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In the above equations the right-hand terms should be taken for further processing.
These obviously depend only on the past values, so that they produce the set of
equations written in matrix form as
y
=
Gu
+
f
,
where,
yyt
' ' '
[
(
1),
yt
(
2)...,
yt
(
n
)]
T
T
uut
[
( ),
ut
(
1),...,
ut
(
n
1)]
T
f
[( ,( 2 ,(
f t
f t
f t
n
]
and
G
is the lower triangular
n
u matrix
ª
º
g
0............0
0
«
»
«
..........0
....................
....
gg
»
G
«
10
.
»
«
»
«
»
g
g
g
¬
¼
n
1
n
2
0
Introducing now the set-point sequence
wwt
[( 1 , ( 2 , (
wt
wt
n
T
and minimizing the expected value of
^
`
T
,
T
J
Eyw yw
(
) (
)
O
uu
or, of
T
T
J
{(
Gu
f
w
) (
Gu
f
w
)
O
u u
}
the projected control increment vector
T
O
1
T
uGG I
(
)
Gwf
(
)
is determined with the unit vector
I
. From the last result, only the first element of
u
is taken as the next control value
T
ut
()
ut
(
1)
g w
(
f
)
where
T
g
is the first row of
O
1
T
T
(
GG
I
)
G
.
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