Name:
Problem 1
For a particular experimental setup, the equations of motion for a ball rolling on a
beam are given by
¨
r
=

sin(
θ
)

0
.
5 ˙
r
+
r
˙
θ
2
¨
θ
=
1
3
r
2
+ 2
h

6
r
˙
r
˙
θ

30
r
cos(
θ
)

2
˙
θ
+
τ
i
where
r
is the ball position,
˙
r
is the ball velocity,
θ
is the beam angle, and
˙
θ
is the beam angular
velocity. The output is the ball position and the input is the torque
τ
applied to the beam.
1. True or False: For each ball position
r
*
there exists an input
τ
*
such that
r
*
is a component of
a valid operating point. Explain.
True. One can take the state vector to be
x
= (
x
1
, x
2
, x
3
, x
4
) = (
r,
˙
r, θ,
˙
θ
)
. To be an operating
point,
x
*
must have
x
*
2
= ˙
r
*
= 0
and
x
*
4
=
˙
θ
*
= 0
. In turn, this implies that
sin(
x
*
3
) = sin(
θ
*
) =
0
and
τ
*
= 30
r
*
cos(
θ
*
)
, which is either
30
r
*
or

30
r
*
.
Henceforth, we will assume that the
angle
θ
stays in the range
±
90
degrees, so that the ball does not fall oF of the beam. In that
case,
θ
*
= 0
and
τ
*
= 30
r
*
.
2. Compute the matrices
A
,
B
,
C
and
D
that would be used to generate the transfer function
from
τ

τ
*
to
r

r
*
for an operating point corresponding to
r
*
= 2.
Write the formula for
the transfer function in terms of
A
,
B
,
C
, and
D
.
(You do not need to compute the transfer
function). How many poles will the transfer function have?
We have
f
(
x, τ
)
=
x
2

sin(
x
3
)

0
.
5
x
2
+
x
1
x
2
4
x
4
1
3
x
2
1
+ 2
[

6
x
1
x
2
x
4

30
x
1
cos(
x
3
)

2
x
4
+
τ
]
h
(
x, τ
)
=
x
1
A
=
∂f
(
x, τ
)
∂x

(
x
=
x
*
,τ
=
τ
*
)
=
0
1
0
0
(
x
*
4
)
2

0
.
5

cos(
x
*
3
)
2
x
*
1
x
*
4
0
0
0
1

6
x
*
2
x
*
4

30 cos(
x
*
3
)
3(
x
*
1
)
2
+2
+
6
x
*
1
(6
x
*
1
x
*
2
x
*
4
+30
x
*
1
cos(
x
*
3
)+2
x
*
4

τ
*
)
(3(
x
*
1
)
2
+2)
2

6
x
*
1
x
*
4
3(
x
*
1
)
2
+2
30
x
*
1
sin(
x
*
3
)
3(
x
*
1
)
2
+2

2

6
x
*
1
x
*
2
3(
x
*
1
)
2
+2
=
0
1
0
0
0

0
.
5

1
0
0
0
0
1

30
14
0
0

2
14