Signals and Systems - Electrical Engineering

12.3 Application to Sampled-Data and Digital Control Systems 723

FIGURE 12.5

Digital implementation of a

continuous-feedback system. ZOH

stands for zero-order hold.

ADC Computer DACZOH Plant

Clock

c(t)

e(t)

y(t)

+

−

Controller

discrete and back from discrete to continuous is very important in obtaining a discrete-time system
from a sampled-data control system.

Consider the relation between a continuous control system and its implementation using a computer
as shown in Figure 12.5. In the continuous-feedback system, the controller responds to an error signal
e(t), which is the difference between a reference input signalc(t)and the system outputy(t), attempt-
ing to change the dynamics of an analog plant. A digital realization of this continuous-feedback
system typically requires that the error signal be converted into a digital signal by means of an ADC
before being fed to a computer implementing the controller (e.g., a PID controller). A DAC with a
zero-order hold that is synchronously connected and has the same sampling period as the previous
ADC is used to generate a signal that will act on the plant. The output of the planty(t)is fed back and
compared with the command signalc(t)to obtain the errore(t). The digital controller is composed
of the ADC, the computer, and the DAC with the zero-order hold, all of which are synchronized by a
common clock.

Why are sampled-data and digital control systems needed? In part, because many systems are inher-
ently discrete—for example, a radar tracking system scans to convert azimuth and elevation into
sampled data. But in general we have that:

n A continuous control system operates in real time, and the amplitude of its signals are allowed to
take any possible value, but its elements are susceptible to degradation with time, and the system
is sensitive to noise and difficult to change since it is hardwired.

n Digital components are less susceptible to aging, environmental variations, and noise. A digital
controller can be modified easily by changing software without changing the hardware. However,
computational speed and resolution (word length) are limitations of digital controllers that can
cause instabilities.

Remarks

n In the following development you need to remember:

1. The Laplace transform of an ideally sampled signal,

xs(t)=

∑

n

x(nTs)δ(t−nTs)