Question

For each system, make an accurate plot of the root locus and find the following: a) The breakaway and break-in points. b) The range of K to keep the system stable. c) The value of K that yields a stable system with critically damped second-order poles. d) The value of K that yields a stable system with a pair of second-order poles that have a damping ratio of 0.707. cally dempe secondes de protest 2) G(s) - K(s+2)(s+1) (s? – 2s + 2)

Answer 1

Answer Given that Ges) = K[S+2) CS+1) (5²-25+2) Diagram R(s) + K(st 2) (St1) (5²-25+2) heb, H05)=1 poles, s=2,1 Zeroes, SE-3,-1 plot pole zero Imaginary We know that, Centroid, s-Epulos - Ezernes P-2 -4 -3 2- Roal -12-21 20 where, p = Number of poles 2. Number of zerors - As asymptopes so controid there are hon't no exist

@ Breakaway and Break-in points ② -> GCE) CH)S=0 - Step 2 - put Squation 6 and solue in (S-2) (5-1) -(5+2) CS+1) [ (3-2)Cs-1) 7° 65° 35+2) -(52+35+2) -0 s 3-3s + P-5²-35-4 =0 - GS=0 s=0 is not a valid break point because doesn't exist. it So (b) to find Range of K first, we need to find the character Equation So, 1+G(s)H(s) = 0 (S-2) (S-1) + K (5+2) (S+1)=0 (S²- 35 + 2) + (5² +35+ 2) = 0 S2 35+ 2 + 5²k + 35k + 2kzo ${(1+k) - 35(1-1) +2(1+) = 0 -

UN OLJ R-H Calculation 2(1tk) s² 11th s |-311+4) so/2C1tk) o o By solving the Above form, ItK>o 3(14 - 3(1-K) >0 2(1+K) >O K>-1, KKI Range oft for the system - 1<x<1 to be stable is So, © jo draustes fonction : C(s) G(s) R(s) 14GCSDHCS) K(S+2) Cs+1) 52(1+K) - 35(1-k) + 2 (144) (4) CS+2) CS+) 5²-3 (1-2 st2 (ltk/ cs) - wn 3²+2 & wost wn

2&wn = -3 (1-6)/Cita) Wnada => 252 = 3(k-1) CK&l) K= 34 So, the value of Kis 34 that can yields a stable System Given & 0.707) &- £ whore cdy 2 & in= 3(k) (441) 24+2 = 3k-3 kestra3 3k-2k - 2+3 [k=5] So, to value of k is 5 that leads a stable system with a pair of second-order poles that have a damping ratio of 0.707