Question

3. Consider a second order system with transfer fuction P(s) = 2-B2 with a = 4000 and ß = 25. Design a compensator assuming unity feedback for the gain and phase margins you apriori specify (try to achieve as high as possible). Compute the poles and zeros of the closed-loor system. Plot the Nyquist plot of your compensator and verify that the Nyquist criterion is satisfied. Plot the step response of the closed-loop system and specify maximum overshoot, peak time, settling time, the damping ratio and natural frequencies of the system. Afterwards You can use MATLAB to verify your results. Find another transfer function with the same magnitude but whose phase is less negative than the phase of P(s).

Answer 1

pole zero plot of closed loop TF:

Plt) tou YEN GCE) Given. hooas ren looe] p)= Sp00, com So0 &+13X1-25) AuYEN PLE)GCE PLER pt) GE) Desgning Cenvoluh a Componsador cwith unido fend bnck PlG) Let, Gls) PG) v00 G (3)= %3D Itee) Go00 %3D s+ 3375 4o00 6+ 58i)G-s8i) t Si 5-58i So, Poles ase phase mrosin =180+ Go00 jw+53)(Sw-5) LGBW) phese mas 3 in- NAN degree) =O HI Gain massin GCOW)/ =| - Lo0o Gainmassin So00 w=25.21 (JwISTi)(w-58i) -4D00 = wSt w= 3364 -5000 wz -636

matlab results:

Im 60 30 -30 Ře

Oscillation frequency:
f = 9.2460666609225[Hz]
Overshoot (in absolute value)
The 1st peak gpk = 2.37 (t =0.054[sec])
The 2nd peak gpk = 0.0008 (t =0.11[sec])
The 3rd peak gpk = 2.37 (t =0.16[sec])
Final value of the step response
g(infinite) = 1.1851851851852

Nyquist plot:

step response:

Im 1. 3600 -4800 -4200 -3600 -3000 -2400 -1800 -1200 -600 600 1200 1800 2400 3000 4200 4800 Re O 0.01HZ 1000HZ

impulse response:

f(t) 2 1. 0.1 0.2 0.3 0.4 0.5 t[s)

F(t) 80 60 40 20 -20 -40 -60 -80 0.2 0.3 0.1 0.4 0.5 tts]