Solution: 7.9.9.3 R + C G c G p Σ - Referring to Figure 1 wecanwrite XX XO Re(s) Im(s) -1-2?3?30 s + 30 s s + 1 s + 2 Figure 1: Accurate root locus K c = jsjjs +1jjs +30j 10js +3:26j : Then all wehavetodoispick some points along the negativereal axis between s = ;30 and s = ;3:26. Table 1 summarizes the search. Thus s -29 -28 -27 -26 -25 K 3.15 6.11 8.87 11.43 13.8 Table 1: SearchforK c =13:8 the third pole is close to s = ;25. The following MATLAB dialogue ver es these computations. EDU>gcgp = zpk([-3.26],[0 -1 -30],138) Zero/pole/gain: 138 (s+3.26) -------------- s(s+1) (s+30) EDU>tc = feedback(gcgp,1) Zero/pole/gain: 138 (s+3.26) ---------------------- (s+25) (s^2 + 6s + 18) EDU> Then the unit step response is C(s)= 1 s + B s +25 + M s +3;j3 + M  s +3+j3 B = (s +25)C(s) j s=;25 = 138(s +3:26) s(s +3;j3)(s+3+j3) j s=;25 = 0:2434 M = (s +3+j3)C(s) j s=;3+j3 = 138(s+3:26) s(s +25)(s+3+j3) j s=;3+j3 = 0:7352 6 ;2:5783 Thus c(t)=[1+0:2434e ;25t +1:4706e ;3t cos(3t; 2:5783)]1(t): The unit step response is shown in Figure 2 Time (sec.) A mp lit u d e Step Response 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 2: Unit step response