Transfer functions pulse

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... 

A unity feedback computer control system, has an open-loop pulse transfer function... 

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 open continuous transfer functions and pulse transfer functions for the plant and compensator are printed in the command window... 

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... 

Thus F(, is a periodic function of s with period ioj. We will use this periodicity property to develop pulse transfer functions in Sec. 18.7. 

We know how to find the z transformations of functions. Let us now turn to the problem of expressing input-output transfer-function relationships in the z domain. Figure 18.9a shows a system with samplers on the input and on the output of the process. Time, Laplace, and z-domain representations are shown. G(2, is called a pulse transfer function. It will be defined below. 

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

We will show how these pulse transfer functions are applied to openloop and closedloop systems in Sec. 18.7. 

We are now ready to use the concepts of impulse-sampled functions, pulse transfer functions, and holds to study the dynamics of sampled-data systems. 

Thus the overall transfer function of the process can be expressed as a product of the two individual pulse transfer functions if there is an impulse sampler between the elements. 

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... 

As we developed in Chap. 18, the openloop pulse transfer function for this process is... 

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. 

This is the ultimate gain if the sampling period is 0.5 min. If a T, = 2 min is used, the openloop pulse transfer function becomes... 

In a digital computer-control system, the feedback controller has a pulse transfer function. What we need is an equation or algorithm that can be programmed into the digital computer. At the sampling time for a given loop, the computer looks at the current process output x, compares it to a setpoint, and calculates a current value of the error. This error, plus some old values of the error and old values of the controller output or manipulated variable that have been stored in computer memory, are then used to calculate a new value of the controller output m,. 

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. ... 

Putting this in terms of a pulse transfer function gives... 

The physical realizability of pulsed transfer functions uses the basic criterion that the current output of a device (digital computer) cannot depend upon future information about the input. We cannot build a gadget that can predict the future. 

This pulse transfer function has a zero at z = a and a pole at z = +1. It cannot produce any phase-angle advance since the pole lies to the right of the zero (a is less than 1). The pole at -I-1 is equivalent to integration (pole at s = 0 in continuous systems) which drives the system to zero steadystate error for step disturbances. 

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

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