P.J. McLellan

Parameter estimation in continuous-time dynamic models using principal differential analysis

Principal differential analysis (PDA) is an alternative parameter estimation technique for differential equation models in which basis functions (e.g., B-splines) are fitted to dynamic data. Derivatives of the resulting empirical expressions are used to avoid solving differential equations when estimating parameters. Benefits and shortcomings of PDA were examined using a simple continuous stirred-tank reactor (CSTR) model. Although PDA required considerably less computational effort than traditional nonlinear regression, parameter estimates from PDA were less precise. Sparse and noisy data resulted in poor spline fits and misleading derivative information, leading to poor parameter estimates. These problems are addressed by a new iterative algorithm (iPDA) in which the spline fits are improved using model-based penalties. Parameter estimates from iPDA were unbiased and more precise than those from standard PDA. Issues that need to be resolved before iPDA can be used for more complex models are discussed.

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.

TEMPERATURE CONTROL OF INDUSTRIAL GAS PHASE POLYETHYLENE REACTORS

Abstract The performance of linear and nonlinear temperature control schemes is assessed for an open-loop unstable gas-phase polyethylene reactor (GPPER), based on speed, damping, robustness and the ability to maintain closed-loop stability in different operating regimes. An existing industrial GPPER model is improved by modelling the temperature states in the external heat exchanger using linear and nonlinear driving force models with varying numbers of heat transfer stages. Differences in heat exchanger models do not produce gain mismatch but do result in phase mismatch. It is shown that the nonlinear error trajectory controller (ETC) exhibits significantly superior responses in terms of speed, damping and robustness compared with an optimally-tuned PID controller. Therefore, substantial benefits could be realized using nonlinear controllers because they can provide good disturbance rejection capabilities and ensure closed-loop stability over a wide range of operating conditions. An approach is presented for tuning ETCs for minimum-phase processes of arbitrary relative degree.

A functional-PCA approach for analyzing and reducing complex chemical mechanisms

In industrial reactive flow systems such as furnaces and gas turbines, there are considerable variations in the temperature and concentrations of species along different spatial directions. Functional principal component analysis (fPCA) can be used to study the temporal (or spatial) evolution of reactions in a reactive flow system, and to develop simplified kinetic models to describe this behaviour. A comprehensive kinetic mechanism for CO oxidation is used to demonstrate application of fPCA to identify important reactions as a function of time. In conventional PCA, the eigenvalue-eigenvector decomposition specifically transforms the variations associated with the time (or spatial directions) and species into loadings that represent only the reactions. However, fPCA produces functional loading vectors (xi) over bar (1)(t) which are functions of time or distance, whose elements are referred to as functional loadings. The functional loading vectors are the eigenfunctions of the covariance matrix associated with the sensitivity trajectories. The functional loadings are used to identify reactions playing a significant role, possibly as a function of time, and are used to develop a reduced kinetic scheme from a detailed kinetic mechanism.

Determining Polymer Chain Length Distributions Using Numerical Inversion of Laplace Transforms

ABSTRACTThe kinetic expressions for a chain growth polymerization mechanism lead to an infinite set of ordinary differential equations that describe the material balance behaviour of living and dead polymer molecules of arbitrary length. One approach for solving these equations is to make a continuous variable approximation in the chain length dimension, thereby converting the ordinary differential equations to a finite set of partial differential equations. The set of partial differential equations can be solved by taking the Laplace transform with respect to the chain length, yielding ordinary differential equations in time, parameterized by the Laplace variable s. This system of ordinary differential equations can be numerically integrated over the desired reaction time with appropriate boundary conditions and the chain length distribution can be recovered by inverting the Laplace transform. Practical application of this methodology for calculating chain length distributions requires numerical solution...

A differential-algebraic perspective on nonlinear controller design methodologies

Abstract Considerable activity has occurred independently in the fields of nonlinear geometric controller design and the solution of differential-algebraic systems of equations. Recently, a differential-algebraic approach to nonlinear controller design has been proposed. In this paper, the formal relationship between these approaches is identified. In particular, it is shown that the index of the nonlinear inversion problem is equal to ρ + 1, where ρ is the relative order of the process. The merits of the primary nonlinear control algorithms are assessed from a differential-algebraic perspective. Error trajectory controllers offer the advantage of being index one differential-algebraic problems with no associated initialization difficulties. In contrast, the nonlinear inversion and sliding mode control designs result in higher-order index problems with initialization restrictions. The restrictions identified for the nonlinear inverse design are related to the concept of functional controllability from the nonlinear systems literature. The relationship between the differential-algebraic design approach and the nonlinear geometric approach is extended to a class of processes described by nonlinear differential-algebraic equations. Finally, the nonlinear controller design problem is analyzed graphically, highlighting the differences between the various approaches.

Parametric sensitivity: A case study comparison

This article presents a comparative analysis of three derivative-based parametric sensitivity approaches in multi-response regression estimation: marginal sensitivity, profile-based approach developed by [Sulieman, H., McLellan, P.J., Bacon, D.W., 2004, A Profile-based approach to parametric sensitivity in multiresponse regression models, Computational Statistics & Data Analysis, 45, 721-740] and the commonly used approach of the Fourier Amplitude Sensitivity Test (FAST). We apply the classical formulation of FAST in which Fourier sine coefficients are utilized as sensitivity measures. Contrary to marginal sensitivity, profile-based and FAST approaches provide sensitivity measures that account for model nonlinearity and are pertinent to linear and nonlinear regression models. However, the primary difference between FAST and profile-based sensitivity is that traditional FAST fails to account for parameter dependencies in the model system while these dependencies are considered in the analysis procedure of profile-based sensitivity through the re-estimation of the remaining model parameters conditional on the values of the parameter of interest. An example is discussed to illustrate the comparisons by applying the three sensitivity methods to a model described by set of non-linear differential equations. Some computational aspects are also explored.

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.

On the computation of optimal singular and bang-bang controls

To avoid difficulties associated with the computation of optimal singular/bang–bang controls, a common approach is to add a perturbed energy term. The efficacy of this perturbation method is assessed here via a direct search iterative dynamic programming procedure. A potential limitation of the strategy is shown from a computational point of view, and some guidelines for selecting the perturbation parameter are provided using numerical examples. It is demonstrated that many gradient-based methods may not be well suited for computing singular/bang–bang controls when perturbation methods are used to solve optimal control problems in chemical process control.

A profile-based approach to parametric sensitivity in multiresponse regression models

Abstract The profile-based sensitivity measure proposed by Sulieman et al. (Technometrics 43(4) (2001) 425) for single-response nonlinear models is extended to the case of multiresponse parameter estimation based on the Box–Draper estimation criterion. The profile-based sensitivity measure for the parameters in the multiresponse case is a vector-valued measure which simultaneously provides complete insight on the sensitivity behaviours of the M predicted responses in the model with respect to a parameter of interest. While providing first-order sensitivity information, the profile-based sensitivity coefficients, unlike the conventional sensitivity coefficients, also account for nonlinear parameter estimate co-dependencies derived from the Hessian of the determinant criterion, and are thus valid over broader ranges of parameter uncertainties. Two examples are discussed to illustrate applications of the extended measure.