Interval constrained state estimation of nonlinear dynamical systems using unscented Gaussian sum filter

Unscented Kalman Filter (UKF) [1] based approaches are popular due to the absence of analytical linearization steps as well as their use of a deterministically chosen but limited set of samples (labeled sigma points). However, UKF is based on an implicit assumption that the conditional state densities at various steps are Gaussian. This assumption of Gaussianity is then carried over to the various extensions of UKF available in literature for incorporating constraints on states. This in turn may lead to potentially inferior performance of the constrained state estimators if the densities are significantly non-Gaussian. To overcome this issue, various attempts have been made that represent the conditional densities as a Gaussian Sum. However, the use of a large number of Gaussians renders these methods computationally demanding. Recently, a novel approach termed as Unscented Gaussian Sum Filter (UGSF) has been proposed that approximates the prior with a Sum of Gaussians using only the sigma points as generated in UKF [2]. It was shown using numerous solutions that UGSF outperforms the UKF while using a similar computational effort as the UKF [2], [3], [4]. In this work, we propose to extend UGSF to constrained UGSF by incorporation of constraints on the states in the estimation process. In particular, we propose to use Interval Constrained Unscented Transformation (ICUT) [5] and probability density function truncation algorithms [6] with the UGSF framework. Implementation on the three state isothermal batch reactor case study [7] shows that the proposed constrained UGSF outperforms constrained UKF.