Digital Signal Processing Reference
In-Depth Information
term
or the
forced response
. Equation (8.29) is sometimes called the
convolution so-
lution
of the state equation.
The solution in (8.29) is quite difficult to calculate, except for the simplest of
systems. The system of the preceding example is now used to illustrate this calculation.
EXAMPLE 8.7
Convolution solution for second-order state equations
In Example 8.5, the input was a unit step function. From (8.29), with
u(t) = 1
for
t 7 0,
the
second term becomes
t
0
≥(t - t)
Bu
(t)dt =
L
t
2e
-
(t -t)
- e
-2(t -t)
e
-(t -t)
- e
-2(t -t)
0
1
B
RB
R
L
dt
-2e
-(t -t)
+ 2e
-2(t -t)
-e
-(t -t)
+ 2e
-2(t -t)
0
t
(e
-(t -t)
- e
-2(t -t)
dt
1
2
e
-2t
e
2t
)
t
(-e
-t
e
t
+ e
-2t
e
2t
)
t
(e
-t
e
t
-
L
0
B
R
=
D
T
=
t
(-e
-(t -t)
+ 2e
-2(t -t)
dt
L
0
2
(1 - e
-2t
)
(-1 + e
-t
) + (1 - e
-2t
)
(1 - e
-t
) -
1
1
2
- e
-t
2
e
-2t
e
-t
- e
-2t
1
+
B
R
B
R
=
=
.
This result checks that of Example 8.5. Only the forced response is derived here. The initial-
condition term of the solution is the function
F(t)
x
(0)
in (8.29), which was evaluated in Ex-
ample 8.5; it is not repeated here.
■
The complete solution to the state equations was derived in this section. This
solution may be evaluated either by the Laplace transform or by a combination of
the Laplace transform and the convolution integral. Either procedure is long, time
consuming, and prone to errors.
As just shown, the state transition matrix can be evaluated with the use of the
Laplace transform. An alternative procedure for this evaluation is now developed.
One method of solution of homogeneous differential equations is to assume as
the solution an infinite power series with unknown coefficients. The infinite series is
then substituted into the differential equation to evaluate the unknown coefficients.
This method is now used to find the state transition matrix as an infinite series.
We begin by considering all system inputs to be zero. Thus, from (8.9), the
state equation may be written as
x
#
(t) =
Ax
(t),
(8.30)
with the solution
x
(t) = F(t)
x
(0),
(8.31)
from (8.29).
