Solution 3.8.8.5 a. The unit step reponse is Y (s) = 16 s(s + 2)(s+4) = A s + B s +2 + C s +4 : Then A = s16 s(s +2)(s +4) j s=0 = 16 (s + 2)(s +4) j s=0 = 2 B = (s + 2)16 s(s +2)(s +4) j s=;2 = 16 s(s +4) j s=;2 = 16 ;2(;2+4) = ;4 C = (s + 4)16 s(s +2)(s +4) j s=;4 = 16 s(s +2) j s=;4 = 2: Thus, the unit step response is y(t)=[2; 4e ;2t +2e ;4t ]1(t): The MATLAB program: t=(linspace(0,5,200))';; ss = 2*ones(200,1);; y2 = 2*exp(-4*t);; 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 1: y(t),y 1 (t), and y 2 (t) y1 = 2 -4*exp(-2*t);; y=y1+y2;; plot(t,y,'k-',t,y1,'k:',t,y2,'k-.') print -deps sm3885a.eps draws the plot shown in Figure 1. b. The unit step reponse is Y (s) = 24 s(s + 2)(s+6) = A s + B s +2 + C s +6 : Then A = s24 s(s +2)(s +6) j s=0 = 24 (s + 2)(s +6) j s=0 = 2 2 B = (s + 2)24 s(s +2)(s +6) j s=;2 = 24 s(s +4) j s=;2 = 24 ;2(;2+6) = ;3 C = (s + 6)24 s(s +2)(s +6) j s=;6 = 24 s(s +2) j s=;6 = 1: Thus, the unit step response is y(t)=[2; 3e ;2t + e ;6t ]1(t): The MATLAB program: t=(linspace(0,5,200))';; ss = 2*ones(200,1);; y2 = exp(-6*t);; y1 = 2 -3*exp(-2*t);; y=y1+y2;; plot(t,y,'k-',t,y1,'k:',t,y2,'k-.') print -deps sm3885b.eps draws the plot shown in Figure 2. c. The unit step reponse is Y (s) = 32 s(s + 2)(s+8) = A s + B s +2 + C s +8 : Then A = s32 s(s +2)(s +8) j s=0 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 -0.5 0 0.5 1 1.5 2 Figure 2: y(t),y 1 (t), and y 2 (t) = 32 (s + 2)(s +8) j s=0 = 2 B = (s + 2)32 s(s +2)(s +8) j s=;2 = 32 s(s +8) j s=;2 = 32 ;2(;2+8) = ; 8 3 C = (s + 8)32 s(s +2)(s +8) j s=;8 = 32 s(s +2) j s=;8 = 2 3 : 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 -0.5 0 0.5 1 1.5 2 Figure 3: y(t),y 1 (t), and y 2 (t) Thus, the unit step response is y(t)=[2; (8=3)e ;2t +(2=3)e ;8t ]1(t): The MATLAB program: t=(linspace(0,5,200))';; ss = 2*ones(200,1);; y2 = (2/3)* exp(-6*t);; y1 = 2 -(8/3)*exp(-2*t);; y=y1+y2;; plot(t,y,'k-',t,y1,'k:',t,y2,'k-.') draws the plot shown in Figure 3. We note: 1. For p 2 =p 1 as small as 2, wecanstill nd p 1 reliably. 2. If p 2 is muchbigger than four than it will be dicult to nd p 2 reliably. 5