Signals and Systems - Electrical Engineering

728 CHAPTER 12: Applications of Discrete-Time Signals and Systems

We thus have

P(z)[C(z)−Y(z)]

︸ ︷︷ ︸

M(z)

Gˆ(z)=Y(z)

from which we obtain

Y(z)=

P(z)C(z)Gˆ(z)

1 +P(z)Gˆ(z)

CallingF(z)=P(z)Gˆ(z)(the feed-forward transfer function consisting of the discretized analog

controller and the ZOH and the plant), we get

Y(z)

C(z)

=

F(z)

1 +F(z)

(12.20)

or the transfer function of the data-sampled system. Notice that this equation looks like the equation

of a continuous-feedback system.

Remarks

n In the equivalent discrete-time system obtained above, the information of the output of the open-loop or

the closed-loop systems in between the sampling instants is not available; only the samples y(nTs)are. This

is also indicated by the use of the Z-transform.

n Depending on the location of the sampler, there are some sampled-data control systems for which we

cannot find a transfer function. This is due to the time-variant nature of the system.

nExample 12.6

Suppose we wish to have a data-sampled system like the one shown in Figure 12.8 that simulates

the effects of an integral analog controller. Let the plant be a first-order system,

G(s)=

1

s+ 1

Let the sampling period beTs=1. DetermineP(z)and find the discrete transfer function of the

sampled-data system whenH(s)=1.

Solution

Ife(t)is the input of an integrator andv(t)its output, lettingt=nTsand approximating the integral

by a sum we have that

v(nTs)=

∑n

k= 0

e(kTs)Ts=

n∑− 1

k= 0

e(kTs)Ts+e(nTs)Ts

=v(nTs−Ts)+Tse(nTs)