Journal of Process Control 24 (2014) 1425–1443 Contents lists available at ScienceDirect Journal of Process Control j ourna l ho me pa ge: www.elsevier.com/locate/jprocont State estimation of nonlinear dynamical systems using nonlinear update based Unscented Gaussian Sum Filter , Krishna Kumar Kottakki, Sharad Bhartiya ∗ , Mani Bhushan ∗ Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India a r t i c l e i n f o Article history: Received 12 December 2013 Received in revised form 19 June 2014 Accepted 19 June 2014 Available online 10 August 2014 Keywords: Nonlinear state estimation Unscented Kalman Filter Sum of Gaussians a b s t r a c t Two attractive features of Unscented Kalman Filter (UKF) are: (1) use of deterministically chosen points (called sigma points), and (2) only a linear dependence of the number of sigma points on the number of states. However, an implicit assumption in UKF is that the prior conditional state probability density and the state and measurement noise densities are Gaussian. To avoid the restrictive Gaussianity assump- tion, Gaussian Sum-UKF (GS-UKF) has been proposed in literature that approximates all the underlying densities using a sum of Gaussians. However, the number of sigma points required in this approach is significantly higher than in UKF, thereby making GS-UKF computationally intensive. In this work, we propose an alternate approach, labeled Unscented Gaussian Sum Filter (UGSF), for state estimation of nonlinear dynamical systems, corrupted by Gaussian state and measurement noises. Our approach uses a Sum of Gaussians to approximate the non-Gaussian prior density. A key feature of this approximation is that it is based on the same number of sigma points as used in UKF, thereby resulting in similar com- putational complexity as UKF. We implement the proposed approach on two nonlinear state estimation case studies and demonstrate its utility by comparing its performance with UKF and GS-UKF. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction Recursive Bayesian estimation has been widely used for state and parameter estimation of dynamic systems. Bayesian estimation techniques make use of prior knowledge, typically in the form of a mathematical model of the system, and available measurements to obtain the conditional posterior probability density function of the states and parameters [1,2]. Since the mean of the conditional posterior density minimizes the mean square error of the estimate [3], it is commonly used as the point estimate. Recursive Bayesian estimation techniques use a two-step procedure to compute the posterior density. These steps are: (i) a prediction step, which involves determination of prior density by using the process model and past measurements, and (ii) an update step, which involves modification (update) of the prior density using the current measurement to obtain the posterior density. For linear dynamical systems with linear measurement functions and Gaussian uncertainties in the states, measurements and the initial conditions, the posterior density is also Gaussian [4], with mean and covariance corresponding to the well-known Kalman Filtering equations [5]. In particular, the update step involves obtaining the posterior mean as a linear combination of the prior mean and the available measurements, thereby resulting in a linear update equation. For nonlinear dynamical systems, analytical solutions to the Bayesian estimation problem are generally not available even if the uncer- tainties (noise) in the states and the measurements are Gaussian. Several extensions of the basic Kalman Filter (KF) have been proposed in literature to address such cases, with the Extended Kalman Filter (EKF) being one of the most popular [6]. The prediction step in EKF involves approximation of the system dynamics by a linearized model, thereby preserving the Gaussianity in the Gaussian prior. Similar to the KF, the update step in EKF is also linear involving approximation of the nonlinear measurement function by a linearized function. In the last two decades, Unscented Kalman Filter (UKF) [7] has emerged as a popular alternative to EKF for state estimation of nonlinear dynamical systems [8–12]. It is based on the premise that “approximating a density function is easier than approximating a nonlinear function” [7]. Similar to EKF, UKF also implicitly assumes the prior density as Gaussian whose mean and covariance are obtained by propagating a  The authors thank the Department of Science and Technology, India, for partial financial assistance under grant 09DST010.  Condensed version of this work has been accepted for DYCOPS-2013. ∗ Corresponding authors. Tel.: +91 22 25767225/22 25767214; fax: +91 22 25726895/22 25726895. E-mail addresses: bhartiya@che.iitb.ac.in (S. Bhartiya), mbhushan@iitb.ac.in (M. Bhushan). http://dx.doi.org/10.1016/j.jprocont.2014.06.013 0959-1524/© 2014 Elsevier Ltd. All rights reserved.