Digital Signal Processing Reference
In-Depth Information
with the state transition matrix
F(t)
given either by (8.36) or by
≥(t) =
l
-1
[≥(s)] =
l
-1
[(s
I
-
A
)
-1
].
[eq(8.27)]
Because of the similarity of (8.36) and the Taylor's series for the scalar expo-
nential
= 1 + kt + k
2
t
2
2!
+ k
3
t
3
Á
,
e
kt
3!
+
(8.37)
the state transition matrix is often written, for notational purposes only, as the ma-
trix exponential
≥(t) = exp
A
t.
(8.38)
The
matrix exponential
is defined by (8.36) and (8.38). An example is given next that
illustrates the calculation of the state transition matrix, using the series in (8.36).
Series solution for second-order state equations
EXAMPLE 8.8
To give an example for which the series in (8.36) has a finite number of terms, we consider
the movement of a rigid mass in a frictionless environment. The system model is given by
f(t) = M
d
2
x(t)
dt
2
,
where
M
is the mass, the displacement, and the applied force. For convenience, we let
We choose the state variables as the position and the velocity of the mass such that
x(t)
f(t)
M = 1.
x
1
(t) = x(t),
x
2
(t) = x
#
(t) = x
#
1
(t),
and
x
#
2
(t) =
$
(t) = f(t),
where the last equation is obtained from the system model. The state equations are then
01
00
0
1
x
#
(t) =
B
R
x
(t) +
B
R
f(t).
Then, in (8.36),
01
00
01
00
01
00
00
00
,
A
2
B
R
B
RB
R
B
R
A
=
=
=
,
and
A
3
=
AA
2
= 0.
