Step response function

The step response function h(x) is the response of a system to an unit step s(x) at the input. 

The unit step function s(x) is defined as a step from 0 to /. The function s(x) is shown in fig. 1 (centre) together with an example of a step response function h(x). 

This gives a step response function of the form shown in Figure 3.16. 

Appendix 7.2. Determination of the Step Response Function of a Second-Order System from its Transfer Function... 

If a mathematical description of the system characteristics is not possible, the system parameters have to be determined experimentally. If the step response function is of a higher order than one (see Fig. 4.36[b]), the response can be approximated by a first order response function with lag time. 

The phenomenological theory of the dielectric relaxation behaviour of linear systems is well-established [1-5]. The fundamental relationship joining the frequency-dependent complex permittivity c(cu) measured at frequency / = (ofln and the transient step-response function t) is the Fourier transform relationship... 

We now turn to the discussion of the so-called Debye model that allows one to account for the relaxation processes with a finite t in the expressions for the electric displacement and the dielectric permittivity. To obtain the explicit dependencies, one should know the step response function which is generally unknown. Debye assumed that, once a constant external electric field is applied, the equilibrium state of an isotropic dielectric is achieved through an exponential relaxation, i.e.. 

Often an unit impulse is not available as a signal to get the impulse response function g(x). Therefore an other characteristic signal, the unit step, is be used. 

That means, the derivation of the measured step response h(x) along the path x delivers the impulse response function g(x) of the system. 

So, a comparison of different types of magnetic field sensors is possible by using the impulse response function. High amplitude and small width of this bell-formed function represent a high local resolution and a high signal-to-noise-characteristic of a sensor system. On the other hand the impulse response can be used for calculation of an unknown output. In a next step it will be shown a solution of an inverse eddy-current testing problem. 

The step response of this transfer function is shown in Fig. 8-19. Note that all curves reach about 60 percent of their final value at t = nX. 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

Determine the values of Wn and ( and also expressions for the unit step response for the systems represented by the following second-order transfer functions... 

From Table 7.2, it can be seen that the discrete and continuous step response is identical. Table 7.3 shows the discrete response x kT) and continuous response x t) to a unit ramp function where Xo t) is calculated from equation (3.39)... 

Since H =, the system has unity feedback, and the closed-loop transfer function and step response is given by... 

The closed-loop transfer function and step response is given by... 

Example 7.3 Transfer Function to z-Transform %Continuous and Discrete Step Response num=[1] den=[1 1 ] ... 

Example 7.4 Open and Closed-Loop Pulse Transfer Functions %Discrete Step Response... 

Script file examp77.m plots the closed-loop step responses of both the continuous system and discrete system (see Figure 7.21). In the latter case the plant pulse transfer function uses zoh, and the compensator is converted into discrete form using... 

The continuous and discrete closed-loop systems are shown in Figures 7.22(a) and (b). The digital compensator is given in equation (7.128). Script file examp78.m produces the step response of both systems (Figure 7.25) and prints the open and closed-loop continuous and pulse transfer functions in the command window... 

Example 3.2 Using the first order Pade approximation, plot the unit step response of the first order with dead time function ... 

First order function unit step response... 

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

Based on what we have obtained in Example 5.7, if we did an open-loop experiment as suggested in Eq. (6-1), our step response would fit well to the function ... 

Empirical tuning with open-loop step test Measure open-loop step response, the so-called process reaction curve. Fit data to first order with dead-time function. 

We can compare the unit step responses of the two transfer functions with... 

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. 

The transfer functions G and H will be imported automatically when the LTI Viewer is launched, and the unit step response plots of the two functions will be generated. 

In this case, the LTI Viewer will display both the unit step response plot and the Bode plot for the transfer function G. We will learn Bode plot in Chapter 8, so don t panic yet. Just keep this possibility in mind until we get there. 

The step () function also accepts state space representation, and to generate the unit step response is no more difficult than using a transfer function ... 

If this is not enough to convince you that everything is consistent, try step o on the transfer function and different forms of the state space model. You should see the same unit step response. 

The second step when determining impacts is the hazard assessment [28]. During the hazard assessment, the impact caused by the exposure to a substance is determined [30]. This is often done using in vitro or in silico testing. The results of the hazard assessment are often presented as dose-response functions. 

For the design of the actively compensated RF pulse, experimental and numerical determination of the response function h(t) of the circuit is necessary. We should also keep in mind that modification to the circuit, such as probe timing, insertion or removal of RF filters, and so on, can alter h(t). In practice, it is convenient to measure the response y t) to a step excitation u(t) instead of that to the impulse excitation. By performing Laplace transformation to... 

Figure 22C summarizes the procedure for calculating the programming pulse shape v(t) from the target pulse shape bi(t) and the step response u(t). In addition to the measurement of y t), we need to perform a number of data operations such as multiplication, division, Laplace and inverse Laplace transformations. All of these functions can be performed on the software. 

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. 

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