Wiley signals and systems e book TLFe BO 410

16.3. Fecdback Systems

395

Figure 16.3: k-ourth-order recursive discrete-time system

feedback c m be described by a transfer function, as showii in Figure 6.15. but the
i~€orn~at~
o n the ~ ~ d i ~
aboiit
transfer
r i ( f ~ ~~n c~t ~~of
o n~
the
s f o r w a d a d feedbark
paths is lost (in Figure 6.15 these are F ( s ) a d Gfs)). T h e feedback principle
is used in illany arms. both natural and technical. We will show thrce typical
problems that can be solved using feedback.

3.1

~~v~~~~~~
a

~~~~~~~

Using ~~~~~a~~

In Exxrnple 6.9 we saw that feedback can iiivert the transfer function of a system.
Figurc 16.4 shows the situation again. but in contrast to Figlire 6.15. thc feedback
path coritains another change of sign. The transfer function of- the closed loop i s

then
(16.28)

The poles ofthe u7hole fcedback system H ( s ) a r the
~ zeros of G(s). Tliej must lie
in the left half of the s-plane, so that stability can be assured. TIP poles of G(.s),
liowever, have no influence on the stability of tkw feedback system.
Coatinuous system that havc no zeros in the right s-plane are called ~~~~~~~~1
phase ~ ~ ~ s ~t o~r ~ en s ~p o. ~discrete
i ~ ~ ~ systems
~ ~ l y with
~
no zeros outside of the
unit circle are also called t i i i ~ i n i phase
~l
systems. For thr iiiverse sptem
(or
___ to be siablc, C ( s ) (01 G ( z ) )has to hc minimal phase and additionally may

G'iz))

&

riot have any zeros on the imaginaiy &xis (OK on tlic unit circle) of the complex
frequency plane.