t05 control system

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Curtin University of Technology
Department of Electrical and Computer Engineering
CS301/CS603
Tutorial 5
1. If v(t) is a step voltage in the RLC circuit shown in Figure 1, find the value R of the resistor
such that a 20% overshoot in voltage will be seen across the capacitor if C = 10−6 F and
L = 1H.
−
v(t)
+
R L
C
i(t)
+
−
y(t)
Figure 1. RLC circuit.
Answer:
The required damping ratio ζ is given by
ζ =
√
(lnMp)2
pi2 + (lnMp)2
= 0.456
As the transfer function for the circuit is
Y (s)
V (s)
=
1/(LC)
s2 + (R/L)s+ 1/(LC)
there hold
ω2n = 1/(LC) = 10
6 and 2ζωn = R/L
leading to
R = 912Ω
2. Consider the satellite attitude control system shown in Figure 2(a). The output of this system
exhibits continued oscillations and is not desirable. This system can be stabilized by use of
tachometer feedback, as shown in Figure 2(b). If K/J = 4, what value of Kh will yield the
damping ratio to be 0.6?
Σ
+
−
K 1
Js2
R Y
Figure 2(a). Unstable satellite attitude control system.
Σ
+
−
Σ
+
−
K
Js
Kh
1
s
R Y
Figure 2(b). Stabilized system.
Answer:
From the closed-loop transfer function
Y (s)
R(s)
=
K
Js2 +KhKs+K
=
4
s2 + 4Khs+ 4
it is seen that
4Kh = 2ζωn and ω
2
n = 4
Substituting 0.6 for ζ into the above gives
Kh = 0.6
3. Consider a unity-feedback system with the open-loop transfer function
G(s) =
1
s(s+ 1)
Obtain the steady-state output of the system when it is subjected to the following input:
r(t) = sin(t+ 30◦)
Answer:
The closed-loop transfer function of the system is
H(s) =
G(s)
1 +G(s)
=
1
s2 + s+ 1
With the input frequency ωo = 1, one has
|H(jωo)| = 1√
(1− ω2o)2 + ω2o
= 1
∠H(jωo) = − tan−1 ωo
1− ω2o
= −90◦
Therefore, the steady-state response to the input r(t) is given by
y(t) = |H(jωo)| sin(ωot+ 30◦ + ∠H(jωo))
= sin(t− 60◦)