Please join the department as Mahshad Valipour defends her PhD thesis on the novel methodologies in state estimation for constrained nonlinear systems under non-Gaussian measurement noise & process uncerta.
Chemical processes often involve changes in operating conditions that may lead non-Gaussian process uncertainties and measurement noises. Moreover, the distribution of system variables subjected to process constraints may not often follow Gaussian distributions. Kalman Filter (KF) and its extension, i.e., Extended Kalman Filter (EKF), are well-known state estimation schemes for unconstrained applications. This thesis initially performed state estimation using this approach for an unconstrained large-scale gasifier that supports efficiency and accuracy offered by KF. However, the underlying assumption in KF/EKF is that all the variables follow Gaussian distributions.
The current research aims to introduce an efficient EKF-based scheme, referred to as constrained Abridged Gaussian Sum EKF (constrained AGS-EKF), to generalize EKF for constrained nonlinear applications featuring non-Gaussian distributions uncertainties and noises. Constrained AGS-EFK uses Gaussian mixture models to approximate the non-Gaussian distributions of the constrained states, uncertainties, and 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 non-Gaussian variables. New modifications in both prior and posterior estimation steps are considered to capture the non-zero mean distribution of the uncertainties and noises, respectively, at no additional computational costs. Moreover, an intermediate step is considered in constrained AGS-EKF that explicitly applies the constraints on the priori estimation of the distributions, which require a relatively small additional computational costs.
Moving Horizon Estimation (MHE) is an optimization-based state estimation approach that can take into account all the constraints and provide the optimal estimated states. An error analysis is provided that shows that
(unconstrained) EKF can provide accurate estimates if it is constantly initialized by a constrained estimation scheme such as MHE. Similar to EKF, MHE assumes that the distributions of uncertainties and noises are zero-mean Gaussian, known a priori, and remain unchanged throughout the operation, i.e., known time-independent distributions. An Extended MHE (EMHE), is presented in this thesis that uses Gaussian mixture models to capture the known time-dependent non-Gaussian distributions of the uncertainties and noises. 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, at cost of a relatively small additional CPU time. Similar to the standard MHE, the application of EMHE is limited to the scenarios where the changes in the distribution of noises and uncertainties are known a priori, which may not be valid if any unscheduled operating changes occur during the plant operation.
Motivated by this aspect, a novel Robust MHE (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 those needed in the standard MHE.
Supervisor: Professor Luis Sandoval
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