Extended Kalman filters (EKF) have been increasingly used for nonlinear state and parameter estimation in the chemical process industries. Unlike EKF, where the first-order linearization is used for error covariance estimation, unscented Kalman filters (UKF), as proposed by Julier and Uhlman [1], pass the mean and covariances of estimated states through a nonlinear transformation. This is achieved by carefully choosing a set of sigma points, which capture the true mean and covariance of the given distribution and result in UKF being capable of estimating the mean and covariances accurately. Performance of UKF can exceed the one for EKF where only first-order accuracy is achieved. Despite UKF's potential for improved state and parameter estimation, there are only few applications in chemical engineering [2, 3, 4].

In this work, UKF has been applied to a highly nonlinear CSTR [5], in order to investigate the performance achieved by the state and parameter estimator based upon a UKF. The performance of an EKF has also been studied and compared with the one for UKF on the described process subject to process uncertainties and measurement noise. In the second part of the work, both EKF and UKF have been applied to estimate process parameters in the dynamic system and a detailed comparison is made.

The case study shows that UKF performs as well as EKF when the described CSTR is operated in a small operating region. However, when the operating region is significantly enlarged, UKF performs better than EKF since the nonlinear behavior becomes more profound in this case and linearization used for computing covariances of the EKF can result in larger errors.

References:

[1] Julier, S.J., Uhlmann, J.K. 2004. Unscented Filtering and Nonlinear Estimation. Proc. of the IEEE 92:401–422.

[2] Rawlings, J. B., and Bakshi, B.R.. 2006. Particle Filtering and Moving Horizon Estimation. Comp. & Chem. Eng. 30:1529-1541

[3] Romanenko, A., Castro, J. A.A.M. 2004. The Unscented Filter as an Alternative to the EKF for Nonlinear State Estimation: a Simulation Case Study. Comp. & Chem. Eng. 28:347–355

[4] Romanenko, A., Castro, J. A.A.M. 2004. Unscented Kalman Filtering of a Simulated pH System. Ind. Eng. Chem. Res. 43:7531-7538

[5] Rajaraman, S., Hahn, J., and Mannan, M.S. 2006. Sensor fault diagnosis for nonlinear processes with parametric uncertainties. J. Hazard Mater.130:1-8.