Environmental Engineering Reference
In-Depth Information
Supposing that no constraints exist on the control signal, the minimum value of
J
can be found through the gradient zero:
=
G
T
G
λ
I
−
1
G
T
u
+
(
r
−
f
).
(5.59)
In order to deal with constraints and profit from efficient optimization programs
such as linear programming (LP) algorithms, the objective function is recast in the
following form:
∑
N
2
N
1
∑
i
J
(
Δ
u
)=
|
y
i
(
t
+
j
|
t
)−
r
(
t
+
j
)|
j
=
=
1
(5.60)
λ∑
N
u
j
1
∑
i
+
|
Δ
u
i
(
t
+
j
−
1
)|
=
=
1
where
N
1
and
N
2
define the costing horizon,
N
u
defines the control horizon and
n
and
m
are the number of outputs and inputs, respectively. Absolute values of the
output tracking error and absolute values of control increments are taken from their
squares, as is usual in GPC. If a set of auxiliary variables μ
i
≥
0andβ
i
≥
0are
defined so that
−
μ
i
≤
y
i
(
k
+
j
|
t
)−
r
(
k
+
j
) ≤
μ
i
−
β
i
≤
Δ
u
i
(
k
+
j
−
1
)
≤
β
i
(5.61)
∑
n
·(
N
1
−
N
2
)
λ
∑
m
·
Nu
0
≤
μ
i
+
1
β
i
≤
ζ
,
i
=
1
i
=
then ζis an upper bound of
J
(
Δ
u
)
and the problem is reduced to minimize the upper
bound ζ [16].
When constraints on output variables
[
y
min
,
y
max
]
, manipulated variables
[
u
min
,
u
max
]
and slew rate of the manipulated variables
[
u
min
,
u
max
]
are taken into account,
the problem can be formulated as a LP problem with
min
c
T
x
subject to
Ax
≤
b
,
x
≥
0
(5.62)
x
with
⎡
⎣
⎤
⎡
⎤
⎦
u
−
u
min
μ
β
ζ
0
0
0
1
⎦
⎣
x
=
c
=
(5.63)
⎡
⎣
⎤
⎦
G
−
IO
0
−
GO O
0
GO O
0
−
GO O
0
IO
−
I
0
A
=
(5.64)
I
0
IO O
0
TO O
0
−
−
IO
−
TO O
0
01
t
1
t
λ
−
1
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