16.3 DIGITAL CONTROL SYSTEMS 805
(d) Sinceξ=F/ 2
√
KJ, it follows that
ξ′=
ξ
√
4. 44
= 0. 4746 ξ= 0. 4746 × 0. 222 = 0. 105
corresponding to which the percent maximum overshoot from Figure 16.2.10 is 72%.
(e) For a 25% overshoot, Figure 16.2.10 shows thatξ= 0 .4. Then from Equation (16.2.32),
ξ=(F+Qo)/ 2
√
KJ,or
0. 4 =
2 × 10 −^4 +Qo
2
√
0. 0135 × 1. 5 × 10 −^5
=
2 × 10 −^4 +Qo
2 × 0. 45 × 10 −^3
Hence,
Qo= 0. 36 × 10 −^3 − 0. 2 × 10 −^3 = 0. 16 × 10 −^3
Then
Ko=
Qo
K′aKm
=
0. 16 × 10 −^3
444 × 2. 7 × 10 −^4
= 0. 0013 ,or 1. 3 × 10 −^3 V·s/rad.
16.3 Digital Control Systems
Significant progress has been made in recent years in discrete-data and digital control systems
because of the advances made in digital computers and microcomputers, as well as the advantages
found in working with digital signals. Discrete-data and digital control systems differ from the
continuous-data or analog systems in that the signals in one or more parts of these systems
are in the form of either a pulse train or a numerical (digital) code. The terms, sampled-data
systems, discrete-data systems, discrete-time systems, and digital systems have been loosely used
in the control literature. However, sampled-data systems usually refer to a general class of systems
whose signals are in the form of pulse data; sampled data refers to signals that are pulse-amplitude
modulated, i.e., trains of pulses with signal information carried by the amplitudes. Digital control
systems refers to the use of a digital computer or controller in the system; digital data usually
refers to signals that are generated by digital computers or digital transducers and are thus in
some kind of coded form. A practical system such as an industrial process control is generally
of such complexity that it contains analog and sampled as well as digital data. Hence the term
discrete-data systems is used in a broad sense to describe all systems in which some form of
digital or sampled signals occur. When a microprocessor receives and outputs digital data, the
system then becomes a typical discrete-data or digital control system.
Figure 16.3.1(a) illustrates the basic elements of a typical closed-loop control system with
sampled data; Figure 16.3.1(b) shows the continuous-data inpute(t) to the sampler, whereas
Figure 16.3.1(c) depicts the discrete-data outpute*(t) of the sampler. A continuous input signal
r(t) is applied to the system. The continuous error signal is sampled by a sampling device, the
sampler, and the output of the sampler is a sequence of pulses. The pulse train may be periodic
or aperiodic, with no information transmitted between two consecutive pulses. The sampler in
the present case is assumed to have a uniform sampling rate, even though the rate may not be
uniform in some other cases. The magnitudes of the pulses at the sampling instants represent
the values of the input signale(t) at the corresponding instants. Sampling schemes, in general,
may have many variations: periodic, cyclic-rate, multirate, random, and pulse-width modulated
