Civil Engineering Reference
In-Depth Information
Therefore, the model of the process plays a decisive role in the controller, and
thus, the selected model must be able of precisely capture the dynamics of the process
and, at the same time, it should be simple to use and understand. Another important
factor of MPC strategies is the optimiser since it provides the future control signals
by minimising the cost function. As MPC is not a unique technique but rather a set of
different methodologies, there are many types of models used in various formulations
(Camacho and Bordons
2004
). The most used ones are: (i) impulse response models
which are usedwithMPC techniques such asModel Algorithmic Control (MAC), and
as a special case in Generalised Predictive Control (GPC) and Extended Prediction
Self Adaptive Control (EPSAC); (ii) the MPC techniques which used step response
models as in the case of Dynamic Matrix Control (DMC), (iii) these methodologies
based on state-space models such as Predictive Functional Control (PFC), and finally,
(iv) MPC strategies which make use of transfer functions such as EPSAC or GPC
among others. Actually GPC, which was presented by Clarke et al. (
1987a
,
b
) has
become one of the most popular MPC methods in both industry and research.
5.2.2.1 Prediction Model
As mentioned previously, the GPC methodology makes use of the transfer func-
tion concept,
G
, where
z
−
1
is the backward shift operator.
Therefore, the output of the system can be expressed as is shown in Eq.
5.3
.
z
−
1
z
−
1
z
−
1
(
)
=
B
(
)/
A
(
)
z
−
1
z
−
1
(
)
(
)
=
(
)
(
)
+
(
)
A
y
k
B
u
k
n
k
(5.3)
z
−
1
where
u
(
k
)
and
y
(
k
)
are the input and output sequences in discrete time
k
,
A
(
)
,
z
−
1
are polynomials in the
z
−
1
as follows
B
(
)
z
−
1
a
1
z
−
1
a
2
z
−
2
a
n
a
z
−
n
a
A
(
)
=
1
+
+
+···+
(5.4)
z
−
1
b
1
z
−
1
b
2
z
−
2
b
n
b
z
−
n
b
B
(
)
=
b
0
+
+
+···+
(5.5)
Hence, the output prediction can be estimated according to Eq.
5.6
z
−
1
B
(
)
ˆ
y
(
k
+
j
|
k
)
=
u
(
k
+
j
|
k
)
(5.6)
z
−
1
A
(
)
This representation is also valid for unstable processes, and besides, it takes advan-
tage of the necessity of a few parameters although prior knowledge of the modelled
process is needed. Moreover, at the same level of importance as the selection of a
certain model of the process is the choice of a model able to accurately represent
any disturbance. One of the most used models is the Auto-Regressive Integrated
Moving Average (ARIMA), in which, disturbances (existing differences between
the measured output and the one estimated by the model) are estimated just as can
be observed in Fig.
5.7
.
