Page 473 - Mechanical Engineers' Handbook (Volume 2)

464 Closed-Loop Control System Analysis

Z{y(k)} Y(z) (43a)
Z{u(k)} U(z) (43b)
By referring to Table 2, the z-transform of Eq. (42) becomes
n
1
m
Y(z) az (Y(z) az y(z) bU(z) bz U(z) bz U(z)
1
0
n
1
1
m
or
m
1
[1 az 1 az ]Y(z) [b bz 1 bz ]U(z)
1
1
0
m
n
which can be written as
Y(z) b bz 1 bz m
m
0
1
(44)
U(z) 1 az 1 az n
1
n
Consider the response of the linear discrete-time system given by Eq. (44), initially at rest
when the input u(t) is the delta ‘‘function’’ (kT),
(kT) 1 k 0
0 k 0
since
Z{ (kT)} (kT)z k 1

k 0
U(z) Z{ (kT)} 1
and
b bz 1 bz m
Y(z) 0 1 m G(z) (45)
1 az 1 az n
n
1
Thus G(z) is the response of the system to the delta input (or unit impulse) and plays the
same role as the transfer function in linear continuous-time systems. The function G(z)is
called the pulse transfer function.
4.6 Zero- and First-Order Hold
Discrete-time control systems may operate partly in discrete time and partly in continuous
time. Replacing a continuous-time controller with a digital controller necessitates the con-
version of numbers to continuous-time signals to be used as true actuating signals. The
process by which a discrete-time sequence is converted to a continuous-time signal is called
data hold.
In a conventional sampler, a switch closes to admit an input signal every sample period
T. In practice, the sampling duration is very small compared with the most significant time
constant of the plant. Suppose the discrete-time sequence is ƒ(kT); then the function of the
data hold is to specify the values for a continuous equivalent h(t) where kT t (k 1)T.
In general, the signal h(t) during the time interval kT t (k 1)T may be approximated
by a polynomial in as follows:
n
h(kT ) a a n 1 n 1 a a 0 (46)
n
1
where 0 T. Since the value of the continuous equivalent must match at the sampling
instants, one requires