Signals and Systems - Electrical Engineering

12.3 Application to Sampled-Data and Digital Control Systems 727

FIGURE 12.8
Closed-loop sampled-data control system.

Plant

ADC

Sampler Computer

DAC

ZOH

c(t) e(t) y(t)

es(t)

Ps(s)

H(s)

G(s)

+

−

Sensor

1 −e−sTs

s

with Laplace transform

Es(s)=Cs(s)−Ys(s) (12.16)

The functionPs(s)corresponds to the discretization of an analog controller, such as a PID controller.
The Laplace transform of the output of the computer is then

Ms(s)=Ps(s)Es(s)=

∑

n

m(nTs)e−snTs (12.17)

or the Laplace transform of a sampled signal. On the other hand, the DAC with ZOH and the plant
have together a transfer function

Gˆ(s)=(^1 −e

−sTs)G(s)

s

Thus, the Laplace transform of the output of the plant is

Ms(s)Gˆ(s)=

∑

n

m(nTs)

[

Gˆ(s)e−snTs

]

(12.18)

Using the time-shifting property, the inverse Laplace transform of the above equation is

∑

n

m(nTs)ˆg(t−nTs)

which when sampled att=kTsgives the convolution sum

∑

n

m(nTs)ˆg(kTs−nTs)=ys(nTs) (12.19)

so thatM(z)Gˆ(z)=Y(z).

Letting z=esTs in Equation (12.16), we obtain E(z)=C(z)−Y(z), and replacing it in
Equation (12.17) gives

M(z)=P(z)E(z)=P(z)[C(z)−Y(z)]