Environmental Engineering Reference
In-Depth Information
5.5.1 Models for Control Design
Process models are the core of many control strategies. They can be used for tuning
a strategy or as an integral part of the strategy itself.
Linear models are widely used to represent the process dynamic around a given
operating point. These models, as described in Chapter 4, have a given structure and
a set of parameters that can be adjusted to represent a specific process. Chapter 4 has
provided a thorough review of techniques for finding both structure and parameters
for models. This section focuses on the model structure that is suitable for predictive
control design.
A basic structure generally used for the design of discrete time controllers is an
ARMAX structure (additional structures are commented upon in Chapter 7):
z
−
1
z
−
d
B
z
−
1
z
−
1
A
(
)
y
(
t
)=
(
)
u
(
t
−
1
)+
C
(
)
w
(
t
),
(5.34)
where
y
(
t
)
and
u
(
t
)
are the model output and input, respectively, and
w
(
t
)
is a white
z
−
1
z
−
1
z
−
1
noise signal. The polynomials
A
(
)
,
B
(
)
and
C
(
)
are defined as follows:
z
−
1
a
1
z
−
1
a
n
a
z
−
n
a
;
A
(
)=
1
+
+···+
z
−
1
b
1
z
−
1
b
n
b
z
−
n
b
B
(
)=
b
0
+
+···+
n
b
≤
n
a
;
(5.35)
z
−
1
c
1
z
−
1
c
n
c
z
−
1
n
c
C
(
)=
1
+
+···+
n
c
≤
n
a
.
Parameters for these polynomials can be estimated using the techniques described
in Chapter 4. However, in the model predictive control framework,
C
z
−
1
is con-
sidered as a design parameter as it has a direct effect on closed loop sensitivity [17].
In order to account for the effect of unmeasurable disturbances normally found
in practice (such as step-like disturbances), the so-called controlled auto-regressive
integrated moving average (CARIMA) structure has been proposed:
(
)
z
−
1
C
(
)
z
−
1
z
−
d
B
z
−
1
A
(
)
(
)=
(
)
(
−
)+
(
),
y
t
u
t
1
w
t
(5.36)
Δ
(
z
−
1
)
z
−
1
z
−
1
. The above equation can also be written as
where Δ
(
)=
1
−
z
−
1
z
−
1
z
−
d
B
z
−
1
z
−
1
z
−
1
A
(
)
Δ
(
)
y
(
t
)=
(
)
Δ
(
)
u
(
t
−
1
)+
C
(
)
w
(
t
)
(5.37)
and in the time domain:
n
a
i
=
1
a
i
Δ
y
(
t
−
i
)=
n
b
i
=
1
b
i
Δ
u
(
t
−
i
−
d
−
1
)+
n
c
i
=
1
c
i
w
(
t
),
(5.38)
where Δ
y
. These models can be
easily extended to systems with more inputs and outputs. For instance, if the system
has a control input
u
(
t
)=
y
(
t
) −
y
(
t
−
1
)
and Δ
u
(
t
)=
u
(
t
) −
u
(
t
−
1
)
(
t
)
and a measurable disturbance
p
(
t
)
acting on the process
input, then a suitable model could be
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