Application of broyden's method to reconciliation of nonlinearly constrained data

Abstract

Process data reconciliation and gross error detection have been the subject of many recent publications, for which Tamhane and Mah (1985) and Mah (1987) have provided a thorough review. Analytical solutions to linear data reconciliation problems can be obtained efficiently, for instance, by matrix projection (Crowe et al., 1983). However, solving general data reconciliation problems with nonlinear constraints invariably requires some type of iterative procedure. In this note, an iterative procedure is developed that makes use of Crowe's matrix projection and for the first time combines a quasi-Newton update with the Gauss-Newton scheme for solving nonlinear data reconciliation problems.Using the Lagrange method, Britt and Leucke (1973) developed the normal equations for nonlinear parameter estimation. More recently, Stephenson and Shewchuk (1986) and Serth et al. (1 987) also based their data reconciliation problems on the normal equations. Although straightforward, the approaches based on solving the system of normal equations suffer from two drawbacks. First, the size of the problem may be unduly large due to the Lagrange multipliers; second, first-order derivatives of the constraints appear in the normal equations and have to be calculated for every iteration.Crowe (1986) extended his method of matrix projection to bilinear constraints with an iterative procedure for determining one of the variables in the bilinear terms. Unfortunately, this procedure is too specific to be useful for more general cases.Knepper and Gorman (1980) proposed a Gauss-Newton iterative algorithm and, in order to reduce computational effort, suggested using old derivatives of constraints until the constraints are satisfied (i.e., the constant-direction approach). HLowever, their algorithm is limited to problems with no more constraints than measured variables and the constant-direction approach is characterized by slow convergence. Knepper and Glorman applied the theory of generalized inverses to solve the linearized subproblem. Crowe (private communication, 1987) presented a more general method based on matrix projection (Crowe et al., f983). The latter not only effectively reduces the problem size but also removes the restriction that no more constraints than measured variables be handled. In this note, a Broyden-type update (Broyden, 1965) is proposed to replace the old derivatives so that the rate of convergence can be improved without repeatedly evaluating the derivatives. Crowe's Iterative SchemeA general data reconciliation problem is defined aswhere f is an m vector of functions, x is an n vector of measured variables, u is an r vector of unmeasured variables, and Z is the variance-covariance matrix of measurements 2, or some weighting matrix. The sizes of these vectors are related by n + r 2 m > r z 0. Linearizing the constraint functions f around Xk and v k gives where Bk and Pk denote, respectively, the ( m x n) and ( m x r ) matrices of the derivatives offwith respect to x and v evaluated a t xk and v k .Following Crowe ...

Data reconciliation of nonnormal observations with nonlinear constraints

“…Although the context of the model is not stated in [14], we assume that all variables have to be nonnegative. The prior of the observation x i is assumed to be the lognormal pdf with meanx i and standard deviation σ i = 1, for i = 1, .…”

“…The second example uses the system of nonlinear constraints analysed in [14], which is a commonly used benchmark example 4 for nonlinear data reconciliation with normally distributed observation errors: and the covariance matrix = I. Although the context of the model is not stated in [14], we assume that all variables have to be nonnegative.…”

“…There is a large variety of methods that can be applied. Some examples in the engineering literature are successive linearization [10], nonlinear programming [1], quasi-Newton methods [14] and genetic algorithms [23]. For textbooks on optimization with nonlinear constraints, see, for instance, [5,6,22].…”

Several techniques to detect and identify systematic biases when process constraints are bilinear

“…Several p^ers have described ways to improve computational speed, which would be especially beneficial for large process networks (Britt and Leucke, 1973;Knepper and Gorman, 1980;Stephenson and Shewchuk, 1986;Pai and Fisher, 1988;Tjoa and Biegler, 1991).…”

“…Crowe (1986) extended his matrix projection technique for linear constraints to the case of bilinear constraints. Pai and Fisher (1988) extended the techniques developed by Crowe to the more general case of nonlinear constraints.…”