This paper presents the computational challenges of state estimation in nonlinear stochastic continuous-discrete time systems. The extended Kalman filter for continuous-discrete time systems is introduced by ad hoc extension of a probabilistic approach, based on Kolmogorov's forward equation, to filtering in linear stochastic continuous-discrete time systems. The resulting differential equations for the mean-covariance evolution of the nonlinear stochastic continuous-discrete time systems are solved efficiently using an ESDIRK integrator with sensitivity analysis capabilities. This ESDIRK integrator for the mean-covariance evolution is implemented as part of an extended Kalman filter and tested on several systems. For moderate to large sized systems, the ESDIRK based extended Kalman filter for nonlinear stochastic continuous-discrete time systems is more than two orders of magnitude faster than a conventional implementation. This is of significance in nonlinear model predictive control applications, statistical process monitoring as well as grey-box modelling of systems described by stochastic differential equations.

In the paper, stochastic differential equations are introduced for modelling of chemical systems. The proposed extended Kalman filter algorithm is a methodology to filter and predict such a system based on noise corrupted measurements at discrete times. This paradigm is ideally suited for state estimation in nonlinear model predictive control as it allows a systematic decomposition of the model into predictable and non-predictable dynamics. The application of the extended Kalman filter as the predictor in grey-box modelling of process systems using the prediction-error approach is emphasized in the presentation. We demonstrate the proposed algorithm to several large-scale process examples. In addition, the methodology is described in detail using the Van der Vusse reaction system as example.