Page 189 - MATLAB an introduction with applications

174 ——— MATLAB: An Introduction with Applications

Example E3.22: The open-loop transfer function of a unity-feedback control system is given by

K
G(s) =
2
( ss ++ 5)
s
(a) determine the value of gain K such that the phase margin is 50º
(b) find the gain margin for the gain K obtained in (a).

Solution:
K
G(s) =
2
( ss ++ 5)
s
The undamped natural frequency is 5 rad/s and the damping ratio of 0.1. 5 from the denominator.
Let the frequency corresponding to the angle of –130º (Phase Margin of 50°) be ω and therefore
1
∠G(jω ) = –130º
1
The Bode diagram is shown in Fig. E3.22 from MATLAB program.
Bode Diagram
20

(dB) 0
Magnitude –40
–20

–60
–90

(deg) –135
–180
Phase –225

–270
10 –1 10 0 10 1
Frequency (rad/sec)
Fig. E3.22
From Fig. E3.22, the required phase margin of 50º and occurs at the frequency ω =1.06 rad/s. The magnitude
of G( jω) at this frequency is then –7 dB. The gain K must then satisfy

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