Specifications

Design Controller Using the Design Tool
3-27
the number of plant outputs, w
y
j
is the weight for output j, and the term [r
j
(k + i) – y
j
(k +
i)] is a predicted deviation for output j at interval k + 1.
The weights must be zero or positive. If a particular weight is large, deviations for that
output dominate S
y
(k). One of the controller's objectives is to minimize S
y
(k). Thus, a
large weight on a particular output causes the controller to minimize deviations in that
output (relative to outputs having smaller weights).
For example, the default values used to produce Plant Outputs for T Setpoint Scenario
with Added Data Markers specify equal weights on each output, so the controller is
trying to eliminate deviations in both, which is impossible. On the other hand, the design
of Improved Setpoint Tracking for CSTR Temperature uses a weight of zero on the
second output, so it is able to eliminate deviations in the first output.
Note The second output is unmeasured. Its predictions rely on the plant model and the
temperature measurements. If the model were reliable, we could hold the predicted
concentration at a setpoint, allowing deviations in the reactor temperature instead. In
practice, it would be more common to control the temperature as done here, using the
predicted reactant concentration as an auxiliary indicator.
You might expect equal output weights to result in equal output deviations at steady
state. Plant Outputs for T Setpoint Scenario with Added Data Markers shows that this
is not the case. The reason is that the controller is trying to satisfy several additional
objectives simultaneously.
Rate Weights
One is to minimize the weighted sum of controller adjustments, calculated according to
S k w u k i
u j
u
j
j
n
i
M
mv
D
D
==
= D + -
{ }
ÂÂ
( ) ( ) 1
2
11
where M is the number of intervals in the control horizon, n
mv
is the number of
manipulated variables,
D + -u k i
j
( )1
is the predicted adjustment in manipulated
variable j at future (or current) sampling interval k + i – 1, and
w
j
uD
is the weight on this
adjustment, called the rate weight because it penalizes the incremental change rather