A method of reconciliation of measured data with nonlinear constraints

摘要

Mathematical models of technological systems at steady state are generally represented by a set of equations, say F(z)=0 where zϵU is a vector variable taking its values in region U and F is a vector of functions. The equations may not determine a unique solution, but only a submanifold UF of a vector space; they are then regarded as a set of constraints due to certain model hypotheses adopted. If the values of certain variables (say x) have been measured, we decompose z=(y,x). Because of possible errors of measurement, it can happen that (y,x)∉UF whatever y is. The problem of reconciliation consists of finding an adjustment of x, thus some x̂ “as close as possible” to x and such that (y,x̂)ϵUF for some y. Using a norm (derived in practice from the covariance matrix of measurement errors), we have to minimize ||x̂-x||. Instead of the classical method of Lagrange multipliers, an approximative method is proposed that is based on a sequence of linearized solutions; the method requires only the evaluation of the Jacobi matrix in each step. Under precisely formulated conditions, the convergence of the successive approximations is proved. The method can also be considered a generalization of Newton's method where matrix inversion is replaced by matrix pseudoinversion (generalized inversion) satisfying the minimum norm condition.