Signals and Systems - Electrical Engineering

12.3 Application to Sampled-Data and Digital Control Systems 725

FIGURE 12.7
Equivalent discrete-time system of the open-loop sampled-data
system.

x(nTs) y(nTs)

F(z)=Y(z)

X(z)

(i.e., a sequence of impulses at times{nTs}with amplitude the sampled valuesx(nTs)), then the output

of the DAC with ZOH is

v(t)=[xs∗hzoh](t)=

∑

n

x(nTs)hzoh(t−nTs) (12.14)

or a piecewise constant signal (see Figure 12.6). Putting together the transfer function of the ZOH

with that of the plant so thatF(s)=Hzoh(s)G(s), we have thatY(s)=F(s)Xs(s).

If we letf(t)=L−^1 [F(s)], then the output of the plant is given by the convolution integral as

y(t)=[xs∗f](t)=

∑

n

x(nTs)[δ∗f](t−nTs)=

∑

n

x(nTs)f(t−nTs)

which is the convolution sum of the discrete input and the sampled-impulse response of the plant

combined with that of the ZOH. ForY(z)=Z[y(nTs)] andX(z)=Z[x(nTs)], we have that when we

sampley(t), then

y(kTs)=y(t)|t=kTs=

∑

n

x(nTs)f(kTs−nTs)

The transfer function of the discrete system is

F(z)=Z[f(nTs)]=

Y(z)

X(z)

(12.15)

which can be obtained by sampling the inverse Laplace transformf(t)=L−^1 [F(s)] and then comput-

ing its Z-transform. We have thus obtained the equivalent discrete-time system to the sampled-data

system shown in Figure 12.7.

nExample 12.5

Consider the open-loop sampled-data system shown in Figure 12.6, where the DAC with ZOH is

synchronized with an ADC, which is just an ideal sampler. LetTs=1 sec/sample be the sampling

period. If the transfer function of the plant is

G(s)=

1

(s+ 1 )(s+ 2 )

find the transfer functionF(z)=Y(z)/X(z).