CALIFORNIA INSTITUTE OF TECHNOLOGY
Control and Dynamical Systems
CDS 101
D. G. MacMynowski
Fall 2010
Problem Set #5
Issued:
1 Nov 10
Due:
8 Nov 10
Note: In the upper left hand corner of the
second
page of your homework set, please
put the number of hours that you spent on this homework set (including reading).
1. Consider the block diagram for the following secondorder system


+
+
1
s
1
s
c
a
b
r
y
z
(a) Compute the transfer function
H
yr
between the input
r
and the output
y
.
(b) Show that the following state space system has the same transfer function, with the
appropriate choice of parameters:
d
dt
x
1
x
2
=
0
1

a
2

a
1
x
1
x
2
+
0
1
r
y
=
b
2
b
1
x
1
x
2
+
dr
Give the values of
a
i
,
b
i
, and
d
that correspond to the transfer function you computed
in (a).
(c) Compute the transfer function
H
zr
between the input
r
and the output
z
. (Hint: It is
not
H
zr
= 1.)
2. Consider the following simplified equations of motion for a cruise control system (these are a
linearization of the equations from Section 3.1 in
˚
Astr¨
om and Murray):
m
dv
dt
=

cv
+
bτ
+
F
hill
,
where
m
= 1000 kg is the mass of the vehicle,
c
= 50 N s/m is the viscous damping coefficient,
b
= 25 is the conversion factor between engine torque and the force applied to the vehicle.
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 Fall '08
 Murray,R

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