Digital Signal Processing Reference
In-Depth Information
s
+
1
¢(s)
2s
+
1
¢(s)
5s + 4
s
2
+ 3s + 2
.
= [3
1]
C
S
=
This transfer function is the same as that used in Example 8.2 to derive the
x
(
t
)-state model.
The following MATLAB statements, when appended to the program in Example 8.13, verify
the results of this example:
[n d]=ss2tf(Av,Bv,Cv,Dv)
Hs=tf(n,d)
■
Similarity transformations have been demonstrated through an example. Certain
important properties of these transformations are derived next. Consider first the
determinant of
(s
I
-
A
v
).
From (8.64),
det (s
I
-
A
v
) = det (s
I
-
P
-1
AP
) = det (s
P
-1
IP
-
P
-1
AP
)
= det [
P
-1
(s
I
-
A
)
P
].
(8.66)
For two square matrices,
det
R
1
R
2
= det
R
1
det
R
2
.
(8.67)
Then (8.66) becomes
det (s
I
-
A
v
) = det
P
-1
det (s
I
-
A
) det
P
.
(8.68)
R, R
-1
R
=
I
.
For a matrix
Then,
det
R
-1
R
= det
R
-1
det
R
= det
I
= 1.
(8.69)
Thus, (8.68) yields the first property:
det (s
I
-
A
v
) = det (s
I
-
A
) det
P
-1
det
P
= det (s
I
-
A
).
(8.70)
The roots of are the
characteristic values,
or the
eigenvalues,
of
A
. (See Appendix G.) From (8.70), the eigenvalues of are equal to those of
A
.
Because the transfer function is unchanged under a similarity transformation, and
since the eigenvalues of
A
are the poles of the system transfer function, we are not
surprised that they are unchanged.
A second property can be derived as follows: From (8.64),
det (s
I
-
A
)
A
v
det
A
v
= det
P
-1
AP
= det
P
-1
det
A
det
P
= det
A
.
(8.71)
The determinant of is equal to the determinant of
A
. This property can also be
seen from the fact that the determinant of a matrix is equal to the product of its
eigenvalues. (See Appendix G.)
A
v
