New Extended Kaiman Filter Algorithms for Stochastic Differential Algebraic Equations

Abstract

We introduce stochastic differential algebraic equations for physical modelling of equilibrium based process systems and present a continuous-discrete paradigm for filtering and prediction in such systems. This paradigm is ideally suited for state estimation in nonlinear predictive control as it allows systematic decomposition of the model into predictable and non-predictable dynamics. Rigorous filtering and prediction of the continuous-discrete stochastic differential algebraic system requires solution of Kolmogorov’s forward equation. For non-trivial models, this is mathematically intractable. Instead, a suboptimal approximation for the filtering and prediction problem is presented. This approximation is a modified extended Kaiman filter for continuous-discrete systems. The modified extended Kaiman filter for continuous-discrete differential algebraic systems is implemented numerically efficient by application of an ESDIRK algorithm for simultaneous integration of the mean-covariance pair in the extended Kaiman filter [1, 2]. The proposed method requires approximately two orders of magnitude less floating point operations than implementations using standard software. Numerical robustness maintaining symmetry and positive semi-definiteness of the involved covariance matrices is assured by propagation of the matrix square root of these covariances rather than the covariance matrices themselves.