Question

Let the open loop transfer function be K/(s(s+1)(s+3)) with unity feedback and ‘K’ as variable determine what is the maximum value of K for which it is stable and what will happen to the stability if we add one real zero ‘z’ between -3 and -1 poles. What is the conclusion derived. Plot the rootlocus with and without zero in MATLAB.

Answer 1

0 Given open loop toranstes funcions K G(S) $(5+1)(5+3) 0, -1,-3 poles Namber of poly p=3 -3 Number of Zen & z=0 not a part of root lour ► 20, . p-z = 3 le 3 branches # Centroid: o-Ehal post of poles - Erooppart of (0-1-3) -0 -H 3 : -1033 Zergk 3 P. Z * Asymptater. ..((P-2)-1) 2 (aq +1) 8180 where q = 0,1,2 P一定 so here q=0,112 P-z = 3 q=0 0 - > 0, = 1(1) xiro = 60° 3 drawn 8, +180° : 1 (2+1) x180 9=1 A symptotal Centrond 3 from 9g = (5) xido = I 300 3 = 952 → 3 # Break away point K = k CE=→1+ S(5+1) (5+3) -> E-s (st)) (543) s(s+45+3) *= -65° +48 +38) : - ( (35°4 85+3). girler breakaway points. de - (35++ 8.5 43) S = -0.4514, S5 -2.215 de ds dk .: 364 X

Sis - 0.4514 is Valod bleak pointi Now drk a present on root loan and of = -2.215 is a present on not a part of root locue. so of not Valod break point. -(65+) -(61-0-4614) ++) - a break away point! 85-0-4514 dtv drk - - Ve so S= -0.4514 ů der + K Sesto (543) jw CE =15+48 + $5) +-0. www - 3 jun= √3 1 76 CE= s²+45 43 stk=0 -33. 2004514 -1 by RH Criterian I 3 - jw K 4 > If 12- K 4 > o then the system is stable s 12-K 4 12 - K>0 ☆ <12. -ky-12 Z on cyo O ks 12 o fol the reystem is stable System Stable. a add (5+2) foransfer function Sz-a ie Zehn to the 2) Now giren in part 6) G CM k (6+2) S (5+1) (51%) pols = 0,-1,-3 Zero -2 P-3 Z = 1 P-2=2+28 700

Es - (0+1+3) - (-2) -22 읓 tos 3 3-1 SA = (aq + 1x i so poz here njw. p-z=2 q=0,1 .. obot o q=0 -> Øn : (1) Xito q=1 = $(3) X180 ) -2 -3: I 270 2 58 ! break pointe K (3+2) s(541) (543) (5(5419 (143) K - J (512) ²4454 35 (512) 7 ㅔ du ds [(-[***] (-48). = (512) - 3 e-[58-754-5-9+5817,33 +587.38 + x ] egt salt, soitas & SF -0.5344 , S2=+2.232420-732 S2 = -2.232rjo.792

by RH Critesian: Ce-s}+48 35 +X542x = 0 +3+48 +(3+K) St 2kso 2 -3 تی 1 3+k 02 s' 4 2K 4(3+k)-2k 4 2K If 4 (3+*)-2x = 0 12+ 4K - 240 K = -6 Rost locey touchy the Imaginary . at korb adal. .: A-Es 45 tak so 457+2(-6)=0 48 12.50 S = 3 ) S=fv3 o Loccus does not touch the rap maginaly undtable by observing the the stability of System i improved. Sonce locus anie. # Stabilelgi k <-6 , system is fol root to cus on - ve real asidi than forft Case ... e local & Shifted ligging more left side to By adding of Zero Overall Stability of System och Domproved