(176d) An Abridged Gaussian Sum Framework to Generalize Extended Kalman Filter for Constrained Nonlinear Systems Under Non-Gaussian Noise

The accuracy and computational efficiency of a state estimation framework plays a key role in feedback control since it will impact the performance and operation of chemical engineering systems. In closed-loop systems featuring complex nonlinear applications, nonlinear model predictive control (NMPC) is often coupled with extended Kalman filter (EKF) due to its computationally efficiency. Although moving horizon estimation is a widely known alternative to EKF for constrained nonlinear applications, it often requires significantly higher computational costs. Current approaches to improve Kalman filter performance for applications featuring constrained state variables and process and measurement noises that follow non-Gaussian distributions often involve the evaluation of second-order gradient information, online solution of optimization problems, or the implementation of sampling-based approaches (i.e. Particle Filter), which come at the cost of a significant increase in the computational effort [1][2][3][4]. Gaussian sum filters (GSF) have received attention to improve EKF performance. This method approximates the non-Gaussian densities using Gaussian mixture models. To evaluate the point estimates, GSF performs a set of EKFs with respect to each Gaussian component in the mixture. The number of Gaussian components and their corresponding number of EKFs in the set of GSF increases exponentially with both the number of constrained states (and non-Gaussian noises) and the degree of nonlinearity in their distributions. Therefore, GSFs may require significantly higher computational costs than that needed by the standard EKF. Moreover, GSFs may lead to biased estimations as some of the EKFs in the set operate at the edge of the process feasibility region. Although studies have shown that GSFs can yield accurate point estimates for systems with constraints on the states, the application of this method have been limited to small-scale systems with Gaussian noises [5][6][7]. Furthermore, studies considering GSFs for constrained systems in chemical engineering are scarce [8][9]. To the best of our knowledge, performing GSFs for applications involving non-Gaussian process and measurement noises are absent from the literature.

In this study, a novel approach referred to as Abridged Gaussian Sum-Extended Kalman Filter (AGS-EKF) is presented to improve EKF performance for processes involving non-Gaussian states (i.e. constrained states) and non-Gaussian noises. To avoid the high computational costs often required in the conventional GSF, AGS-EKF estimates the states based on the overall probability distributions that describe the Gaussian mixture models of non-Gaussian noises and constrained states. This makes AGS-EKF capable of considering all the possible noises and the values of states simultaneously, i.e., without the need of performing EKF calculations on the individual Gaussian components in GSF. Hence, AGS-EKF only requires performing EKF once using the mean and covariance of the overall Gaussian mixture model of the corresponding non-Gaussian variable (i.e., states, process and/or measurement noise). Consequently, AGSF-EKF not only reduces the computational costs incurred when using GSF but it also avoids biased estimations that may occur in the GSF estimation.

To handle the non-Gaussian process and measurement noises, the proposed AGS-EKF uses a modified version of EKF formulation to consider non-zero mean noises [10]. This modified EKF formulation requires the same computational costs as needed in the standard EKF. Moreover, the proposed AGS-EKF scheme includes an additional step that takes into account the constraints on the states. In this additional step, AGS-EKF uses the prior distribution of the states obtained by the prior estimation step in EKF, as well as the information of the state constraints/bounds, to approximate the non-Gaussian distribution of the states by a Gaussian mixture model. The main characteristics of the Gaussian mixture model (i.e., mean value of the states and covariance matrix) are used to estimate the constrained prior distribution of the states, which is then used to estimate the posterior estimation step in EKF. Expectation-maximization is a computationally efficient method used to approximate the Gaussian mixture model of the non-Gaussian densities of the states. Therefore, the proposed AGS-EKF does not increase the complexity in the calculations (as in previous constrained-EKFs that require the solution of optimization problems) or the computational effort (as in the case of sampling-based approaches). Multiple case studies arising in chemical engineering such as the Williams-Otto reactor and a Wastewater treatment plant have been used to demonstrate the benefits of the proposed approach when compared to standard EKF and conventional GSF under open-loop and closed-loop operation using an NMPC strategy. The results show that AGS-EKF offers higher accuracy in the estimation than that observed in GSF and standard EKF while still requiring similar CPU times to those needed by standard EKF. When tested in closed-loop, AGS-EKF is able to provide in short CPU times accurate estimates to the NMPC framework. Thus, this approach returned notable improvements in both performance and efficiency of the online control when compared to the case of using standard EKF or GSF as the state estimator in closed-loop. Hence, AGS-EKF can be used for online monitoring and control of practical industrial-scale applications where constrained states and noises exhibit non-zero mean nonlinear probability distributions.

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