Nonlinear Data Reconciliation

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. 

The most general mathematical statement of an optimization problem is 

The necessary conditions for an optimal solution of problem (5.29) are equivalent (Edgar and Himmelblau, 1988) to those for optimizing the Lagrange function defined as 

These conditions are the well known Kuhn-Tucker (KT) conditions Lineal dependency of the gradients  

The sufficient conditions for obtaining a global solution of the nonlinear programming problem are that both the objective function and the constraint set he convex. If these conditions are not satisfied, there is no guarantee that the local optima will be the global optima. 

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

Islam, K., Weiss, G., and Romagnoli, J. A. (1994). Nonlinear data reconciliation for an industrial pyrolysis... 

Wongrat, M., Srinophakun, T. Srinophakun, P., 2005. Modified genetic algorithm for nonlinear data reconciliation. Comput. Chem. Eng. 29, 1059. 

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. 

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. 

Liebman, M. J. T. F. Edgar and L. S. Lasdon. Efficient Data Reconciliation and Estimation for Dynamic Processes Using Nonlinear Programming Techniques. Comput Chem Eng 16(10/11) 963-986 (1992). 

Dynamic Data Reconciliation Using Nonlinear Programming Techniques 148... 

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 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. 

The use of Q-R orthogonal factorizations is presented as an alternative methodology for performing data reconciliation for bilinear systems. Finally, we briefly describe current techniques for tackling the general nonlinear problem. 

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. 

Tjoa, I., and Biegler, L. (1991), Simultaneous strategies for data reconciliation and gross error detection of nonlinear systems. Comput. Chem. Eng. 15,679. 

In this chapter, the data reconciliation problem for dynamic/quasi-steady-state evolving processes is considered. The problem of measurement bias is extended to consider dynamic situations. Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation using nonlinear programming techniques will be discussed. 

DYNAMIC DATA RECONCILIATION USING NONLINEAR PROGRAMMING TECHNIQUES... 

If the most recent available measurements are at time step c, then a history horizon HAt can be defined from (tc — HAt) to tc, where At is the time step size. In order to obtain enough redundant information about the process, it is important to choose a horizon length appropriate to the dynamic of the specific system (Liebman et al., 1992). As shown in Fig. 5, only data measurements within the horizon will be reconciled during the nonlinear dynamic data reconciliation run. 

Finally, an approach for nonlinear dynamic data reconciliation using nonlinear programming techniques was discussed. This formulation involves the optimization of an objective function through the adjustment of estimate functions constrained by differential and algebraic equalities and inequalities. 

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. 

The most straightforward approach for solving nonlinear EVM problems is to use nonlinear programming to estimate zy and 6 simultaneously. In the traditional weighted least squares parameter estimation formulation there are only n optimization variables corresponding to the number of unknown parameters. In contrast, the simultaneous parameter estimation and data reconciliation formulation has (pM + n)... 

Kim et al. (1990) proposed a nested, nonlinear EVM, following ideas similar to those of Reilly and Patino-Leal (1981). In this approach, the parameter estimation is decoupled from the data reconciliation problem however, the reconciliation problem is optimized at each iteration of the parameter estimation problem. 

In this section the extension of the use of nonlinear programming techniques to solve the dynamic joint data reconciliation and parameter estimation problem is briefly discussed. As shown in Chapter 8, the general nonlinear dynamic data reconciliation (NDDR) formulation can be written as ... 

In this chapter, the general problem of joint parameter estimation and data reconciliation was discussed. First, the typical parameter estimation problem was analyzed, in which the independent variables are error-free, and aspects related to the sequential processing of the information were considered. Later, the more general formulation in terms of the error-in-variable method (EVM), where measurement errors in all variables are considered in the parameter estimation problem, was stated. Alternative solution techniques were briefly discussed. Finally, joint parameter-state estimation in dynamic processes was considered and two different approaches, based on filtering techniques and nonlinear programming techniques, were discussed. 

If in addition to the binary assumption on P x, sensor errors are normally distributed and independent across the data sets, the problem becomes our typical nonlinear least squares data reconciliation problem ... 

Tjoa and Biegler (1991) used this formulation within a simultaneous strategy for data reconciliation and gross error detection on nonlinear systems. Albuquerque and Biegler (1996) used the same approach within the context of solving an error-in-all-variable-parameter estimation problem constrained by differential and algebraic equations. 

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. 

In the second example, that of an industrial pyrolysis reactor, simplified material and energy balances were used to analyze the performance of the process. In this example, linear and nonlinear reconciliation techniques were used. A strategy for joint parameter estimation and data reconciliation was implemented for the evaluation of the overall heat transfer coefficient. The usefulness of sequential processing of the information for identifying inconsistencies in the operation of the furnace was further demonstrated. 

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