Digital Signal Processing Reference
In-Depth Information
≥(z) = z(z
I
-
A
)
-1
The matrix is called the
resolvant
of
A
[3]. Note that for an
N
th-
order system, the state transition matrix is an matrix. The inverse
z
-transform
of a matrix, as in (13.32), is defined as the inverse transform of the elements of the
matrix. A computer algorithm is available for calculating
Finding the inverse
z
-transforms indicated in (13.32), in general, is difficult,
time consuming, and prone to error. A more practical procedure for calculating the
state vector
x
[
n
] is a recursive computer solution, as described earlier. An example
is now presented to illustrate the calculation of a state transition matrix.
N * N
z(z
I
-
A
)
-1
[3].
Transition matrix for a second-order system
EXAMPLE 13.5
We use the system of Example 13.4. From this example, the state equations are given by
01
-65
0
1
B
R
B
R
x
[n + 1] =
x
[n] +
u[n]
and
y[n] = [3
1]
x
[n].
To find the state transition matrix, we first calculate the matrix
(z
I
-
A
):
10
01
01
-65
z
-1
B
R
B
R
B
R
z
I
-
A
= z
-
=
.
6
z - 5
To find the inverse of this matrix, we calculate its adjoint matrix (see Appendix G):
z - 51
-6
Adj(z
I
-
A
) =
B
R
.
z
The determinant of
(z
I
-
A
)
is given by
det(z
I
-
A
) = z(z - 5) - (-1)(6)
= z
2
- 5z + 6 = (z - 2)(z - 3).
As we will show later, this determinant is always equal to the denominator of the transfer
function. The inverse of a matrix is the adjoint matrix divided by the determinant:
z(z - 5)
(z - 2)(z - 3)
z
(z - 2)(z - 3)
z(z
I
-
A
)
-1
=
D
T
z
2
(z - 2)(z - 3)
-6z
(z - 2)(z - 3)
3z
z - 2
+
-2z
z - 3
-z
z - 2
+
z
z - 3
=
D
T
.
6z
z - 2
+
-6z
z - 3
-2z
z - 2
+
3z
z - 3
The state transition matrix is the inverse
z
-transform of this matrix. Thus, from Table 11.2,
3(2)
n
- 2(3)
n
- (2)
n
+ (3)
n
≥[n] =
B
R
.
6(2)
n
- 6(3)
n
- 2(2)
n
+ 3(3)
n
