Digital Signal Processing Reference
In-Depth Information
and
y(t) =
C
v
v
(t) + D
v
u(t),
(8.63)
where the subscript indicates the transformed matrices. The matrices for the vector
x
(
t
) are not subscripted. From (8.61), (8.62), and (8.63), the transformed matrices
are given by
A
v
=
P
-1
AP
,
B
v
=
P
-1
B
,
C
v
=
CP
, and
D
v
= D.
(8.64)
An example is given to illustrate the derivations.
Similarity transformation for a second-order system
EXAMPLE 8.13
Consider the system of Example 8.12. The state equations for the state vector
v
(
t
) will be de-
rived. From Example 8.12,
01
-2
0
1
x
#
(t) =
Ax
(t) +
B
u(t) =
B
R
x
(t) +
B
R
u(t)
-3
and
y(t) =
Cx
(t) = [4
5]
x
(t),
with the similarity transformation
11
12
2 -1
-11
P
-1
B
R
B
R
=
Q
=
Q
P
=
.
From (8.64), the system matrices for
v
(
t
) are given by
11
12
01
-2
2 -1
-11
A
v
=
P
-1
AP
=
B
RB
RB
R
-3
-2
-2
2 -1
-11
-20
-3
=
B
RB
R
=
B
R
;
-4
-5
-1
11
12
0
1
1
2
B
v
=
P
-1
B
=
B
RB
R
B
R
=
;
2 -1
-11
B
R
C
v
=
CP
= [4
5]
= [3
1].
The transformed state equations are then
-20
-3
1
2
v
#
(t) =
A
v
v
(t) +
B
v
u(t) =
B
R
v
(t) +
B
R
u(t)
-1
and
(8.65)
y(t) =
C
v
v
(t) = [3
1]
v
(t).
