Data Reconciliation A Filtering Approach

Let us now consider a system modeled by the following system of equations  

The first term in the bracket stands for the mean, and the second for the spectral density matrix. For the continuous formulation, the covariances for the model and observation errors are given as 

The distinctive feature of the dynamic case is the time evolution of the estimate and its error covariance matrix. Their time dependence is given by 

These differential equations depend on the entire probability density function pi s., t) for x(r). The evolution with time of the probability density function can, in principle, be solved with Kolmogorov s forward equation (Jazwinski, 1970), although this equation has been solved only in a few simple cases (Bancha-Reid, 1960). The implementation of practical algorithms for the computation of the estimate and its error covariance requires methods that do not depend on knowing p(s, t). 

An often-used method consists of expanding f in Eq. (8.1) as a Taylor series about a certain vector that is close to x(t). In particular, if a first-order expansion is carried out on the current estimate of the state vector, we obtain 

---

DYNAMIC DATA RECONCILIATION A FILTERING APPROACH 8.2.1. Problem Statement... 

Jang, S. S., Josepth, B and Mukai, H. (1986). Comparison of two approaches to on-line parameter and state estimation problem of non-linear systems. Ind. Eng. Chem. Process Des. Dev. 25, 809-814. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York. Liebman, M. J., Edgar, T. F., and Lasdon, L. S. (1992). Efficient data reconciliation and estimation for dynamic process using non-linear programming techniques. Comput. Chem. Eng. 16, 963-986. McBrayer, K. F., and Edgar, T. F. (1995). Bias detection and estimation on dynamic data reconciliation. J Proc. Control 15, 285-289. 

The filters tuning is a crucial issue due the need to quantify the accuracy of the model in terms of the process noise covariance matrix for process characterized by structural uncertainties which are time-varying. Thus, approaches to time-varying covariances were studied and included to a traditional EKF and an optimization-based state estimators constrained EKF (CEKF) formulations. The results for these approaches have shown a significant improvement in filters performance. Furthermore, the performance of these estimators as a transient data reconciliation technique has been appraised and the results have shown the CEKF suitability for this proposes. 

---