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~~ - -~ 3.86678 ~ - 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> =$ Q ( z >has zeroh will1 jzj ==+H ( z ) has poles with IZI + 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 := 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,