Wiley signals and systems e book TLFe BO 162

147 7.3 Higher-Qrclcr Syslriris ~ x a ~ 7.9 n ~ ~ e \Vr arc' looking for tlic output signal y(i) of a given input g r d I ( t ) : + 0.I g = s, g ( ) = 8, A first-order systeln for 1, x ( ! ) - U9.1 siril , t b 0, for (7.99) The svskni firriction is N(s)= - _ 'j + 0.1 R.e{s) > -0.1 (7.100) by splitting X ( s ) H ( s )into paItial fractioiis: 10.1 Y w ( s )= X ( s ) N ( s )= (93 t r,js +0.1) A -s f0.l c Bs 52 +C +1 (7.102) The cwfieintts arc o h t a i n ~ deither as in Exarnple 5, or bv equating cnefIicwnl i anti solution ol a linear x eqimtho svstem The result ii: il = 10, 13 - -10, C = Iriversr trausformatioii yields the external part: T h e general form of the internal part is obtairrrtl from the dcnomiiiatoi of t tic systcni funrtioa: (2 1;nt ( s ) = s + 0.1 -0 pint(t) = ae-".Lf , t >0 (7.1114) Thc cc~riipletesoltit iorr is thc sum 01 both parts: If fillfills the ( ~ ~ ~ € e r t-'cpation ~ ~ r i ~ iand ~ l the initid conditions for CI = T h r Figures 7.7, 7.8 and 7.9 show Iht-' inteinal part, the exlmrlal part and thr cornplrtcx solut,iorr