Question

For each of the following feedback systems a. Sketch the Root Locus b. Indicate if there are break-in and/or break-away points, and how many c. Indicate if there are asymptotes and how many d. Use hand calculations to compute the break-in/break-away points e. Use hand calculations to compute the asymptotes SYSTEMI 1 G(s) = (s2 + 5s + 4)(s2 + 5s +6) H(s) = (s + 0.5) SYSTEM II G(s) = (s2 – 3s + 2) (52 +3s + 2) H(s) = 1

Answer 1

Root Locus is plotted an dmarked with apprpriate information and results are:

System 1 has poles at -1 , -2 , -3 ,-4 but no zeroes therefore number of asymptotes=no of poles-no of zeroes=4 as shown in figure. Two break way points at -1.38 and -3.61

System 1 has poles at -1 and -2 , zeroes at 1 and 2 , number of asymptotes=0, Break away point at -1.41414.
Firstly let us understand the procclure to plot the root loci cliagram / root locus CRLD) RLD is a plot of loci of roots of charelorother equation while gain he is varied from oho KG HCS) - Closed loop tranfere puelon is kG(s) ItG(S)H(s) - Charecherches equelon Open loop traufur huilem = Ges) H) Devery branch of RLD starts at a pole and ends at zero No of brancher of RLD s NSP of PZ2 z of zzP where P = No of poles IP-21 = No of aynplotes zs No of zeroes 2) centroid is the intersection of asymphones and eilways lies on real canis o. (Sum of real part of poles -sum of real part of zeres of a csucs) P-Z Asymptotes guide root loci to zeroes which be at infimby

P-2 3 Angle of anyuktoles (Oi) e-Ql+d) 1 = 0, 1, 2, ... P-2) - (6422 for syrken I 0 15:30 1.514:*** for syelem I, no no lesymfates ? Of is not presene 4) On real anis to right side of any selon if sum of total no. of open loop poles 4 zeroes in oeld, RLD ereits in that stutto, secton Syhen I Rool inay? Sugehen un - 4 3 - 2 -2 -1 While Dank lines have RLD in that patts While Pack lines haree RLO in that pater 5) Break in 1 Breate awary pouts Of there are multiple roots on RLD, there outs break in 1 bway pont At a valid break pont, aryle of alus is odo multiple of ± 188

Syllen I dk q ds (6675 ds +553437325:46) 3 - 4.5 - 30s -70s - 500 1 S= -3.61,-1.38, -2.5 . There are 2 breakaway ponto -3.61 & 21.38 (Both le on RO) Routann Inan a = - 438 b=-3-67 e= -3.33 092 OS! 014 deal -4 013 tires -Asynphle Path anis Noof aryaptton = 4 Two break away & tradu ponts -1.38 € -3.61 Bryle of 0,, - TV asynhtres 4 4