Digital Signal Processing Reference
In-Depth Information
for a single-input single-output system. The Laplace transform of the first equation
in (8.45) yields [see (8.24)]
s
X
(s) =
AX
(s) +
B
U(s).
(8.46)
Because we are interested in the transfer function, the initial conditions are ignored.
Collecting terms for
X
(
s
) yields
(s
I
-
A
)
X
(s) =
B
U(s);
(8.47)
thus,
X
(
s
) is given by
X
(s) = (s
I
-
A
)
-1
B
U(s).
(8.48)
The Laplace transform of the output equation in (8.45) yields
Y(s) =
CX
(s) + DU(s).
(8.49)
From (8.48) and (8.49), the input-output relationship for the system is given by
Y(s) = [
C
(s
I
-
A
)
-1
B
+ D]U(s).
(8.50)
Because the system transfer function is defined by the equation
from (8.50), we see that the transfer function is given by
Y(s) = H(s)U(s),
Y(s)
U(s)
=
C
(s
I
-
A
)
-1
B
+ D =
C
≥(s)
B
+ D.
H(s) =
(8.51)
(1 * n), (s
I
-
A
)
-1
Because
C
is
is
(n * n),
and
B
is
(n * 1),
the product
C
(s
I
-
A
)
-1
B
is
(1 * 1),
or a scalar, as required. An example is given to illustrate
this result.
Transfer function from state equations
EXAMPLE 8.10
Consider the system of the earlier examples with the transfer function
Y(s)
U(s)
=
5s + 4
H(s) =
+ 3s + 2
.
s
2
The state equations were found in Example 8.2 to be
01
-2
0
1
x
#
(t) =
B
R
B
R
[eq(8.19)]
x
(t) +
u(t);
-3
y(t) = [4
5]
x
(t).
(s
I
-
A
)
-1
The resolvant matrix
was calculated in Example 8.5. Then, from (8.51) and Ex-
ample 8.5, with D = 0,
