Digital Signal Processing Reference
In-Depth Information
13.1
STATE-VARIABLE MODELING
In Chapters 9 and 10, we introduced the modeling of discrete-time systems by dif-
ference equations. If the discrete-time systems are linear and time invariant (LTI),
we can represent these systems by transfer functions. Let the notation
z
[
#
]
denote
the
z
-transform. For an LTI system with an input of
U(z) =
z
[u[n]]
and output of
Y(z) =
z
[y[n]],
we can write, from Chapter 10,
Y(z) = H(z)U(z),
(13.1)
where is the system transfer function. This discrete-time system can be repre-
sented by either of the two block diagrams of Figure 13.1. The impulse response
h
[
n
] and the transfer function are related by We use the
variable
u
[
n
] to denote the input in (13.1), since we use the variable
x
[
n
] to denote
state variables. The use of
u
[
n
] as the symbol for the general input function can lead
to confusion, because that symbol is also used for the discrete-time unit step function.
However, this is the notation commonly used for the input signal in state-variable
models.
We now introduce state variables by an example; then a general development
will be given. Consider a system modeled by the second-order linear difference
equation with constant coefficients:
H(z)
H(z)
H(z) =
z
[h[n]].
y[n + 2] - 0.7y[n + 1] + 0.9y[n] = 2u[n].
(13.2)
In this equation,
u
[
n
] is the input and
y
[
n
] is the output. We commonly write differ-
ence equations for state models in terms of advances rather than delays. Of course,
we can also write (13.2) as
y[n] - 0.7y[n - 1] + 0.9y[n - 2] = 2u[n - 2]
(13.3)
by replacing
n
with in (13.2).
Ignoring the initial conditions, from Table 11.4 we find that the
z
-transform of
(13.2) yields
(n - 2)
(z
2
- 0.7z + 0.9)Y(z) = 2U(z).
Hence, the transfer function is given by
Y(z)
U(z)
=
2
H(z) =
- 0.7z + 0.9
.
(13.4)
z
2
u
[
n
]
y
[
n
]
U
[
z
]
Y
(
z
)
h
[
n
]
H
(
z
)
H
(
z
)
[
h
[
n
]]
Figure 13.1
LTI system.
