Transfer function, identifiability

This is a problem that we have to revisit many times in later chapters. For now, draw the block diagram of the dye control system and provide the necessaiy transfer functions. Identify units in the diagram and any possible disturbances to the system. In addition, we do not know where to put the photodetector at this point. Let s just presume that the photodetector is placed 290 cm downstream. 

Another set of experimental output is used to validate the model. To this aim, the transfer function identified for the sensor 7.5 x 1 x 0.17 mm was employed to estimate the voltage output of the sensor at different frequencies, which are shown in Fig. 11. The close correspondence between the experimental and estimated voltage outputs demonstrates that the transfer function model of the sensor is accurate enough to estimate its electrical output. 

Fitting Dynamic Models to E erimental Data In developing empirical transfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been pubhshed. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The i s in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first order with deadtime and have the transfer func tion ... 

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. 

Figure 2. Experimental trial used to Identify transfer function. In this experiment, the reactant flow rate was deliberately varied and the reactant temperature measured on-line in the pilot plant. This allowed us to identify the proper time series model.

After this exercise, let s hope that we have a better appreciation of the different forms of a transfer function. With one, it is easier to identify the pole positions. With the other, it is easier to extract the steady state gain and time constants. It is veiy important for us to leam how to interpret qualitatively the dynamic response from the pole positions, and to make physical interpretation with the help of quantities like steady state gains, and time constants. 

The use of block diagrams to illustrate cause and effect relationship is prevalent in control. We use operational blocks to represent transfer functions and lines for unidirectional information transmission. It is a nice way to visualize the interrelationships of various components. Later, they are crucial to help us identify manipulated and controlled variables, and input(s) and output(s) of a system. 

To derive the state space representation, one visual approach is to identify locations in the block diagram where we can assign state variables and write out the individual transfer functions. In this example, we have chosen to use (Fig. E4.6)... 

Another advantage of frequency response analysis is that one can identify the process transfer function with experimental data. With either a frequency response experiment or a pulse experiment with proper Fourier transform, one can construct the Bode plot using the open-loop transfer functions and use the plot as the basis for controller design.1... 

To help understand MATLAB results, a sketch of the low and high frequency asymptotes is provided in Fig. E8.9. A key step is to identify the comer frequencies. In this case, the comer frequency of the first order lead is at 1/5 or 0.2 rad/s, while the two first order lag terms have their comer frequencies at 1/10, and 1/2 rad/s. The final curve is a superimposition of the contributions from each term in the overall transfer function. 

What is the phase angle of the minimum phase function (s + 3)/(s + 6) versus the simplest nonminimum phase function (s - 3)/(s + 6) Also try plot with MATLAB. The magnitude plots are identical. The phase angle of the nonminimum phase example will go from 0° to -180°, while you d see a minimum of the phase angle in the minimum phase function. Thus for a transfer function that is minimum phase, one may identify the function from simply the magnitude plot. But we cannot do the same if the function is nonminimum phase. 

MATLAB is object-oriented. Linear time-invariant (LTI) models are handled as objects. Functions use these objects as arguments. In classical control, LTI objects include transfer functions in polynomial form or in pole-zero form. The LTI-oriented syntax allows us to better organize our problem solving we no longer have to work with individual polynomials that we can only identify as numerators and denominators. 

Now, go to the LTI Viewer window and select Import under the File pull-down menu. A dialog box will pop out to help import the transfer function objects. By default, a unit step response will be generated. Click on the axis with the right mouse button to retrieve a popup menu that will provide options for other plot types, for toggling the object to be plotted, and other features. With a step response plot, the Characteristics feature of the pop-up menu can identify the peak time, rise time, and settling time of an underdamped response. 

Frequency response testing mesures the amplitude, r, and the phase angle, . These involve the parameters of the transfer function so this mode of testing can identify two such parameters. The relationship depends on the properties of complex numbers. 

G( ) is the transfer function relating 0O and 0X. It can be seen from equation 7.18 that the use of deviation variables is not only physically relevant but also eliminates the necessity of considering initial conditions. Equation 7.19 is typical of transfer functions of first order systems in that the numerator consists of a constant and the denominator a first order polynomial in the Laplace transform parameter s. The numerator represents the steady-state relationship between the input 0O and the output 0 of the system and is termed the system steady-state gain. In this case the steady-state gain is unity as, in the steady state, the input and output are the same both physically and dimensionally (equation 7.16h). Note that the constant term in the denominator of G( ) must be written as unity in order to identify the coefficient of s as the system time constant and the numerator as the system... 

Finally, the overall major tiers (0 to 4) can be identified according to the characteristics given below. Note that further subtiering, as indicated in Table 10.3, is possible from Tier-2 onward. For example, in matrix and media extrapolation, one may apply statistical modeling to derive transfer functions that describe the entry of substances in organisms as a function of matrix characteristics. These models can be simple so as to address gross problems, such as the function... 

The general problem of building a model for an actual process begins with a flow description where we qualitatively appreciate the number of flow regions, the zones of interconnection and the different volumes which compose the total volume of the device. We frequently obtain a relatively simple CFM, consequently, before beginning any computing, it is recommended to look for an equivalent model in Table 3.4. If the result of the identification is not satisfactory then we can try to assimilate the case with one of the examples shown in Figs. 3.26-3.28. If any of these previous steps is not satisfactory, we have three other possibilities (i) we can compute the transfer function of the created flow model as explained above (ii) if a new case of combination is not identified, then we seek where the slip flow can be coupled with the CFM example, (iii) we can compare the created model with the different dispersion flow models. 

In some situations where one or more of the latex properties are measured either directly or indirectly through their correlation with surrogate variables and where extreme nonlinearities such as the periodic generation of polymer particles does not occur, one can use much simpler modehng and control techniques. Linear transfer function-type models can he identified directly from the plant reactor data. Conventional control devices such as PID controllers or PID controllers with dead-time compensation can then be designed. If process data is also used to identify... 

For the SCC of type II an example of a RTD modelled is shown in Figure 7, The model used is the dispersion model (sec Esq. 6). The values of the model parameters determined arc a Bodenstein number of 8.8 and a mean residence time of 0.6 s. It clearly shows that the model for the RTD explains the frequency response measurement up to a frequency of 2 Hz, At the frequency of 2 Hz the signal-to-noise ratio of 100 is reached. Any mixing processes which affect the transfer function above this frequency cannot be identified. 

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