Assessment and Future Directions of Nonlinear Model Predictive ...

New Extended Kalman Filter Algorithms for SDAEs 361C k = ∂H∂x (ˆx k|k−1)R k|k−1 = R k + C k P k|k−1 C k′K fx,k = P k|k−1 C kR ′ −1k|k−1and the filtered mean estimate and covariance(3b)(3c)(3d)ˆx k|k =ˆx k|k−1 + K fx,k e k(3e)P k|k = P k|k−1 − K fx,k R −1k|k−1 K′ fx,k(3f)As w k ⊥v k by assumption, ŵ k|k =0andQ k|k = Q k . The one-step ahead predictionequations of the extended Kalman filter are the mean-covariance evolutionequationsˆx k+1|k = F (ˆx k|k , ŵ k|k )P k+1|k = A k P k|k A ′ k + B k Q k|k B k′in which A k ∂x (ˆx k|k, ŵ k|k )andB kprediction of the measurements is= ∂Fŷ k+1|k = H(ˆx k+1|k )(4a)(4b)= ∂F∂w (ˆx k|k, ŵ k|k ). The one-step ahead2.2 Continuous-Discrete Time SDE SystemMost physical systems are modelled in continuous-time using conservation equation.For deterministic systems this gives rise to a system of ordinary differentialequations, while it for stochastic systems gives rise to a system ofstochastic differential equations [5, 6].When measurements at the discrete-times{t k : k =0, 1,...} are added to that system we have a continuous-discrete timestochastic systemdx(t) =F (x(t))dt + σ(t)dω(t)y(t k )=H(x(t k )) + v(t k )(4c)(5a)(5b)In this notation {ω(t)} is a standard Wiener process, i.e. a Wiener processwith incremental covariance Idt, the additive measurement noise is distributedas v(t k ) ∼ N(0,R k ), and the initial states are distributed as x(t 0 ) ∼N(ˆx 0|−1 ,P 0|−1 ).The filter equations in the extended Kalman filter for the continuous-discretetime system (5) are equivalent to the filter equations for the discrete-time system(2), i.e. (3) constitutes the filter equations. The mean and covariance of the onestepahead prediction, ˆx k+1|k =ˆx k (t k+1 )andP k+1|k = P k (t k+1 ), are obtainedby solution of the mean-covariance system of differential equationsdˆx k (t)= F (ˆx k (t)) (6a)dtdP k (t)dt=( )∂F∂x (ˆx k(t))P k (t)+P k (t)( ) ′ ∂F∂x (ˆx k(t)) + σ(t)σ(t) ′ (6b)