Nonlinear empirical models

Transfer function models are linear in nature, but chemical processes are known to exhibit nonhnear behavior. One could use the same type of optimization objective as given in Eq. (8-26) to determine parameters in nonlinear first-principle models, such as Eq. (8-3) presented earlier. Also, nonhnear empirical models, such as neural network models, have recently been proposed for process applications. The key to the use of these nonlinear empirical models is naving high-quality process data, which allows the important nonhnearities to be identified. 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

In contrast to the mechanistic models, empirical models make no a priori assumptions about the importance of single descriptors and the type of relationship (e.g., linear or nonlinear) between the input and the observed data. Still, many empirical models either employed descriptors that had been previously found to be important or the descriptors in the final model were replaced by such that could be easily interpreted from a mechanistic point of view. 

NN applications, perhaps more important, is process control. Processes that are poorly understood or ill defined can hardly be simulated by empirical methods. The problem of particular importance for this review is the use of NN in chemical engineering to model nonlinear steady-state solvent extraction processes in extraction columns [112] or in batteries of counter-current mixer-settlers [113]. It has been shown on the example of zirconium/ hafnium separation that the knowledge acquired by the network in the learning process may be used for accurate prediction of the response of dependent process variables to a change of the independent variables in the extraction plant. If implemented in the real process, the NN would alert the operator to deviations from the nominal values and would predict the expected value if no corrective action was taken. As a processing time of a trained NN is short, less than a second, the NN can be used as a real-time sensor [113]. 

This transformation of a physico-chemical model into an empirical model was previously discussed by Weyland et al. [27]. A ternary nonlinear blending term is often added to improve the descriptive power of this equation. Then, a special cubic mixture model is obtained ... 

For continuous process systems, empirical models are used most often for control system development and implementation. Model predictive control strategies often make use of linear input-output models, developed through empirical identification steps conducted on the actual plant. Linear input-output models are obtained from a fit to input-output data from this plant. For batch processes such as autoclave curing, however, the time-dependent nature of these processes—and the extreme state variations that occur during them—prevent use of these models. Hence, one must use a nonlinear process model, obtained through a nonlinear regression technique for fitting data from many batch runs. 

Mathematical models can also be classified according to the mathematical foundation the model is built on. Thus we have transport phenomena-bas A models (including most of the models presented in this text), empirical models (based on experimental correlations), and population-based models, such as the previously mentioned residence time distribution models. Models can be further classified as steady or unsteady, lumped parameter or distributed parameter (implying no variation or variation with spatial coordinates, respectively), and linear or nonlinear. 

The nonlinear viscoelastic models (VE), which utilize continuum mechanics arguments to cast constitutive equations in coordinate frame-invariant form, thus enabling them to describe all flows steady and dynamic shear as well as extensional. The objective of the polymer scientists researching these nonlinear VE empirical models is to develop constitutive equations that predict all the observed rheological phenomena. 

The process of research in chemical systems is one of developing and testing different models for process behavior. Whether empirical or mechanistic models are involved, the discipline of statistics provides data-based tools for discrimination between competing possible models, parameter estimation, and model verification for use in this enterprise. In the case where empirical models are used, techniques associated with linear regression (linear least squares) are used, whereas in mechanistic modeling contexts nonlinear regression (nonlinear least squares) techniques most often are needed. In either case, the statistical tools are applied most fruitfully in iterative strategies. 

As opposed to the models mentioned above, which are based on both transport and hydrophobic binding, Franke (161) proposed an empirical model to account for the nonlinear behavior based on hydrophobic binding only. This model distinguishes between binding to a hydrophobic surface and binding to a hydrophobic pocket. Both binding processes are characterized by a linear dependence... 

A nonlinear local isotherm model is clearly required for description of sorption reactions between the TCB and the shale isolate. A variety of conceptual and empirical models for representing nonlinear sorption equilibria, exists (2). The Langmuir model is one of the ideal limiting-condition-type models cited earlier. It is predicated on a uniform surface affinity for the solute and prescribes a nonlinear asymptotic approach to some maximum sorption capacity. 

Artificial neural networks are able to derive empirical models from a collection of experimental data. This applies in particular to complex, nonlinear relationships between input and output data. 

At present, there exist several dozens of rheological (mostly empirical) models of nonlinear viscous fluids. This is due to the fact that for the vast variety of fluid media of different physical nature, there is no rigorous general theory, similar to the molecular kinetic theory of gases, which would enable one to calculate the characteristics of molecular transport and the mechanical behavior of a medium on the basis of its interior microscopic structure. 

Franke developed another empirical model to bridge the gap between so many linear relationships and a nonlinear model (Figure 12). He considered binding of ligands at a hydrophobic protein surface of limited size as being responsible for nonlinear lipophilicity-activity relationships and formulated two equations, one for the linear left side (eq. 82) and the other one for the right side, the nonlinear part (eq. 83 log P = critical log P value, where the linear relationship changes to a nonlinear one) [435]. 

The most common nonlinear empirical model is a second order polynomial of the design variables, often called a quadratic response surface model, or simply, a quadratic model. It is a linear plus pairwise interactions model added with quadratic terms, i.e. design variables raised to power 2. For example, a quadratic model for two variables is y = 0 + b-yX-y +12X2+ bi2XiX2 + + 22XI. In general, we use the notation that b,- is the... 

Scientifically, a mechanistic model is always to be preferred if it can be constructed. In practice, however, empirical modeling is frequently sufficient, so that the nonparametric nonlinear methods are also important tools. 

In 1952, Alan Hodgkin and Andrew Huxley pubHshed a paper showing how a nonlinear empirical model of the membrane processes could be constructed [Hodgkin and Huxley, 1952]. In the five decades since their work, the Hodgkin-Huxley (abbreviated HH) paradigm of modeling cell membranes has been enormously successful. While the concept of ion channels was not estabhshed when they performed their work, one of their main contributions was the idea that ion-selective processes existed in the membrane. It is now known that most of the passive transport of ions across cell membranes is accompHshed by ion-selective channels. In addition to constructing a nonhnear model, they also estabhshed a method to incorporate experimental data into a nonlinear mathematical membrane model. 

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