S-shape step response

General Second-Order Element Figure 8-3 illustrates the fact that closed-loop systems can exhibit oscillatory behavior. A general second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a unit step input, the transient responses shown in Fig. 8-16 result. As can be seen, when t, < 1, the response oscillates and when t, < 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response most exhibit an S-shaped step response. 

FIG 11.23. Feedforward control without dynamic compensation produces an S-shaped step response. 

Equations (8-23) and (8-24) can be multiphed together to give the final transfer function relating changes in ho to changes in as shown in Fig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R Ro and two time constants, R A and RoAo. For two equal tanks, a step change in fi produces the S-shaped response in level in the second tank shown in Fig. 8-14. 

Figure 23 Calculation of the shape of the actively compensated pulse can be carried out on the software. (A) shows the real (red line) and the imaginary (green line) component of an example of the target pulse shape t>,(f). Its leading and the trailing edges have a cosine shape with a transition time of 1.25 xs in 50 steps, and the width of the plateau is 5 ps. (B) Laplace transformation B(s) multiplied by the Laplace transformed step function U(s). (C) It was then divided by the Laplace transformation Y(s) of the measured step response y(t) of the proton channel of a 3.2-mm Varian T3 probe tuned at 400.244 MHz to obtain V(s). (D) Finally, inverse Laplace transformation was performed on V(s) to obtain the compensated pulse that results in the RF pulse with the target shape. Time resolution was 25 ns, and o = 20 was used for the Laplace and inverse Laplace transformations.

Next, bi(t) was Laplace transformed into B(s), and then multiplied by the Laplace transformation U(s) of the step function u(t). The result B(s)U(s) is displayed in Figure 23B. In this example, the step response y(t) was measured for the 1H channel of a Varian 3.2 mm T3 probe tuned at 400.244 MHz with a time resolution of 25 ns, and Laplace transformed into Y(s). By dividing B(s)U(s) by Y(s), the function plotted in Figure 23C was obtained, from which, by performing inverse Laplace transformation, the programming pulse shape v(t) was finally obtained, as shown in Figure 23D. The amplitude and the phase of the complex function v(t) give the intensity and the phase of the transient-compensated shaped pulse. 

Kobayashi (22) performed computer simulations via Eq. (4) as applied to his differential fixed-bed reactor. The model gas-phase reaction X Y is considered to pass in series through elementary steps to adsorbed X, an adsorbed intermediate in, adsorbed Y, to give finally Y. The forward and backward rate parameters were adjusted to simulate various mechanisms with their rate-determining steps. The shapes of the response curves for Y for typical mechanisms arc classified as instantaneous, monotonic, overshoot, S shaped, false start, and complex. This paper is a good so urce of ideas for the interpretation of transient responses. These ideas are illustrated by application to the oxidation of ethylene over a silver catalyst (23). The response curves last more than 100 min because the temperature is only 91°C and the bed contains 261 g of catalyst the flow rate is 160 ml/min. 

The response of the overdamped multicapacity system to step input change is S-shaped (i.e., initially changes slowly and then it picks up speed). This is in contrast to a first-order response which has the largest rate of change at the beginning. This sluggishness or delay is also known as transfer lag and is characteristic of multicapacity systems. 

The rule developed in Example 27.1 for the sampling rate of a first-order response can be extended to cover a large class of overdamped systems. Figure 16.9 shows the experimental response of an overdamped process to an input step change. The S-shaped response of Figure 16.9 can be approximated by the response of a first-order plus dead time system,... 

Other stimuli that can be used are a random input and a sinusoidal input. The response of a step input is an S-shaped curve see Fig. 11.20 top. The response of a pulse input is a bell-shaped curve see Fig. 11.20 bottom. The ideal pulse input is of infinitely short duration such an input is called a delta function or impulse. The normalized response to a delta function is called the C curve. Thus, the total area under the curve equals unity. 

Since each curve at a given potential requires initial conditions of unperturbed concentration profiles for the species, different ways to renew the diffusion layer between two subsequent steps have been proposed. In principle, every curve is recorded at a constant potential potential step values between two subsequent curves should be small enough to allow easy interpolation of the i values collected. The result is an S-shaped i vs. E voltammetric curve. Figure 10.10 reports the waveform and the relevant response obtained only a few chronoamperometric curves and relevant sampled currents are shown for clarity. 

The tests generally involve some form of maze but the simplest is the passive avoidance test. In this the animal learns that in a certain environment it will be punished with an electric shock for some particular action, like stepping onto a special part of the floor of the test chamber. The test of memory is how long the rat avoids (remains passive to) making the movement that will initiate the shock. Of course, drugs that reduce the animal s anxiety also modify the response. Using a maze in its simplest T shape, the animal is placed at the base of the vertical arm and a food reward at the end of one of the horizontal arms. Clearly the animal has to learn which arm contains the reward. Memory is assessed by the time taken for a food-deprived animal to reach the reward and the number of false arm entries. This simple system can be made more complex by introducing many more arms and branches but the principle is the same. 

The second step of the dose-response assessment is an extrapolation to lower dose levels, i.e., below the observable range. The purpose of low-dose extrapolation is to provide as much information as possible about risk in the range of doses below the observed data. The most versatile forms of low-dose extrapolation are dose-response models that characterize risk as a probability over a range of environmental exposure levels. Otherwise, default approaches for extrapolation below the observed data range should take into account considerations about the agent s mode of action at each tumor site. Mode-of-action information can suggest the likely shape of the dose-response curve at these lower doses. Both linear and nonlinear approaches are available. 

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