Question

K(s+1) 2. (20 %) Let G(S)=52's +9) " in Figure 2. Plot the root locus. (Run a numerical simulation and verify your solution). C(s) G(s) Figure 2:

Answer 1

Desktop Window t o What's New Help Current Folder. View MATLAB 7.10.0 (R2010) File Edit Debug Parallel :D Ph Shortcuts How to Add >> num=[1 1]. >> den=[1 9 0 0]: >> x=tf (num, den) Insert File Edit UaQ Tools Desktop Window Help 2 .1@1000 Root Locus Transfer function: s + 1 -------- 93 + 9 942 >> rlocus (x) fx >> Imaginarazis Real Axis LUMMAMM WU12 WW WWW Www

k(st) Set - - kls+1) žista) Number of Pdee & zerols & location no of poles ; n=3 they are Pz=-9 no. of Beroes; m=1 they are Z,=-1 Pico P20 Pis s-plane pole ti -zero. Reet locus on Real Qu'sGas acts A place a test stint on real anis in between s=0 and 5=-1. from the right side of the test point, number of peles & bestel ale two which is even. t. a restlocus Branch doesnt exists on realaxis in between s ound s=-2. * place a test point on real arts in between s=-9 and 5=-2. from the right of the test point, number of peles & Zerols are Three, which is odd. in a rootlos branch exists on red One's in between s=-2 ands=-a. * Place a test point on real axis in between s=-9 and 5=-00. from the right of the test point, number of poles & Beroes are four, which is even. .. a nest locus branch does not enests on real ans in between s=-9 and sa-ao- 9 fů liwan soplane . L -Loc ics=-9 in" STATIITTTTTT aq 4 6 8 -IV -10 -8 -6 -4 T -21

Centroid and Angle of anyuaptades, Centroid; c= Sum of Polos - Jumefrees 0-90(-1) - --4 Mugle of Aryumpledes, A-A = +180129+1) nam tsoloth too 2=0,1,2 --n-m - = 3--> ..9=0, 1,2 2=0;A-A: Liso (67) fet q-'; A.As 1190(2+'+9700 108 2:2; A.As I 180(2+), tuso-1969 ri 3- 1524503 3- T- - ISO augle u ^^ Il grow fü of Asyustedes 4 6 -159-5 109-5 -6 A-* -6 /T-4 -219 2 centruid , tas at s=-4 Breakin and Breakaway points, . The characteristic equation is 't KACS=0 - it K(s+) * = -(05) - 6 = dk r 1. Cstor (St. XC3f7188) tsa37o - 353 185 +35+185-5295=0 - 253 + 125 +185=0 - Sco and s'+68+9-0 ® =) s=-3,-3- positive reed : at 320; place in ko noso at 52-3, Plac 1-2748 1 *=-(-3²496-820 L -3+1 = 275 positive real .

... are have a break away point ats=0 C and beat point at 5=-1 to angle of amical and Angle of departure, fine alo Complex pole and complex zero in the trausfertinction Rost locus cooking posut on Imaginaly anist from equation $]+9$+k(SH) = 0 =) 3+ qə +ks+k=0 put sajo r. =) (Sw33t 9 (ſwit kjw tk=otjo -) -303-910? +kibotk Fotjo -> (6-90)+jckw-43)=o+jo imaginate Imaginaly part so two-w?co 0 Real part=0 K-90=0 =) E= 90 =) k =W² =) t =96 - 8 and w orad feca ... crossing point on imaginaly anish s=Ijw Nocroting pinton is scoothnest=1s=10] u a'. AloCroth'

And Rootlocus. vapupunts to oglaw PS Su- breakaway ats=o. Ash 4 2 "J 3 4 5 6 -6-5-- 6 cerid ots--4