Models frequency response

The best model in terms of fitting the frequency response results (obtained in MATLAB by fast-Fourier-transforming the input and output signals) is the third-order with a dead time equal to four sampling periods. Fig. 14.9h compares the model frequency response with the data. 

It is usually important to get an accurate fit of the model frequency response to the experimental frequency response near the critical region where the phase angle is between 135 and — 180 . It doesn t matter how well or how poorly the approximate transfer function fits the data once the phase angle has dropped below — 180 . So the fitting of the approximate transfer function should weight heavily the differences between the model and the data over this frequency range. 

A recent addition to the model-based tuning correlations is Internal Model Control (Rivera, Morari, and Skogestad, Internal Model Control 4 PID Controller Design, lEC Proc. Des. Dev., 25, 252, 1986), which offers some advantages over the other methods described here. However, the correlations are similar to the ones discussed above. Other plant testing and controller design approaches such as frequency response can be used for more complicated models. 

An important difference between analysis of stability in the. v-plane and stability in the frequency domain is that, in the former, system models in the form of transfer functions need to be known. In the latter, however, either models or a set of input-output measured open-loop frequency response data from an unknown system may be employed. 

In the limit of small pressure perturbations, any kinetic equation modeling the response of a catalyst surface can be reduced to first order. Following Yasuda s derivation C, the system can be described by a set of functions which describe the dependence of pressure, coverage amplitude, and phase on T, P, and frequency. After a mass balance, the equations can be separated Into real and Imaginary terms to yield a real response function, RRF, and an Imaginary response function, IRF ... 

We first illustrate the idea of frequency response using inverse Laplace transform. Consider our good old familiar first order model equation, 1... 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

The R-X plot shows the most variation in the subthreshold region, while the G-B plot shows the most variation above threshold. One sees from the G-B plot that the high frequency response of the diode is independent of bias (>1 MHz). To fit the data, one models each material phase or interface as a parallel R-C combination. These combinations are then added in series, and an overall series resistance and series inductance are added. For the data in Figure 10.6, three R-C elements are used. One R-C element is associated with the Schottky barrier. Another is associated with the high frequency bias-independent arc, which we believe is associated with the capacitance of the alkoxy-PPV. The thinness of the film... 

There are a number of ways to obtain the frequency response of a process. Experimental methods, discussed in Chap. 14, are used when a mathematical model of tbe system is not available. If equations can be developed that adequately describe the system, the frequency response can be obtained directly from the system transfer function. 

It is usually important to get an accurate fit of the frequency response of the model to the experimental frequency only near the critical region where the phase... 

The important feature of the ATV method is that it gives transfer function models that fit the frequency-response data very well near the important frequencies of zero (steadystate gains) and the ultimate frequency (which determines closedloop stability). 

Instead of converting the step or pulse responses of a system into frequency response curves, it is fairly easy to use classical least-squares methods to solve for the best values of parameters of a model that fit the time-domain data. 

AC impedance measurements were also made in bulk paints. A Model 1174 Solartron Frequency Response Analyser (FRA) with a Thompson potentiostat developed ac impedance data between 10 KHz and 0.1 Hz at the controlled corrosion potential The circuit has been described in the literature( ). 

Sections 3.2.1—3.2.3 have referred specifically to the system illustrated in Fig. 6. However, the approach in these sections is quite general and can therefore be used in situations where the system transfer function G(s) is other than that given by eqn. (7). For the case of the ideal PFR responses, G(s) is exp(— st) and impulse, step and frequency responses are simply these respective input functions delayed by a length of time equal to r. The non-ideal transfer function models of Sect. 5 may be used to produce families of predicted responses which depend on chosen model parameters. 

Even though the use of Turner s structures to represent packed beds may be too complex, the overall concept of utilizing frequency-response experiments to construct detailed models is very interesting and might And use for other situations. 

Next, we will use the op-amp circuit created in the previous section to demonstrate an AC Sweep. We created an op-amp with frequency dependence in the previous section. We will now show how the frequency response varies with feedback. In the circuit below, the op-amp model is used as a non-inverting amplifier with gain 1 + (Rf/R4) ... 

The nominal value of pF is 200. The range of pF is 200 150. The transistor is equally likely to have a value anywhere in this range. The name of the model is QBf. The text NPN specifies the model as an NPN bipolar transistor. This model is included in class.lib. A limitation of this model is that none of the other transistor parameters have been specified. This model is almost ideal. Its limitations will become apparent when you observe that the high-frequency response does not roll off, even at frequencies beyond 1 GHz. 

Treffer s model system consists of signal power P (in watts) falling on the input aperture of the modulator (i.e., the spectrophotometer) that modulates the input power as a function of time by a factor M(t) such that 0 < M(t) < 1. The modulation function M(t) is not the modulation of the signal due to chopping but modulation of the signal due to scanning. The modulated signal falls on a detector with responsivity R (in volts per watt) (Kruse et al., 1962 Stewart, 1970) and flat frequency response. The idealized instantaneous... 

The results show the transient effects due to a 15 kW step decrease to the generator (from 60 to 45 kW), which occurred at time zero. Figure 8.13 compares the dynamics of the shaft speed and shows that both models are in good agreement with the experimental data with regards to frequency. This is not surprising since the system volume will have the primary influence on the frequency response, and the mod-... 

Fabry and Tasell, 1990] Fabry, D. and Tasell, D. V. (1990). Evaluation of an articulation-index based model for predicting the effects of adaptive frequency response hearing aids. J. Speech and Hearing Res., 33 676-689. 

Diffuse Reflectance Spectroscopy Instead of analyzing the frequency response of the fight to extract chemical information as is done in reflectance-mode absorption spectroscopy, diffuse reflectance spectroscopy extracts the bulk absorption and scattering coefficients by fitting the spectrum to a particular model.76... 

Physical state space models are more attractive for use with the LQP (especially when state variables are directly measurable), while multivariable black box models are probably better treated by frequency response methods (22) or minimum variance control (discussed later in this section). 

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