Question

Consider the unity feedback system is given below R(S) C(s) G() with transfer function: G(s) = K s(s + 1)(s + 2)(8 + 6) a) Find the value of the gain K, that will make the system stable. b) Find the value of the gain K, that will make the system marginally stable. c) Find the actual location of the closed-loop poles when the system is marginally stable.

Answer 1

+ K o Yo 9K <.224 R(8) Gls) Cos) G(8) $18+1)(8+2)(8+6) here characteristics Equation is given by, + 6(8)=0 k 1 + 8(5+1)(8+2)(8+6) 818+1)(8+2)(8+6) +ko 84+993 + 2082 +128 +k=0 Routh hurwitz Criteria, २० k 9 12 9x20 -12x) k 9 s' Sex 9k ki for system stable there is no sign changes in column al Routh hurwitz assay hence, k>0, Ş¢x 12 °96 56 84 T3 56 sz X12 56 so first K< 24.888888

hence for stable system OSK <24.888888 b zero column, for mozginally stable system there in first Sex 56 hence X 12 -9k Co 9k = 224 => k= 24.888888 Anower for k= 24.888888 Auscillary Equation is, Sa+k=0 o Soss 56 g²==k 24.88888 S= j 1.154700538 si +9 83 +20 82 +12 stk SG 82 tk 224 84 +983 + 2032 +128+ 9 56 82 + + 224 9 3 56 Il x9 8+1=0 t مل 3 56

fi 49 13 + 208² 4125 4224 XII ti S6 g 2 + 224 3 १ g. روم 2 of مابی) 9.83 + 56 3 JA 933 + 128 56 82 +224 3 9 56 82 + 224 9 56 36 32 + 36x9 8+1=0 FO g2_4856 82 + 9 $ + 56 3 fa 2 80-3.941694261 8 = -5,758305739 hence boles are ka 224 for 24-88888 9 $= -3.241694261 s= 53758305739 + 1.1547 s = - Î .1. 1547