
The output y
k
is expres sed in terms of the state and input according to
y
k
¼ b
m
x
nmþ1,k
þþb
2
x
n1,k
þ b
1
x
n,k
þ b
0
w
k
(4:512)
¼ b
m
x
nmþ1,k
þþb
2
x
n1,k
þ b
1
x
n,k
þ b
0
[u
k
a
n
x
1,k
a
2
x
n1,k
a
1
x
n,k
](4:513)
y
k
¼
a
n
b
0
x
1,k
a
n1
b
0
x
2,k
a
1
b
0
x
n,k
þ b
0
u
k
, m ¼ 0
a
n
b
0
x
1,k
a
n1
b
0
x
2,k
a
mþ1
b
0
x
nm,k
þ(b
m
a
m
b
0
)x
nmþ1,k
þþ(b
1
a
1
b
0
)x
n,k
þ b
0
u
k
, m ¼ 1, ..., n 1
(b
n
a
n
b
0
)x
1,k
þ ( b
n1
a
n1
b
0
)x
n1,k
þþ(b
1
a
1
b
0
)x
n,k
þ b
0
u
k
, m ¼ n
8
>
>
>
<
>
>
>
:
(4:514)
In the general case of a linear discrete-time system with r inputs and p outputs, the discrete-time
state equations are of the form
x
kþ1
¼ Ax
k
þ Bu
k
, y
k
¼ Cx
k
þ Du
k
(4:515)
where the system matrix A is n n, the input matrix