An approximate expectation maximisation algorithm for estimating parameters in nonlinear dynamic models with process disturbances

Abstract

Stochastic terms are included in fundamental dynamic models of chemical processes to account for disturbances, input uncertainties and model mismatch. The resulting equations are called stochastic differential equations (SDEs). An approximate expectation maximisation (AEM) algorithm using B‐splines is developed for estimating parameters in SDE models when the magnitude of the disturbances and model mismatch is unknown. The AEM method is evaluated using a two‐state nonlinear continuous stirred tank reactor (CSTR) model. The proposed algorithm is compared with two other maximum‐likelihood‐based methods (continuous time stochastic modelling (CTSM) [Kristensen and Madsen, Continuous Time Stochastic Modelling: CTSM 2.3 User's Guide, 2003; Kristensen et al., Automatica 2004; 40: 225] and extended approximate maximum likelihood estimation (AMLE) [Varziri et al., Can. J. Chem. Eng. 2008; 86: 828]). For the CSTR examples studied, the AEM algorithm provides more accurate estimates of model parameters, unknown initial conditions and disturbance intensities. SDE models and associated parameter estimates obtained using AEM will be helpful to engineers who subsequently implement on‐line state estimation and process monitoring schemes because the two types of uncertainties that are considered (i.e. measurement noise and stochastic process disturbances) are consistent with the error structure used in extended Kalman filters.