EE 429
PROBLEM SET 1
DUE: 5 Feb 2008
Reading assignment: Ch 1 and Ch 2
The goal of the Frst problem set is to develop your ability to:
1. Convert from one method of representing a dynamic system to another method of representation. ±or example,
given an ODE (DE) representation, Fnd a corresponding statespace representation.
2. Determine the ZOH discretetime equivalent model of a continuoustime process using statespace techniques.
Problem 1:
(20 points)
A singleinput singleoutput (SISO) system is represented by the secondorder ordinary diﬀerential equation
d
3
y
dt
3
+4
d
2
y
dt
2
+8
dy
dt
+2
y
(
t
)=2
u
(
t
)
,
where
y
(
t
) is the system output and
u
(
t
) is the system input. At time
t
= 0 the initial conditions for the system are
y
(0) = 3, ˙
y
(0) = 8, and ¨
y
(0) = 0.
1. (2 points) Determine the system transfer function
Y
(
s
)
/U
(
s
).
2. (9 points) Represent the system by an allintegrator block diagram. State the initial condition for each inte
grator.
3. (9 points) ±ind a statespace representation for the system using the statevariables
x
1
=
y
and
x
2
=˙
y
, and
x
3
=¨
y
. Specify the initial state vector
x
(0).
Problem 2:
(20 points)
A singleinput singleoutput (SISO) system is represented by the ordinary diﬀerential equation
¨
y

y
u
u,
where
y
(
t
) is the system output and
u
(
t
) is the system input.
1. (2 points) Determine the system transfer function
Y
(
s
)
/U
(
s
). Are there any polezero cancelations?
2. (2 points) What are the poles of the transfer function (
do not
include poles canceled by zeros !)?
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 Spring '08
 SCHIANO
 Signal Processing, tk, initial state vector

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