Digital Signal Processing Reference
In-Depth Information
13.3
SOLUTION OF STATE EQUATIONS
We have developed procedures for writing the state equations for a system, given
the system difference equations, a transfer function, or a simulation diagram. In this
section, we present two methods for finding the solution of state equations.
Consider the state equations
x
[n + 1] =
Ax
[n] +
Bu
[n]
and
(13.20)
y
[n] =
Cx
[n] +
Du
[n].
We assume that the initial state vector
x
[0] is known and that the input vector
u
[
n
]
is known for
n G 0.
In a recursive manner, we write, for
n = 1,
x
[1] =
Ax
[0] +
Bu
[0]
and for
n = 2,
x
[2] =
Ax
[1] +
Bu
[1]
=
A
(
Ax
[0] +
Bu
[0]) +
Bu
[1]
=
A
2
x
[0] +
ABu
[0] +
Bu
[1].
In a like manner, we can show that
x
[3] =
A
3
x
[0] +
A
2
Bu
[0] +
ABu
[1] +
Bu
[2];
x
[4] =
A
4
x
[0] +
A
3
Bu
[0] +
A
2
Bu
[1] +
ABu
[2] +
Bu
[3].
We see from this pattern that the general solution is given by
Á
x
[n] =
A
n
x
[0] +
A
n- 1
Bu
[0] +
A
n- 2
Bu
[1] +
+
ABu
[n - 2] +
Bu
[n - 1]
n- 1
=
A
n
x
[0] +
a
A
(n- 1 -k)
Bu
[k],
(13.21)
k= 0
A
0
where
=
I
.
We define the
state-transition matrix
≥[n]
from this solution:
≥[n] =
A
n
.
(13.22)
This matrix is also called the
fundamental matrix
. From (13.21), the solution can
then be expressed as
n- 1
k= 0
≥[n - 1 - k]
Bu
[k];
x
[n] =≥[n]
x
[0] +
a
(13.23)
