Transfer functions parameters estimated

An appropriate transfer function model can be obtained from the step response by using the parameter estimation methods of Chapter 7. For processes that have monotonically increasing step responses, such as the responses in Fig. 12.15, the models in Eqs. 12-40 and 12-43 are appropriate. Then, any of the model- 

The parameters of each individual transfer function 0, s) can be estimated separately from the tomographic measurements and the complete transfer function for the column is calculated with the relations for parallel flow regimes and the combination of axial sections given in Table 2.3. 

Identification of the theoretical and experimental transfer functions in order to estimate the effective diffusivity De is obtained by minimizing a relative error function taken between the two transfer functions. The Rosenbrock method of optimization has been used. All the measurements have been made at room temperature and something close to normal atmospheric pressure. The only parameter that changes is the carrier gas flow rate. 

Unlike the transfer-function-based technique, the time-domain technique does not require the central values of the nominally constant parameters to be determined from a minimization exercise. Nevertheless, we will expect the modeller to use sensible estimates, which may be expressed as a condition similar to inequality (24.52)  

In addition to the aforementioned slope and variance methods for estimating the dispersion parameter, it is possible to use transfer functions in the analysis of residence time distribution curves. This approach reduces the error in the variance approach that arises from the tails of the concentration versus time curves. These tails contribute significantly to the variance and can be responsible for significant errors in the determination of Q)L. 

When working with regression-based techniques for process model identification, one of the challenging tasks is to determine the most appropriate process model structure. In a linear model context, this would be information such as the number of poles and zeros to be included in the transfer function description. If the structure of the system being identified is known in advance, then the problem reduces to a much simpler parameter estimation problem. 

It s seen from this example why the use of moments is a popular way to determine the model parameters—Eqs. (c) and (d) are relatively simple compared to the complete solution of Eq. 12.S.b-3,4 by attempting to invert the transfer function Eq. 12.S.b-ll, for example. The method appears to work well in chromatographic columns and has been widely used. However, for a broader range of systems that may be of interest in chemical reaction engineering, moments will often not provide the best parameter estimates. Section 12.5.C will discuss this further. 

As an alternative to nonlinear regression, a number of graphical correlations can be used quickly to find approximate values of ti and T2 in second-order models. The accuracy of models obtained in this way is often sufficient for controller design. In the next section, we present several shortcut methods for estimating transfer function parameters based on graphical analysis. 

It is obvious that a drawback of this method is that the reaction should be almost fully understood in order to produce a reliable transfer function. Another drawback is the fact that for more complex reactions the number of variables is very high. For a reliable fit the less important processes have to be ignored or variables have to be fixed in other words, the important parameters of the system have to be estimated prior to the fitting procedure. In many cases the interpretation of measurements on coatings will be extremely complex due to the unknown geometries and local variation in electrolyte concentrations. 

CPM of multivariable control systems has attracted significant attention because of its industrial importance. Several methods have been proposed for performance assessment of multivariable control systems. One approach is based on the extension of minimum variance control performance bounds to multivariable control systems by computing the interactor matrix to estimate the time delay [103, 116]. The interactor matrix [103, 116] can be obtained theoretically from the transfer function via the Markov parameters or estimated from process data [114]. Once the interactor matrix is known, the multivariate extension of the performance bounds can be established. 

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