The control objective of the jacketed reactor case study, shown below and
discussed in detailed
in this article, is
disturbance rejection. More specifically, we seek a controller
design that will minimize the impact on reactor operation when the
temperature of the liquid entering the cooling jacket changes.
As labeled in the graphic (click for large view), the important variables for this
case study include:
CO = signal to valve that adjusts cooling jacket liquid flow rate
(controller output, %)
PV = reactor exit stream temperature (measured process variable,
oC)
SP = desired reactor exit stream temperature (set point, oC)
D
= temperature of cooling liquid entering the jacket (major
disturbance, oC)
Controller Design and Tuning Recipe
As with any control project, we follow our
controller design and tuning recipe:
1. Establish the design level of operation (the normal or expected
values for set point and major disturbances)
2. Bump the process and collect controller output (CO) to process
variable (PV) dynamic process data around this design level
3. Approximate the process data behavior with a first order plus
dead time (FOPDT) dynamic model
4. Use the model parameters from step 3 in rules and correlations to
complete the controller design and tuning.
Step 1: Design Level of Operation (DLO)
The details and discussion for our DLO are
presented in this article and are summarized:
▪
Design PV and SP = 90 oC with approval for brief dynamic testing
of
±2 oC
▪
Design D = 43 oC with spikes up to 50 oC
Step 2: Collect Data at the DLO
The point of bumping a process is to generate and collect dynamic
process data. To be of value, this data must clearly
reveal the cause and effect relationship between how changes in the CO
signal force a response in the measured PV.
• Data Must Be Wire Out to Wire In
The data must be
collected from the controller's viewpoint, since the
controller will be making all decisions once in automatic mode. The
controller only knows about the state of the process from the PV signal arriving
on the "wire in" from the sensor. It can only impact the process with
the CO signal it sends on the "wire out" to the final control element (e.g.,
a valve). All devices, mechanisms and instruments that affect the signal in the complete "wire-out
to wire-in" CO to PV loop must be accounted for in the recorded data.
• Process Should Be At Steady State
To further isolate the pure cause and effect relationship, we must wait
until the CO, PV and major disturbances have settled out and are reasonably constant before
bumping the process. This provides us a clean slate and some confidence that
observed PV responses during the dynamic test are a direct result of the CO
bumps. In the perfect world, this steady state will be near our DLO.
• Center Data Around the DLO
Like most processes with streams comprised of gases, liquids, powders,
slurries and melts, the jacketed stirred reactor displays a
nonlinear or changing behavior as operating level changes.
In fact, it is this nonlinear character that leads us to
specify a design level of operation in the first place. To the extent possible
in our manufacturing environment, we should collect our process data centered around the DLO.
A model fit of such data (step 3) will then average out the nonlinear effects
and provide a controller equally balanced to address process movement both
up and down.
• The PV Response Should Dominate the Noise
The CO bump must be far enough and fast enough to
force a clear "cause and effect" response in the PV. And this PV response
must clearly dominate the
measurement noise.
Two popular open loop (manual mode) methods for generating dynamic process
response data as
described above include the step test and the doublet test. We show
data for the jacketed reactor collected around the DLO using both methods in step 3
below.
Step 3: Fit a FOPDT dynamic model to Process Data
The design level of operation (DLO) includes the normal or expected value of
both the PV and the major disturbances. The primary disturbance (D) of
interest in this study is cooling jacket inlet temperature. While not shown in the
plots below, D is steady at its design value of 43 oC during
these bump tests.
◊ Step Test
Step tests have value because we can
analyze the plot data by hand to compute the first order plus dead time
(FOPDT) model parameters Kp, Tp and
Өp. It is important that
we have proper values for these model parameters because they are used in
the rules and correlations of step 4 from the recipe to
complete the controller design and tuning (examples
here and
here).One disadvantage of a step test is
that it is conducted in manual mode. Opening a control loop is not always
possible on an industrial process. As shown in the plot below, the PV is away from the DLO
for an extended period of time. This can create profitability concerns from
off-spec production and perhaps even safety concerns if constraints become
violated. Closed loop or automatic mode
testing using set point bumps minimizes many of these concerns and is
discussed later.
To collect process data in manual mode that will “average out” the nonlinear effects
around our design level of operation, we
start the test at steady state with the PV on one side of the DLO.
Then, as shown in the plot below, we step the CO so that the measured PV moves across to
settle on the other side of the DLO.
We acknowledge that it may be unrealistic to attempt such a precise step
test in some production environments. But we should understand that the
motivation is to obtain an approximating FOPDT model of the dynamic response
behavior that averages out the changing nature of the process as it moves
across the expected range of operation.
A FOPDT model fit of the CO to PV data using
Control Station software is shown as the yellow trace in the plot above.
It clearly tracks the measured PV data quite closely. The computed model
parameters are listed below the plot.
Such results can be obtained in a few mouse clicks, and the visual confirmation
that "model equals data" gives us confidence that we indeed have a
meaningful description of the dynamic process behavior. This, in turn, gives
us confidence that the subsequent controller design will provide the
performance we desire.
◊ Doublet Test
A doublet test, shown below, is two CO pulses performed in rapid
succession and in opposite direction. The second pulse is implemented as
soon as the process has shown a clear response to the first pulse that
dominates the noise in the PV. It is not necessary to wait for the process
to respond to steady state for either pulse.
A FOPDT model fit of the CO to PV data using
the software is shown as the yellow trace in the plot above. It also tracks
the measured PV data quite closely.
A doublet test offers attractive benefits, including that:
▪ it starts
from and quickly returns to the DLO,
▪ it produces data above and below
the DLO to "average out" the nonlinear effects,
▪ the PV
stays close to the DLO, minimizing off-spec production and safety concerns.
For these reasons, a doublet is preferred by many practitioners for open
loop testing.
|
The FOPDT Model
By approximating the dynamic behavior of the jacketed reactor process with a first order plus dead time (FOPDT) model, we
quantify
those essential features that are fundamental to control.
| Aside: the FOPDT dynamic model is a linear, ordinary
differential equation describing how PV(t) responds over time to changes
in CO(t): 
Where in this case study: t [=] min,
PV(t) [=] °C, CO(t – Өp) [=] %
|
The FOPDT model describes how the PV
will respond to a change in CO via the:
▪
Process gain, Kp, that describes the direction and how far the PV will travel,
▪
Time constant, Tp, that states how fast the PV moves after it begins its response,
▪
Dead time, Өp,
that is the delay that occurs from when CO is changed until when the PV begins its response.
Based on both the step and doublet test model fits shown above, we
conclude that, when operating near the DLO, the jacketed reactor process
dynamics are described:
· Kp =
