Simple and Robust Formulations of the Unscented Kalman Filter and Model Predictive Control

Abstract

This thesis focuses mainly on state estimation by the Unscented Kalman filter (UKF), and partly on robust nonlinear model predictive control (RNMPC). Our aim is to refine or develop algorithms which are simple to implement, fast, robust, and accurate enough to facilitate practical usage. Our aim does not necessarily align with theoretically, but impractical, optimal state estimators or controllers. For the UKF, we have focused on i) a simple noise estimation algorithm and ii) numerical robustness in implementation. Our noise estimation method considers parametric uncertainty entering the model nonlinearly (e.g. the Cv-value of a valve) and estimates additive noise using the generalized unscented transformation. The framework of parametric uncertainty is far from new, but our proposed method outperforms the benchmark methods in either estimation accuracy or computational cost in theory and in a case study. A robust numerical implementation addresses the issue of finite numerical accuracy in a computer, and it may be the difference between successful estimation and filter divergence. We propose the normalized UKF (NUKF) which propagates standard deviations and correlations to remedy the issue of propagating ill-conditioned covariance matrices. This is numerically beneficial since entries in correlation matrices are bounded between (-1,1] and standard deviation is the square-root of variance. While other common square-root filters in literature propagates (more obscure) quantities as Cholesky factors, our NUKF propagate meaningful statistics which may be used to check for filter divergence online. The UKF relies on the unscented transformation (UT) to estimate relevant statistics using an approximate integration rule. If the input random variable is Gaussian, the UT is numerically unstable for moderate state dimensions due to a negative weight in the integration rule. We address this issue by reformulating a high-dimensional unstable UT to a sum of one-dimensional stable UTs. RNMPC has received much attention in academia the last years. We are, however, not aware of any industrial RNMPC implementation, which may be because the current algorithms are either i) very complex and hard to implement (e.g. tube-based NMPC), or ii) slow and somewhat theoretically unsatisfying (multistage NMPC). We propose a new methodology in RNMPC, using n-steps-ahead Monte Carlo simulations to estimate the required back-off from the constraints, where n is a (short) uncertainty horizon. Our new formulation is faster, easier to implement and more robust than the academia-standard multi-stage NMPC in a case study.