Question

7. Consider a unity feedback control system with open-loop transfer function G(s) = k 5 s + 2)(52 + 4s + 5) Find the value of gain K > 0 for which the root locus crosses the imaginary axis.

Answer 1

7. Consider with a unity ler a unity feedback Control system open loop transfer function GIS); is (S) ) s(s+2) (5445+5) find the value of gain k root locus crosses the o for which the imaginary anis. To find value of k 20 for which rootlocus crosses the imaginary anis, first we have to find the value of 'k for which root locus intersect's imaginary aniso This can be fooled by using Routh Hurwitz criteria. Given open loop tf = n(s)= KISH) s(5+2) (5245+5) feedback H(S)=1 (assuming negative feedback) Characteristic equation CE- Ithis) H())=0 Elt kishi). - X S18+z)g?rusts 20 -) Ce: 5446+135% (10#k)S+K=0

54765341352+10 +1 )s kso RH table al s 16 13 k lorko SP 178–(18+) K O y not possible because all element's in a row cannot be zero ata time . s! |68-X) (1044)_GK o y 68-K y possible case Sick for entire row to be zero y not possible because to is initial - Starting for root locus to intersect imaginary anis at least one row of RH table should be - point entirely zero. (all element's of a row) 168-) (107k) – OK -) (6**) (10 4k) –ck=0 (63,) (10+ k) =OK 168-K) (104) – 30k > 228-680 30

on solving Kz 11+3.189, 11-3889 - 39030194 -17-30194 Remo for noot locus ockcd so k must be positive So for K= 39, 30194 - the root locus intersect's & Imaginary anis for k> 39-30194 the root locus crosses imagimary arcis.