Question

Please sketch the Root Loci of the system below and show intermediate steps. Thanks!

Problem 2. [5 points] Utilizing the Routh's stability criterion, determine the range of K for stability for the given characteristic equation s+2s3 (4+K)s2 +9s25 0, and verify the analysis by selecting K values for stable and unstable regions, respectively, and by observing time responses with Simulink simulations. Note that the associated open-loop transfer function can be derived such that s +2s3 +4s+925+Ks2-0+K G() 0 where G(5 And the associated block diagram is R(s) C(s) G(s)

Answer 1

AN5 open loop hansfei funchon -S CG(5) H(5) s+35 454 45+2 No of pole(P) NO.of bianches P (!) angle 다 a3ymptoden. (2 )T D-t (29 () Centoid: Poles冫 0.8326± , . ОЗ 35 i 小8230+ ) .3667; 7 р-т 1-0 .991

y 25 6 0.50 o z k-I 0 25 (250 o 6.05 6 x605625 D 足 25-o s" ( 5,56)

(5) Angle of departume for pole 0.8326 2.0335 othe pole 0.8326+2.033si 1323)6.2 for the pole- 0 5326+20335- (-16320-1-30691) Pd> 150+ 2x67-78-18-5).9jjo )2455%. -165.9))

Ro lous 2-120 (024, 1035) O3) 2 -).ho<ㅡㅡㅡㅡ-X s ㄧ Co.t324,-Qo3bO

Root Locus 15 10 5 아 10 15 -0.5 Real Axis (seconds) -2 -1 0 0.5