Question

Question 4 a) A simplified form of the open loop transfer function of a chemical process as shown in Figure 4 is 1 G(S) = S(S + 3)(S2 + 25 + 2) where H(S) = 1. i. ii. Construct the root locus. (5 marks) From the root locus in (a), show the range of gain K at the root loci so that the system will stable. Discuss your answer. (2 marks) b) State two important parameters that you got from Bode plot method and explain how you got it theoretically. Next, state the differences between Bode plot and Root Locus method. (3 marks) [10 MARKS] U(s) Y($) KG(s) HS)

Answer 1

201: → 1 G18) ► -3 (a) 8(8+3) 682+28+2) open loop foles are $= 0, $ $3 →-ti & Sy=-1-8 No. of Asymptots are (P=2) Angle of asymptotes is given as (2k+1) 30 K = 0,1,2,3 (P-z) Oo 44 , 374 4 55/4 ) 70/4 Centroid of Asymtotes EP - Ez (P-z) > o ) 0-3-1-1 4 - -1.25 ) -1125 -3 goot loci cuts gw axis at a point gives as 1 +KG (8)H(8) => $(8 +3)( 82+28+2) +k -28 +382+68+63 tk =0 84+593 +882 +68 + 20 20 ${**+ 282+28 +382

Roeth Hurwitz criterion 84 1 k 3 SE 5 6 ४२ 6.8 K s 40.8-5k o 6.9 so K 40,8-sk 6.8 K = 40.8/5 38.16 w 6.8 6.8w? +8.16 = 0 W = tj. 19:16 W = +0.42 Hence root loci cult imaginary axes at w = 30.42 € W=3042 and the value of kat that points is 0.16 Breakin er freakaway point :- I +G4C8)H( 8) k 50 84458² - 8g²-68 dk -483-158²-168-6 ds dk ds 483 +15 82 +168 +6 = 0 si 8.28 s2 =-073-0 by = -0.73 +0.344 "X -X K 7 0.34€

Hence Summary :- centroid o = -1.25 breakawaylin {= -2.28, 83-0.13+6034, 68=-0.13-1034 ✓ break ih break away Asynteles coogle My Bly, 594 & uly Cut Magiñasy ase's W= + j0.42 & K= 8:16] Stability Ke Range: 1 According to Routh table shoon in figure forevious page:- K>0 40.8-5K 70 56 < 40.8 K<8.16 Hen for stalle systemos k < 8.16 البال Asymboles Kao j0:43 (k = 8.16 X 0.34 42.28 (K20) * -3 -1,25 70.34 -0.75V ko ejo,u2 10 (k-o) Ck=8.16) Stability region of k<8:16)

the frequency phase = -180° margin = 20 (b) Bodle troo importcent parametou are gain margint phase Margin. Gain Margin - It is calculated at phase crossover frequency at which se becomes -180°. Upc can be calculated as LGCjWpc) Glain log Ličajeral] Phase, margin: It is the phase of system at gain crossover frequency (Wge) the frequency at which gain becomes unity 1 GCjwgel phase-margin 100-d a 1809– Lo juege) Boole flots are used to gain insight on the frequency response of a transfer function Root locus plot is used to lenderstand how the foles of fayetem vary based on system farameters Contraller gain model farameters

Root Locus 6 0.81 0.7 0.56 0.38 0.2 4 0.89 0.95 0.988 7 6 5 3 2 Imaginary Axis (seconds-') 0.988 0.95 -4 0.89 0.81 0.7 0.56 0.38 0.2 -6 -8 -6 -4 0 2 4 6 Real Axis (seconds-1)