Process Control
Reprinted with permission from CEP (Chemical Engineering Progress), June 2008.
Copyright © 2008 American Institute of Chemical Engineers (AIChE).
Improve Control
of Liquid
Level Loops
Use this tuning recipe
for the classic integrating
process control challenge.
Robert Rice
Douglas J. Cooper
Control Station, Inc.
B
ecause most processes are self-regulating, it can
sometimes be challenging to tune a controller for an
integrating process. The principal characteristic of a
self-regulating process is that it naturally seeks a steadystate operating level if the controller output and disturbance
variables are held constant for a sufficient period of time.
For example, a car’s cruise control is self-regulating.
By holding the fuel flow to the engine constant (assuming
the car is traveling on flat ground on a windless day), the
car is maintained at a constant speed. If the fuel flowrate
PV
Self-Regulating
CO
PV tracks up and
down with CO
PV
IntegratingBehavior
CO
PV at new value
when CO returns
Time
■ Figure 1. Integrating processes are characterized by the process
variable moving to a new value when the controller output returns to
its starting value. In an ideal self-regulating process, the process variable returns to its original value when the controller output is stepped
back down.
54
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June 2008
CEP
is increased by a fixed amount, the car will accelerate and
then settle at a different constant speed.
The temperature of a process stream exiting a heat
exchanger is also self-regulating. If the shellside cooling
fluid flowrate is held constant and there are no significant
external disruptions, the tubeside exit stream temperature
will settle at a constant value. If the cooling flowrate is
increased, allowed to settle, and then returned it to its
original value, the tubeside exit stream temperature will
move to a new operating level during the increased
flowrate and then return to its original steady-state.
Tanks that have a regulated exit flow stream do not naturally settle at a steady-state operating level. This is a
common example of what process control practitioners
refer to as a non-self-regulating (or integrating) process.
Integrating processes can be remarkably challenging to
control. This article explores their distinctive behaviors.
Armed with this knowledge, you may come to realize
that some of your facility’s more-difficult-to-control level,
temperature, pressure and other loops have such character.
Integrating (non-self-regulating)
behavior in manual mode
The top plot of Figure 1 shows the open-loop (manual
mode) behavior of a self-regulating process. In this idealized response, the controller output (CO) signal and measured process variable (PV) are initially at steady state. The
CO is stepped up from this steady state and then back
down. As shown, the PV responds to the step, and ultimately returns to its original operating level.
The bottom plot of Figure 1 shows the open-loop
response of an ideal integrating process. The distinctive
behavior occurs when the CO returns to its original
value and the PV settles at a new operating level.
Level control in a surge tank for a single-valve kegging (SVK) system
urge tanks are designed to counteract fluctuaSingle Valve Kegging System
tions in flow characteristics that would otherDraft
wise disrupt upstream or downstream systems.
Beer
Surge tanks are often installed between two
Storage
process systems with incompatible flow patterns
Beer Pump
to provide flow smoothing. The “wild stream” has
flow control requirements that are difficult to influence, and the controller then adjusts the controlled
Beer
FC
LIC
Surge
stream to maintain the liquid level in the tank.
Tank
The primary objective of a surge tank is to
absorb the fluctuations of the wild stream without
significantly impacting the controlled stream. To
Beer Valve
best achieve this result, the level in a surge tank
Racker Pump
should be allowed to swing between an upper
and a lower level limit. The more the tank is
Surge Tank Performance
70
allowed to swing, the larger the surge capacity of
the tank. Often, however, these swinging tanks
Upper Constraint
are viewed as poor performers and are then
60
tuned for tight performance, counteracting the
intended design objective.
50
A major beer brewer uses an SVK system to
fill several lanes of kegs (top). Because the kegLower Constraint
40
filling lanes are operated in an on/off fashion, the
Aggressively Tuned PI Controller
Conservatively Tuned PI Controller
wild stream flowrates requested by the SVK system can quickly vary from 0 to 180 gal/min
40
depending on the number of kegs being filled at
20
any point in time.
140
As shown in the figure on the bottom, adjust120
ing the flow of beer pumped from the large stor100
age tanks controls the level in the surge tank.
80
60
Due to the sensitive nature of beer and of the
40
analytical instrumentation involved, a surge tank
20
is installed to dampen the large demand fluctua0
0
5
10
15
20
25
30
tions required by the keg-filling system. By allowTime,
h
ing the surge tank to swing more freely between
its constraints, the control changes sent to the
■ Tuning a control system for a beer-keg filling line (top) to allow the surge tank to fluctuate more
large storage tank are reduced.
between its constraints (bottom) reduces the control changes sent to the storage tank.
S
FT02
CT01
FLOW
CO2
CT04
FT01
FLOW
TT01
PT01
AT01
PT02
CT02
PRES
BALL
PSI
O2
TT02
AT02
PT03
TEMP
pH
PRES
CT03
BALL
Wild-Stream
Flow, gal/min
CO, %
Level PV / SP, %
TEMP
O2
LIC
The integrating behavior plot is somewhat misleading, as
it implies that for such processes, a steady controller output
will produce a steady process variable. While this is possible with idealized simulations like that used to generate the
plot, such “balance point” behavior is rarely found in integrating processes in industrial operations.
More realistically, if left uncontrolled, the lack of a balance point means that the process variable of an integrating
process will naturally tend to drift up or down, possibly to
extreme and even dangerous levels. Consequently, integrating processes are rarely operated in manual mode for long.
P-only control behavior is different
To appreciate the difference in controlled behavior for
integrating processes, first consider the proportional, or
P-only, control of an ideal self-regulating simulation. As
shown in Figure 2 (p. 56), when the setpoint (SP) is initially at the design level of operation (DLO) in the first
moments of operation, then PV equals SP (the DLO is
where the setpoint and process variable are expected to be
during normal operation when the major disturbances are
at their normal or typical values).
The setpoint is then stepped up from the DLO on the
left side of the plot. The simple P-only controller is unable
to track the changing SP, and a steady error, called offset,
results. The offset grows as each step moves the SP farther
away from the DLO.
Midway through the process, a disturbance occurs, as
shown in the middle of the plot. (Its size was predetermined for this simulation to eliminate the offset.) When
the SP is then stepped back down (on the right) the offset
shifts, but again grows in a similar and predictable pattern.
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June 2008
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55
65
60
55
50
60
55
50
45
65
3) … shifting
the offset
1) Offset grows …
2) … then disturbance
load changes …
50
100
200
300
400
500
600
Time
D, %
CO, % PV and SP, %
■ Figure 2. P-only control of an ideal self-regulating process shifts
the offset caused by disturbances.
65
60
55
50
1) No offset …
Controller output behavior is telling
The CO plots in Figures 2 and 3 demonstrate an interesting feature that distinguishes self-regulating from integrating
process behavior. In the self-regulating process plot, the
average CO value tracks up and then down as the SP steps
up and then down. In the integrating process plot, the CO
spikes with each SP step, but then in a most unintuitive fashion, returns to the same steady value. It is only the change in
the disturbance flow that causes the average CO to shift midway through the plot, where it then remains centered around
the new value for the remainder of the SP steps.
3) … producing
sustained offset
60
50
40
54
2) ...then disturbance
load changes …
52
50
40
80
120
160
Time
56
54
52
50
48
CO, %
60
Kc = 0.3
Modest oscillation
Kc = 1.2
PV oscillates
55
50
45
57
54
51
48
45
Kc = 1
PV oscillates
Kc = 4
Kc = 8
Overshoot but PV oscillates
no oscillation
100
80
60
40
20
150
300
450
600
750
75
Time
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June 2008
150
225
300
Time
■ Figure 4. Increasing controller gain (Kc) for the PI control of an ideal
self-regulating process causes the process variable response to move
from a sluggish to an oscillatory response behavior.
56
PI control behavior is different
The dependent, ideal form of a proportional-integral (PI)
controller (1) is one of numerous algorithms that are widely
employed in industrial practice:
K
CO = CObias + K c e(t ) + c ∫ e(t )dt
(1)
Ti
CO, %
PV and SP, %
■ Figure 3. Unlike an ideal self-regulating process, P-only control for
an ideal integrating process shifts the baseline operation of the
process, producing a sustained offset even as the setpoint returns to
its original value.
Kc = 0.3
No oscillation
With this as background, consider an ideal integrating
process simulation under P-only control. Even under simple
P-only control, as shown on the left of Figure 3, the process
variable is able to track the setpoint steps with no offset. This
behavior can be quite confusing, as it does not fit the expected behavior of the more-common self-regulating process.
This happens because integrating processes have a natural accumulating character (and is, in fact, why “integrating
process” is used as a descriptor for non-self-regulating
processes). Since the process integrates, it appears that the
controller does not need to.
Yet the setpoint steps in the right of Figure 3 show this is
not completely correct. Once a disturbance shifts the baseline
or balance-point operation of the process (shown roughly at
the midpoint in the plot), an offset develops and remains constant even as SP returns to its original design value.
PV and SP, %
D, %
CO, % PV and SP, %
Process Control
CEP
■ Figure 5. For PI control of an integrating process, oscillatory
response behavior can occur both when the controller gain (Kc) of a
PI controller is too small and when it is too large.
Figure 4 shows an ideal self-regulating process simulation that is controlled using this PI algorithm. Reset time,
Ti, is held constant throughout the simulation while controller gain, Kc, is doubled and then doubled again. As Kc
increases, the controller becomes more active, and, as
expected, this increases the tendency of the PV to display
oscillating (underdamped) behavior.
For comparison, consider PI control of an ideal integrating process simulation as shown in Figure 5. Ti, is
again held constant while Kc, is increased. A counter-intuitive result is that as Kc becomes small and as it becomes
large, the PV begins displaying an underdamped (oscillating) response behavior. While the frequency of the oscillations is clearly different between a small and large Kc,
when seen together in a single plot, it is not always obvious in what direction the controller gain needs to be
adjusted to settle the process, in particular, when seeing
such unacceptable performance on a control room display.
A tuning recipe provides benefit
One of the biggest challenges for practitioners is recognizing that a particular process shows integrating behavior prior
to starting a controller design and tuning project. This, like
most things, comes with training, experience and practice.
Once in automatic mode, closed-loop behavior of an
integrating process can be unintuitive, and even confounding. Trial-and-error tuning methods can lead one in circles
trying to understand what is causing the unacceptable
control performance.
A formal controller design and tuning procedure for integrating processes helps overcome these issues in an orderly
and reliable fashion. Best practice is to follow a formal recipe
when designing and tuning any PID controller. A recipebased approach causes less disruption to the production
schedule, wastes less raw material and utilities, requires less
personnel time, and generates less off-specification product.
The controller design and tuning recipe for integrating
processes contains four steps, as follows (2):
1. Establish the design level of operation (the normal or
expected values for the setpoint and major disturbances).
2. Bump the process, and collect dynamic process data of
the process variable response to changes in controller output.
3. Approximate the process data behavior with a firstorder-plus-dead-time integrating (FOPDT integrating)
dynamic model.
4. Use the model parameters generated in step 3 and the
correlations in Table 1 to complete the controller tuning.
It is important to recognize that real processes are
more complex than the simple FOPDT integrating model.
In spite of this, the model does provide an approximation
of process behavior that is sufficiently rich in dynamic
Table 1. Use tuning correlations for
PI and PID controllers for integrating processes.
PI
PID
Kc
2Tc + θ p
Ti
Td
1
K *p (Tc + θ p )2
2Tc + θ p
2Tc + θ p
1
*
K p (Tc + 0.5θ p )2
2Tc + θ p
0.25θ p 2 + Tcθ p
2Tc + θ p
information to yield reliable and predictable control
performance when used with the rules and correlations
in Step 4 of the recipe.
The FOPDT integrating model
The FOPDT dynamic model commonly used to approximate self-regulating dynamic process behavior has the form:
dPV (t )
Tp
(2)
+ PV (t ) = K p CO(t − θ p )
dt
where Kp is the steady-state process gain, Tp is the overall
process time constant, and θp is the process dead time. Yet
this model cannot describe the kind of integrating process
behaviors explored above. These dynamic behaviors are
better described with the FOPDT integrating model form:
dPV (t )
= K *p CO(t − θ p )
dt
(3)
It is interesting to note when comparing these two models that individual values for the familiar process gain, Kp,
and process time constant, Tp, are not separately identified
for the FOPDT integrating model. Instead, an integrator
gain, Kp*, is defined that has units of the ratio of the process
gain to the process time constant, or:
K *p [=]
Kp
Tp
or
K *p [=]
PV
CO × time
(4)
Tuning correlations for integrating processes
The FOPDT integrating model parameters Kp* and θp can
be computed using a graphical analysis of plot data, or in an
industrial setting by automated analysis using a commercial
software package. Once the model parameters are known, the
tuning values for the dependent, ideal PI form, Eq. 1, as well
as the popular PID algorithm form, can be calculated:
dPV
K
CO = CObias + K c e(t ) + c ∫ e(t )dt − K cTd
(5)
dt
Ti
For integrating processes there is no identifiable
process time constant in the FOPDT integrating model.
Thus, dead time, θp, is used as the baseline marker of time
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57
Process Control
Disturbance
Flow, L/min
15.3
2.5
D
Setpoint, m
Tank
Level, m
4.0
LC
4.01
Controller
Output, %
70.0
Discharge
Flow, L/min
17.8
■ Figure 6. Simulated pumped-tank level control in automatic mode
uses a throttling valve to adjust the process variable, the liquid level
in the tank.
for tuning. Specifically, θp is used as the basis for computing the closed-loop time constant, Tc.
Building on the popular internal model control (IMC)
approach to controller tuning, the closed-loop time constant is computed as Tc = 3θp (3).
The controller tuning correlations for integrating
processes use this Tc, as well as the Kp* and θp from the
FOPDT integrating model fit, in the correlations of Table 1.
A simulated example — the pumped-tank
A pumped-tank simulation illustrates the design and tuning of a controller for an integrating process. As shown in
Figure 6, the process has two liquid streams feeding the top
of the tank and a single exit stream pumped out of the bottom. The measured process variable (PV) is the liquid level
in the tank. To maintain the liquid level, the controller output (CO) signal adjusts a throttling valve at the discharge of
a constant-pressure pump to manipulate the flowrate out of
the bottom of the tank. This approximates the behavior of a
centrifugal pump operating at relatively low throughput.
Note that a pump strictly regulates the discharge
flowrate out of the tank. As a consequence, the physics do
not naturally work to balance the system when any of the
stream flowrates change. This lack of a natural balancing
behavior is why the pumped tank is classified as an integrating process. If the total flow into the tank is more than
the flow pumped out, the liquid level will rise and continue to rise until the tank fills or a stream flow changes. If
June 2008
… and tank drains
2
0
Exit flow increases …
80
75
70
20
CO
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CO, %
SP
PV
58
4
Level, m
Brine Feed
Flow, L/min
CEP
25
30
35
40
45
Time, min
■ Figure 7. With the simulated level control in manual mode,
the liquid level falls as the controller increases the flowrate out of the
bottom of the tank.
the total flow into the tank is less than the flow pumped
out, the liquid level will fall and continue to fall.
Figure 7 is a plot of the pumped-tank behavior with the
controller in manual mode (open-loop). The CO signal is
stepped up, increasing the discharge flowrate out of the
bottom of the tank. The flow out becomes larger than the
total feed into the top of the tank and, as shown, the liquid
level begins to fall. As the situation persists, the liquid
level continues to fall until the tank is drained. The sawtoothed pattern occurs when the tank is empty because the
pump briefly surges every time enough liquid accumulates
for it to regain suction.
Figure 7 does not show that if the controller output were
to be decreased enough to cause the flowrate out to be less
than the flowrate in, the liquid level would rise until the
tank was full. If this were a real process, the tank would
overflow and spill, creating safety and profitability issues.
Graphical modeling of integrating process data
The graphical method of fitting an FOPDT integrating
model to process data requires a data set that includes at
least two constant values of controller output, CO1 and
CO2. As shown in Figure 8 for the pumped tank, both must
be held constant long enough that a slope trend in the PV
response (tank liquid level) can be visually identified.
An important difference between the traditional process
reaction curve graphical technique for self-regulating
processes and integrating processes is that integrating
processes need not start from a steady-state value before a
bump is made to the CO. The graphical technique discussed
here is only concerned with the slopes (or rates of change)
in PV and the constant controller output signal that caused
each PV slope.
The FOPDT integrating model describes the PV behavior
at each value of constant controller output, CO1 and CO2, as:
Slope2
Level, m
4.8
4.4
Slope1
4.0
75
CO1
70
(27, 5.2)
5.2
CO, %
CO, %
Level, m
5.2
CO2
65
4.8
(36, 4.6)
4.4
(24, 4.8)
4.0
75
CO1 = 65
70
CO2 = 75
65
20
25
30
35
40
20
25
Time, min
Level, m
(6))
4.8
4.4
θP = 1 min
4.0
2
1
CO, %
Subtracting and solving for Kp* yields:
Slope2 − Slope1
=
CO2 − CO1
40
5.2
and
dPV
dPV
−
dt
dt
2
K *p =
CO2 − CO1
35
■ Figure 9. The slopes are calculated from bump test data to
compute the integrator gain, Kp*.
dPV
= K *p CO1 (t − θ p )
dt 1
= K *p CO2 (t − θ p )
30
Time, min
■ Figure 8. To perform a manual-mode bump test of the pumped-tank
process, the controller outputs must be held constant long enough to
show the slope trend in the PV response.
dPV
dt
(31, 5.2)
(7)
75
70
65
20
Graphical modeling of pumped-tank data
Computing integrator gain. The values of the open-loop
data from the pumped-tank simulation in Figure 8 are displayed in Figure 9. The CO is stepped from 71% down to
65%, causing the liquid level (the PV) to rise. The controller output is then stepped from 65% up to 75%, causing
a downward slope in the liquid level.
The slope of each segment is calculated as the change in
tank liquid level divided by the change in time. From the
plot data, Slope1 is calculated to be 0.13 m/min and Slope2
as –0.12 m/min. Using the slopes with their respective CO
values yields the integrator gain, Kp* = –0.025 m/%-min.
Computing dead time. The dead time, θp, is calculated as
the difference in time from when the CO signal was stepped
and when the measured PV starts to exhibit a clear response
to that change. From the plot in Figure 10, the pumped-tank
dead time is estimated be θp = 1.0 min.
PI control study
Now the controller design and tuning recipe for integrating processes can be used to design and test a PI controller.
Determining bias value, CObias. A commercial controller
25
30
35
40
Time, min
■ Figure 10. The difference in time from when the CO signal is
stepped and when the measured PV starts to show a clear response
to that change provides an estimate of the dead time from the bump
test data.
is normally put into practice using bumpless transfer —
that is, when switching to automatic control, SP is initialized to the current value of PV and CObias to the current
value of CO. By choosing the current operation as the
design state at switchover, the controller needs no corrective actions and it can smoothly engage.
Controller gain, Kc, and reset time, Ti. The first step in
using the IMC correlations listed in Table 1 is to compute
Tc, the closed-loop time constant. Tc describes how active
the controller should be in responding to a setpoint change
or in rejecting a disturbance. For integrating processes, the
design and tuning recipe suggests:
Tc = 3θp = 3 × 1.0 min = 3 min
The PI controller gain, Kc, and reset time, Ti, are
computed as:
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59
t the top of a petroleum-refinery distillation column
(below, top), vapor enters a condenser and flows as liquid into a reflux drum. The liquid then exits the drum and
either returns to the column as reflux or exits the unit as distillate. The control strategy design for the column is to maintain a fixed distillate flow and adjust the level of the reflux
drum through manipulation of the reflux flowrate returning to
the top of the column.
Distillation columns are very sensitive unit operations
with very slow response times (long time constants). If the
level controller is tuned aggressively for tight setpoint tracking, large and rapid reflux flow changes could dramatically
impact column efficiency and stability. Thus, the reflux drum
needs to be tuned for conservative control actions while
maintaining the level constraints.
Using the tuning procedure outlined in this article, the
reflux drum level not only tracks closer to setpoint — it does
so with 95% less controller output movement.
A
D, %
Level control in a distillation column reflux drum
CO, % PV and SP, %
Process Control
5.0
4.8
4.6
4.4
Accept some PV overshoot …
… to get disturbance rejection
80
70
60
inlet flow disturbance
4
3
2
1
10
20
30
40
50
60
70
80
Time
Sample Time, T = 1 s
■ Figure 11. A PI controller provides setpoint tracking and
disturbance rejection.
Kc =
1 2Tc + θ p
⋅
K *p (Tc + θ p )2
Ti = 2Tc + θ p
(8)
Substituting the Kp*, θp and Tc identified above into
these tuning correlations, we compute:
Condenser
LIC
Distillation
Column
Kc =
Reflux Drum
FIC
L
FIC
D
Distillate Valve
Reflux Valve
Level PV / SP, %
Reflux Drum Level Performance
60 Upper Constraint
50
40
Reflux Flow, %
Ti = 2( 3) + 1 = 7 miin
Recall that the P-only control of an integrating process
(Figure 3) can provide a rapid setpoint response with no
overshoot until a disturbance changes the balance point of
the process. As labeled in Figure 11, the PI control setpoint response now includes some overshoot.
The benefit of integral action is that when a disturbance
occurs, a PI controller can reject the upset and return the
process variable to its setpoint. This is because the constant summing of integral action continues to move the
controller output until the controller error is driven to zero.
Thus, PI control requires accepting some overshoot during
setpoint tracking in exchange for the ability to reject disturbances. In many industrial applications, this is
CEP
considered a fair trade.
Lower Constraint
30
90
Aggressively Tuned PI Controller
Conservatively Tuned PI Controller
Literature Cited
80
70
60
50
40
30
20
0
4
8
12
16
20
24
28
Time, h
■ Using the tuning recipe for reflux drum (top) level control improves the
performance (bottom) with 95% less controller
output movement.
60
1 2( 3) + 1
= –18 m/%
–0.025 ( 3 + 1)2
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1. Cooper, D. J., ed., “Practical Process Control,”
www.controlguru.com (2008).
2. Rice, R., and D. J. Cooper, “A Rule-Based Design
Methodology for the Control of Non-Self-Regulating
Processes,” Proc. ISA Expo 2004, ISA CD Vol. 454,
TP04ISA076 (2004).
3. Arbogast, J. E., and D. J. Cooper, “Extension of
IMC Tuning Correlations for Non-Self-Regulating
(Integrating) Processes,” ISA Transactions, 46,
pp. 303 (2007).
Glossary and Nomenclature
CO
CObias
DLO
e(t)
FOPDT
IMC
Kc
Kp*
PV
SP
SVK
T
Tc
Ti
Tp
θp
= controller output signal
= controller bias or null value
= design level of operation
= current controller error, defined as SP – PV
= first-order-plus-dead-time model
= internal model control
= controller gain, a tuning parameter
= integrator gain
= measured process variable
= setpoint
= single-valve kegging
= sample time
= closed-loop time constant
= reset time, a tuning parameter
= overall process time constant
= process dead time
ROBERT RICE, PhD, is director of solutions engineering at Control Station, Inc., a
provider of process control solutions (One Technology Dr., Tolland, CT 06084;
Phone: (860) 872-2920 x101; E-mail: bob.rice@controlstation.com;
Website: www.controlstation.com). He has extensive field experience in both
regulatory and advanced controls and has published papers on a wide array
of topics associated with automatic process control, including multi-variable
process control and model predictive control. He has led the development
and support of LOOP-PRO Product Suite, a PID diagnostic and optimization
toolkit, and is a trainer for the company’s portfolio of practical process control
training workshops. Prior to joining Control Station, he was an engineer with
PPG Industries. He received his BS in chemical engineering from Virginia
Polytechnic Institute and State Univ. and both his MS and PhD in chemical
engineering from the Univ. of Connecticut.
DOUGLAS J. COOPER, PhD, is founder and chief technology officer of Control
Station, Inc. (Phone: (860) 872-2920; E-mail:
doug.cooper@controlstation.com) and a professor of chemical, materials and
biomolecular engineering at the Univ. of Connecticut. He is also the author
and editor of controlguru.com, an e-book of industry best practices for
improving process control. He is a recognized specialist in the fields of
advanced process modeling, monitoring and control; intelligent technologies
and adaptive process control; and software tools for process control system
analysis, tuning and training. Prior to forming Control Station, he held
research positions with Arthur D. Little and Chevron. He received his BS in
chemical engineering from the Univ. of Massachusetts, Amherst, MS in
chemical engineering from the Univ. of Michigan, and PhD in chemical
engineering from the Univ. of Colorado.
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