Question

Question 8: Consider the closed-loop magnetic levitation system shown in Figure below. With Ko = 1, determine the conditions on the system parameters (a, K, Z, p) to guarantee closed-loop system stability. Question 9: Considering following transfer function, find the range of K for which system will be stable K(s +2) T(S) 54 +353-352 + (K + 3) + (2K-4)

Answer 1

G(S) = K (S+z). H18)=1 (8+p) (8292 IND SINDS 14 for Stability find Routz Hoodz coiteria- char. @gan I+G68) HCB) = 0 => 1+ Kc8+2) . (8+1)(89² = 8² ps²-sa²-Pa²+ k&tke=0 -> 83 +232 +BOK a?) + kz-la=0 R-H.coiteria- 83 11 (K-a i 82 IP kz-pa² 81 (k-a?) - (K2-P03) o so kz-pa² for stability - column of RH shoud be greater than jezo - P²O • k-a² - kz-pa2 >o k-a²> K2-pa² 2 KP-Pa²>kz-pa² 2 kp > kz = [PZ K2-pa² >o = K2>Pa² - K>Pq² Р

TCS) = K(S+2) 84 +383_38?+ (K43)8+(264) char.eq"84+383–382+CK+3) H2K-4)

R.H criteria- I - unft o K+3 3 o - CK+3) 2K-4 o Kt3 & 3(2K-4) 3+k+3 so | 2K-4 I. k+3+ 9 (2K-4) >O 12+k • 2K-4 > 0 K> 27 e 3 -3 - (k+3) >0 - 3+k+3 co * 12 tk co 12K+K2+36 +3k +18K-36 > & K²+23KO KCK+23) >0 K>0 ; K>-23 » K>O 2 o i there is no suck K72 K>2 I i k value exist so K7-23 that system will be Stable -12 7 K & K<-12 at means systeem is unstable
in this problem, we varify route Hertz criteria to check stability.