Wiley signals and systems e book TLFe BO 409

16. Stability and Feedback Systems

3!)4

Figure 16.3 s11ou.s a f o u x t ~ ~ - rwxsive
~ ~ ~ ~ e ciiscrete
r
syst,eni. In order to determine its stability we f i i ~ s tread from the block diagram:

(16.25)
and froin here obtain the syskm fiinction

Using the bilinear t r ~ i s f o r ~ i L t ~the
o n ,de~1omiK~alv~
&(2 ) becomes the partial fraction Qis):
O(s)=

Q(%)

-1.875

(.q4

c-

- 1.466'7~~
- 3 . 2 -~ 3.86678
~
- 1)
.
(s - 114

(16.27)

W e mush T ~ O W test the polyrioniial h parenhhis in the ~ u ~ ~ of~g (ds ) ~
to o r
Ilstcxtain whether or not it is a lIuiwitz p ~ l y t i ~The
~ ~necesiq,'
~ l .
ronditioii for i~
Elurwitz pdynomial (16.17) is not fulfilled as nut all o f t I-tt twfficients arc positive.
We can now make the following dednctions:

+ Q ( s ) ~ i a szeros with Re{s> 2 0.
=$

Q ( z >has zeroh will1 jzj 2 1.

==+H ( z ) has poles with IZI 2 1.

+ T h r a system is unstable.
Of tourbe, t h zeros
~
of the deriominator polynomial &(z) can be found using a
coniputer, Tlw result>is

There is a zero at := 2 wl-iich is oiitside the tiriit circle, am1 t h i s confirm the
result we obtained using the bilitiear ~ ~ a nand
s the
~ ~Wurwit-z
r ~ tcst,.