Applications of filtering theory for the inversion of temporal chemical systems

Abstract

This paper presents a method for inverting temporal experimental data from chemical models to obtain estimates of unknown parameters. Most of the models under consideration are deterministic and we assume that the measurements obtained from experimental observations are represented as the solution of a differential equation containing the variables of the model. To incorporate any extraneous laboratory effects that are not included in the model, we assume that these equations are perturbed by a white noise process so that the measurements become time-dependent stochastic variables. A particular measurement is then equivalent to a realization of these variables and applying stochastic estimation theory this realization can be used to obtain estimates of the unknown parameters in the model. As an example of this estimation method, we consider chemical kinetics models with various observational equations and construct an estimator for the unknown reaction rate constants. We also show the estimators for the structural constants in a laser model depending on the representation of the experimental data. In some cases the observations are simulated numerically and we present the parameter estimates as a function of time. The efficiency of the estimation process is calculated as the ratio of the a posteriori variance of the parameter estimator and the Rao–Cramer lower bound. Some issues in the numerical implementation of the filtering equations are discussed and a comparison is made between the minimum least-square estimation method and the filtering method.