First-order systems pulse transfer function

Several important features should be noted. The first-order process considered in Example 19.1 gave a pulse transfer function that was also first-order, i.e., the denominator of the transfer function was first-order in z. The second-order process considered in this example gave a sampled-data pulse transfer function that had a second-order denominator polynomial. These results can be generalized to an Nth-order system. The order of s in the continuous transfer function is the same as the order of z in the corresponding sampled-data transfer function. 

Figure 29.6 Pulse transfer functions for (a) pure integrator (b) first-order lag system.

Parametric models are more or less white box or first principle models. They consist of a set of equations that express a set of quantities as explicit functions of several independent variables, known as parameters . Parametric models need exact information about the inner stmcture and have a limited number of parameters. For instance, for the description of the dynamics, the order of the system should be known. Therefore, for these models, process knowledge is required. Examples are state space models and (pulse) transfer functions. Non-parametric models have many parameters and need little information about the inner stmcture. For instance, for the dynamics, only the relevant time horizon shoirld be known. By their stmcture, they are predictive by nature. These models are black box and can be constructed simply from experimental data. Examples are step and pulse response functions. 

Fig. 5.5. Pulse transfer function derivation for a first-order lag system.

We prepared three bifunctional redox protein maquettes based on 12 16-, and 20-mer three-helix bundles. In each case, the helix was capped with a Co(III) tris-bipyridyl electron acceptor and also functionalized with a C-terminal viologen (l-ethyl-V-ethyl-4,4 -bipyridinium) donor. Electron transfer (ET) was initiated by pulse radiolysis and flash photolysis and followed spectrometrically to determine average, concentration-independent, first-order rates for the 16-mer and 20-mer maquettes. For the 16-mer bundle, the a-helical content was adjusted by the addition of urea or trifluoroethanol to solutions containing the metal-loprotein. This conformational flexibility under different solvent conditions was exploited to probe the effects of helical secondary structure on ET rates. In addition to describing experimental results from these helical systems, this chapter discusses several additional metalloprotein models from the recent literature. 

Show that the concentration cA of reactant A in an isothermal continuous stirred tank reactor exhibits first-order dynamics to changes in the inlet composition, cA/. The reaction is irreversible, A - B, and has first-order kinetics (i.e., r = kcA). Furthermore (a) identify the time constant and static gain for the system, (b) derive the transfer function between cA and cA (c) draw the corresponding block diagram, and (d) sketch the qualitative response of cA to a unit pulse change in cAj. The reactor has a volume V, and the inlet and outlet flow rates are equal to F. 

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