MECH 6323 HW 2

MECH 6323 HW 2 
yes
Due Thursday 02/14/2022 (at the beginning of the class)
1. Consider the negative feedback interconnection:
+ C(s) P(s)
r e u y
−
(a) If possible, give an example of P and C transfer functions such that 1
1 + P C and P
1 + P C
are stable, but C
1 + P C is not.
(b) If possible, give an example of P and C transfer functions such that C
1 + P C and P
1 + P C
are stable, but 1
1 + P C is not.
(c) If possible, give an example of P and C transfer functions such that 1
1 + P C and C
1 + P C
are stable, but P
1 + P C is not.
2. For the following transfer functions, obtain the asymptotes of the Bode plots and compare them
with the Bode plots generated in MATLAB. Include the results from MATLAB.
(a) 100s + 100
s
2 + 110s + 1000
(b) 10s
s
2 + 3s
(c) −100s
(s + 1)2(s + 10)
(d) 30 
s + 10
s
2 + 3s + 50
(e) 4

s
2 + s + 25
s
3 + 100s
2

(f) 10
s
2(1 + 0.2s)(1 + 0.5s)
3. For the following two Bode plots,
(a) Determine the breakpoints and the transfer function.
(b) Determine the gain cross-over frequency ωc and the phase cross-over frequency ω180.
1
Bode plot 1:
Bode plot 2:
2
4. Consider the feedback interconnection of Problem 1 with the PI controller C(s) = 10(s + 3)
s
and
plant P(s) = −0.5(s
2 − 2000)
(s − 3)(s
2 + 50s + 1000).
(a) Is the feedback system stable? Why?
(b) Use the Bode plot of the open loop transfer function L(s) = P(s)C(s) to find the phase
cross-over frequencies ω0 such that ∠L(jω0) = ±180°
. Use this information to compute the
gain margin(s) of the feedback system. Check your answers using the allmargin command
in MATLAB.
(c) For each gain margin g0 obtained in the previous part, construct the closed-loop using the
perturbed loop transfer function g0L(s) and verify that the closed-loop has poles at ±jω0.
(d) Compute kS − Tk∞ and the corresponding frequency ωp where the peak gain of S − T is
achieved.
(e) What is the symmetric disk margin m for this plant and controller? Verify your answer
using dm = diskmargin(P∗C). Note that the diskmargin command uses the convention
m = dm.DiskMargin/2.
(f) Construct an α on the boundary of Disk 1 − m
1 + m
,
1 + m
1 − m


such that the perturbed closedloop Sα :=
1
1 + αL(s)
has a pole at jωp. Verify your construction by forming Sα and
demonstrating that it has a pole at jωp.
Hint: Assume kS − Tk∞ = 1/m at frequency ωp. Then there exists a complex number S(jωp) − T(jωp) = 1/z where |z| = m. Algebraically show that α =
1 + z
1 − z
satisfies
1 + αL(jωp) = 0 and this α is in the symmetric disk defined by m.