Open-Loop Sampled-Data System
12.3.1 Open-Loop Sampled-Data System
Consider the system shown in Figure 12.6. Assume the discrete-time signal x(nT s ) coming from a computer is used to drive an analog plant with a transfer function G(s). To change the state of the plant, x(nT s ) is converted into a continuous signal that holds the value of the sample for the duration of the sample period T s . This can be implemented using a DAC with a zero-order hold (ZOH), which holds the value of x(nT s ) until the next sample arrives at (n + 1)T s . Furthermore, to allow the output signal to be possibly processed by a computer, assume the output of the plant y(t) is also sampled to get y(nT s ) . We are interested in the transfer function that relates the discrete input x(nT s ) to the discrete output y(nT s ) where T s is the sampling period chosen according to the maximum frequency present in the analog input x(t).
As we saw in Chapter 7, the transfer function of a zero-order hold is
which corresponds to an impulse response
(12.12) or a pulse of duration T s and unit amplitude. If the sampled signal is written as
h zoh ( t) = u(t) − u(t − T s )
s ( t) =
x(nT s )δ( t − nT s )
x(nT s )
DAC
Plant
y(t ) y(nT s )
FIGURE 12.6 ZOH signal Open-loop sampled-data system for an
analog plant G(s). The output of the DAC t with a ZOH is illustrated.
0 T s 2T s 3T s 4T s 5T s 6T s
12.3 Application to Sampled-Data and Digital Control Systems 725
FIGURE 12.7 x (nT Equivalent discrete-time system of the open-loop sampled-data ) s
(i.e., a sequence of impulses at times {nT s } with amplitude the sampled values x(nT s ) ), then the output of the DAC with ZOH is
v(t) X = [x
s ∗h zoh ](t) =
x(nT s ) h zoh ( t − nT s )
or a piecewise constant signal (see Figure 12.6). Putting together the transfer function of the ZOH with that of the plant so that F(s) =H zoh ( s)G(s), we have that Y(s) = F(s)X s ( s).
If we let f (t) =L −1 [F(s)], then the output of the plant is given by the convolution integral as
X y(t) X = [x
s ∗ f ](t) =
x(nT s ) [δ ∗ f ](t − nT s ) =
x(nT s ) f (t − nT s )
which is the convolution sum of the discrete input and the sampled-impulse response of the plant combined with that of the ZOH. For Y(z) = Z[y(nT s ) ] and X(z) = Z[x(nT s ) ], we have that when we sample y(t), then
y(kT X
s ) = y(t)| t =kT s =
x(nT s ) f (kT s − nT s )
The transfer function of the discrete system is
Y(z)
F(z) = Z[ f (nT s ) ] =
X(z)
which can be obtained by sampling the inverse Laplace transform f (t) =L −1 [F(s)] and then comput- ing its Z-transform. We have thus obtained the equivalent discrete-time system to the sampled-data system shown in Figure 12.7.
■ Example 12.5
Consider the open-loop sampled-data system shown in Figure 12.6, where the DAC with ZOH is synchronized with an ADC, which is just an ideal sampler. Let T s = 1 sec/sample be the sampling period. If the transfer function of the plant is
1 G(s) = ( s + 1)(s + 2)
find the transfer function F(z) = Y(z)/X(z).
C H A P T E R 12: Applications of Discrete-Time Signals and Systems
Solution
The combined transfer function of the ZOH and the plant is
G(s)(1 −e −s )
G(s)
G(s)e −s
F(s) =
so that if we find the inverse Laplace transform of ˆ G(s) = G(s)/s, call it ˆg(t), then
f (t) = ˆg(t) − ˆg(t − 1)
The inverse of ˆ G(s) = G(s)/s is obtained by partial fraction expansion
1 A B ˆG(s) = C
G(s)
s(s + 1)(s + 2)
so that
ˆg(t) = [0.5 − e −t + 0.5e −2t ]u(t)
Sampling f (t) = ˆg(t) − ˆg(t − 1) with a sampling period T s = 1 gives
f (n) = ˆg(n) − ˆg(n − 1)
where ˆg(n) = [0.5 − e −n + 0.5e −2n ]u(n). The Z-transform of f (n) is then the transfer function
Y(z)
F(z) =
= ˆG(z)(1 − z −1 )