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