Digital Signal Processing Reference
In-Depth Information
H(s) =
C
(s
I
-
A
)
-1
B
s + 3
s
2
+ 3s + 2
1
s
2
+ 3s + 2
0
1
B
R
= [4
5]
≥
¥
-2
s
s
2
s
2
+ 3s + 2
+ 3s + 2
1
s
2
+ 3s + 2
s
5s + 4
s
2
+ 3s + 2
.
= [4
5]
≥
¥
=
s
2
+ 3s + 2
This transfer function checks the one given.
■
Although (8.51) 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 [3]. For many practical systems, the system
differential equations are written from the laws of physics. State equations are then
written from these differential equations. Then, a digital-computer algorithm such as
the one mentioned is used to calculate the transfer function. Most system analysis
and design software packages have programs for finding a state model from a trans-
fer function and for finding the transfer function from a state model. Almost all
analysis and design software use transfer functions or state models. The following
MATLAB program solves Example 8.10:
(s
I
-
A
)
-1
A=[0 1;-2 -3];B[0;1];C=[4 5];D=0;
[n,d]=ss2tf(A,B,C,D)
Hs=tf(n,d)
We saw in Section 7.7 that bounded-input bounded-output (BIBO) stability can be
determined from the transfer function of an LTI system. The transfer function of
(8.51) can be expressed as a rational function:
Á
b
n
s
n
+
+ b
1
s + b
0
H(s) =
C
(s
I
-
A
)
-1
B
+ D =
.
(8.52)
+
Á
+ a
1
s + a
0
s
n
From Section 7.7, this system is BIBO stable, provided that all poles of are in
the left half-plane. The poles of the transfer function are the zeros of the denomina-
tor polynomial in (8.52).
The transfer function
H(s)
H(s)
can be expressed as
adj(s
I
-
A
)
det(s
I
-
A
)
C
(s
I
-
A
)
-1
B
+ D =
C
B
R
B
+ D.
(8.53)
