Digital Signal Processing Reference
In-Depth Information
In a like manner,
A
n
=
A
2
A
n- 2
=
0
; n G 3.
Thus, the state transition matrix is, from (8.36),
10
01
01
00
1 t
01
B
R
B
R
B
R
≥(t) =
I
+
A
t =
+
t =
,
and the states are given by
1 t
01
x
1
(0) + tx
2
(0)
x
2
(0)
B
R
B
R
x
(t) =≥(t)
x
(0) =
x
(0) =
.
This example was chosen to give a simple calculation. In general, evaluating the matrix expo-
nential is quite involved. The calculation of
≥(t)
in this example is easily checked with the
use of Laplace transforms. (See Problem 8.18.)
■
The series expansion of is well suited to evaluation on a digital computer
if is to be evaluated at only a few instants of time. The series expansion is also
useful in the analysis of digital control systems [5, 6]. However, as a practical matter,
the time response of a system should be evaluated by simulation, such as is given in
Example 8.6.
In this section, two expressions for the solution of state equations are derived.
The first, (8.26), expresses the solution as a Laplace transform, while the second,
(8.29), expresses the solution as a convolution. The state-transition matrix is found
either by the Laplace transform, (8.27), or by the series (8.36).
≥(t)
≥(t)
8.4
PROPERTIES OF THE STATE-TRANSITION MATRIX
Three properties of the state-transition matrix will now be derived. First, for an un-
forced system, from (8.29),
x
(t) =≥(t)
x
(0) Q
x
(0) =≥(0)
x
(0);
(8.39)
hence, the first property is given by
≥(0) =
I
,
(8.40)
where
I
is the identity matrix. This property can be used in verifying the calculation
of
≥(t).
The second property is based on time invariance. From (8.39), with
t = t
1
,
x
(t
1
) =≥(t
1
)
x
(0).
(8.41)
Suppose that we consider
t = t
1
to be the initial time and
x
(t
1
)
to be the initial con-
ditions. Then, at
t
2
seconds
later than
t
1
,
from (8.39), (8.41), and the time invariance
of the system,
