Question

find

Consider the Transfer Function Shown Below: G(S) = (s +2) s(s + 3)(s + 5)2 a. Plot the magnitude and phase plots for each element of the above transfer function. (1 b. Plot the Bode magnitude and phase plots of the system in the given logarithmic paper. Use the plotted Bode plots to estimate the gain and phase margins of the system. (10 P d. Is the system stable or not? Explain why? (5 Pts) C.

Answer 1

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part b and c

as we have plotted manually the bode plot above

we can plot the same in matlab for accurate plots as below

Bode Diagram Gm = 51.1 dB (at 5.79 rad/s), Pm = 89.6 deg (at 0.0267 rad/s) man -50 Magnitude (dB) -100 -150 -90 -135 Phase (deg) -180 -225 -270 102 10-1 10° 101 102 Frequency (rad/s)

the gain margin becomes 51.1dB

phase margin is 89.6 deg

use the code below

g = tf([1 2],conv([1 5],[1 5]))
g2= tf([1],[1 3 0])
margin(g*g2);

part d

for stability the gain margin should be above 0 dB

As the gain margin is above 0 dB we can say that the system is stable for closed loop resonse

in summary

we have plotted the bode for each element

we have then plotted the bode for total function

we have used matlab for accurate plots

we have then found the phase and gain margin using the bode plots

we have finally decided the stability of the system