Environmental Engineering Reference
In-Depth Information
In order to obtain a prediction, it is necessary to know the future value for the
disturbance. A common practice is to assume that it will have a constant value equal
to the last measured value [17]. Taking this observation into account, Equation 5.49
can be written as:
y
=
Gu
+
f
,
(5.54)
where
f
is a vector composed of both controlled and control variables known at time
t
.
5.5.2 Generalized Predictive Control
In this section, a brief summary of a model-based control algorithm is given. A
thorough description can be found in [16] and [18]. A practical presentation, with
only a few equations but deep insight, is given in [19]. Chapter 7 provides a survey
of current industrial control products based on model-predictive approaches.
The generalized predictive control (GPC) algorithm applies a control sequence
that minimizes a multistage cost function of the form:
N
1
y
)
2
∑
N
2
J
=
(
t
+
j
|
t
)−
r
(
t
+
j
j
=
1
Δ
u
)
(5.55)
2
λ
∑
N
u
j
+
(
t
+
j
−
1
=
is an optimum
j
-step-
ahead prediction of system output on data up to time
t
.
N
1
and
N
2
are the minimum
and maximum costing horizons,
N
u
is the control horizon, λ is the weight of future
control actions and
r
subject to Δ
u
(
t
+
j
−
1
)=
0for
j
>
N
u
.Thevariable
y
(
t
+
j
|
t
)
is the desired trajectory.
Using the predictive model (5.54) with
(
t
+
j
)
T
=[
(
+
|
),...,
(
+
|
)]
y
y
t
N
1
t
y
t
N
2
t
T
=[
(
),...,
(
+
−
)]
u
Δ
u
t
Δ
u
t
N
u
1
(5.56)
T
=[
(
+
),...,
(
+
)]
f
f
t
N
1
f
t
N
2
and
⎡
⎣
⎤
⎦
g
0
0
...
0
g
1
g
0
...
0
G
=
(5.57)
.
.
.
.
.
.
g
N
2
−
N
−
1
−
1
g
N
2
−
N
−
1
−
2
...
g
0
Equation 5.55 can be expressed as follows:
=
Gu
r
Gu
r
T
J
+
f
−
+
f
−
(5.58)
λ
u
T
u
+
,
T
.
where
r
=[
r
(
t
+
N
1
),...,
r
(
t
+
N
2
)]
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