(242b) Combined Data Reconciliation and Parameter Estimation

Process measurements are fundamentally contaminated by errors during the measurement, processing and transmission of the measured signal [1-3]. It should be emphasized that errors in measured data often lead to considerable deterioration in plant performance. A simplified understanding of techniques used to process measurement data can be split into three basic steps, namely variable classification, gross error detection and data reconciliation. Through utilizing spatial or temporal redundancies, data reconciliation adjusts the process measurements to improve its accuracy.  The most prevalent form of dynamic data reconciliation is the widely-established Kalman filter, which is the optimal estimator for linear systems in the presence of Gaussian noise [4]. Modifications to Kalman filter have been developed to handle nonlinear systems; a common modification leads to the extended Kalman filter (EKF). However, Kalman filter approaches are limited in relevance because its performance is directly related to the quality of approximations made in the state and covariance estimates. Most chemical engineering processes often function dynamically in highly nonlinear regions. Under such conditions, linearization may generate errors and the extended Kalman filter may produce biased estimates. Moreover, it may be difficult to tune the extended Kalman filter to achieve a satisfactory performance. In addition, the Kalman filter approaches do not support the inclusion of variable bounds and inequality constraints. The inclusion of inequality constraints is important because process models may be described in terms of inequalities. Despite its shortcomings, the established Kalman filter approaches are the yardstick for comparison for other dynamic data reconciliation techniques [3,5]. Our objective is to identify and improve upon existing dynamic data reconciliation techniques to overcome the limitations of the extended Kalman filter.

Our approach is to utilize a nonlinear dynamic data reconciliation approach, which is similar to the approach undertaken by Liebman et al [3]. This approach does not introduce linearization errors and therefore can handle processes with strong nonlinearities. Moreover, the nonlinear approach does not depend on any assumption of measurement error distribution. Inclusion of inequality constraints and variable bounds are also supported. Firstly, a moving window approach is used, so that only measurements within the window will be reconciled. This has the advantage of constraining the optimization problem to one of a fixed dimension. Secondly, the data reconciliation, which is a nonlinear optimization problem, generally does not have an analytical solution. Many solvers are available from literature [3, 6-11], and in this work nonlinear sequential programming [7,8] is utilized since it is not limited to linear or bilinear systems, and is not restricted to any form of objective function. Thirdly, for the choice of the optimization objective functions, two new objective functions, namely the Logistic probability density and Lorentz probability density functions are proposed in this paper. These two functions have the advantage of being statistically robust, which is an advantage over the conventional weighted least squares function. Fourthly, we also propose extending the approach to the combined data reconciliation and parameter estimation, which is expected to generate more accurate state and parameter estimates. This is confirmed by the findings of MacDonald and Howat [12].

The extended Kalman filter and the proposed nonlinear approach are implemented and verified via two case studies, namely a simulation case study of two continuously stirred tank reactors (CSTRs) in series, which introduce strong process nonlinearities, and more importantly an experimental case study of a heat exchanger. The results obtained from the case studies demonstrated that the extended Kalman filter is unable to handle strong process nonlinearities and the estimates diverged away from the true values and attained a steady steady bias (Fig. 1). In contrast, the proposed approach demonstrated that it is capable of handling strong process nonlinearities as the estimates remained close to the true values. The results obtained from the simulation case study demonstrated a reduction of approximately 50% error of the nonlinear approach over the extended Kalman filter (Fig. 2).

Fig. 1.  Estimates using EKF and NDDR

Fig. 2. Percentage error using EKF and NDDR

References:

[1]     Romagnoli, J. and Sanchez, M., ?Data Processing and Reconciliation for Chemical Process Operations?, Academic Press (2000)

[2]     Narasimhan, S. and Jordache, C., ?Data Reconciliation and Gross Error Detection: An Intelligent Use of Process Data?, Gulf Publishing Company (2000)

[3]     Liebman, M.J, Edgar, T.F., Lasdon, L.S., ?Efficient Data Reconciliation and Estimation for Dynamic Processes using Nonlinear Programming Techniques?, Comp. Chem. Eng. 16, 963-986 (1992)

[4]     Gelb, A., ?Applied Optimal Estimation?, MIT Press, Cambridge, Massachusetts (1974)

[5]     Henson, M.A. and Seborg D.E., ?Nonlinear Process Control?, Prentice Hall (1997)

[6]     Swartz, C.L.E., ?Data Reconciliation for Generalized Flowsheet Applications?, 197^th National Meeting, American Chemical Society, Dallas, TX (1989)

[7]     Crowe, C.M., ?Reconciliation of process flow rates by matrix Projection. Part II: The nonlinear case?, AIChE. J. 32, 616-623 (1986)

[8]     Tjoa, I. and Biegler, L., ?Simultaneous Strategies for Data Reconciliation and Gross Error Detection of Nonlinear Systems?, Comp. & Chem. Eng. 15, 679-690 (1991)

[9]     Gill, P., Murray, W.A., Saunders, M.A., and Wright, M.H., ?User Guide for NPSOL (Version 4.0): A FORTRAN program for Nonlinear Programming, Technical Report? SOL 86-2, Stanford University, Department of Operation Research, Stanford, CA.

[10]  Finlayson, B.A., ?The Method of Weighted Residuals and Variational Principles?, Academic Press, New York (1972)

[11]  Renfro, J.G., Morshedi and Asbjornsen, O.A., ?Simultaneous optimization and solution of systems described by differential/algebraic equations.?, Comp. Chem. Eng. 11, 503-517 (1987)

[12]  MacDonald, R.J. and Howat, C.S., ?Data Reconciliation and Parameter Estimation in plant performance analysis?, AiChE. 34, 1-8 (1988)