Question

2) For the system below, find the phase margin if the system is stable for time delays of 0.1 and 0.2 second. Please use Matlab to plot the frequency response of G(s) (10 pts) Delay 50 (s +2s +5) G(s)

Answer 1

Sol: for delay = 0.1 H (S) -= HCS) - (5+2) (S+5)+5 Reo (CH3 CSHG] 5 H(5) $ +15+15 bode plot for above transfer function is

Bode Diagram Gm = Inf dB (at Inf rad/sec), Pm = Inf Magnitude (dB) Phase (deg) T- 10 100 101 102 Frequency (rad/sec)

In The above plot maginitude plot is not touches to o and phase is not touches the - 180 degrees axis so gain margin and phase margins are infinate, for delay = 0.2 HCS)= oll come ) 1401( (S123675) ſ 50 50 10 H (S) = (5+2) (5+5)+10 HCS) = _10 $+75 + 20.

Bode Diagram Gm = Inf dB (at Inf rad/sec), Pm = Inf Magnitude (dB) Phase (deg) - 180 101 100 702 10 10 Frequency (rad/sec)

in the above plot maginitude plot is not touches to 0 and phase is not touches the -180 degrees axis so gain margin and phase margins are infinte

MATLAB code for G(s) is

>> n=[0 50];
>> d=[1 7 10];
>> z=tf(n,d)

Transfer function:
      50
--------------
s^2 + 7 s + 10

>> bode(z)
>>

Bode Diagram Gm = Inf dB (at Inf rad/sec), Pm = 57.9 deg (at 6.05 rad/sec) Magnitude (dB) Phase (deg) 18OIUIIIIIIIIIIIIIIIIIIIIIIIIII IIII IIIIIIII IIIIIII ITTn . . 102 101 10" 102 103 Frequency (rad/sec)