Assessment and Future Directions of Nonlinear Model Predictive ...

360 J.B. Jørgensen et al.In this contribution, we propose an extended Kalman filter (EKF) for stochasticcontinuous-time systems sampled at discrete time. The system evolution isdescribed by stochastic index-1 differential algebraic equations and the outputmeasurements at discrete times are static mappings contaminated by additivenoise. In the implementation, the special structure of the resulting EKF equationsare utilized such that the resulting algorithm is computationally efficientand numerically robust. By these features, the proposed EKF algorithm can beapplied for state estimation in NMPC of large-scale systems as well as in algorithmsfor grey-box identification of stochastic differential-algebraic systems [3].2 Extended Kalman FiltersThe extended Kalman filter has been accepted as an ad hoc filter and predictorfor nonlinear stochastic systems. It is ad hoc in the sense that it does not satisfyany optimality conditions but adopts the equations for the Kalman filter of linearsystems, which is an optimal filter. In this section, we present the extendedKalman filter for a stochastic difference-algebraic system (discrete time) andshow how this method is adapted to a stochastic differential-algebraic system(continuous-discrete time).2.1 Discrete-Time SystemConsider the stochastic difference-algebraic systemx k+1 = f(x k , z k , w k )0=g(x k , z k )y k = h(x k , z k )+v k(1a)(1b)(1c)in which w k ∼ N(0,Q k ), v k ∼ N(0,R k )andx 0 ∼ N(0,P 0|−1 ). (1a) and (1b)represent the system dynamics while (1c) is a measurement equation. Assumefurther that ∂g∂zis non-singular. Then according to the implicit function theorem,the algebraic variables, z k , are an implicit function of the state variables, x k , i.e.z k = χ(x k ). Consequently, the stochastic difference-algebraic system (1) may berepresented as a stochastic difference system with a measurement equationx k+1 = F (x k , w k )=f(x k ,χ(x k ), w k )y k = H(x k )+v k = h(x k ,χ(x k )) + v k(2a)(2b)This is the stochastic difference system to which the discrete-time extendedKalman filter applies. The extended Kalman filter for (2) consists of the equationsdescribed in the following [4]. The filter part of the extended Kalman filterconsists of the innovation-computatione k = y k − ŷ k|k−1 ,(3a)the feedback gain, K fx,k , computation