Solutions Manual to Accompany
Homework
For Linear Control
System Engineering
Liu Sijiu
HIT 651
Copyright 2006
P1-2 Redesign the turbine speed control system discussed in Sample
Problem 1.1, but replace the fly-ball governor with the tachometer
shown in Fig.1.2. A tachometer consists basically of a small DC
motor operated in reverse as a generator, the shaft being rotated
continuously, producing a DC voltage proportional to shaft speed.
Solution
2
P1-4.
Shown in Fig.P1.4 is a water-level control system comprising a tank, inlet pipe, slide valve, and float.
Details of the operation of the flow valve are
also shown in the figure. Draw a block diagram
of the feedback control system and identify the
main elements, writing down the mathematical
transfer functions where appropriate.
Solution
P1-5
A
method
of
producing
a
displacement
proportional to an input displacement but with a much
larger output force is shown in Fig.P1.5. The input
displacement x causes the movement of the spool valve
to produce a differential flow to the hydraulic actuator.
Draw a block diagram of the system, label the principal
parts of the control loop, and identify as many of the
transfer functions as possible. State any assumptions
made in the analysis.
Solution
3
P2-1
A system unknown transfer function is shown in Fig.P2.1. If a unit impulse applied at the input produces
at the output a signal described by the time function
c( t )
=
2e
−
3t
, determine the unknown transfer function.
Solution
c( t )
=
2e
−
3t
=>
C
2
=
R s
+
3
P2-2.
Find the solution of the differential equation
d
2
x dx
dz
+
+
8x
=
+
3z
2
dt
dt
dt
When
z( t )
=
e
−
2 t
and all other initial conditions are zero.
⎛
⎞
⎜
⎟
⎛
1
s
+
C
2
⎞
s
−
11
1
s
+
3
⎜
1
−
⎟⋅
1
⎟ ⋅
C
1
=
Solution
Z(s)
=
=⎜
⋅
⎟
⎜
s
+
2
−
2
31
⎟
10
1
⎜
s
+
2
s
+
2 s
2
+
s
+
8
⎝
s
+
s
+
8
⎠
(s
+
)
2
+ ⎟
⎜
4
⎠
2
⎝
y( t )
1
−
t
⎛
=
0.1e
−
2 t
−
0.1e
2
⎜
cos
⎜
⎝
31
23
31
t
−
sin
2
2
31
⎞
t
⎟
⎟
⎠
P2-3
For the system shown in Fig.P2.3., determine the relationship between voltage and current, express this
relationship in the form of a transfer function and determine the current as a function of time when the voltage
is a step change from zero to 10V.
Solution
1
Cs
,
=
R
+
sL
+
1 / Cs LCs
2
+
RCs
+
1
10
Cs
10
⋅
10
−
6
10
Y(s)
=
=
=
s LCs
2
+
RCs
+
1 10
−
6
s
2
+
10
−
3
s
+
1 s
2
+
10
3
s
+
10
6
y( t )
=
10
−
2
2
3
e
−
500 t
sin(
3
1000 t )
=
0.01154
⋅
e
−
500 t
sin(866 t )
2
P2-8
For the system shown in Fig2.8 determine the closed –loop
transfer function C/R.
Solution
⎛
G
G
1
C
=
⋅ ⎜
1
+
2
R 1
+
G
1
H
⎜
G
1
⎝
⎞
G
1
+
G
2
⎟=
⎟
1
+
G H
1
⎠
P2-9
For the single input system shown in Fig2.9, find the
transfer function of output to input C/R.
Solution
G
1
G
2
G
3
G
1
G
2
=>
1
+
G
2
G
3
H
2
1
+
G
2
G
3
H
2
+
G
1
G
2
H
1
G
1
G
2
G
3
C
=
R 1
+
G
2
G
3
H
2
+
G
1
G
2
H
1
+
G
1
G
2
G
3
H
3
4
P3.2
Determine the output of the open-loop system G(s)=a/(1+sT) to the input r(t)=t. Sketch
both input and output as function of time, and determine the steady-state error between the
input and output. Compare the result with that given by Fig.3.7.
a
a
=
Solution:
c(s)
=
R (s)
⋅
c( t )
=
a
⋅
( t
−
T
+
Te
−
t / T
)
(1
+
sT) s
2
⋅
(1
+
sT)
e
=
r
−
c
=
t
−
a ( t
+
T
−
Te
−
t / T
)
As
t
→ ∞
,
e
ss
=
t
−
at
−
aT
= ⎨
⎧
T for a
=
1
⎩∞
for a
≠
1
1
, where the time
1
+ τ
s
P3.5
An open-loop first-order system is characterized b the transfer function
G (S)
=
constant is
τ =
5s
. Calculate the steady-state error when the system input is r(t)=1+6t. Confirm the result by
using the final-value theorem.
Solution:
By the superposition theorem, the system output c(t) could be considered as a sum of a step
response and a ramp response. That is
e( t )
=
r
−
c
=
(6 t
+
1)
−
6
⋅
( t
−
5
+
e
−
t / 5
)
−
(1
−
e
−
t / 5
)
=
30
−
5e
−
t / 5 , yields
e(
∞
)
=
30
By the final value theorem for Laplace’s transform,
E (s)
=
R (s)
−
C(s)
=
R (s)
⋅
[
1
−
G (s)
]
=
(
1
⎛
1
⎞
(6
+
s )
⋅
5
+
)
⋅ ⎜
1
−
⎟=
s
2
s
⎝
5s
+
1
⎠
s
⋅
(5s
+
1)
(6
+
s)
⋅
5
Lim e( t )
=
Lim s
⋅
E(S)
=
Lim s
⋅
=
30
t
→∞
s
→
0
s
→∞
s
⋅
(5s
+
1)
6
P3.7
One definition of the bandwidth of a system is the frequency range over which the amplitude of the output
signal is greater than 70% of the input signal amplitude when a system is subjected to a harmonic input. Find a
relationship between the bandwidth and time constant of a first-order system. What is the phase angle at the
bandwidth frequency?
Solution:
1
=
1
+ τ
s
1
1
+ ω
2
τ
2
b
=
0.707
≈
2
or
1
+ ω
2
τ
2
=
2
b
2
i.e.
ω
b
=
1 /
τ ∠
G ( j
ω
)
= −
tan
−
1
(
ωτ
)
= −
tan
−
1
(1)
= −
45
°
, phase lag is
45
°
at the bandwidth frequency.
P3.8
Figure P3.8 shows the experimentally obtained voltage output of an unknown system subjected to a step
input of +10V. Determine the transfer function of the system and locate its pole on the complex plane.
Solution:
System appears to be first order. So suppose the transfer function is as follow:
G (s)
=
C
=
K
, then
c( t )
=
10K
⋅
(1
−
e
−
t /
τ
)
R 1
+ τ
s
From final value theory
10 K
=
10
⋅
K
=
2.5V
, hence K=0.25
s
τ
s
+
1
t
→∞
s
→
0
s
→
0
1
Now consider time constant
，
τ =
t
⋅
=1.4983
⎛
1
−
1 c ( t )
⎞
−
ln
⎜
⎟
⎝
10K
⎠
Lim c( t )
=
Lim s
⋅
C(s)
=
Lim s
⋅
Matlab:
K=0.25, c=[1.22, 1.84, 2.16, 2.33, 2.41], t=1:5, tao=t./(-log(ones(1,5)-C/10/K))*ones(5,1)/5
5
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