Environmental Engineering Reference
In-Depth Information
objective function minimizes the error between predicted outputs and the set-points
during the prediction horizon as well as the control effort Δ
u
during the control
horizon. The function may be expressed as
N
p
∑
N
u
j
=
1
[
2
2
J
=
N
1
[
w
(
t
+
j
)−
y
(
t
+
j
|
t
)]
+
λ
Δ
u
(
t
+
j
|
t
)]
,
(7.1)
j
=
(
+
|
)
+
where
y
t
j
t
is the expected value of the predicted output at instant
t
j
with
(
+
)
+
(
)=
known history up to time
t
,
w
t
j
is the set-point at instant
t
j
,andΔ
u
t
u
the manipulated variable at instant
t
. Parameter λ weights
the control in the optimization problem.
N
1
,
N
u
and
N
p
define the prediction and
control horizons.
The optimization process may involve hard or soft constraints. For linear uncon-
strained systems this optimization problem is tractable and convex and can be solved
analytically, but in general applications it is common to take into account constraints
or non-linearities in the process, and in such cases the optimization problem must
be solved using iterative numerical methods [39].
A fundamental element in MPC is the model used to characterize the dynamic
behavior of the process. The origins and formulations of such models are diverse,
but may be classified as follows:
(
t
)−
u
(
t
−
1
)
with
u
(
t
)
•
phenomenological or first principle models, in the vast majority of cases non-
linear and continuous time;
•
models obtained through numerical adjustments based on operating data using
discrete time series, either linear or non-linear.
Notable among the non-linear models are neural networks, which are used to
numerically approximate a highly complex non-linear function by interconnecting
simpler processing elements such as adders, multipliers and sigmoid functions. As
with linear time series models, the neural model must be calibrated by adjusting its
parameters to the operating data, a task generally performed by a backpropagation
gradient algorithm [40].
Numerous alternative versions of MPC algorithms have been developed based
on the foregoing concepts. They differ from one another as regards the model used
to represent the process and disturbances, the cost function to be minimized, the
constraints applied to each variable and the optimization algorithm employed. MPC
has achieved widespread recognition over the last 25 years in process industries,
where it is currently utilized in more applications than any other advanced control
technique. Even though the great majority of processes display non-linear behavior,
very few MPC developments rely on non-linear models. If such a model is used, or
if the objective function is not quadratic or subject to constraints, the optimization
problem will not be convex, difficulties will arise regarding the existence of local
optima and computational complexity will be much greater.
The popularity of MPC among academics is evidenced by the large number of
topics and articles published on the subject. Unlike many other process control tech-
niques, however, MPC is also widely accepted in the industry. This is amply attested
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