Krishna Kumar Kottakki

An Improved Gaussian Sum Unscented Kalman Filter

Abstract Gaussian sum Unscented Kalman Filter (GS-UKF) is a recently improved estimator for state and parameter estimation of nonlinear dynamical systems. GS-UKF makes use of Unscented Kalman Filter (UKF) as a local filter at each of its individual Gaussians to obtain the local statistics. The UKF based local filters use a limited number of deterministically chosen samples, known as sigma points, to approximate the higher order moments of non-Gaussian prior. However, UKF has an inherent assumption of Gaussian statistics in approximation of non-Gaussian prior. This can undermines the true performance of UKF as well as GS-UKF. In this work, we propose to use Unscented Gaussian Sum Filter (UGSF), at each Gaussian of Gaussian Sum. UGSF uses the same UKF sigma points but approximates the prior with the sum of Gaussians, which can approximate any arbitrary density. We thus label our proposed approach as GS-UGSF. We show the utility of proposed GS-UGSF approach using a nonlinear case study from literature and compare the performance of GS-UGSF with GS-UKF.

Nonlinear Update Based Unscented Gaussian Sum Filter

Abstract Unscented Kalman Filter (UKF) is a popular method for state and parameter estimation of nonlinear dynamic systems. An attractive feature of UKF is that it utilizes deterministically chosen points (called sigma points), and the number of such points depends linearly on the dimension of the state space. However, an implicit assumption in UKF is that the underlying probability densities are Gaussian. To mitigate the Gaussianity assumption, Gaussian Sum-UKF has been proposed in literature that approximates all underlying densities using a sum of Gaussians. For accurate approximation, the number of sigma points required in this approach is significantly higher than UKF, thereby making the Gaussian Sum-UKF computationally intensive. In this work, we propose an alternate approach labeled unscented Gaussian Sum Filter (UGSF) that leverages the ability of Sum of Gaussians to approximate an arbitrary density, while using the same number of sigma points as in UKF. This is achieved by making suitable design choices of the various parameters in the Gaussian Sum representation. Thus, our approach requires similar computational effort as in UKF and hence does not suffer from the curse of dimensionality. We implement the proposed approach on a nonlinear state estimation case study and demonstrate its superior performance over UKF.

Optimization Based Constrained Unscented Gaussian Sum Filter

Abstract Bayesian state estimation for constrained nonlinear systems involves two main issues: (i) appropriate representation of the underlying non-Gaussian densities, and (ii) use of appropriate techniques to bound the estimates so that they satisfy the give constraints. In this work we present a new constrained state estimation approach, named as Optimization based Constrained Unscented Gaussian Sum Filter. The proposed approach uses Unscented Gaussian Sum Filter to represent the non-Gaussian densities and an optimization approach to incorporate the state constraints. Benefits of the proposed approach are illustrated by applying it to a benchmark case study.

Constrained Unscented Gaussian Sum Filter for state estimation of nonlinear dynamical systems

Abstract This work presents a novel constrained Bayesian state estimation approach for nonlinear dynamical systems. The proposed approach uses the recently proposed Unscented Gaussian Sum Filter to represent the underlying non-Gaussian densities as sum of Gaussians, and explicitly incorporates constraints on states during the measurement update step. This approach, labeled Constrained-Unscented Gaussian Sum Filter (C-UGSF), can thus model non-Gaussianity in constrained, nonlinear state estimation problems. Its applicability is demonstrated using three nonlinear, constrained state estimation case studies taken from literature, namely, (i) a gas phase batch reactor, (ii) an isothermal batch process, and (iii) a continuous polymerization process. Results demonstrate superior estimation performance along with a significant improvement in computational time when compared to Unscented Recursive Nonlinear Dynamic Data Reconciliation (URNDDR), which is a popular nonlinear, constrained state estimation approach available in literature.

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.

Projection based constrained nonlinear state estimation using Gaussian sum filters

Constraint handling is an integral part of any practical state estimation procedure. Most current approaches to constrained state estimation ensure that the point estimate is feasible with respect to the constraints and are based on popular unconstrained approaches such as sampling based approaches such as Unscented Kalman Filter (UKF) [1], which use deterministically chosen set of sigma points. However, UKF is based on an implicit assumption that the conditional state densities at various steps are Gaussian. In a recent work [2], a new filter termed as Unscented Gaussian Sum Filter (UGSF) was proposed that represents the prior density as a Gaussian sum. It was shown, using simulation studies, that the UGSF outperforms UKF thereby demonstrating the advantage of using a non-Gaussian prior. 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) [3] and projection algorithm [4] with the UGSF framework. Implementation on the three state isothermal batch reactor case study [5] shows that the proposed constrained UGSF using the projection method outperforms constrained UKF while maintaining the computation effort similar to the constrained UKF version.