Microcontroller based applied digital control (D. Ibrahim, 2006)

Equations (6.34) and (6.35) tell us that if at least one of the continuous functions has been sampled, then the z-transform of the product is equal to the product of the z-transforms of each function (note that [e*(s)]* = [e*(s)], since sampling an already sampled signal has no further effect). G(z) is the transfer function between the sampled input and the output at the sampling instants and is called the pulse transfer function. Notice from (6.35) that we have no information about the output y(z) between the sampling instants.

6.3.1 Open-Loop Systems

Some examples of manipulating open-loop block diagrams are given in this section.

Example 6.15

Figure 6.18 shows an open-loop sampled data system. Derive an expression for the z-transform of the output of the system.

Solution

For this system we can write

y(s) = e*(s)KG(s)

or

y*(s) = [e*(s)KG(s)]* = e*(s)KG*(s)

and

y(z) = e(z)KG(z).

Example 6.16

Figure 6.19 shows an open-loop sampled data system. Derive an expression for the z-transform of the output of the system.

Figure 6.18 Open-loop system

Figure 6.19 Open-loop system

Solution

The following expressions can be written for the system: y(s) = e*(s)G1(s)G2(s)

or

y*(s) = [e*(s)G1(s)G2(s)]* = e*(s)[G1G2]*(s)

and

y(z) = e(z)G1G2(z),

where

G1G2(z) = Z {G1(s)G2(s)} = G1(z)G2(z).

For example, if

G1(s) = 1 s

and the output is given by

z(1 − e−aT )

y(z) = e(z) (z − 1)(z − e−aT ) .

Example 6.17

Figure 6.20 shows an open-loop sampled data system. Derive an expression for the z-transform of the output of the system.

Figure 6.20 Open-loop system

From (6.37) and (6.38),

y*(s) = e*(s)G*1 (s)G*2 (s),

then

6.3.2 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. Some examples are given below.

Example 6.18

A unit step signal is applied to the electrical RC system shown in Figure 6.21. Calculate and draw the output response of the system, assuming a sampling period of T = 1 s.

Solution

The transfer function of the RC system is

1

G(s) = . s + 1

158 SAMPLED DATA SYSTEMS AND THE Z-TRANSFORM

y(nT)

1.571

1.552

1.503

1.367

1.0

T

Figure 6.22 RC system output response

From the z-transform tables we find

y(nT ) = 1.582 − 0.582(0.368)n .

The first few output samples are

y(0) = 1,

y(1) = 1.367, y(2) = 1.503, y(3) = 1.552, y(4) = 1.571,

and the output response (shown in Figure 6.22) is given by

y(nT ) = δ(T ) + 1.367δ(t − T ) + 1.503δ(t − 2T ) + 1.552δ(t − 3T ) + 1.571δ(t − 4T ) + . . . .

It is important to notice that the response is only known at the sampling instants. For example, in Figure 6.22 the capacitor discharges through the resistor between the sampling instants, and this causes an exponential decay in the response between the sampling intervals. But this behaviour between the sampling instants cannot be determined by the z-transform method of analysis.

Example 6.19

Assume that the system in Example 6.17 is used with a zero-order hold (see Figure 6.23). What will the system output response be if (i) a unit step input is applied, and (ii) if a unit ramp input is applied.

Figure 6.23 RC system with a zero-order hold

The block diagram of a closed-loop sampled data system is shown in Figure 6.29. Derive an expression for the output function of the system.