Digital Signal Processing Reference
In-Depth Information
second one, an algebraic equation, is called the
output equation
. The state equation
is a first-order matrix difference equation, and the state vector
x
[
n
] is its solution.
Given knowledge of
x
[
n
] and the input vector
u
[
n
], the output equation, which is
algebraic, yields the output
y
[
n
].
In the state equation in (13.9), the only variables that may appear on the left
side of the equation are and the only variables that may appear on the
right side are and Only and may appear in the output
equation (no or ). Valid equations that model an LTI system can
be written without following these rules; however, those equations will not be in the
standard form.
The standard form of the state equations, (13.9), allows for more than one
input and more than one output. Systems with more than one input or more than
one output are called
multivariable systems
. For a single-input system, the matrix
B
is an column matrix and the input is the scalar
u
[
n
]. For a single-output
system, the matrix
C
is a row matrix and the output is the scalar
y
[
n
]. An
example is now given to illustrate a multivariable system.
x
i
[n + 1],
x
i
[n]
u
i
[n].
y
i
[n], x
i
[n],
u
i
[n]
x
i
[n + 1]
u
i
[n + 1]
(N * 1)
(1 * N)
State variables for a third-order discrete system
EXAMPLE 13.1
Consider the system described by the coupled difference equations
y
1
[n + 2] + 2y
1
[n] + 3y
2
[n] = u
1
[n] + 9u
2
[n]
and
y
2
[n + 1] + 4y
2
[n] - 6y
1
[n + 1] = 5u
1
[n],
where and are the input signals and and are the output signals. We
define the states as the outputs, and, where necessary, the advanced outputs. Thus,
u
1
[n]
u
2
[n]
y
1
[n]
y
2
[n]
x
1
[n] = y
1
[n];
x
2
[n] = y
1
[n + 1] = x
1
[n + 1];
x
3
[n] = y
2
[n].
From the system difference equations, we write
y
1
[n + 2] = x
2
[n + 1] =-2y
1
[n] - 3y
2
[n] + u
1
[n] + 9u
2
[n]
=-2x
1
[n] - 3x
3
[n] + u
1
[n] + 9u
2
[n];
y
2
[n + 1] = x
3
[n + 1] =-4y
2
[n] + 6y
1
[n + 1] + 5u
1
[n]
= 6x
2
[n] - 4x
3
[n] + 5u
1
[n].
We rewrite the state equations in the following order:
x
1
[n + 1] = x
2
[n];
x
2
[n + 1] =-2x
1
[n] - 3x
3
[n] + u
1
[n] + 9u
2
[n];
x
3
[n + 1] = 6x
2
[n] - 4x
3
[n] + 5u
1
[n].
The output equations are
y
1
[n] = x
1
[n]
