Digital Signal Processing Reference
In-Depth Information
s
IX
(s) -
AX
(s) = (s
I
-
A
)
X
(s) =
x
(0) +
BU
(s).
(8.25)
This additional step is necessary, since the subtraction of the matrix
A
from the
scalar
s
is not defined; we cannot factor
X
(
s
) directly in (8.24). Equation (8.25) may
now be solved for
X
(
s
):
X
(s) = (s
I
-
A
)
-1
x
(0) + (s
I
-
A
)
-1
BU
(s).
(8.26)
The state vector
x
(
t
) is the inverse Laplace transform of this equation.
To develop a general relationship for the solution, we define the
state transi-
tion matrix
≥(t)
as
≥(t) =
l
-1
[≥(s)] =
l
-1
[(s
I
-
A
)
-1
].
(8.27)
This matrix is also called
the fundamental matrix
. The matrix
is called the
resolvant
of
A
[3]. Note that for an
n
th-order system, the
state transition matrix is of order
The inverse Laplace transform of a matrix, as in (8.27), is defined as the in-
verse transform of the elements of the matrix. Solving for in (8.27) is in general
difficult, time consuming, and prone to error. A more practical procedure for calcu-
lating the state vector
x
(
t
) is by computer simulation. Next, an example is presented
to illustrate the calculation in (8.27).
≥(t)
≥(s) =
(s
I
-
A
)
-1
(n * n).
≥(t)
State transition matrix for a second-order system
EXAMPLE 8.4
We use the system of Example 8.2, described by the transfer function
Y(s)
U(s)
=
5s + 4
H(s) =
+ 3s + 2
.
s
2
From Example 8.2, the state equations are given by
01
-2
0
1
x
#
(t) =
B
R
B
R
[eq(8.19)]
x
(t) +
u(t);
-3
y(t) = [4
5]
x
(t).
To find the state transition matrix, we first calculate the matrix
(s
I
-
A
):
10
01
01
-2
s
-1
B
R
B
R
B
R
s
I
-
A
= s
-
=
.
-3
2
s + 3
We next calculate the adjoint of this matrix (see Appendix G):
s + 31
-2
B
R
Adj(s
I
-
A
) =
.
s
