Digital Signal Processing Reference
In-Depth Information
(z
I
-
A
)
-1
The resolvant matrix
was calculated in Example 13.5. Then, from (13.44) and
Example 13.5, with
D = 0,
H(z) =
C
(z
I
-
A
)
-1
B
z - 5
1
z
2
z
2
0
1
- 5z + 6
- 5z + 6
T
B
R
= [3
1]
D
-6
z
2
- 5z + 6
z
z
2
- 5z + 6
1
z
2
z + 3
z
2
- 5z + 6
.
- 5z + 6
z
= [3
1]
D
T
=
z
2
- 5z + 6
This transfer function checks the one given.
■
Although (13.44) does not appear to be useful in calculating the transfer func-
tion for higher-order systems, relatively simple computer algorithms exist for evalu-
ating the resolvant matrix
(z
I
-
A
)
-1
[3].
A MATLAB program that performs the
calculations of Example 13.8 is given by
A=[0 1;-6 5],B=[0;1],C=[3 1],D=0
[num,den]=ss2tf(A,B,C,D)
Hz=tf(num,den,-1)
We saw in Section 11.6 that bounded-input bounded-output (BIBO) stability can be
determined from the transfer function of an LTI system. From (13.13), the transfer
function of (13.44) can be expressed as a rational function:
+
Á
+ b
N- 1
z + b
N
b
0
z
N
H(z) =
C
(z
I
-
A
)
-1
B
+ D =
.
(13.45)
Á
z
N
+
+ a
N- 1
z + a
N
From Section 11.6, this system is BIBO stable, provided that all poles of are
inside the unit circle, where the poles of the transfer function are the zeros of the
denominator polynomial in (13.45).
The transfer function
H(z)
H(z)
can be expressed as
adj(z
I
-
A
)
det(z
I
-
A
)
C
(z
I
-
A
)
-1
B
+ D =
C
B
R
B
+ D.
(13.46)
Hence, the denominator polynomial of
H(z)
is the determinant of
(z
I
-
A
);
the
poles of the transfer function are the roots of
det(z
I
-
A
) = 0.
(13.47)
This equation is then the system characteristic equation. Note that stability is a
function only of the system matrix
A
and is not affected by
B
,
C
, or
D
. In fact,
