Reconciliation linear

Crowe, C.M., Recursive Identification of Gross Errors in Linear Data Reconciliation, AJChE Journal, 34(4), 1988,541-550. (Global chi square test, measurement test)... 

Note that nonhnear constraints can be treated in this manner through linearization. Consequently, adjustments to the measurements are required. The result from the reconciliation process is this set of adjusted measurements,... 

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... 

Several researchers [e.g., Tjoa and Biegler (1992) and Robertson et al. (1996)] have demonstrated advantages of using nonlinear programming (NLP) techniques over such traditional data reconciliation methods as successive linearization for steady-state or dynamic processes. Through the inclusion of variable bounds and a more robust treatment of the nonlinear algebraic constraints, improved reconciliation performance can be realized. 

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. 

Crowe, C. M. (1986). Reconciliation of process flow rates by matrix projection. Part II The non-linear case. AIChE J. 32, 616-623. 

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). 

Let us consider the system of linear balance equations described by Eq. (3.8). In the presence of measurement errors the balance equations are not satisfied exactly, and any general data reconciliation procedure must solve the following least squares 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. 

In this chapter the use of Q-R factorizations with the purpose of system decomposition and instrumentation analysis, for linear and bilinear plant models, is thoroughly investigated. Simple expressions are provided using subproducts of Q-R factorizations for application in data reconciliation. Furthermore, the use of factorization procedures... 

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. 

Two situations arise in linear data reconciliation. Sometimes all the variables included in the process model are measured, but more frequently some variables are not measured. Both cases will be separately analyzed. 

Linear Data Reconciliation with All Measured Variables... 

To illustrate the application of data reconciliation to linear systems, we will consider the problem presented by Ripps (1965). Four mass flows are measured, two entering and two leaving a chemical reactor. Three elemental balances are considered ... 

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,... 

The operation of a plant under steady-state conditions is commonly represented by a non-linear system of algebraic equations. It is made up of energy and mass balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization 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. 

In this sense, the application of Q-R factorizations constitutes an efficient alternative for solving bilinear data reconciliation. Successive linearizations and nonlinear programming are required for more complex models. These techniques are more reliable and accurate for most problems, and thus require more computation time. 

The linear/linearized data reconciliation solution deserves some special attention because it allows the formulation of alternative strategies for the processing of information. In this chapter the mathematical formulation for the sequential processing of both constraints and measurements is analyzed. 

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. 

In the following sections, different approaches to the solution of the preceding problem are briefly described. Special attention is devoted to the two-stage nonlinear EVM, and a method proposed by Valko and Vadja (1987) is described that allows the use of existing routines for data reconciliation, such as those used for successive linearization. 

Step 3 At fixed <9,+l perform the data reconciliation for each j = 1,2, M using successive linearization ... 

Liebman, M. J., and Edgar, T. F. (1988). Data reconciliation for non-linear processes. Prepr. AIChE Annu. Meet., Washington, DC. 

The first case study consists of a section of an olefin plant located at the Orica Botany Site in Sydney, Australia. In this example, all the theoretical results discussed in Chapters 4,5,6, and 7 for linear systems are fully exploited for variable classification, system decomposition, and data reconciliation, as well as gross error detection and identification. 

The second case study corresponds to an existing pyrolysis reactor also located at the Orica Botany Site in Sydney, Australia. This example demonstrates the usefulness of simplified mass and energy balances in data reconciliation. Both linear and nonlinear reconciliation techniques are used, as well as the strategy for joint parameter estimation and data reconciliation. Furthermore, the use of sequential processing of information for identifying inconsistencies in the operation of the furnace is discussed. 

Plant raw data PROCESS with raw data Reconciled data without bias deletion Linear reconciliation with bias deletion Nonlinear reconciliation with bias deletion PROCESS with nonlinear reconciliation... 

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