EE 482
PROBLEM SET 3
DUE: 26 Feb 2008
Reading assignment: Ch 3
Problem 11:
(15 points) Consider an
n
th
order system with statespace representation
x
(
k
+ 1)
=
Ax
(
k
) +
Bu
(
k
)
y
(
k
)
=
Cx
(
k
) +
Du
(
k
)
.
1. (10 points) Assuming that the initial conditions
y
(0)
,
· · ·
, y
(
n

1) and input
u
(
k
) are known for
k
≥
0, find an
expression for the initial statevector
x
(0) and place your answer in the form
Y
=
Qx
(0) +
G,
and specify the elements of
Y
,
Q
, and
G
.
2. (5 points) What condition must hold so that we can solve for a unique value of
x
(0)?
Problem 12:
(25 points)
Consider a phaselead compensator with transfer function representation
G
c
(
s
) =
K
s/ω
o
+ 1
s/aω
o
+ 1
,
where
K, ω
o
>
0 and
a >
1.
In this problem you will design a series of sampleddata systems, each with sample
period
T
, that approximates the dynamic behavior of the continuoustime compensator.
1. (5 points) Using an appropriate substitution from the handout for Laboratory #1, find the transfer function
representation
G
BR
c
(
z
) using the backward rectangular integration method, and express your answer in the
form
G
BR
c
(
z
) =
¯
K
z
+
¯
b
o
z
+ ¯
a
o
.
Specify the parameters
¯
K,
¯
b
o
and ¯
a
o
in terms of
T, K, ω
o
and
a
.
2. (5 points) Using an appropriate substitution from the handout for Laboratory #1, find the transfer function
representation
G
TR
c
(
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 Spring '08
 SCHIANO
 Digital Signal Processing, Nyquist–Shannon sampling theorem, ωo

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