Transfer functions parameters estimated

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. 

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. 

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

Physical Models versus Empirical Models In developing a dynamic process model, there are two distinct approaches that can be taken. The first involves models based on first principles, called physical or first principles models, and the second involves empirical models. The conservation laws of mass, energy, and momentum form the basis for developing physical models. The resulting models typically involve sets of differential and algebraic equations that must be solved simultaneously. Empirical models, by contrast, involve postulating the form of a dynamic model, usually as a transfer function, which is discussed below. This transfer function contains a number of parameters that need to be estimated from data. For the development of both physical and empirical models, the most expensive step normally involves verification of their accuracy in predicting plant behavior. 

The parameters of transfer functions may be estimated from the results obtained by analysis of a sufficient number of specimens (preferably human) spanning the relevant range of concentrations in all participating laboratories. Sometimes functions obtained by simple linear regression suffice using y=Uo + iX, the constant term Uq compensates for systematic shifts among the methods, whereas the coef-... 

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. 

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) ... 

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. 

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. 

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. 

This representation can also be seen as a system model in which the given biosignal is assumed to be the output of a linear time-invariant system that is driven by a white noise input e(/t). The coefficients or parameters of the AR model a, become the coefficients of the denominator polynomial in the transfer function of the system and therefore determine the locations of the poles of the system model. As long as the biosignal is stationary, the estimated model coefficients can be used to reconstruct any length of the signal sequence. Theoretically, therefore, power spectral estimates of any desired resolution can be obtained. The three main steps in this method are... 

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 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. 

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-... 

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