Question

H(s) where: 6619) = * G6-ss 6) = 32 – 0.3s +10. HG) =s+3 Determine: a. the range of K for closed-loop stability b. the value of K when one of the closed-loop poles is: $3.4 c. if a second-order approximation is valid for the value of K found in (b.)

Answer 1

Ge(s): k. Geurs) - Gps) 1 st5 s²-0.5stlo H (S): S+3 Ges) = Ge(s) Gucs) Gp (s) Close Loop Trasfer fxn= a(s) I+G(S) H15) (5+52 (52-0.55+10) t k (513) (5+5) (52_0.55+10) 30.552 tlos755-2.55 +50 + ks +3k (33+4.55 (7.5 +k)5 +3 6+ टापक्रम ख 4 , 5 = 0 463) = 53+4.55€ +(7.5+E) 5+ 3k+50 =0 Cseanned with Cam Ssanner

Routh Hurwitz critersia s² s 8 (7.5+1) 4.5 3k + 50 4.5(7.5+k) - Bk+50) 4.5 o so 3kt so 3kt so o 7.5 + k >o o 3k + 50 k > - 7.5 ko -50 3 k> -16,66 4.5 (7.5+k) > 3k +50 33.75 +4.5k > 3k + 50 1.5k > 16.25 k>10.833 k>-0.75 o k>- 16.66 oko 10. 833 10.833 close Loop Stability Range of k 10.833 < kc s scanned with conscanner

GUAR D Q- b o 915) = 53 +4.552 +(7.5+ks + 3k+50 = 0 © given one pole af Let other 2 Roots s=-4. are at =-2 $-ß. then 968) = (s+y) (Ste) (s+B) so (8+) ( 5+(+375 +2B) = 0 53+ (Q+B) s? taßs+452+ Ulat B) s + 4 & ß. Po S3 + (x+3+4) 52 + (u2+uß tabs + YKB co compare egn @ $ 7 2+ B + 4 = 4.5 42B = 3k + 50 (2+B = 0.5 ) Y(Q+B) +2ß= 7.5+k ho 2+2 B=7.5+ *B)= 5.5+ks cu 15.5+k) = 3k+50 22+uk = 3k + 50 F = 50 - 22 k= 287 'When one closed Loop poles at s=-4 the value of csecanect with form seadister R o n 3 - aina

is valid the if and order value of k approximation found in b Hy=$+3 GIS ) = -0,5 s +10 #4 st5 46) - - en -0.55 +10 -6.55410 5 st 1 s2-0.55+10 by and order approximation. a(s) = 5( 52-0.55 +10). ULTE = दड) ta' (S)HLS) 5(52-0.55710) It K(5+3). 5(52-0.55 +10) CLTR = - k 5s^– 9:5S + 5pts+31< 9(9) = 555-9,55 +ks + 3k+50 =0 . s? +(0.5+k)st 3k +10 =0 @ give -Poto -5=-प, Tet other pole will be IC scanned with SamScanner a

then 9(s) = (s+47(5+) = s²+(at2)st 42 so compare eq" @ $ 5) 2 + 4= 0.5 + k पर .k +10 a = -3.5 + k 5 _ _ _ cu 1-3.5+ k) = 3k +10 -14+ 4k Bk +10 5 k = 24 - ks 120H if and approximation X is valid them K= 120 when one pole at $3-4 1C Scanned with CamScanner