ME 132 Fall 2009
Solutions to Homework 4
1. (
§
3.3, Problem 13)
(a) The steady state gains are
γ
u
→
y
=
G
=
−
cb
2
a
γ
d
→
y
=
H
=
−
cb
1
a
(b) By plugging in the controller and output equations, we see that
˙
x
=
ax
+
b
1
d
+
b
2
[(
K
1
+
K
2
)
r
−
K
2
y
−
K
2
n
]
= (
a
−
b
2
K
2
c
)
x
+
b
2
(
K
1
+
K
2
)
r
+
b
1
d
−
b
2
K
2
n
so the constants for the closed loop dynamics are
A
=
a
−
b
2
K
2
c
B
1
=
b
2
(
K
1
+
K
2
)
B
2
=
b
1
B
3
=
−
b
2
K
2
For the output equations, the constants can be found by inspection:
C
1
=
c
D
11
=
D
12
=
D
13
= 0
C
2
=
−
cK
2
D
21
=
K
1
+
K
2
D
22
= 0
D
23
=
−
K
2
(c) The system is stable if and only if
A <
0, or
a
−
b
2
K
2
c <
0
The closedloop steady state gain from
r
→
y
is
γ
r
→
y
=
−
C
1
B
1
A
+
D
11
=
cb
2
(
K
1
+
K
2
)
b
2
K
2
c
−
a
The closedloop steady state gain from
d
→
y
is
γ
d
→
y
=
−
C
1
B
2
A
+
D
12
=
cb
1
b
2
K
2
c
−
a
1
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(d)
i. All of the closedloop steady state gains are given below:
γ
r
→
y
=
−
C
1
B
1
A
+
D
11
=
−
cb
2
K
1
a
=
−
cb
2
1
G
a
= 1
γ
d
→
y
=
−
C
1
B
2
A
+
D
12
=
−
cb
1
a
γ
n
→
y
=
−
C
1
B
3
A
+
D
13
=
−
cb
2
K
2
a
= 0
γ
r
→
u
=
−
C
2
B
1
A
+
D
21
=
−
cK
2
b
2
(
K
1
+
K
2
)
a
+
K
1
+
K
2
=
1
G
γ
d
→
u
=
−
C
2
B
2
A
+
D
22
=
−
cK
2
b
1
a
= 0
γ
n
→
u
=
−
C
2
B
3
A
+
D
23
=
cb
2
K
2
2
a
−
K
2
= 0
ii. The steady state gain from
r
→
y
is 1. This means that, in the absence of
disturbances and noise, the controller achieves perfect tracking of the reference
signal in the steady state.
iii. Treating
K
1
as a fixed number and recalling the definition of sensitivity,
S
γ
r
→
y
b
2
:=
∂γ
r
→
y
∂b
2
b
2
γ
r
→
y
=
∂
∂b
2
parenleftbigg
−
cb
2
K
1
a
parenrightbigg
b
2
γ
r
→
y
=
−
cK
1
b
2
a
−
a
cK
1
b
2
= 1
Sensitivity ranges from 0 (least sensitive) to 1 (most sensitive). This result
implies that the “feedforward” controller’s performance in tracking
r
is ex
tremely sensitive to parameter uncertainty in the plant. This is easily seen
from letting
˜
K
1
=

a
c
˜
b
2
for
˜
b
2
negationslash
=
b
2
, which causes
γ
r
→
y
negationslash
= 1.
iv. The openloop steady state gain from
d
→
y
is the same as the closedloop
steady state gain from
d
→
y
.
This means that there is no disturbance
rejection for the feedforward controller. Intuitively, this makes perfect sense
because the controller has no measurements of the output, so there is no way
of detecting an unknown disturbance.
v. The steady state gain from
d
→
u
is 0. The fact that the controller is not
compensating for the disturbance makes the result in the previous part un
surprising.
vi. The steady state gain from
n
to both
y
and
u
are 0. This makes sense because
the noise is modeled as entering the system through the sensor, which is
inconsequential because
K
2
= 0.
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 Spring '08
 Tomizuka
 k2, steady state gain

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