Digital Signal Processing Reference
In-Depth Information
where boldface type denotes vectors and matrices. In these equations,
x
[n] = (N * 1) state vector for an Nth-order system;
u
[n] = (r * 1) vector composed of the system input signals;
y
[n] = (p * 1) vector composed of the defined output signals;
A
= (N * N) system matrix;
B
= (N * r) input matrix;
C
= (p * N) output matrix;
D
= (p * r)
matrix that represents the direct coupling between the
input and the output.
Expanding the vectors in (13.9) yields
x
1
[n + 1]
x
2
[n + 1]
o
x
N
[n + 1]
x
1
[n]
x
2
[n]
o
x
N
[n]
x
[n + 1] =
D
T
,
x
[n] =
D
T
,
(13.10)
u
1
[n]
u
2
[n]
o
u
r
[n]
y
1
[n]
y
2
[n]
o
y
p
[n]
u
[n] =
D
T
,
and
y
[n] =
D
T
.
As stated earlier, it is standard notation to denote the input functions as
u
i
[n].
We
illustrate the
i
th state equation in (13.9):
x
i
[n + 1] = a
i1
x
1
[n] + a
i2
x
2
[n] +
Á
+ a
iN
x
N
[n]
Á
+ b
i1
u
1
[n] +
+ b
ir
u
r
[n].
(13.11)
The
i
th output equation is
Á
y
i
[n] = c
i1
x
1
[n] + c
i2
x
2
[n] +
+ c
iN
x
N
[n]
Á
+ d
i1
u
1
[n] +
+ d
ir
u
r
[n].
(13.12)
We now define the
state
of a system:
The state of a system at any time is the information that, together with all inputs for
determines the behavior of the system for
n
0
n G n
0
,
n G n
0
.
It will be shown that the state vector
x
[
n
] of the standard form of the state-variable
equations, (13.9), satisfies this definition.
We refer to the two matrix equations of (13.9) as the
state equations
of the sys-
tem. The first equation, a difference equation, is called the
state equation
, and the
