Question 6 Consider a closed-loop control with the following loop transfer function: 1 L(s) = Ge(s)G(s) = s(4s + 1)(6s + 1) (a) State L(jw), and then determine the phase crossover frequency w₁, i.e., angular frequency at which the phase angle of L(jw) = -180°. Assume that w > 0. (b) Calculate the gain margin of the system, and comment on the system stability. (c) Sketch the bode magnitude plot of L(jw) (i.e., |L(jw)|dB vs log w). Clearly indicate the corner frequencies, the asymptotic lines and their gradients, and the approximated curve. (d) If it is later found that system L(s) has a new transfer function of (6s+1), with the corresponding Bode plot shown in Fig. Q6. Comment, based on Bode analysis, the stability of the closed-loop system. Bode Diagram 40 20 0 -20 -40 -60 -90 Magnitude (dB) Phase (deg) -135 -180 10-² 10-1 Frequency (rad/s) Fig. Q6 10⁰ 10¹

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Question 6
Consider a closed-loop control with the following loop transfer function:
1
L(s) = Ge(s)G(s) =
s(4s + 1)(6s + 1)
(a) State L(jw), and then determine the phase crossover frequency @₁, i.e., angular frequency at
which the phase angle of L(jw) = -180°. Assume that w > 0.
(b) Calculate the gain margin of the system, and comment on the system stability.
(c) Sketch the bode magnitude plot of L(jw) (i.e., |L(jw)|dB vs log w). Clearly indicate the corner
frequencies, the asymptotic lines and their gradients, and the approximated curve.
with the corresponding
(d) If it is later found that system L(s) has a new transfer function of
s(6s+1)'
Bode plot shown in Fig. Q6. Comment, based on Bode analysis, the stability of the closed-loop
system.
Bode Diagram
40
20
0
Magnitude (dB)
Phase (deg)
-20
-40
-60
-90
-135
-180
10-²
10-1
Frequency (rad/s)
Fig. Q6
10⁰
10¹
Transcribed Image Text:Question 6 Consider a closed-loop control with the following loop transfer function: 1 L(s) = Ge(s)G(s) = s(4s + 1)(6s + 1) (a) State L(jw), and then determine the phase crossover frequency @₁, i.e., angular frequency at which the phase angle of L(jw) = -180°. Assume that w > 0. (b) Calculate the gain margin of the system, and comment on the system stability. (c) Sketch the bode magnitude plot of L(jw) (i.e., |L(jw)|dB vs log w). Clearly indicate the corner frequencies, the asymptotic lines and their gradients, and the approximated curve. with the corresponding (d) If it is later found that system L(s) has a new transfer function of s(6s+1)' Bode plot shown in Fig. Q6. Comment, based on Bode analysis, the stability of the closed-loop system. Bode Diagram 40 20 0 Magnitude (dB) Phase (deg) -20 -40 -60 -90 -135 -180 10-² 10-1 Frequency (rad/s) Fig. Q6 10⁰ 10¹
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