Civil Engineering Reference
In-Depth Information
C
(
z
−
1
)
n
(
k
)
=
)
ε(
k
)
(5.7)
z
−
1
D
(
z
−
1
ε(
)
(
)
where
k
is a zero mean noise and
D
polynomial explicitly includes the
z
−
1
. For the GPCmethodology,
C
z
−
1
z
−
1
=
−
(
)
=
(
)
=
integrator
are
set, with this configuration themodel is called ControlledAuto-Regressive Integrated
Moving Average (CARIMA). This model is considered appropriate for two different
kinds of disturbances: randomchangeswhich happen at random instant andBrownian
motion (Camacho and Bordons
2004
). Furthermore, as it includes an integrator, an
offset-free control can be achieved.
1
1 and
D
5.2.2.2 Cost Function and Control Law
The quadratic cost function in the GPC strategy accounts for the error between the
predicted trajectory of the output of the process and an established reference, as well
as, the control effort necessary to reach this reference. This cost function can be
formulated as shown in Eq.
5.8
.
N
u
N
j
=
1
˃(
j
=
1
ˉ(
)
ˆ
)
2
]
2
J
=
j
y
(
k
+
j
|
k
)
−
w
(
k
+
j
+
j
)
[
u
(
k
+
j
−
1
)
(5.8)
where
y
ˆ
(
k
+
j
|
k
)
is the prediction of the output of the system estimated at instant
k
is a sequence of
future control increments obtained through the cost function minimisation,
N
is the
prediction horizon,
N
u
is the control horizon. It is important to highlight that, if the
process has a dead time
d
there is no reason to start in
j
+
j
with the information available at instant
k
,
u
(
k
+
j
−
1
)
1 in the lower bound of
the left summation of Eq.
5.8
, since the process output will not begin to evolve until
instant
k
=
+
d
. Thus, in these cases, that is, in processes with delay, the usual choice
is
j
are weighting sequences that penalise
the future tracking errors and control efforts, respectively, along the horizons.
Another important characteristic of MPC strategies is that they allow us to include
constraints in the algorithmat the design stage. As it was shown previously by Eq.
5.8
,
the control increments calculated by the GPC strategy are obtained by minimising a
quadratic function which can be also expressed in vectorial form as in Eq.
5.9
.
=
d
+
1. The coefficients
˃(
j
)
and
ˉ(
j
)
T
u
T
J
=
˃(
ˆ
y
−
w
)
(
ˆ
y
−
w
)
+
ˉ
u
(5.9)
for all
j
in the GPC cost
function. Moreover, the sequence of predicted outputs can be estimated according to
Eq.
5.10
.
In the previous equation usually,
˃(
j
)
=
˃
and
ˉ(
j
)
=
ˉ
y
ˆ
=
f
+
G
u
(5.10)
