to successfully monitor and control chemical plants. Kalman Filter (KF) and its extension, i.e., Extended Kalman Filter (EKF), are well-known model-driven state estimation schemes for unconstrained applications. The present thesis initially performed state estimation using this approach for an unconstrained large-scale gasifier that supports the efficiency and accuracy offered by KF. However, the underlying assumption considered in KF/EKF is that all state variables, input variables, process uncertainties, and measurement noises follow Gaussian distributions. The existing EKF-based approaches that consider constraints on the states and/or non-Gaussian uncertainties and noises require significantly larger computational costs than those observed in EKF applications. The current research aims to introduce an efficient EKF-based scheme, referred to as constrained Abridged Gaussian Sum Extended Kalman Filter (constrained AGS EKF), that can generalize EKF to perform state estimation for constrained nonlinear applications featuring non-zero mean non-Gaussian distributions. Constrained AGS-EFK uses Gaussian mixture models to approximate the non-Gaussian distributions of the constrained states, process uncertainties, and measurement noises. In the present abridged Gaussian sum framework, the main characteristics of the overall Gaussian mixture models are used to represent the distributions of the corresponding non-Gaussian variable. Constrained AGS-EKF includes new modifications in both prior and posterior estimation steps of the standard EKF to capture the non-zero mean distribution of the process uncertainties and measurement noises, respectively. These modified prior and posterior steps require the same computational costs as in EKF. Moreover, an intermediate step is considered in the constrained AGS-EKF framework that explicitly applies the constraints on the priori estimation of the distributions of the states. The additional computational costs to perform this intermediate step is relatively small when compared to the conventional approaches such as Gaussian Sum Filter (GSF). Note that the constrained AGS-EKF performs the modified EKF (consists of modified prior, intermediate, and posterior estimation steps) only once and thus, avoids additional computational costs and biased estimations often observed in GSFs. Moving Horizon Estimation (MHE) is an optimization-based state estimation approach that provides the optimal estimations of the states. Although MHE increases the required computation costs when compared to EKF, MHE is best known for the constrained applications as it can take into account all the process constraints. This PhD thesis initially provided an error analysis that shows that EKF can provide accurate estimates if it is constantly initialized by a constrained estimation scheme such as MHE (even though EKF is unconstrained state estimator). Despite the benefits provided by MHE for constrained applications, this framework assumes that the distributions the process uncertainties and measurement noises are zero-mean Gaussian, known a priori, and remain unchanged throughout the operation, i.e., known time-independent distributions, which may not be accurate set of assumptions for the real-world applications. Performing a set of MHEs (one MHE per each Gaussian component in the mixture model) more likely become computationally taxing and hence, is discouraged. Instead, the abridged Gaussian sum approach introduced in this thesis for AGS-EKF framework can be used to improve the MHE performance for the applications involving non-Gaussian random noises and uncertainties. Thus, a new extended version of MHE, i.e., referred to as Extended Moving Horizon Estimation (EMHE), is presented that makes use of the Gaussian mixture models to capture the known time-dependent non-Gaussian distributions of the process uncertainties and measurement noises use of the abridged Gaussian sum approach. This framework updates the Gaussian mixture models to represent the new characteristics of the known time-dependent distribution of noises/uncertainties upon scheduled changes in the process operation. These updates require a relatively small additional CPU time; thus making it an attractive estimation scheme for online applications in chemical engineering. Similar to the standard MHE and despite the accuracy and efficiency offered by the EMHE scheme, the application of EMHE is limited to the scenarios where the changes in the distribution of noises and uncertainties are known a priori. However, the knowledge of the distributions of measurement noises or process uncertainties may not be available a priori if any unscheduled operating changes occur during the plant operation. Motivated by this aspect, a novel robust version of MHE, referred to as Robust Moving Horizon Estimation (RMHE), is introduced that improves the robustness and accuracy of the estimation by modelling online the unknown distributions of the measurement noises or process uncertainties. The RMHE problem involves additional constraints and decision variables than the standard MHE and EMHE problems to provide optimal Gaussian mixture models that represent the unknown distributions of the random noises or uncertainties along with the optimal estimated states. The additional constraints in the RMHE problem do not considerably increase the required computational costs than that needed in the standard MHE and consequently, both the present RMHE and the standard MHE require somewhat similar CPU time on average to provide the point estimates. The methodologies developed through this PhD thesis offers efficient MHE-based and EKF-based frameworks that significantly improve the performance of these state estimation schemes for practical chemical engineering applications.