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PULSE TRANSFER FUNCTION AND MANIPULATION OF BLOCK DIAGRAMS

The pulse transfer function is the ratio of the z-transform of the sampled

output and the input at the sampling instants. Suppose we wish to sample a

system with output response given by :

y(s)=e*(s)G(s)

The output signal is sample to obtain:

[e*(s)]*= [e*(s)], since sampling is already sampled, signal has no further effect

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OPEN-LOOP SYSTEMS

Figure 3.1 shows an open-loop sampled data system.

Figure 3.1

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Figure 3.2

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Example 3.1:

Figure 3.3 shows an open-loop sampled data system. Derive an expression for the

z-transform of the output of the system.

Solution

Figure 3.3

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Example 3.2:

Figure 3.4 shows an open-loop sampled data system. Derive an expression for the

z-transform of the output of the system.

Figure 3.4

Solution

The following expressions can be written for the system:

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OPEN-LOOP TIME RESPONSE

The open-loop time response of a sampled data system can be obtained by finding

the inverse z-transform of the output function.

Example 3.3

A unit step signal is applied to the electrical RC system shown in Figure 3.5. Calculate

and draw the output response of the system, assuming a sampling period of T=1 s.

Figure 3.5

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Solution:

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From the z-transform tables :

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The open-loop block diagram of a system with a zero-order hold is shown in Figure 3.6.

Calculate and plot the system response when a step input is applied to the system,assuming that T=1 s.

Example 3.4

Figure 3.6

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CLOSED-LOOP SYSTEMS

The block diagram of a closed-loop sampled data system is shown in Figure 3.7.

The expression of the transfer function by referring to this system can be write

as shown in the following derivation:

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Example 3.5

The block diagram of a closed-loop sampled data system is shown in Figure 3.8.

Derive an expression for the output function of the system.

Solut ion

For the system in Figure 3.8 we can write y(s) = e(s)G(s)

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Example 3.6

The block diagram of a closed-loop sampled data control system is shown in

Figure 3.9. Derive an expression for the transfer function of the system.

Solut ion

The A/D converter can be approximated with an ideal sampler. Similarly, the

D/A converter at the output of the digital controller can be approximated with a

zero-order hold. Denoting the digital controller by D(s) and combining the zero-

order hold and the plant into G(s), the block diagram of the system can be

drawn as in Figure 3.10.

Figure 3.9

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For this system can write

Note that the digital computer is represented as D*(s). Using the above two

equations, we can write

* *

( ) ( ) ( ) ( ) ....................... 1

( ) ( ) ( ) ( ) ....................... 2

e s r s H s y s

y s e s D s G s

Figure 3.10

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and, from (1),

The sampled output is then

In z-transform format:

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Closed-Loop Time Response

The closed-loop time response of a sampled data system can be obtained by

finding the inverse z-transform of the output function.

Example 3.7

A unit step signal is applied to the sampled data digital system shown in Figure

3.11. Calculate and plot the output response of the system. Assume that T=1 s.

Figure 3.11

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Solut ion

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Example 3.8

Figure 3.12 shows a digital control system. When the controller gain K is

unity and the sampling time is 0.5 seconds, determine:a) the open-loop pulse transfer function

b) the closed-loop pulse transfer function

c) the difference equation for the discrete time response

d) a sketch of the unit step response assuming zero initial conditionse) the steady-state value of the system output

Figure 3.12

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Solution:

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(c)

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(d)

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OPEN-LOOP SYSTEMS CONTAINING

DIGITAL FILTERS

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Example 3.9

Let us determine the step response of the system shown in Figure 3.13.

Suppose that the filter is described by the difference equation

m(kT) = 2e(kT)e[(k1)T]

and thus

In addition, suppose that:

Figure 3.13

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Example 3.10

In figure 3.15, let the input be a unit step, to=0.4T, and 1

( 1)

Tse

G ss s

Figure 3.15

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By the power-series method:

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Example 3.11

Consider the system of Figure 3.16. The delay is 1 ms (to= 10-3s) and T =

0.05 s. Thus, for this system,

Figure 3.16

Thus, for this system,

0.05 0.001 0.049

mT T T

mT T T

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NONSYNCHRONOUS SAMPLING

The output of this system can be derived using the modified z -transform

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Example 3.11

Find C(z) for the system of figure 3.17, which contains nonsynchronous

sampling.

Solution:

For the system, T=0.05 and hT=0.01

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