Assessment and Future Directions of Nonlinear Model Predictive ...

New Extended Kalman Filter Algorithms forStochastic Differential Algebraic EquationsJohn Bagterp Jørgensen 1 , Morten Rode Kristensen 2 , Per Grove Thomsen 1 ,and Henrik Madsen 11 Informatics and Mathematical Modelling, Technical University of Denmark,DK-2800 Kgs. Lyngby, Denmark{jbj,pgt,hm}@imm.dtu.dk2 Department of Chemical Engineering, Technical University of Denmark, DK-2800Kgs. Lyngby, Denmarkmrk@kt.dtu.dkSummary. We introduce stochastic differential algebraic equations for physical modellingof equilibrium based process systems and present a continuous-discrete paradigmfor filtering and prediction in such systems. This paradigm is ideally suited for stateestimation in nonlinear predictive control as it allows systematic decomposition of themodel into predictable and non-predictable dynamics. Rigorous filtering and predictionof the continuous-discrete stochastic differential algebraic system requires solutionof Kolmogorov’s forward equation. For non-trivial models, this is mathematically intractable.Instead, a suboptimal approximation for the filtering and prediction problemis presented. This approximation is a modified extended Kalman filter for continuousdiscretesystems. The modified extended Kalman filter for continuous-discrete differentialalgebraic systems is implemented numerically efficient by application of an ESDIRKalgorithm for simultaneous integration of the mean-covariance pair in the extendedKalman filter [1, 2]. The proposed method requires approximately two orders of magnitudeless floating point operations than implementations using standard software.Numerical robustness maintaining symmetry and positive semi-definiteness of the involvedcovariance matrices is assured by propagation of the matrix square root of thesecovariances rather than the covariance matrices themselves.1 IntroductionThe objective of state estimation in nonlinear model predictive control is toreconstruct the current state from past and current measurements. This stateestimate is called the filtered state and is used as initial condition for predictionof the mean evolution in the dynamic optimization part of nonlinear modelpredictive control. While there is little or no difference in the way the predictionsare accomplished, extended Kalman filtering (EKF) and moving horizonestimation (MHE) approaches have been suggested to compute the filtered stateestimate for systems described by index-1 differential algebraic equations. However,mainly discrete-time stochastic systems or deterministic continuous-timesystems with stochastics appended in an ad hoc manner have been applied toindex-1 differential algebraic systems.R. Findeisen et al. (Eds.): Assessment and Future Directions, LNCIS 358, pp. 359–366, 2007.springerlink.com c○ Springer-Verlag Berlin Heidelberg 2007