Reconciliation problem

The authors are indebted to E. McCarney, C. A. Royer, D. Segel, M. Kataoka, Y Ka-matari, and G. V. Semisotnov for providing unpublished or previously published data and T. C. B. McLeish, G. Fredrickson, D. Shortle, R. Pappu, and G. Rose for many stimulating conversations regarding the reconciliation problem. ... 

The use of this extended planning model will only be problematic if extra reference points, e.g., initial tank storage levels, have to be considered. This may lead to overdetermination of the model (i.e., conflicting level values for a given point in time) and it may be necessary to solve a data reconciliation problem. ... 

Historically, treatment of measurement noise has been addressed through two distinct avenues. For steady-state data and processes, Kuehn and Davidson (1961) presented the seminal paper describing the data reconciliation problem based on least squares optimization. For dynamic data and processes, Kalman filtering (Gelb, 1974) has been successfully used to recursively smooth measurement data and estimate parameters. Both techniques were developed for linear systems and weighted least squares objective functions. 

The steady-state linear model data reconciliation problem can be stated as... 

Extended Kalman filtering has been a popular method used in the literature to solve the dynamic data reconciliation problem (Muske and Edgar, 1998). As an alternative, the nonlinear dynamic data reconciliation problem with a weighted least squares objective function can be expressed as a moving horizon problem (Liebman et al., 1992), similar to that used for model predictive control discussed earlier. 

Joint Parameter Estimation-Data Reconciliation Problem 166... 

SOME ISSUES ASSOCIATED WITH A GENERAL DATA RECONCILIATION PROBLEM... 

Finally, approaches are emerging within the data reconciliation problem, such as Bayesian approaches and robust estimation techniques, as well as strategies that use Principal Component Analysis. They offer viable alternatives to traditional methods and provide new grounds for further improvement. 

It is our goal in this book to address the problems, introduced earlier, that arise in a general data reconciliation problem. It is the culmination of several years of research and implementation of data reconciliation aspects in Argentina, the United States, and Australia. It is designed to provide a simple, smooth, and readable account of all aspects involved in data classification and reconciliation, while providing the interested reader with material, problems, and directions for further study. 

Chapter 5 deals with steady-state data reconciliation problem, from both a linear and a nonlinear point of view. Special consideration is given, in Chapter 6, to the problem of sequential processing of information. This has several advantages when compared with classical batch processing. 

In Chapter 11 some recent approaches for dealing with different aspects of the data reconciliation problem are discussed. A more general formulation in terms of a probabilistic framework is first introduced and its application in dealing with gross error is discussed in particular. In addition, robust estimation approaches are considered, in which the estimators are designed so they that are insensitive to outliers. Finally, an alternative strategy that uses Principal Component Analysis is reviewed. 

Because of the complexity of integrated processes and the large volume of available data in highly automated plants, classification algorithms are increasingly used nowadays. They are applied to the design of monitoring systems and to reduce the dimension of the data reconciliation problem. 

The idea of process variable classification was presented by Vaclavek (1969) with the purpose of reducing the size of the reconciliation problem for linear balances. In a later work Vaclavek and Loucka (1976) covered the case of multicomponent balances (bilinear systems). 

It should be noted here that this formulation of the problem is totally equivalent to that of previous chapter, since the data reconciliation problem is only a special case of the general estimation problem, where we directly measure the process variables. 

The authors (Meyer et al., 1993) introduced a variant method derived from Kretsovalis and Mah (1987) that allows chemical reactions and splitters to be treated. It leads to a decrease in the size of the data reconciliation problem as well as a partitioning of the equations for unmeasured variable classification. 

The unmeasured determinable variables in set NA2 are then substituted by their corresponding expressions as function of the measured variables and set NA2 is obtained. After this is accomplished, sets NA1 and NA2 contain only measured variables, which are then redundant. The corresponding equations constitute the set of constraints in the reconciliation problem. 

This chapter is devoted to the analysis of variable classification and the decomposition of the data reconciliation problem for linear and bilinear plant models, using the so-called matrix projection approach. The use of orthogonal factorizations, more precisely the Q-R factorization, to solve the aforementioned problems is discussed and its range of application is determined. Several illustrative examples are included to show the applicability of such techniques in practical applications. 

Crowe et al. (1983) proposed an elegant strategy for decoupling measured variables from the linear constraint equations. This procedure allows both the reduction of the data reconciliation problem and the classification of process variables. It is based on the use of a projection matrix to eliminate unmeasured variables. Crowe later extended this methodology (Crowe, 1986, 1989) to bilinear systems. 

Returning to our reconciliation problem, the Q-R decomposition of matrix A2 allows us to obtain Qu and Ru matrices and the permutation matrix IIU, such that... 

The following illustrative examples will allow us to understand the implications of the previous results on the reconciliation problem. 

In the following sections, the use of the Q-R decomposition approach is discussed within the framework of the general multicomponent and energy (bilinear) reconciliation problem. In this case the classification of the measured and unmeasured process variables involves a sequence of steps. 

A method for decomposing unmeasured process variables from the measured ones, using the Q-R orthogonal transformation, was discussed before for the linear case. A similar procedure is applied twice in order to resolve the nonlinear reconciliation problem. 

All unmeasured variables are eliminated from the constraints by using simple Q-R transformations and a linear reconciliation problem results. The zero columns of Ga... 

In this chapter, the use of projection matrix techniques, more precisely the Q-R factorization, to analyze, decompose, and solve the linear and bilinear data reconciliation problem was discussed. This type of transformation is selected because it provides a very good balance of numerical accuracy, flexibility, and computational cost (Goodall, 1993). 

In this chapter we concentrate on the statement and further solution of the general steady-state data reconciliation problem. Initially, we analyze its resolution for linear plant models, and then the nonlinear case is discussed. 

Let us first define the models to be used in our formulation of the data reconciliation problem. 

The data reconciliation problem can be generally stated as the following constrained weighted least-squares estimation problem ... 

The assumption that all variables are measured is usually not true, as in practice some of them are not measured and must be estimated. In the previous section the decomposition of the linear data reconciliation problem involving only measured variables was discussed, leading to a reduced least squares problem. In the following section,... 

In this section we will explore the applicability of different techniques for solving the nonlinear data reconciliation problem. 

Orthogonal factorizations may be applied to resolve problem (5.3) if the system of equations cp(x, u) = 0 is made up of linear mass balances and bilinear component and energy balances. After replacing the bilinear terms of the original model by the corresponding mass and energy flows, a linear data reconciliation problem results. 

The procedure is not complex, and the required computation time is low. However, its use is restricted to data reconciliation problems with linear and bilinear constraints, and variable bounds cannot be handled. 

Effect of the a priori classification/decomposition on the solution of the reconciliation problem. 

From the classification it was found that, for this specific problem, there are 10 redundant and 6 nonredundant measured variables, and all the unmeasured process variables are determinable. Symbolic manipulation of the equations allowed us to obtain the three redundant equations used in the reconciliation problem ... 

In this chapter we have stated the data reconciliation problem and explored some available techniques for solving it. It must be noted that the performance of reconciliation methods is strongly dependent on the particular system. 

Now, because of the manner in which the balances arise, the total set of algebraic equations can be partitioned into two arbitrary subsystems. The first contains (m — a) equations and the second contains the remaining a equations, where a is an arbitrary number 1 < a < m. Note that the cases a = 0 or a = m correspond to the overall reconciliation problem. 

This chapter discussed the idea of exploiting the sequential processing of information (both constraints and measurements), to allow computations to be done in a recursive way without solving the full-scale reconciliation problem. 

A more systematic approach was developed by Romagnoli and Stephanopoulos (1981) and Romagnoli (1983) to analyze a set of measurement data in the presence of gross errors. The method is based on the idea of exploiting the sequential processing of the information (constraints and measurements), thus allowing the computations to be done in a recursive way without solving the full-scale reconciliation problem. 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

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