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MEM 255: Introduction to Controls
Summer 0607
Home work # 5
Due: 08/02/07
1.
Consider a system with
u
as its input and
y
as its output, described by the
following differential equation.
(Items in ‘blue’ are matlab commands)
2
12
22
12
2
18
40
y
y
y
y
u
u
u
+
+
+
=
+
+
a.
Determine the transfer function
Y(s)/U(s)
.
b.
Determine the partial fraction expansion of
Y(s)/U(s)
.
c.
Obtain the state space representation (the one from class notes) of the system,
and determine the
{A,B,C,D}
matrices. Let
x
denote the states. Verify that
1
( ) /
( )
(
)
.
Y s
U s
C sI
A
B
D

=

+
d.
Generate a random nonsingular (3 by 3) matrix
T
r
[Tr = rand(3,3)]
. Compute
1
1
{
;
;
;
}
r
r
r
r
r
r
r
r
A
T AT B
T B C
CT D
D


=
=
=
=
. This amounts to rewriting the
state space model in (c) above using the states
1
.
r
r
x
T x

=
; known as a
“
similarity transformation
.” Verify that
1
( ) /
( )
(
)
.
r
r
r
r
Y s
U s
C sI
A
B
D

=

+
Hence observe that
transfer function is invariant under similarity
transformations
.
e.
Construct the matrix
2
r
r
r
r
r
T
B
A B
A B
=
, and determine
1
1
2
2
2
2
{
;
;
;
}
r
r
r
r
A
T A T B
T B C
C T D
D


=
=
=
=
. This is another transformation
x
r
to
x
2
.
T
is called the
controllability matrix
. Verify that the transfer function
remains the same. Can you observe any relationship between the states
x
2
and
x
of (c)?
f.
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This note was uploaded on 06/03/2008 for the course MEM 255 taught by Professor Yousuff during the Spring '08 term at Drexel.
 Spring '08
 Yousuff

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