Empirical Model Development

The alternative model development paradigm is based on developing relations based on process data. Input-output models are much less expensive to develop. However, they only describe the relationships between the process inputs and outputs, and their utility is limited to features that are included in the available data sets. There are numerous well-established techniques for linear input-output model development. Methods for development of linear models are easier to implement and more popular. Since most monitoring and control techniques are based on the linear framework, use of linear models is a natural choice. The design of experiments to collect data and the amount of data available have an impact on the accuracy and predictive capability of the model developed. Data collection experiments should be designed such that all key features of the process are excited in 

As manufacturing processes have become increasingly instrumented in recent years, more variables are being measured and data are being recorded more frequently. This yields data overload, and most of the useful information may be hidden in large data sets. The correlated or redundant information in these process measurements must be refined to retain the essential information about the process. Process knowledge must be extracted from measurement information, and presented in a form that is easy to display and interpret. Various methods based on multivariate statistics, systems theory and artificial intelligence are presented in this chapter for data-based input-output model development. 

ANNs (Section 3.6.f) provide one framework for nonlinear model development. Extensions of PCA and PLS to develop nonlinear models have 

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Dynamic Model A key feature of MPC is that a dynamic model of the process is used to predict future values of the controlled outputs. There is considerable flexibility concerning the choice of the dynamic model. For example, a physical model based on first principles (e.g., mass and energy balances) or an empirical model developed from data could be employed. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonlinear 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. 

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

Multivariate analysis Empirical models developed to relate multiple spectral intensities from many calibration samples to known analyte concentrations, resulting in an optimal set of calibration parameters. 

Liu and Cohen suggested in their paper on hypothetical PC3N4 [12] that on the microscopic level, for ideal systems, hardness is determined by the bulk modulus . To estimate the bulk modulus they used an empirical model developed earlier [26], where the bulk modulus scales as a homopolar energy gap divided by the volume of the bond charge. The resulting relation gives the dependence of the bulk modulus B (in GPa) as a function of the bond length d (in A) and the empirical parameter A ... 

Often recommended is the empirical model developed by Onda et al. (1968). His correlation of the effective interfacial area is... 

The empirical model illustrated in Table 5.2 for density is compared with the experimental data and the density factor model derived previously as shown in Fig. 5.10. The trend lines between the experimental data and empirical model show very good correlation whereas the density factor analysis tend to underestimate the values. The density empirical model gives reasonable estimation of the composites density in terms of the amount of constituent material for both the gelatin and SDS in the ranges between 0 and 50 % for gelatin and 0-0.66 % for SDS respectively. Similarly, the models developed for compressive strength and modulus also showed close proximity between the experimental and empirical values of as shown in Figs. 5.11 and 5.12, respectively. Thus, it can be concluded that the empirical models developed via ANOVA in terms of process variables offer close estimation of the GSA and GSA-SDS composites experimental values. 

A typical thermodynamic property formulation is based on an equation of state that allows the calculation of all thermodynamic properties of the fluid, including properties such as entropy that cannot be measured directly. In this case the term equation of state is used to refer to an empirical model developed for calculating fluid properties such as those reported by Jacobsen et al and Span et al The equation of state is based on one of four fundamental relations internal energy as a function of volume and entropy enthalpy... 

J. Amphlett, R. Baumert, R. Mann, B. Peppley, P. Roberge, T. Harris, 1995. Performance modeling of the Ballard Mark IV solid polymer electrolyte fuel cell II. Empirical model development. Journal of the Electrochemical Society, 142 9-15. 

The effect of fuel contaminants can be observed in the following example (see Fig. 6.15) (Wessel, 2007). In this case, unexpected performance loss of 20% over a three-hour time period was observed in fuel cell stacks powering a bus. It was determined that small amounts of CO ( 4 ppm) were present in the fuel. Empirical models developed based on single-cell small-scale testing were used to predict the performance loss due to the presence of the CO, and a close correlation was observed, with 113 V loss from the nominal bus stack performance of 660 V predicted at 200 A after three hours with four ppm CO. The performance loss due to the CO is completely recoverable, but the incident resulted in a service call due to the unexpected contamination in the fuel. 

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