EE 429
PROBLEM SET 6
DUE: 5 May 2008
Reading assignment: Chapter 7 Sections 7.1 through 7.6 and Chapter 8
Problem 25:
(16 points)
Consider a discretetime plant with the statespace representation
x
(
k
+1) =
±
0
.
50
.
5
01
.
5
²
x
(
k
)+
±
0
2
²
u
(
k
)
y
(
k
)=
(
12
)
x
(
k
)
.
1. (2 points) Compute the openloop plant transfer function
G
p
(
z
Y
(
z
)
U
(
z
)
,
and determine whether or not the openloop system is stable. What are the openloop zero(s)?
2. (6 points) In this section of the problem you will design a closedloop system using output feedback
u
(
k
k
o
y
(
k
r
(
k
)
,
where
r
(
k
) is a reference signal for the feedback system and
k
o
is a scalar gain.
•
(2 points) Determine the closedloop transfer function
G
of
(
z
Y
(
z
)
R
(
z
)
in terms of the controller parameter
k
o
.
•
(1 point) Compare the zero(s) of the openloop transfer function
G
p
(
z
) and the closedloop transfer
function
G
of
(
z
). Does output feedback aﬀect the location of the openloop plant zero(s)?
•
(2 points) Using the Jury stability test, determine for what range of controller gains
k
o
, if any, the closed
loop system stable.
•
(1 point) Is it possible to arbitrarily place all the poles of the closedloop system using output feedback?
Justify your answer using one or two short sentences.
3. (8 points) Now consider designing a controller using fullstate feedback
u
(
k
K
s
x
(
k
r
(
k
)
,
where
r
(
k
) is a reference signal for the feedback system and
K
s
is a 1by2 constant gain vector.
•
(2 points) Compute the controllability matrix
P
and determine its rank.
Is the openloop system
completelycontrollable?
•
(4 points) Using the gain vector
K
s
=
(
k
1
k
2
)
,
determine the closedloop transfer function
G
sf
(
z
Y
(
z
)
R
(
z
)
in terms of the gain parameters
k
1
and
k
2
.
•
(1 point) Compare the zero(s) of the openloop transfer function
G
p
(
z
) and closedloop transfer functions
G
sf
(
z
). Does state feedback aﬀect the location of the openloop plant zero(s)?
•
(1 point) Based on the form of the closedloop transfer function
G
sf
(
z
), is it possible to arbitrarily place
all the poles of the closedloop system using fullstate feedback? Justify your answer using one or two
short sentences.
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 Spring '08
 SCHIANO
 Ks, closedloop transfer function

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