Signals and Systems - Electrical Engineering

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

Solution

The combined transfer function of the ZOH and the plant is

F(s)=

G(s)( 1 −e−s)

s

=

G(s)

s

−

G(s)e−s

s

so that if we find the inverse Laplace transform ofGˆ(s)=G(s)/s, call itgˆ(t), then

f(t)=gˆ(t)−ˆg(t− 1 )

The inverse ofGˆ(s)=G(s)/sis obtained by partial fraction expansion

Gˆ(s)=G(s)

s

=

1

s(s+ 1 )(s+ 2 )

=

A

s

+

B

s+ 1

+

C

s+ 2

=

0.5

s

−

1

s+ 1

+

0.5

s+ 2

so that

gˆ(t)=[0.5−e−t+0.5e−^2 t]u(t)

Samplingf(t)=gˆ(t)−gˆ(t− 1 )with a sampling periodTs=1 gives

f(n)=gˆ(n)−gˆ(n− 1 )

wheregˆ(n)=[0.5−e−n+0.5e−^2 n]u(n). The Z-transform off(n)is then the transfer function

F(z)=

Y(z)

X(z)

=Gˆ(z)( 1 −z−^1 )

=( 1 −z−^1 )

(

0.5

1 −z−^1

−

1

1 −e−^1 z−^1

+

0.5

1 −e−^2 z−^1

)

=0.5−

( 1 −z−^1 )

1 −e−^1 z−^1

+

0.5( 1 −z−^1 )

1 −e−^2 z−^1 n

12.3.2 Closed-Loop Sampled-Data System

Consider the feedback system shown in Figure 12.8 where for simplicity we assumeH(s)=1 (i.e., no

feedback sensor). An equivalent block diagram is obtained by moving back the sampler at the input.

Consider finding the transfer function of the sampled input commandcs(t)and the sampled output

of the systemys(t). The above open-loop development can be used to find the transfer function of the

feedback system.

The sampled error signal is

es(t)=cs(t)−ys(t)