Assessment and Future Directions of Nonlinear Model Predictive ...

366 J.B. Jørgensen et al.applies (10a)-(10b) as predictor with ˆx k|k and ẑ k|k as consistent initial conditions[7, 8]. For such an NMPC application numerical robustness and efficiency of theextended Kalman filter is of course important. However, numerical robustnessand efficiency of the extended Kalman filter is even more significant when it isapplied in systematic grey-box modelling of stochastic systems[3]. In such applications,numerical computation of e.g. the one-step ahead maximum-likelihoodparameter estimate requires repeated evaluation of the negative log-likelihoodfunction for parameters, θ, set by a numerical optimization algorithm during thecourse of an optimization. Compared to currently practiced grey-box identificationin stochastic models, the algorithm proposed in this paper is significantly(more than two orders of magnitude for a system with 50 states) faster and hasbeen extended to continuous-discrete time stochastic differential-algebraic systems(9). The proposed systematic estimation of the deterministic and stochasticpart of the EKF-predictor represents an alternative to output-error estimationof the drift terms and covariance matching for the process and measurementnoise covariance.References[1] Kristensen, M. R., Jørgensen, J. B., Thomsen, P. G. & Jørgensen, S. B. An ESDIRKmethod with sensitivity analysis capabilities. Computers and Chemical Engineering28, 2695–2707 (2004).[2] Jørgensen, J. B., Kristensen, M. R., Thomsen, P. G. & Madsen, H. Efficient numericalimplementation of the continuous-discrete extended kalman filter. Submittedto Computers and Chemical Engineering (2006).[3] Kristensen, N. R., Madsen, H. & Jørgensen, S. B. Parameter estimation in stochasticgrey-box models. Automatica 40, 225–237 (2004).[4] Kailath, T., Sayed, A. H. & Hassibi, B. Linear Estimation (Prentice Hall, 2000).[5] Åström,K.J. Introduction to Stochastic Control Theory (Academic Press, 1970).[6] Jazwinski, A. H. Stochastic Processes and Filtering Theory (Academic Press, 1970).[7] Kristensen, M. R., Jørgensen, J. B., Thomsen, P. G. & Jørgensen, S. B. Efficientsensitivity computation for nonlinear model predictive control. In Allgöwer, F. (ed.)NOLCOS 2004, 6th IFAC-Symposium on Nonlinear Control Systems, September01-04, 2004, Stuttgart, Germany, 723–728 (IFAC, 2004).[8] Kristensen, M. R., Jørgensen, J. B., Thomsen, P. G., Michelsen, M. L. & Jørgensen,S. B. Sensitivity analysis in index-1differential algebraic equations by ESDIRKmethods. In 16th IFAC World Congress 2005 (IFAC, Prague, Czech Republic,2005).