D. W. Bacon

Modeling Ethylene/Butene Copolymerization with Multi-site Catalysts: Parameter Estimability and Experimental Design

Mechanistic models for ethylene copolymerization with Ziegler–Natta catalysts contain many kinetic rate constants. Precise estimates of key parameters are required if the models are to be used to obtain reliable predictions of polymer properties for a specified catalyst over a range of polymerization conditions. In this paper, a novel method is developed to assist modelers in assessing whether their model parameters will be estimable from existing or proposed experimental data, and in determining subsets of influential parameters that can be estimated from the data when the complete set of model parameters cannot be estimated. The effects of using different types of data, varying the number of observations per experimental run, and increasing the number of experimental runs on the number of estimable parameters are considered. A method for comparison of the relative effectiveness of different experimental designs for estimating key parameters in gas‐phase ethylene copolymerization is also presented.

Prospects for reducing correlations among parameter estimates in kinetic models

Abstract Highly correlated parameter estimates are often encountered in experimental investigations, making interpretation difficult. A sequential design strate

Error trajectory descriptions of nonlinear controller designs

Abstract Research in nonlinear process control is rapidly expanding with an increasing number of seemingly diverse control algorithms appearing. In this review an alternate method of formulating nonlinear control laws is presented. This formulation, which uses the concept of a tracking error trajectory in a differential geometric setting, provides a framework for direct comparison of a number of recently proposed nonlinear process controllers. In addition, the inclusion of integral action in nonlinear control laws can be motivated in terms of the estimation of an output disturbance. Finally, a more general approach to nonlinear controller design is suggested.

Time series methods for dynamic analysis of multiple controlled variables

Research results in controller performance monitoring for multivariate systems have mainly focused on the problem of estimating the control invariant component of the closed-loop output covariance (Harris, T.J., Boudreau, F. and MacGregor, J.F. Performance assessment of multivariable feedback controllers. Automatica, 32:1505-1518, 1996; Huang, B., Shah, S.L. and Kwok, K.E. Good, bad or optimal? performance assessment of multivariable processes. Automatica, 33(6):1175-1183, 1997b.). The contributions of this paper lie on the dynamic analysis side of multivariate controller performance monitoring, where no a priori information is available, yet assessment of dynamic interactions between loops is of interest. Using results from the statistics, identification, and econometrics literature, graphical methods for analyzing the dynamic performance of vectors of tracking error variables are presented. The multi-output dynamic analysis problem can be simplified considerably by treating the tracking error trends as a vector process of endogenous stochastic variables and using a vector autoregressive (VAR) structure to model the dynamic relationships. Once such a model has been estimated, a host of post-estimation diagnostics, such as multivariate impulse response analysis, can be used to interpret the dynamic interactions between the tracking error variables. These methods will be discussed in detail and demonstrated on simulated and industrial control system data. Crown Copyright # 2002 Published by Elsevier Science Ltd. All rights reserved.

Using An Hyperbola as a Transition Model to Fit Two-Regime Straight-Line Data

In a recent paper [1] a general form of transition model was suggested to describe data which appear to follow two different straight line relationships on opposite sides of an undetermined join point. An alternative model is now considered, the familiar hyperbola, parameterized in a geometrically meaningful form. The two models are fitted to two sets of experimental data for purposes of comparison. In one of the examples account is taken of autocorrelated errors using a procedure suggested by Sredni [13].

Further Consideration of Heteroscedasticity in Fitting Kinetic Models

Box and Hill [6] recently proposed a method for using power transformation weighting in least squares analysis to account for changing variance. Such an approach can be useful when the original data are heteroscedastic but adequate weight estimates are not available, and when the original data are homoscedastic but heteroscedasticity is induced by the data analyst in linearising a nonlinear model. Several aspects of their proposal are examined for practical implications in fitting chemical kinetic models and a more direct algorithm is recommended for fitting nonlinear models to heteroscedastic data. Methods for testing model adequacy and assessing parameter precision in such situations are also discussed.

Statistical assessment of chemical kinetic models

Abstract Straight line behaviour on Arrhenius-type plots and the physical plausibility of the signs of the estimated model parameters are traditional criteria for assessing the adequacy of kinetic rate models fitted to experimental data. Several statistical objections to independent application of each of these criteria are presented and a more direct model assessment procedure is described which overcomes these objections. Using an example from the chemical kinetics literature as an illustration, it is shown that substantially different conclusions about the performance of individual models may be reached using this direct approach rather than a more traditional three-stage assessment procedure.

Singularities in Multiresponse Modelling

Care is required when analysing multiple response data if misleading results are to be avoided. Box et al. [4] have warned of errors in analysis resulting from linear relationships among the response data, and have provided a detection procedure. This is effective for most situations; however, we have encountered two cases that require additional consideration. This paper extends the work of Box et al. to include these cases. The key issue is the detection of singularities in the multiresponse dispersion matrix, and a procedure for this is presented. The situation is also discussed in which linear dependencies in the data may be ignored to advantage. Illustrations are from chemical kinetics investigations.

Accounting for Heteroscedasticity in Experimental Design

Power transformation weighting has been found to be a powerful technique for accounting for heteroscedasticity in model fitting. In this paper the transformation weighting concept is used in developing sequential design criteria for precise parameter estimation in heteroscedastic situations. Criteria are proposed for precise estimation of the model and transformation parameters together and for precise estimation of the model parameters alone. Implementation of the criteria is illustrated with two examples from chemical kinetics.

A Profile-Based Approach to Parametric Sensitivity Analysis of Nonlinear Regression Models

Predictions from a nonlinear regression model are subject to uncertainties propagated from the estimated parameters in the model. Parameters exerting the strongest influence on model predictions can be identified by a sensitivity analysis. In this article, a new parametric sensitivity measure is introduced, based on the profiling algorithm developed by Bates and Watts for constructing likelihood intervals for the individual parameters in nonlinear regression models. In contrast with traditional sensitivity coefficients, this profile-based sensitivity measure accounts for both correlation structure among the parameters and model nonlinearity. It also provides sensitivity information over wide ranges of parameter uncertainties. Application of the proposed approach is illustrated with three examples.