The pulse transfer function

which is converted into a difference equation 

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) 

In Table 7.3 the difference between Xo(kT) and Xo(t) is due to the sample and hold. It should also be noted that with the discrete response x kT), there is only knowledge of the output at the sampling instant. 

---

Blocks in Cascade In Figure 7.9(a) there are synchronized samplers either side of blocks G (s) and G2(s). The pulse transfer function is therefore... 

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... 

Figure 7.13 shows the general form of a digital control system. The pulse transfer function of the digital controller/compensator is written... 

The pulse transfer function is defined as the first term in Eq. (18.56). 

Pulse transfer functions for modified z transforms are defined in the same way as for regular z transforms. For a system with input m, and output x, ), the pulse transfer function is... 

Find the pulse transfer functions in the z domain (HBGm z) for the systems is... 

These algorithms are basically difference equations that relate the current value of m to the current value of e and old values of m and e. These difference equations can be derived from the pulse transfer function. ... 

Example 20.11. A second-order opienloop unstable process has the pulse transfer function given below ... 

A comparison of equations 7.209 and 7.210 establishes that, in order to derive the pulse transfer function G (s) from the continuous transfer function G(s), we must star both sides of equation 7.209 by writing ... 

It is interesting to find out what the process output would have been if a sequence of impulses entered a continuous process directly without passing through a hold element. Consider the case of a first-order lag but without the hold element (Figure 29.8a). Then the pulse transfer function is given by... 

Assuming zero-order hold, the pulse transfer function for the hold-process combination was developed in Example 29.3, case b, and is given by eq. (29.15) ... 

With the use of the pulse transfer function, such systems can be converted to discrete systems as above, and consequently the same rule applies for their stability. In particular, if Gp(s) is the transfer function of a continuous process, its pulse transfer function is given by... 

The pulse transfer function HGp(z) is given by (assuming that T = I)... 

Discrete-Time Analysis of Continuous Systems. The Pulse Transfer Function... 

HGp(z) is called the pulse transfer function and can be related to the continuous transfer functions of the hold and the process, H(s) and Gp(s), respectively. Let us now develop this relationship. 

Comparing eqs. (29.12) and (29.8), we conclude that the pulse transfer function is given by... 

Once the pulse transfer function for a process is known, the sampled values of its output can be found by inverting eq. (29.8) ... 

Consider N continuous processes in series with transfer functions G 1(5), Gi(s),..., Gn s). The pulse transfer function of the train of continuous systems is characterized by G G2- Gn(z) and is computed from the equation... 

Define the pulse transfer function for a continuous system. Why is it needed, and why is its continuous transfer function G(s) inadequate ... 

Find the pulse transfer function of a pure integrator with and without the hold element. 

Compute the pulse transfer function of the following second-order process with and without a zero-hold element ... 

When a zero-order hold is used in a sampled-data system with 7, = 0.2, t = 1, and T2 = 5, the pulse transfer function is... 

The pulse transfer function of this process with a zero-order hold is [see Chapter 14, Eq. (14.93)]... 

This function is called the pulse transfer function and as shown above, it can be derived directly from a discrete equation. The relationship y k)/u k) is identical to the relationship y z)/u(z). y(k) stands for the magnitude of signal y at an arbitrarily sample moment kT, whereas (z) stands for the total sampled signal as defined. 

In case of a first-order time lag of Eqn. (5.10), the pulse transfer function can be found as follows (Fig. 5.5). 

---