Transfer functions in series

Figure 5.169. Transfer function consisting of three transfer functions in series.

G. GENERAL TRANSFER FUNCTIONS IN SERIES. The historical reason for the widespread use of Bode plots is that, before the use of computers, they made it possible to handle complex processes fairly easily. A complex transfer function can be broken down into its simple elements leads, lags, gains, deadtimes, etc. Then each of these is plotted on the same Bode plots. Finally the total complex transfer function is obtained by adding the individual log modulus curves and the individual phase curves at each value of frequency. 

A process consists of two transfer functions in series. The fint, relates the manipulated variable M to the variable and is a steady state gain of 1 and two filSt-order lags in series with equal time constants of 1 minute. 

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.

It is frequently required to examine the combined performance of two or more processes in series, e.g. two systems or capacities, each described by a transfer function in the form of equations 7.19 or 7.26. Such multicapacity processes do not necessarily have to consist of more than one physical unit. Examples of the latter are a protected thermocouple junction where the time constant for heat transfer across the sheath material surrounding the junction is significant, or a distillation column in which each tray can be assumed to act as a separate capacity with respect to liquid flow and thermal energy. 

In the styrene—butadiene copolymer application, a series of quantitative ANN models for the as-butadiene content was developed. For each of these models, all of the 141 X-variables were used as inputs, and the sigmoid function (Equation 8.39) was used as the transfer function in the hidden layer. The X-data and Y-data were both mean-centered before being used to train the networks. A total of six different models were built, using one to six nodes in the hidden layer. The model fit results are shown in Table 8.7. Based on these results, it appears that only three, or perhaps four, hidden nodes are required in the model, and the addition of more hidden nodes does not greatly improve the fit of the model. Also, note that the model fit (RMSEE) is slightly less for the ANN model that uses three hidden nodes (1.13) than for the PLS model that uses four latent variables (1.25). 

Comparing this with equation (3) shows that this can be considered as the output of a first order transfer function in response to a random input sequence. More generally, most stochastic disturbances can be modelled by a general autoregressive-integrated moving-average (ARIMA) time series model of order (p,d,q), that is,... 

Clearly, the final MDS diagram is partially dependent on the parameters of the noise imposed on the system. It is possible that frequency domain approaches to time series analysis [10] may help in a study of the role of frequency transfer functions in the control of chemical networks. We have assumed that all species involved in the mechanism may be identified and measured. For systems with many species this may be difficult. When there are missing species, CMC may still be performed on the measurable subset of species. The effects of the other species are subsumed into the correlations among the known species, and a consistent diagram can be constructed. The MDS diagram, then, may not be an obvious representation of the underlying mechanism. In fact, due... 

We can see that, with the control signal specification in Equation (6.11), a lead-lag element has been added in series to the open-loop process transfer function G s) to form the desired closed-loop transfer function in Equation (6.12). 

In the previous chapter the behavior of elementary transfer functions has been discussed in the frequency domain. Systems that are more complex are often composed of a series and/or parallel connection of these elementary transfer functions. In chapter 1 the use of diagrams was introduced to show the coherence between systems. In this chapter the behavior of the different processes will be explained in the time domain based on these diagrams, covering the entire range from elementary first-order lumped systems to complex distributed systems. In the following chapters 11-16 the behavior of different process units will be described based on this general process behavior. 

The terms on the right are the transfer functions. With the two units in series. 

In this example, the inner loop is solved first using feedback. The controller and integrator are cascaded together (numpl, denpl) and then series is used to find the forward-path transfer function (numfp, denfp ). Feedback is then used again to obtain the closed-loop transfer function. 

It has been demonstrated that group 6 Fischer-type metal carbene complexes can in principle undergo carbene transfer reactions in the presence of suitable transition metals [122]. It was therefore interesting to test the compatibility of ruthenium-based metathesis catalysts and electrophilic metal carbene functionalities. A series of examples of the formation of oxacyclic carbene complexes by metathesis (e.g., 128, 129, Scheme 26) was published by Dotz et al. [123]. These include substrates where double bonds conjugated to the pentacarbonyl metal moiety participate in the metathesis reaction. Evidence is... 

The positive results obtained at production scale give us confidence in the validity of our approach. Derivation of a simple scaling factor enabled us to conduct a series of experiments in a small pilot plant which would have been expensive and time-consuming on a production scale. Time series analysis not only provided us with estimates of the process gain, dead time and the process time constants, but also yielded an empirical transfer function which is process-specific, not one based on... 

One final note While the techniques used here were applied to control temperature In large, semi-batch polymerization reactors, they are by no means limited to such processes. The Ideas employed here --designing pilot plant control trials to be scalable, calculating transfer functions by time series analysis, and determining the stochastic control algorithm appropriate to the process -- can be applied In a variety of chemical and polymerization process applications. 

A complex transfer function may consist of processes in series, as shown in Fig. 2.25. This then gives the form... 

Example 4.7 We ll illustrate the results in this section with a numerical version of Example 4.5. Consider again two CSTR-in-series, with V] = 1 m3, V2 = 2 m3, k] =1 min-1, k2 =2 min-1, and initially at steady state, x, = 0.25 min, x2 = 0.5 min, and inlet concentration cos = 1 kmol/m3. Derive the transfer functions and state transition matrix where both c0 and q are input functions. 

If the optimum size heat exchanger for the initial plant is installed, an additional exchanger either in parallel or in series will be required when the plant is expanded. This may be the best option when the heat transfer involves condensation and subcooling. The exchanger can be designed to perform both functions initially, and then when the plant is expanded an aftercooler can be installed and the initial equipment can act only as a condenser. 

In photosystem I, absorption of a photon leads to an excited state that functions as a reducing agent. The electrons are passed from one species to another with several intermediate species that include ferrodoxin (a protein containing iron and sulfur) before finally reducing C02. In photosystem II, electrons are transferred to a series of intermediates, of which a cytochrome bf complex is one entity. Ultimately, the transfer of electrons leads to the reaction... 

The transfer function of a process and measuring element connected in series is given by ... 

A unity feedback control loop consists of a non-linear element N and a number of linear elements in series which together approximate to the transfer function ... 

A particular vessel behavior sometimes can be modelled as a series or parallel arrangement of simpler elements, for example, some combination of a PFR and a CSTR. Such elements can be combined mathematically through their transfer functions which relate the Laplace transforms of input and output signals. In the simplest case the transfer function is obtained by transforming the linear differential equation of the process. The transfer function relation is... 

The Gain and phase angle will be found for several ideal stirred tanks in series. For such a series of vessels, the overall transfer function is the product of the individual transfer functions, that is,... 

A deep stirred vessel is provided with two impellers on a single shaft. Feed is between the impellers. A plausible model for such a vessel is two CSTR s partly in parallel and partly in series, followed by a PER, as indicated on the sketch. Let a be the fraction of the total feed that goes to the first CSTR, and let 0 and j be the fractions of the volume occupied by each of the CSTRs. Find the transfer function of the whole vessel and the response to impulse input. 

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