Unit 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). 

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... 

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 ... 

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 correction circuit of Figure 2 uses an RC-network in the feedback loop of an operational amplifier. The network of Figure 3 has been calculated by Laplace transformation. The network function F] (t), which is the time response at the Uj-port to the unit step function applied at the U -port, simulates the square root decay function F(t)eq.(5). The coefficient a can be adjusted by a calibrated potentiometer. The circuit has the unit step response G(t) and performs a deconvolution for F(t) according to... 

This is a high order process with severe nonminimum phase behaviour. Its noise-free unit step response is shown in Figure 2.5. This process step response is sampled with an interval At = 1.5 sec. The key to success with the Laguerre model for such a complicated process is to find the optimal time scaling factor p for a given model order N, particularly when the model order is small. Figure 2.6 shows a 3-dimensional plot of the loss function V — for iV = 1,2,..., 10 and 0 < p < 0.1, where the coefficients... 

To determine the effects of pole and zero locations, simulate the unit step responses of the discrete transfer functions shown below for the first six sampling instants, k = 0 to A = 5. What conclusions can you make con-... 

Unit step response of a transfer function step... 

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. 

Find the pulse transfer function and hence calculate the response to a unit step and unit ramp. T = 0.5 seconds. Compare the results with the continuous system response Xo t). The system is of the type shown in Figure 7.9(b) and therefore... 

Table 7.2 shows the discrete response x ikT) to a unit step function and is compared with the continuous response (equation 3.29) where... 

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)... 

Example 2.10 What is the time domain response C (t) in Eq. (2-27) if the change in inlet concentration is (a) a unit step function, and (b) an impulse function ... 

It can be synthesized with the MATLAB function feedback (). As an illustration, we will use a simple first order function for Gp and Gm, and a PI controller for Gc. When all is done, we test the dynamic response with a unit step change in the reference. To make the reading easier, we break the task up into steps. Generally, we would put the transfer function statements inside an M-file and define the values of the gains and time constants outside in the workspace. 

A forcing function, whose transform is a constant K is applied to an under-damped second-order system having a time constant of 0.5 min and a damping coefficient of 0.5. Show that the decay ratio for the resulting response is the same as that due to the application of a unit step function to the same system. 

Now the unit step function can be expressed as a limit of the first-order exponential step response as the time constant goes to zero. 

Inversion of the Laplace transformation gives the time function. If we have a transfer function the unit step function is C [G(,y s] and the impulse response is C" [G(,)]. 

We do not lose generality by considering such a unit step function. Because the differential equation is linear, by making superposition of the step function, the response from any surface contour can be treated. The Laplace transform of a step function is... 

When identifying the hypothetical system S we need u. The weighting function found in Example 5.6 is substituted for the input of the hypothetical system. This input does not contain an impulse or a unit step component, and hence we set DC = 0 and US = 0. The response of the hypothetical system equals the "Observed" response. The program is the one used in Example 5.6, only tha data lines are changed as follows ... 

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. 

III.28 Sketch, qualitatively, the response of systems with the following transfer functions. Assume unit step input changes. 

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