Page 475 - Mechanical Engineers' Handbook (Volume 2)

466 Closed-Loop Control System Analysis

h(kT) ƒ(kT)
Hence Eq. (46) can be written as
n
h(kT ) a a n 1 a ƒ(kT) (47)
n n 1 1
If an nth-order polynomial as in Eq. (47) is used to extrapolate the data, then the hold circuit
is called an nth-order hold. If n 1, it is called a first-order hold (the nth-order hold uses
the past n 1 discrete data ƒ[(k n)T]). The simplest data hold is obtained when n 0
in Eq. (47), that is, when
h(kT ) ƒ(kT) (48)
where 0 T and k 0, 1, 2,.... Equation (48) implies that the circuit holds the
amplitude of the sample from one sampling instant to the next. Such a data hold is called a
zero-order hold. The output of the zero-order hold is shown in Fig. 17.

Zero-Order Hold
Assuming that the sampled signal is 0 for k 0, the output h(t) can be related to ƒ(t)as
follows:
h(t) ƒ(0)[u (t) u (t T)] ƒ(T)[u (t T) u (t 2T)]
s
s
s
s
ƒ(2T)[u (t 2T)] u (t 3T)]
s
s
ƒ(kT){u (t kT) u [t (k 1)T]} (49)

k 0 s s
Since L u (t kT) e kTs /s, the Laplace transform of Eq. (49) becomes
s
L[h(t)] H(s) ƒ(kT) e kTs e (k 1)Ts

k 0 s

1 e Ts
ƒ(kT)e kTs (50)
s k 0
The right-hand side of Eq. (50) may be written as the product of two terms:
H(s) G (s)F*(s) (51)
h0

Figure 17 Input ƒ(kt) and output h(t) of the zero-order hold.