Assessment and Future Directions of Nonlinear Model Predictive ...

364 J.B. Jørgensen et al.using the state sensitivities, Φ xx (t, s). In conclusion, the one-step ahead predictedstate mean, ˆx k+1|k = ˆx k (t k+1 ), and covariance, P k+1|k = P k (t k+1 ), incontinuous-discrete time SDAE systems (9) is accomplished by solution of solutionof (10). The one-step ahead prediction of the measurement isŷ k+1|k =ŷ k (t k+1 )=h(ˆx k (t k+1 ), ẑ k (t k+1 )) = h(ˆx k+1|k , ẑ k+1|k ) (11)in which ẑ k+1|k =ẑ k (t k+1 ). The filter equations of the extended Kalman filterfor (9) are identical to the filter equations (3) for the discrete-time system (1).In particular, it should be noted that C k is computed using the implicit functiontheorem, i.e.C k = ∂H∂x (ˆx k|k−1) =[ (∂h∂x)−( ∂h∂z)( ∂g∂z) −1 ( ) ] ∂g∂x(ˆx k|k−1 ,ẑ k|k−1 )(12)3 Numerical ImplementationThe efficiency of the numerical implementation of the extended Kalman filter forstochastic continuous-discrete time differential algebraic equations stems fromefficient integration of the mean-covariance pair describing the evolution of themean and covariance of the system.3.1 ESDIRK Based Mean-Covariance IntegrationThe system (10) is integrated using an ESDIRK method with sensitivity computationcapabilities [1, 8]. The ESDIRK method for integration of (10a)-(10b)consists of solution of the equationsX 1 = x n Z 1 = z n g(X 1 ,Z 1 ) = 0 (13a)([ ] [ ]) ⎛[ ][ ] ⎞Xi f(Xi ,Z i ) xn ∑i−1f(Xj ,ZR(X i ,Z i )= − τ n γ− ⎝j )+ τ n a ij⎠ =00 g(X i ,Z i ) 00j=1(13b)with X i = x(T i ), Z i = z(T i ), T i = t n + τ n c i ,andi =2, 3, 4. It is assumedthat (x 0 ,z 0 ) is consistent, i.e. g(x 0 ,z 0 ) = 0. Note that in the applied ESDIRK∑method: t n = T 1 , t n+1 = T 4 and x n+1 = x n + τ 4n j=1 b jf(X j ,Z j ) with b j = a 4jfor j =1, 2, 3, 4. The method is stiffly accurate implying that x n+1 = X 4 andz n+1 = Z 4 . By construction, the pair (x n+1 ,z n+1 ) is consistent with the algebraicrelation, i.e. g(x n+1 ,z n+1 ) = 0. The integration error estimate, e n+1 ,usedbythe step-length controller in the ESDIRK integration method is4∑e n+1 = τ n d j f(X j ,Z j )j=1(13c)