Question

3. Design a PI or PD controller for the system G(8) = s(s+10) to meet the following specifications • Zero steady state error for unit step reference input • tr< 0.12 - . %OS < 10%. (a) Determine the low frequency gain, crossover frequency and phase margin necessary to meet the specifications. (b) Decide if C(s) needs an integrator. Plot the Bode plot of either G(s) or G(s)/s, depending on your choice. (c) Use sisotool (or iteration) to choose a gain and place a zero to meet the specifications. (d) Write the transfer function of the resulting controller C(s), where C(s) includes the integrator, if used. State whether this is a PI or PD controller. (e) Plot the bode plot of the loop gain C(8)G(s), the Bode plot of the closed loop transfer function, and the closed loop step response.

Answer 1

SOLUTION:::::
Solution :- 1. Given Data, Step: li- q (s) = 5(+10) Here, meet following Specification Eero Steady slate error for unit step input R(S) = 's for unit step unit Now. Ossi S.RCS) - s-70 HG(5) R(S) Css = t 8.116 * Fer 570 it's(s+10) c. . t stOS . Sto 22unst par S-0 = - Now Put 5-0 => [(59 = 0) Ess is zero for unit step reference ile so, G(s) = 25 4105 - as Second order Equation $+2Ęwnstws one Compare with

Step 2 :- TT-0 Rise time tr = wn Rise time tr = TO (n=1 2Ęwn = yor tr. = 8x1=5 : tr&=5 tre 180-P 1-JT-(5)? 180-240 tr = 51-25 tr&o. 10 sec - я е - 9.687.<10% G(S) = scstio) | G632 - WW2716

. G(s) =-909_tanywe) for phase cross over frequency |G(s) I ope ope w/ pc to -1800 - - wpc Sw?pc +10 -180+90 - - - Pan" Conto) -900.- tant coo) Wpc=10 I phase cross over. - frequency = 10 (a) now, we have to find gain cross over frequency 1G(s) |wgc = wge I wagct 10 I s uge Jugé tlo wge = (uge +10) = 1 wge? t lowage -1 = 10

Wgc, = 0.9 rad (sec Wg C2 = -10.09 rad/sec L 10 / phase margin = 180 +0 at Wge = 0.09 q=-909-Tari" (0:09) Q = -90-0.51 Q : -90.51 phase margin = 186 - 90.51 I phase margin = 84.490 at Wgc =-10.09 a = -90°- Pano ( $ = -1350 [phase margin = 180-1350 -1350 -u50 For 6. ©and @ :- c(s) needs as integrator G66) - Scato)

Initial Slopes - 20dB x1 dec final slope - - 20db rool der x (P-2) = -20dB . dec (2-0) .=-40dBldec . Magnitede - 1. WJ w2110 Gain cross over frequency :0.09 and 10.09 phase cross Over free . 100 HBO = ao 20 +100 K: -20 log (20) »k= -26 dB This is the gain

rad/sec Gain dB Initial slope -20 dB/sec final slope -40dB/sec @! Resulting controller CCs), for PI+PD controller I+ TDS wit 7 (s(s+2&twn) (63) This is the only GC5) = 565+10) which is equal to = win /s(s+2&wn) it is pp controller