Assessment and Future Directions of Nonlinear Model Predictive ...

362 J.B. Jørgensen et al.with initial conditions ˆx k (t k )=ˆx k|k and P k (t k )=P k|k . The one-step aheadprediction of the measurements isŷ k+1|k =ŷ k (t k+1 )=H(ˆx k (t k+1 )) = H(ˆx k+1|k ) (7)The mean-covariance pair may be solved by standard ODE solvers in whichthe lower triangular part of the covariance matrix differential equation (6b)is appended to the mean differential equation (6a). For stiff systems in whichimplicit ODE solvers are needed, assuming that a dense linear algebra solver isused, the computational costs of solving this system has complexity O(m 3 )inwhich m = n + n(n +1)/2 andn is the state dimension. For large scale systemsthis corresponds to computational complexity O(n 6 ). Even if sparse solvers forthe linear algebra are applied, the mean-covariance pair cannot be solved inreasonable time by standard ODE solvers [2]. As a consequence, the extendedKalman filter for continuous-discrete time systems is not applicable to large-scalesystems using a standard implicit ODE solver for solution of (6). It can easilybe demonstrated [2] that solution of (6) is equivalent to solution of the systemof differential equationsdˆx k (t)dtdΦ(t, s)dt= F (ˆx k (t)) ˆx k (t k )=ˆx k|k (8a)( )∂F=∂x (ˆx k(t)) Φ(t, s) Φ(s, s) =I (8b)along with the integral equation∫ tP k (t) =Φ(t, t k )P k|k Φ(t, t k ) ′ +t kΦ(t, s)σ(s)σ(s) ′ Φ(t, s) ′ ds (8c)The advantage of the formulation (8) is that very efficient solvers exist [1, 7, 8]forintegration of the states (8a) along with the state sensitivities (8b). Subsequentcomputation of the covariance (8c) by quadrature is relatively cheap computation.Extended Kalman filters for continuous-discrete time systems based onsolution of (8) rather than (6) are more than two orders of magnitude faster fora system with 50 states [2].2.3 Continuous-Discrete Time SDAE SystemMany systems in the process industries are modelled by systems of index-1 differentialalgebraic equations rather than systems of ordinary differential equations.Index-1 differential algebraic equations arise when some physical phenomena aredescribed as equilibrium processes. This is the case for phase-equilibrium modelsof separation processes, e.g. distillation columns. The extension of filtering andprediction in continuous-discrete time stochastic differential equation (SDE) systems(5) to filtering and prediction in continuous-discrete time stochastic index-1