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300 Process Control
Abstract
This section is an introductory reference to process control. It discusses feedback
control algorithms and controller tuning in depth. The unique requirements of level
controller tuning are covered separately in Section 331. The importance of under-
standing the various forms of the proportional-integral-derivative (PID) control
algorithm and the impact on various tuning rules is analyzed.
The benefits and application of common multiple-loop control configurations suchas cascade, ratio, and feedforward are described. The control objectives analysis
(COA) process is described. COA is a proven methodology for gathering the neces-
sary information to ensure that a process control system will meet plant objectives
for optimal performance, and provides a sound basis for control loop design.
An introduction to advanced control and optimization is given. Finally, resources
and references are provided to allow the reader to pursue more advanced topics
about process control.
Contents Page
310 Overview of Process Control and Optimization 300-3
311 Technology Hierarchy
312 Operational Benefits
313 Economic Benefits
320 Basic Control 300-9
321 Control Loops
322 Feedback Controllers
323 Types of Control Algorithms
324 On/Off Control
325 PID Controller Modes
326 Discrete Form of PID Equation
327 Honeywell and Yokogawa PID Control Algorithms
328 Typical Closed-Loop Controller Response
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330 Controller Tuning 300-27
331 Classical Tuning Methods
332 Forms of the PID Equation
333 Model-Based Tuning Methods
334 Typical Tuning Constants for Common Loops
340 Multiple-Loop Control 300-54
341 Cascade Control
342 Ratio Control
343 Feedforward Control
350 Control Objectives Analysis (COA) 300-67
351 Summary
352 COA Products
353 COA Participants
360 Advanced Control 300-70
361 Overview
362 Steps in MPC Implementation
363 MPC Technology Vendors
364 ChevronTexaco’s Use of Advanced Control
370 Online Optimization 300-90
371 Introduction
372 Online Optimization Cycle
373 Online Optimization Technology Vendors
374 ChevronTexaco’s Use of Online Optimization
380 Resources 300-93
381 Process Control Services
382 Support for Projects
390 References 300-96
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310 Overview of Process Control and Optimization
311 Technology Hierarchy
Control and optimization technology is typically implemented in a hierarchy
(Figure 300-1).
Basic and Intermediate Regulatory Controls
At the lowest level in the hierarchy are the basic and intermediate level controls.
• The Basic Regulatory Controls (BRC) consists of the simple control loops
provided to ensure safe, efficient regulation of the process. Examples include
simple single-loop control of flows, pressures, levels, and temperatures, as well
as simple cascades and ratios.
• The Intermediate Regulatory Controls (IRC) are somewhat more compli-
cated than BRC loops and include such control strategies as steam drum level
control, boiler combustion control, fuel gas BTU control, feedforward control,separation factor control for distillation columns, and furnace pass balancing.
The basic and intermediate loops are typically implemented in a Distributed Control
System (DCS) such as provided by Honeywell or Yokogawa. These loops nomi-
nally operate once per second. At this level in the technology hierarchy, PID
(proportional, integral, derivative) controllers are typically used.
Fig. 300-1Technology Pyramid
ONLINE PROCESS OPTIMIZATION
(e. g. Invensys / SimSci ROMeo)
ADVANCED PROCESS CONTROL
(e. g. AspenTech DMCplus or Honeywell RMPCT)
PROCESS
PLANNING & SCHEDULING
BASIC & INTERMEDIATE REGULATORY CONTROL
(e. g. Honeywell DCS or Yokogawa DCS)
300-1
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Advanced Process Control
Advanced Process Control (APC) as practiced in ChevronTexaco consists of Multi-
variable, Model-Predictive Control (MPC) such as Honeywell’s RMPCT or Aspen-
Tech’s DMCplus™.
MPC is layered on top of the BRC and IRC loops and is an effective tool to increaseunit profitability. MPC typically runs once per minute and typically resides in a
computing module direct-connected to the DCS.
In general, MPC maximizes economic benefits by ensuring smoother operation
(reduced impact of process disturbances) and by providing consistent operation at
optimal constraints. Typically, the MPC controller finds new ways to run the
process. The optimum steady-state constrained operating point is determined at each
control cycle. Thus, the process is continuously pushed towards the most profitable
operation.
Online Process Optimization
An online optimizer, which often encompasses the scope of several MPC control-lers, can be layered on top of MPC to bring additional opportunities for economic
benefits. Online optimization is based on optimizing a rigorous non-linear steady-
state model of the process in real time. An economic objective function is solved
and an optimal set of targets are sent to the MPC for implementation in the process.
The larger scope of the optimizer and it’s use of non-linear models increase the
probability of finding the true economic optimum. Whereas MPC will always find a
solution at set of constraints, online optimization has the potential to find a solution
between constraints.
Typically, two or three optimization cycles can be completed per day.
Planning and Scheduling
In the planning and scheduling layer, production targets and product qualities are set
to satisfy supply and logistics constraints.
312 Operational Benefits
Tighter control shifts the target closer to the plant constraint or specification. This
can result in significant benefits to the operation such as:
• increased throughput,
• increased yield,
• maximum production of a more valuable product, and• lower energy costs.
This section illustrates how improved control allows the process to run closer to
constraints or setpoints. Figure 300-2 shows typical performance data from a control
loop. The controller attempts to keep the controlled variable at the target. However
due to disturbances and other factors, the controlled variable deviates from the
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target. The target has to be positioned away from the constraint or specification to
achieve an acceptable level of performance.
An improved controller configuration, better controller tuning or the use of
advanced control can reduce the standard deviation. Advanced control can typically
reduce the standard deviation by a factor of two or three (Figure 300-3).
Reducing the standard deviation brings improved stability to the process, which can
be beneficial in reducing or eliminating upsets (Figure 300-4).
Fig. 300-2 Typical Data and Distribution Plot, Controlled Loop
Fig. 300-3 Reduced Standard Deviation With Improved Control
C o n t r o l l e d
V a r i a b l e
Constraintor Specification
Time, days Normalized Frequency
of Occurance
C o n t r o l l e d
V a r i a b l e
Target
300-2
µ−3σ µ−2σ µ−1σ µ µ+1σ µ+2σ µ+3σ
σ = 1
σ = 1/2
σ = 1/4
0.0
0.5
1.0
1.5
N o r m a l i z e d
F r e q u e n c y o f O c c u r r e n c e
Controlled Variable Measurement
Constraint /Specification
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Figure 300-5 quantifies several aspects of the previous curves, which are assumed to
be normal distribution curves. As such, there will always be a small percentage of
“off-spec” data, no matter how far the target is from the constraint/specification.
For example, to limit the “off-spec” data to 2.5%, the setpoint (or target) must be
two standard deviations from the constraint/specification, assuming a one sigma
Fig. 300-4 Shifting Target
Fig. 300-5 Potential Shift in Target
µ−3σ µ−2σ µ−1σ µ µ+1σ µ+2σ µ+3σ
σ = 1
σ = 1/2
σ = 1/4
0.0
0.5
1.0
1.5
N o r m a l i z e d
F r e q u e n c y o f O c c u r a n c e
C o n s t r a i n t / S p e c i f i c a t i o n
Target(mean)
Controlled Variable Measurement
300-4
Reduction in Standard Deviation
0.0 0.5σ 1.0σ0.0
+1.0σ
+2.0σ
+3.0σ
S t a n d a r d D e v i a t i o n o f T a r g e t
f r o m C
o n s t r a i n t / S p e c i f i c a t i o n
% of Data ExceedingConstraint / Specification
0.1%
2.5%
5.0%
10.0%
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variation in the data. But, if we were able to reduce the standard deviation in half
due to improved control, we could move the setpoint one standard deviation closer
to the constraint/specification.
313 Economic Benefits
Industry Benchmark
For new plants where plant data is not available, the benefits of applying MPC to a
particular facility are best determined by comparison with industry benchmarks. The
Solomon Associates report, 1994 worldwide study of process control and on-stream
analyzers in the refining industry is the most complete and widely recognized
benchmark. The Solomon numbers have been used throughout the industry both to
benchmark the performance of existing applications and to justify future applica-
tions.
Fifty refineries participated in the study (30 US, 10 Europe and 10 other) including
ChevronTexaco’s Pascagoula, Richmond and Salt Lake refineries.The study focused on key activities involved in the following:
• Planning how the refinery units should operate to maximize profitability,
• Setting operating targets to meet the plan and operating objectives,
• Controlling the processes to meet those targets, and
• Monitoring actual performance.
Economic incentives were reported for advanced control and on-line optimization,
and were based on reported actual applications.
The numbers reflect typical incentives for advanced control and optimization above
a base level of performance achieved by regulatory (DCS) controls. For example, an
atmospheric distillation unit with a throughput of 100,000 Bbl/Day would have a
Mid-range Incentives
(US Cents Per Barrel of Process Throughput)
Process Unit
Advanced
Control
Online
Optimization Total
Atmospheric Distillation
Vacuum Distillation
Coking
Catalytic Cracking
Hydrocracking
Reforming
Alkylation
Isomerization
Heavy Oil Hydroprocessing
Gasoline Blending
10
10
20
18
18
15
15
8
15
10
5
4
7
10
10
7
7
3
7
8
15
14
27
28
28
22
22
11
22
18
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mid-range incentive of $3,650,000/year for advanced control. Since these are mid-
range estimates, actual incentives at specific sites could differ substantially.
There is some evidence the Solomon averages are strongly affected by plants that
gain feed max benefits. Typically, only one or two units in a refinery are a bottle-
neck to production or are required by economics to run at maximum feed rate.
Note Feed maximization benefits are substantially larger than yield and energy
saving benefits.
Relative Costs / Benefits of Controls
Figure 300-6 gives a rough idea of the relative costs and benefits of implementing
the various levels of technology.
• The relatively high cost for the basic regulatory controls (BRC) reflects the cost
of the infrastructure that is required (e.g., distributed control system, instrumen-
tation and control valves).
• Once the infrastructure is there, more advanced applications can be added for a
relatively low cost (relative to the benefits that can be achieved).
• Advanced control and online optimization applications offer the possibility of
very large benefits for a relatively small incremental cost.
Typically, the biggest “bang for the buck” comes from advanced control (e.g.,
AspenTech’s DMCplus or Honeywell’s RMPCT).Depending on the scope of the application and the type of process, costs can range
from $100,000 to $1,000,000, with payout times of from one month to a year.
Fig. 300-6 Costs & Benefits -BRC-IRC-AC-OPT
R e l a t i v e
C o s t
Relative Benefits
IRC
BRC
0 1000
100
AdvancedControl
Online
Optimization
300-6
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320 Basic Control
321 Control Loops
Process control is fundamental to most industrial processes. Although control tech-
nology has evolved greatly in arriving at today’s microprocessor and digital imple-
mentations, all control methods rely on the same basic structure, called a “control
loop.”
Basic control loops have six main elements:
• Controlled variable: The process variable being controlled.
• Setpoint: The value at which a controlled variable must be maintained.
• Controller: A device or software algorithm that keeps the controlled variable at
the setpoint.
• Final control element: The control valve or other device adjusted by the
controller to keep the controlled variable at its setpoint.
• Manipulated variable: A condition (variable) that is being adjusted by the
controller to cause the controlled variable to change.
• Disturbance: A process condition that changes the value of the controlled vari-
able.
Types of Control Loops
Control loops can be either “manual” or “automatic.”
• A manual control loop requires a human being to observe the value of the
controlled variable. If this variable is not at the setpoint, the human observer
adjusts a manipulated variable.
• An automatic control loop employs a controller to keep the controlled vari-able at the setpoint.
Feedback Control Loops. Figure 300-7 shows a typical feedback control loop. In
the process furnace, a temperature controller monitors the outlet temperature
(controlled variable) of the furnace. If the outlet temperature is not at the desired
value (setpoint), the controller changes the fuel flow (manipulated variable) by
changing the position of the fuel gas control valve (final control element). A typical
disturbance would be the furnace feed rate. This type of control is called a closed
loop feedback control system. Perfect feedback control is impossible in all cases
since the controlled variable must deviate from the setpoint before any control
action takes place.
Feedforward Control Loops. In contrast, feedforward control uses a measured
disturbance to generate a corrective action which minimizes the deviations of the
controlled variable from its setpoint (outside of any feedback action). Perfect feed-
forward control is (theoretically) possible in some cases. But, practically speaking,
there will always be errors.
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Use of Control Loops
In practice, feedforward control is always implemented in conjunction with feed-
back control. Figure 300-8 is a simplified sketch showing combined feedforward
plus feedback control loop.
Note also that because of control valve non-linearity, feedforward control normally
would be used in conjunction with a furnace outlet temperature to fuel gas flowcascade feedback control configuration.
Fig. 300-7 Typical Feedback Control Loop
Fig. 300-8 Simple Feedforward+Feedback Furnace Control
Temperature
Setpoint
Temperature
Transmitter
Furnace Outlet
Temperature
Temperature
Comtroller
Control Valve
Fuel Gas
Supply
Burners
Feed
Stream
Furnace
TC301
Fuel Gas
Furnace
Controlled
Variable
Feedforward
Feed
TC
Feedback
Disturbance
Variable
OutletTemperature
Manipulated
Variable
FI
FFC
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322 Feedback Controllers
A block diagram of a feedback controller is shown in Figure 300-9.
There are two key elements: the comparator and the control algorithm. The setpoint
(the desired value of the controlled variable) is compared with the actual measured
variable to form an “error.” As shown in the block diagram, error is usually defined
as follows: Error(t) = Setpoint(t) - Measurement(t) (Eq. 300-1)
Note There is inconsistency in the industry on the above definition; error is just as
often defined as measurement minus setpoint.
Direct vs Reverse Controllers
All commercial controllers are consistent on one related issue:
• a “direct” controller is one whose output increases when the measurement
increases and
• a “reverse” controller is one whose output decreases when the measurement
increases.
323 Types of Control Algorithms
In the control algorithm, the controller calculates an output which tends to drive the
error to zero, thus keeping the measurement at the setpoint target.
• For single-loop control, the controller output signal is sent to the control valve
(final control element).
• For cascade (multiple-loop) control, the controller output becomes the setpoint
of the secondary controller.
The control algorithm is typically one of the following:• On/Off
• Proportional Control Mode (P)
• Proportional plus Integral Control Mode (PI)
• Proportional plus Integral plus Derivative Control Mode (PID)
Fig. 300-9 Feedback Controller Block Diagram
+
-
Controller
Output, %Error, %
Measurement, %
Setpoint, % Control
Algorithm
303
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These algorithms will now be discussed (along with some less-commonly used vari-
ations).
324 On/Off Control
On/Off control. This is the simplest mode of automatic control. It has only twooutputs:
• “on” (100%)
• “off” (0%).
It only responds to the sign of the error, that is, whether it is above or below the
setpoint.
On/Off control is not generally suitable for continuous automatic feedback control
because it results in constant cycling of the controlled variable.
On/Off with “differential gap” control. This is a refinement of on/off control.
Instead of changing output from on (100%) to off (0%) at a single setpoint, differen-tial gap action changes output at high and low limits called boundaries. As long as
the measurement remains between the boundaries, the controller holds the last
output. A typical application of differential gap control is the operation of a dump
valve or pump to keep a vessel level within an acceptable range.
325 PID Controller Modes
PID control is the most widely used continuous controller type in industry. There
are three control “modes”:
• Proportional: Controller output changes by an amount related to the size of the
error.
• Integral: Controller output changes by an amount related to the size and dura-
tion of the error.
• Derivative: Controller output changes by an amount related to the rate-of-
change of the error.
Most control applications use proportional plus integral control.
Proportional-plus-integral-plus-derivative is sometimes used for temperature control
with delays (dead time) of several minutes.
Proportional only control is sometimes used in non-critical services such as draining
vessels.
Proportional Control (P) Mode
In proportional control, there is a linear relationship between the error (setpoint
deviation) and the controller output. Below is the control algorithm:
CO(t) = K C ⋅ E (t) (Eq. 300-2)
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where:
CO(t) = Controller output [=] %
K C = Controller Gain [=] %/% (dimensionless)
E(t) = Error [=] %
t = Time [=] minutes
The controller gain, K c, is also called the controller “sensitivity.” It represents the
proportionality constant between the control valve position and controller error.
Figure 300-10 shows the relationship between the controller output (valve position)
and error that is characteristic of proportional control.
The valve position changes in exact proportion to the amount of error, not to its rate
or duration. The response is almost instantaneous, and the valve returns to its initial
value when the error returns to its original value.
Figure 300-11 shows how controller gain affects valve opening for constant change
in error.
High controller gains result in a larger response.
Proportional Band. Another way of characterizing a proportional controller is to
describe its proportional band. The proportional band is the percent change in value
of the controlled variable necessary to cause full travel of the final control element.
Fig. 300-10 Proportional Mode Output is Proportional to Error (Open loop)
Fig. 300-11 Proportional Mode Plots Step Response (Open loop)
Time, Minutes0
Error
Controller
Output
304
K C =1.5
K C =1
K C =0.5
Time, Minutes0
Error
305
Controller
Output
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The percent proportional band, PB, is related to its gain as follows:
K C = 100 / PB (Eq. 300-3)
Both proportional band and gain are expressions of proportionality. Manufacturers
may call their adjustments gain, sensitivity, or proportional band.
The “throttling range” is a term used to define the error range over which the control
valve can throttle the flow it’s adjusting. Beyond that range, the valve is either wide
open or closed (saturated).
Bias. Bias is the amount of output from a proportional controller when the error is
zero. The equation previously given for proportional control implies that when the
error is zero, controller output is zero. (In that case, the valve would be either fully
open or fully closed and provide no throttling action). Adding a bias provides this
throttling action (that is, the nominal valve position when the error is zero). The
final equation for proportional control then becomes:
(Eq. 300-4)
where:
B = Bias (percent of full output)
Typically, manufacturers set the bias at 50%. To prevent a process bump, the control
system can usually be configured to set the bias such that the valve will not move
when the controller is switched from manual to automatic.
Figure 300-12 shows controller output (control valve position) versus error at
different proportional bands (and controller gains) with a 50% bias. At zero error,
the controller output is 50% of full range for any proportional band.
Offset. A controller’s error is the difference between its setpoint and measurement.
In a proportional only controller, a change in setpoint or load introduces a perma-
nent error called offset.
CO t ( ) K C E t ( ) B 100( ) PB------------- E t ( ) B+⋅=+⋅=
Fig. 300-12 Proportional Mode Gain
-50% 0% +50%0%
50%
100%
Error
C o n t r o l l e r O u t p u t
( C o n t r o l V a l v e )
K C =2
K C =1
K C =0.5
"Throttling Range"
PB=50PB=100
PB=200
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It is impossible for a proportional only controller to return the measurement exactly
to its setpoint, because proportional output only changes in response to a change in
the error, not to the error’s duration. For example, consider Figure 300-13, in which
we assume that a proportional only controller controls the outlet temperature of a
furnace and that the temperature is initially at the setpoint.
If the feed rate to the furnace increases, more fuel will be needed. This disturbance
represents a load change to the furnace. To get more fuel, the fuel valve must be
opened more. As is suggested by the equation for proportional action, the only way
that the valve can be at some value other than its starting point is for an error to
exist. Thus, the proportional controller alone cannot return the outlet temperature to
its setpoint. As mentioned, some controllers allow the operator to adjust the bias
until the value of the error (or offset) is zero.
The proportional only controller is the easiest continuous controller to tune. It
provides rapid response and is relatively stable. If tight control is not required, proportional only control can be used.
Integral Control Mode
Integral (reset) action is the result of an integration of controller error with time.
(Eq. 300-5)
where:
CO(t) = Controller output [=] %
K I = Integral mode gain [=] 1/minutes
E(t) = Error [=] %
t = Time [=] minutes
CO0 = Initial controller output [=] %
Fig. 300-13 P-Only Offset (Closed Loop)
Time, Minutes0
Furnace
Outlet
Temperature
Furnace
Feed Rate
Offset
Setpoint
307
CO t ( ) K I E t ′( )
0
t
∫ dt ' CO0+⋅=
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With integral action, controller output is proportional to both the size and duration
of the error. As long as a deviation from setpoint exists, the controller continues to
drive its output in the direction that reduces the deviation.
The rate of change of controller output is proportional to the magnitude of the error.
(Eq. 300-6)
Figure 300-14 illustrates the open loop response of integral action.
Integral action responds to the sign, size, and duration of the error:
• TIME 0 — A constant error appears. The integral action drives the output
higher at a constant rate proportional to the size of the error
• TIME A — The size of the error doubles. The integral action drives the output
higher twice as fast.
• TIME B — The sign of the error changes. The integral action drives the output
in the other direction.
• TIME C — The error goes to zero. The integral action stops, holding the
existing output.
• TIME D — The error ramps down at a constant rate. The integral action drives
the output down at an ever increasing rate.
• TIME E — The error returns to zero. The integral action stops, holding theexisting output.
Integral action is normally used in conjunction with proportional action; it is rarely
used by itself.
Fig. 300-14 Integral Mode Response (Open Loop)
dCO t ( )dt
------------------ K I E t ( )⋅=
Time, Minutes0
Error
Integral
Mode
Output
0
A B C D E308
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Proportional Plus Integral (PI) Control
Proportional plus integral control is the recommended control action for most appli-
cations. Often called PI control, it combines proportional action and integral action
in one controller. The resulting control action has the fast response and stability of
proportional action, but no offset. In eliminating offset, integral action serves as an
automatic bias adjustment.
The output from a proportional plus integral controller may be expressed as follows:
(Eq. 300-7)
where:
CO(t) = Controller output [=] %
K C = Controller gain [=] %/% (dimensionless)
E(t) = Error [=] %
τ I = Integral (reset) time [=] minutes
t = Time [=] minutes
CO0 = Initial controller output [=] %
Note that the effective gain for the integral mode in the above (standard) equation
for a PI controller is K C / τ I . The overall controller gain K C affects both the propor-tional and integral action.
On some controllers, integral settings are in repeats, meaning repeats per minute; on
others, settings are in minutes, meaning minutes per repeat. One setting is the recip-rocal of the other. Decreasing the integral time increases the amount of integral
action and visa versa. Integral time is also called “reset time.”
Figure 300-15 shows how the PI algorithm responds to a step change on error (open
loop/no feedback from the process):
CO t ( ) K C E t ( )1
τ I ---- E
0
t
∫ t '( )dt '⋅+ CO0+⋅=
Fig. 300-15 PI Step Response (Open Loop)
K C ·A
Time, Minutes0
Error
Controller
Output
0
CO0
P
I
A
Integral (Reset) Time, Minutes
I
K C ·A
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Integral time is quantified as the time required for the controller output to change by
an amount equal to the change caused by the initial “proportional kick.” In other
words, it is the time required for the contribution of the integral mode to “repeat”
the contribution of the proportional mode.
Reset (Integral) WindupA basic problem with integral controllers is that integral action continues as long as
an error exists. Consider the following example (Figure 300-16) based on the
furnace temperature control loop illustrated in the introductory section
The temperature controller responds to the disturbance in feed rate by opening the
control valve. But if the control valve capacity is not large enough, it may saturate
before the furnace outlet temperature (controlled variable) has returned to the
setpoint. A persistent error (offset) will then be present. The integral mode keeps
increasing its output to try to eliminate the offset, but there will be no effect on the
process. This effect is called reset (integral) windup.
If at some later time the feed rate (disturbance) returns to its original value, the
furnace outlet temperature (controlled variable) will drift up to the setpoint due tothe decreased load on the system. The integral action cannot start unwinding until
the error changes sign (when the temperature crosses the setpoint). Then, the
temperature controller output starts un-winding. Since there is no valve movement
until the controller output drops below 100%, furnace outlet temperature over-
shoots the setpoint significantly.
Fig. 300-16 Integral Windup - Furnace TC
Time, minutes
100%
Temperature
Controller
Output
(%)
Furnace Outlet
Temperature
(DegF)
Offset
Controller
Un-winds
Control Valve
Wide Open
Reset Windup
Feed Rate
Disturbance
(MBD)
Large Overshoot
Valve Starts
Moving
310
Setpoint
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All industrial implementations of the PID algorithm have provisions for preventing
reset windup. For standard control loop configurations such as single loop control or
cascade control, anti-windup is generally built in. More complicated, non-standard
control structures may require some custom user configuration.
Let’s look at the performance of the same control system with anti-windup included
(Figure 300-17).
There is no difference in the first part of the plot. But with no reset wind-up, the
temperature controller can start closing the control valve immediately when the
disturbance returns to its initial value. As a result, there is substantially less over-
shoot in the furnace outlet temperature.
Derivative Control Mode
With derivative action (also called rate action), the controller output is proportional
to the rate of change of the error.
(Eq. 300-8)
where:
CO(t) = Controller output [=] %
K D = Derivative mode gain [=] minutes
Fig. 300-17 Integral Anti-Windup - Furnace TC
Time, minutes
100%
Offset
Controller Starts Closing
Valve Immediately
Control Valve
Wide Open
No Reset Windup
Less Overshoot
Temperature
Controller
Output
(%)
Furnace Outlet
Temperature
(DegF)
Feed Rate
Disturbance
(MBD)
311
Setpoint
CO t ( ) K D dE t ( )dt ------------- CO0+⋅=
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E(t) = Error [=] %
t = Time [=] minutes
CO0 = Initial controller output [=] %
The equation shows that the faster the change in error, the faster the change incontroller output and control valve position. By the same token, if the error remains
constant, even with a large error, the derivative controller output would not change
(Figure 300-18).
This makes the use of derivative action by itself impractical.
Proportional Plus Derivative (PD) Control
Derivative action is normally combined with proportional action or proportional
plus integral action. We will first examine proportional plus derivative:
(Eq. 300-9)
where:
CO(t) = Controller output [=] %
K C = Controller gain [=] %/% (dimensionless)
E(t) = Error [=] %
t = Time [=] minutes
τ D = Derivative time [=] minutes
CO0 = Initial controller output [=] %
Note that the effective gain for the derivative mode in the above (standard) equa-
tion for a PI controller is K C ⋅ τ D. The overall controller gain K C affects both modes.
Fig. 300-18 Derivative Mode Response (Open Loop)
Time, Minutes
0
0
Error
Derivative
Mode
Output
312
CO t ( ) K C E t ( ) τ DdE t ( )
dt -------------+ CO0+⋅=
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Figure 300-19 shows how the PD algorithm responds to a ramp change on error
(open loop/no feedback from the process).
In this case, the derivative time is the time for the proportional contribution to
“repeat” the initial derivative kick. Notice that derivative action introduces a “lead”
(or anticipatory) element into the controller.
Derivative Filters. Note that derivative action would produce a “spike” if the error
were to undergo a step change. However, in all “real” implementations of the deriv-
ative function, the derivative is filtered. The filter time constant is ατ D, with alphatypically ranging from 1/6 to 1/10. Use of a derivative filter limits the size of the
derivative spike on sudden changes (Figure 300-20).
Since derivative action is proportional to the rate of change of error, it cannot be
used with controlled variables with high noise levels. Although derivative action is
Fig. 300-19 PD Ramp Response (Open Loop)
Fig. 300-20 Derivative Filter
Time, Minutes
0
0
Error
Derivative
Mode
Output
312
M K C /
D Theoretical
M K C M
Input
Step
M
Gain Filter Devivative
Practical
314
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sometimes difficult to tune because of its extreme sensitivity to measurement noise
and other high frequency disturbances, it does have some applications.
Most importantly, it is used with proportional and integral action in temperature
processes that have large time lags. Derivative action also can be very helpful in
controlling processes that have significant dead time, but tuning it can be tricky.
Derivative on Measurement Option
A commonly used option for the derivative mode is “derivative on measurement”
rather than “derivative on error.” Use of a derivative filter eliminated the infinite
controller impulse for step changes, yet a finite jump, called the “derivative kick”
still occurs for step changes in setpoint, when derivative on error is used. The deriv-
ative can be separated into parts as shown below:
(Eq. 300-10)
When the setpoint is not changing, its derivative is zero, and we can use the
following expression for derivative.
(Eq. 300-11)
Use of the derivative on measurement option is recommended to eliminate the
derivative kick on setpoint changes. Control loop performance would be identical
for either the “derivative on error” or “derivative on measurement” option, when the
setpoint is constant.
Proportional Plus Integral Plus Derivative (PID) Control
The complete PID control algorithm includes all three controller modes previously
discussed.
(Eq. 300-12)
where:
CO(t) = Controller output [=] %
K C = Controller gain [=] %/% (dimensionless)
E(t) = Error [=] %
t = Time [=] minutes
τ I = Integral (reset) time [=] minutes
dE t ( )dt
-------------d SP t ( ) M t ( ) – [ ]
dt ---------------------------------------
dSP t ( )dt
-----------------dM t ( )
dt --------------- – = =
K – C τ DdM t ( )
dt ---------------⋅
CO t ( ) K C E t ( )1
τ I ---- E t '( ) t 'd
0
t
∫ τ DdE t ( )
dt -------------+⋅+ CO0+⋅=
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τ D = Derivative time [=] minutes
CO0 = Initial controller output [=] %
Figure 300-21 shows the open-loop response of the PID controller to a step change
in error (no feedback from the process).
Note how the individual control modes (P, I, and D) combine to form the complete
controller output. The “real” controller response includes the derivative filter
discussed earlier.
Figure 300-22 shows the open loop response of the PID controller to a ramp changein error
Fig. 300-21 PID Step Response (Open Loop)
Fig. 300-22 PID Ramp Response (Open Loop)
K C ·A
Time, Minutes0
Error
Controller
Output
0
CO0
P
I
A
Integral (Reset) Time, Min.
I
K C ·A
Filtered DerivativeD
315
Theoretical Derivative
Time, Minutes0
Error
Controller
Output
0
CO0
P
D
Derivative Time, Min.
D
K C · D·B
1
B
K C ·B
2· I · t 2
K C ·B·t I
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As stated previously, derivative on measurement is a recommended option. The PID
equation then becomes:
(Eq. 300-13)
Derivative on measurement results in smoother control because the measurement
cannot change as rapidly as the setpoint. However, excessive measurement noise
could still rule out the used of derivative action.
326 Discrete Form of PID Equation
We have used the continuous form of the PID equation in these notes. For example,
the ideal form of the PID is as follows:
(Eq. 300-14)
However, with microprocessor-based implementations of the algorithm in distrib-
uted control systems (DCS), programmable logic controllers (PLC), and supervi-
sory control and data acquisition systems (SCADA), discrete approximations are
used. For example, here is the discrete (incremental) equivalent of the above equa-
tion.
(Eq. 300-15)
Or
(Eq. 300-16)
327 Honeywell and Yokogawa PID Control Algorithms
Honeywell uses Laplace domain notation (“s” variable) in their documentation eventhough the algorithm is implemented discretely. Below is how Honeywell docu-
ments their Equation “A” (Non-interactive) advanced process manager (APM) PID
algorithm:
(Eq. 300-17)
CO t ( ) K C E t ( )1
τ I
---- Et ' t 'd
0
t
∫ τ D – dM t ( )
dt ---------------⋅+ CO0+⋅=
CO t ( ) K C E t ( )1τ I ---- E t '( ) t 'd
0
t
∫ τ DdE t ( )dt -------------+⋅+ CO0+⋅=
∆COn K C ∆ E n∆t s
τ I
------- E n
τ D
∆t s
-------∆ ∆ E n( )++
⎩ ⎭
⎨ ⎬⎧ ⎫
=
COn COn 1 – K C E n E n 1 – – ( ) ∆t s
τ I ------- E n
τ D∆t s------- E n 2 E n 1 – E n 2 – – – ( )+ +
⎩ ⎭⎨ ⎬⎧ ⎫
= –
CV S K 1 T 1 s+
T 1 s------------------⎝ ⎠
⎛ ⎞ T 2 s1 aT 2 s+----------------------⎝ ⎠
⎛ ⎞+ PVP S SPP S – [ ]⋅ ⋅=
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where:
CV S , PVP S , SPP S [=]%
K [=]%/% (Controller Gain)
T 1 [=] minutes (Reset Time)
T 2 [=] minutes (Derivative Time)
a = 0.1 (Derivative Limit Factor)
Honeywell also has “interactive” versions of the PID equation.
Below is how Yokogawa documents their Centum CS3000 PID Equation (Non-
interactive):
(Eq. 300-18)
where:
MV n , E n [=] Eng Units
K S = Scale Conversion Factor
PB [=] % (Proportional Band)
TI [=] seconds (Reset Time)
TD [=] seconds (Derivative Time)
∆T [=] seconds (Control Period)
(Effective Derivative Limit Factor = 0.125)
Yokogawa does not have an “interactive” PID alternative.
328 Typical Closed-Loop Controller Response
Finally we compare typical closed-loop controller response for various combina-
tions of control modes. For a setpoint change the expected closed-loop response
would be as shown in Figure 300-23.
∆ MV n K S 100
PB--------- ∆ E n
∆T TI ------- E n
TD
∆T --------∆ ∆ E n( )+ +
⎩ ⎭⎨ ⎬⎧ ⎫
⋅=
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Notice that both proportional-only (1) and proportional-plus-derivative (2) control
have offset. Integral action is required to eliminate offset. Integral-only control (3)
slowly brings the controlled variable to the setpoint with a relatively long period of
oscillation. Proportional-plus-integral control (4) responds more quickly with a
shorter period. Finally, proportional-plus-integral-derivative control (5) potentially
provides the best performance. But, recall that excessive measurement noise could
preclude the use of derivative action.
For a disturbance the expected closed-loop response would be as follows
(Figure 300-24).
The ordering, in terms of controller performance, are the same.
Fig. 300-23 Typical PID Response (Closed Loop)
Fig. 300-24 Typical PID Response (Closed Loop) with Disturbance
Time, Minutes
Controlled
Variable
1
2
345
Setpoint
Offset
317
0
Time, Minutes
Controlled
Variable
Offset
1
2
3
45
Setpoint
No Control
318
0
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330 Controller Tuning
Introduction
Numerous methods are available to tune a controller to function in a specific loop.
This section discusses some of the classical tuning methods commonly used.
Several of the references, particularly Chien and Fruehauf, 1990, should be
consulted for more advanced model-based tuning methods. Consider the following
standard block diagram for a single-loop control system (Figure 300-25).
where:
CV SP ≡ Controlled variable (CV) setpoint [=] EU CV
EU ≡ Engineering units
K M ≡ Controlled variable transmitter gain [=] %/ EU CV
CV SP% ≡ Controlled variable %-setpoint [=] %
K C ≡ Controller gain [=] dimensionless (%/%)
GC ≡ Controller dynamics (integral, derivative)
K V ≡ Control valve gain [=] EU MV /%
GV ≡ Control valve dynamics
MV ≡ Manipulated variable, [=] EU MV
K P ≡
Process gain [=] EU CV
/ EU MV
G P ≡ Process dynamics
D ≡ Disturbance [=] EU D
K D ≡ Disturbance gain [=] EU CV / EU D
G D ≡ Disturbance dynamics
Fig. 300-25 Single-loop Feedback Control Block Diagram (no s)
Controlled Variable
Transmitter
Process
CV
(EU)
CV SP
(EU)
+
+
D
(EU)
Control Valve
K M K C GC K P GP K V GV
K M
GM
+
-
Controller
CV SP%
K D G
D
MV
(EU)
CV M
(%)
318a
CO
(%)
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CV ≡ Controlled variable [=] EU CV
G M ≡ Controlled variable transmitter dynamics
CV M ≡ Controlled variable measurement [=] %
A properly tuned controller ideally would achieve all of the following goals:• Good disturbance rejection
• Rapid, smooth response to setpoint changes
• Minimal control valve movement
• High degree of robustness (insensitive to process changes)
A high performance control loop would have rapid, smooth responses to setpoint
changes and disturbances with minimal control valve movement. A robust control
loop would have good performance for a wide range of process conditions.
However, it is not possible to achieve all of these goals simultaneously. There areinherent conflicts and tradeoffs that must be considered:
• Performance and robustness need to be balanced. Conservative controller
settings (low proportional gain and long integral time) sacrifice performance in
order to achieve robustness.
• There is also a trade-off between tuning for good setpoint response and for
good disturbance rejection (with standard PID controllers). Tuning for good
setpoint response typically yields sluggish disturbance response. Tuning for
good disturbance rejection typically yields oscillatory setpoint response.
All of these issues must be considered when tuning a controller.
331 Classical Tuning Methods
Most common process control loops (flow, temperature, composition, gas pressure,
etc.) can be tuned using either the Ziegler-Nichols (Z-N) ultimate sensitivity or reac-
tion curve methods described below.
Level control loops are the exception; special tuning rules have been developed for
levels (refer to “Tuning Level Controllers” on page 300-33).
Note Direct Synthesis/Internal model control tuning methods ( Section 333 ) are
now accepted as the successor to Z-N tuning rules discussed in this section.
Z-N Ultimate Sensitivity Method (Closed-loop Tuning)The Z-N ultimate sensitivity method is a closed-loop tuning method; the controller
is kept in automatic.
1. First, the controller is changed to “proportional-only” by turning off the inte-
gral and derivative modes.
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2. Then the controller gain is increased in small steps, each time changing the
setpoint if required to induce cycling (Figure 300-26).
3. This is repeated until the controller measurement cycles with constant ampli-
tude (Figure 300-27).
The final controller gain setting is called the ultimate gain, denoted K CU . The
period of oscillation at the ultimate gain is called the ultimate period, measured
in minutes and denoted P U .
4. The ultimate controller gain and the ultimate period are then used to calculate
tuning constants per the following table:
The ultimate controller gain and the ultimate period are then used to calculate tuning
constants per the following table:
Fig. 300-26 Ziegler-Nichols Cycling Plots
Fig. 300-27 Ziegler-Nichols Ultimate Gain and Period
Controlled
Variable
Time, Min.
Controlled
Variable
Time, Min.
Increase
Controller
Gain
319
Controlled
Variable Time, Min.
(K C K
CU )
P U
(Minutes)
320
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This method was the first systematic method developed for tuning industrial
controllers.
Shortcomings. Note that the Z-N tuning objective was “quarter amplitude
damping” (the response oscillates with each peak being one quarter that of the
previous peak).
• Thus, the tuning is aggressive; it is not robust. It is generally recommended that
the controller gain be reduced to provide more robustness.
• Other disadvantages for Z-N include the fact that the process must be brought
to the stability limit (cycling) and that the procedure is very time consuming forslow processes.
Advantages. On the other hand, the Z-N procedure is simple and the tuning “rules”
are easy to remember.
Advanced tuning methods address most of these shortcomings. They are generally
“model-based” and address robustness (directly or indirectly). Model-based tuning
will be described in Section 333.
Z-N Process Reaction Curve Method (Open-loop Tuning)
Ziegler-Nichols also developed an open-loop tuning method. The controller remains
in manual while response tests are made. To perform this test:
1. Put the controller in manual.
2. Change the controller valve position by a small amount and record the
controlled variable.
The controlled variable response curve is called the “process reaction curve.”
Refer to Figure 300-28.
3. Determine the maximum slope, S, of the response curve by drawing a line
through the point of inflection on the curve.
4. The point that this line crosses the initial value of the controlled variable
measurement is used to determine θ P .5. The quantity ∆ X is the size of the controller output step and ∆Y is the final
steady-state response of the controlled variable.
Prop. Gain, %/%
Integral Time,
Min.
Derivative Time,
Min.
P 0.50 KCU - -
PI 0.45 KCU PU / 1.2 -
PID 0.60 KCU PU / 2.0 PU / 8.0
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These values will be used to fit the response curve to a first-order lag plus dead timemodel.
(Eq. 300-19)
The model parameters are determined as follows. The quantity θ P is the dead time
(minutes) and is determined graphically as explained above. The dead time is the
delay between a change in valve position and the resulting change in the controlled
variable. The process time constant is the time required for the controlled variable to
reach 63% of its final value. It can be determined graphically as sketched on the
response plot or calculated from the following equation:
τ P = ∆Y/S [=] minutes
Finally, the process steady-state gain is calculated from the following equation:
K P = ∆Y/ ∆ X [=] % / %
An alternative approach to fitting the model, which is more accurate for noisy
processes, is illustrated below (Figure 300-29).
The process steady-state gain is found as before. The dead time and time constant
are calculated from the following equations:
τ P = 1.5 ⋅ (t 63% - t 28%) [=] minutes
θ P = t 63% - τ P [=] minutes
Fig. 300-28 Reaction Curve — Model-Identification Method #1
Time, Minutes0
Controlled
Variable (%)
1
Maximum Slope, S
Y 1st-Order Lag
+ Dead Time
Approximation
X Controller
Output (%)
P
P X Y K P 322
( )( ) ( ) P P P t CO K t CV
dt
t dCV θ τ −⋅=+
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Having estimated a process model, we then apply the Ziegler-Nichols reaction curve
tuning rules:
As with the ultimate sensitivity tuning method, the controller objective function is
quarter amplitude damping. To reduce the oscillatory behavior, simply reduce the
recommended controller gain by 50 to 100%.
Note that the controller gain is proportional to the ratio of the time constant to thedead time, so be cautious about applying this method when the dead time is small!
Fig. 300-29 Process Reaction Curve — Model-Identification Method #2
Prop. Gain, %/%
Integral Time,
Min.
Derivative Time,
Min.
P (1.0/K P )⋅(τP /θP ) - -
PI (0.9/K P )⋅(τP /θP ) 3.3⋅θP -
PID (1.2/K P )⋅(τP /θP ) 2.0⋅θP 0.5⋅θP
"Process
Reaction
Curve" 0.63 Y
0.28 Y
%63t %28t Time, Minutes0
Controlled
Variable (%) Y
X Controller
Output (%)
X Y K P 323
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Typical Z-N Tuning Results
Figure 300-30 shows typical Z-N tuning results for a setpoint change and then a
disturbance.
Note that the response is oscillatory for both common forms of the PID algorithm.
Refer to “Forms of the PID Equation” on page 300-44 for more information.
Tuning Level Controllers
The level process has some unusual dynamic characteristics and unique control
objectives that require us to develop specialized controller tuning rules. Consider
the surge vessel shown in Figure 300-31.
Fig. 300-30 Typical Z-N Tuning Results for a Setpoint Change and then a
Disturbance
Fig. 300-31 Level Process Surge Vessel
Q In
Q Out
Pump
A
L
LI
324
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Level Control Objectives. Ideally, we would maintain a constant level, and mini-
mize the effect of inflow disturbances on downstream units. However, these are
conflicting objectives. To maintain constant level, outflow would have to mimic
every inflow change. To smooth the outflow, the level would have to change to
absorb the inflow fluctuations.
As a result, two distinct types of level control have evolved:
1. Averaging level control (flow-smoothing)
2. Tight level control
In most cases, averaging level control is more appropriate. As long as the level stays
within a defined range, we can take advantage of a vessel’s “surge” capacity to
smooth out the flow. Averaging level control takes advantage of whatever surge
volume is provided in the vessel. The degree of effectiveness in smoothing the flow
depends on the size of the surge volume relative to the magnitude of the flow distur-
bances.
We will investigate the level process and develop recommendations for propor-tional-integral (PI) controller tuning.
The Level Process. The dynamic response characteristics of the level process can
be determined by writing a dynamic material balance (inflow-outflow = rate of
accumulation):
(Eq. 300-20)
where:
Q In(t) = Inflow [=] GPM
QOut (t) = Outflow [=] GPM
V(t) = Volume [=] Gallons
t = Time [=] Minutes
The volume can be calculated from the measured level as follows (assuming the
cross-sectional area is constant):
(Eq. 300-21)
where:
k = 7.481 Gal / Ft3
A = Cross-sectional area [=] Ft2
L(t) = Level [=] Ft
Q In t ( ) QOut t ( ) – dV t ( )
dt -------------=
V t ( ) k A L t ( )⋅ ⋅=
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then,
or,
where
The quantity “C” is called the “capacitance” of the vessel. It is effectively the
volume per foot of level. Since “C” is a constant, it can be moved outside of the
derivative term.
(Eq. 300-22)
Typically, the pump head is large compared to the static head provided by the level,
and thus, changes in level have very little effect on outflow (The process is non self
regulating). That is,
QOut ≠ f(L) (Eq. 300-23)
We can now solve for dL(t)/dt and integrate.
(Eq. 300-24)
Because of the form of this equation, level is known as an “integrating process.”
The response to a step change in net inflow is shown in Figure 300-32.
Fig. 300-32 Level Process Step Response (Open Loop)
Q In t ( ) QOut t ( ) – dkAL t ( )
dt --------------------=
Q In t ( ) QOut t ( ) – dCL t ( )
dt -----------------=
C k A =[ ]GalFt
--------⋅≡
Q In t ( ) QOut t ( ) – CdL t ( )
dt -----------------=
( ) ( ) ( )[ ] 00
1 Lt d t Qt Q
C
t L
t
Out In +′′−′⋅=
∫
Time, Minutes
QNet
/C [=] Ft/Min
0
Q In
(t)-Q Out
(t)
Level, L(t)
0
L0
1
QNet
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Unlike most processes, the level process is non self-regulating; it does not come to
steady state. For the level process to be at steady state, the net inflow must be zero.
Notice that the slope of the ramp response is Q Net/C. Thus, the capacitance of the
vessel can be determined by introducing a known imbalance between inflow and
outflow and measuring the slope of the level response.
Slope = Q Net /C (Eq. 300-25)
Solving for C
(Eq. 300-26)
Level Control Configurations. There are two common level control configura-
tions:
1. single-loop control (Figure 300-33)
and
2. level-to-flow cascade control (Figure 300-34)
Fig. 300-33 Level Control Configurations (Single-Loop Control)
Fig. 300-34 Level Control Configurations (Cascade Control)
C Q Net
Slope-------------- =[ ]
Gal Min ⁄ Ft Min ⁄
---------------------- Gal
Ft--------= =
Q In
Q Out
LC
FI
Q In
Q Out
LC
FC
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Level Control Response Equations. The closed-loop response equations for both
single-loop and cascade configurations are second-order (and identical) when we
assume the following:
• A proportional plus integral controller is used.
• Both configurations have the same maximum flow (valve max or flowcontroller setpoint max).
• For the single-loop case, the valve’s installed characteristic is linear.
The following second-order differential equation describes the dynamic response of
the outflow to a change in the inflow.
(Eq. 300-27)
where:
[=] minutes
∆ H T = Level transmitter span [=] Ft
The degree of “flow smoothing” between the inflow and outflow depends on the
values of the parameters in this equation.
Note that the “measurable” volume (within the level transmitter range) is given by
Vol Meas = C ⋅ ∆ H T (Eq. 300-28)
Then
where H = Vol Meas/ F Max [=] minutes
The quantity H is the vessel “residence time” based on the maximum outflow F Max.
In other words, it is the time to fill the measurable volume (that is, within the level
transmitter range) with an inflow of F Max and with the outflow valve closed.
The following equation describes the level setpoint-to-level response:
(Eq. 300-29)
τ H τ I d
2QOut
dt 2
------------------ τ I dQOut
dt --------------- QOut τ I
dQ In
dt ------------ Q In+=+ +
τ H C ∆ H T ⋅
K C F MAX ⋅--------------------------≡
τ H 1
K C -------
Vol Meas
F Max--------------------
1
K C ------- H =[ ]Minutes⋅=⋅=
τ H τ I d
2 L
dt 2
--------- τ I dL
dt ------ L τ I =
dLSP
dt ------------ LSP ++ +
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This equation has exactly the same form and parameters as for the inflow to outflow
response. The following equation describes the inflow-to-level response:
(Eq. 300-30)
This equation tells us how much the level will vary as the inflow changes. Note that
the left-hand side of this equation (known as the “characteristic equation”) has
exactly the same form and parameters as for the previous two response equations.
Level Control Period and Damping. We will now compare the equations derived
for the level control system with the standard equation for a second-order system.
(Eq. 300-31)
where:
Y(t) = Dependent variable
X(t) = Independent variable
τn = Natural time constant
= Damping coefficient
K = Steady-State Gain
t = Time
The response of a second order system to a step change in the independent variableis shown in Figure 300-35. The shape of the response varies from a smooth,
“S-shaped” curve to a highly oscillatory one depending on the value of the damping
coefficient .
Comparing the level control system’s equations with the standard form for the
second-order equation we can find the closed-loop period of oscillation, T
(minutes/cycle), and the damping factor, (dimensionless) for the level control
system:
(Eq. 300-32)
τ H τ I d
2 L
dt 2
--------- τ I dL
dt ------ L
τ H τ I C
-----------⎝ ⎠⎛ ⎞=
dQ In
dt ------------+ +
τ2n
d 2Y t ( )
dt 2
---------------- 2ζτndY t ( )
dt ------------- Y t ( ) K X t ( )⋅=+ +
ζ
ζ
ζ
Max
Meas
C
I
F
Vol
K T ⋅⋅
−
= τ
ζ
π
2
1
2
Meas
MAX C I
Vol
F K ⋅⋅=
τ ζ
2
1
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These equations show how the level controller tuning parameters affect the period
and degree of damping of the closed-loop response. A close examination reveals
several important (and surprising) facts about level control systems.
Note that increasing the level controller integral time, τ I , increases level controlstability (i.e., ) and increases control loop period, T. Both of those results are
expected. However, note that increasing level controller gain, K C , decreases control
loop period, but also increases stability (i.e., ). The latter result is exactly oppo-
site of what one would typically expect.
In real-world level control systems, increases in K C eventually will result in an
unstable system because other lags in the system (that we didn’t model) will becomesignificant. (The fact that increasing controller gain initially increases stability, but
ultimately destabilizes the system makes level controllers “conditionally stable”
systems.)
These observations show that tuning level controllers is non-intuitive.
Averaging Level Control Tuning. Page Buckley of Dupont (1964) developed a
tuning approach for averaging level control that has been applied throughout
ChevronTexaco. First, he proposed that the closed-loop response be critically
damped ( = 1). This will produce a smooth, non-oscillatory response.
Recall that
(Eq. 300-33)
Setting = 1 and solving for τ I yields
Fig. 300-35 Step Response General Second-Order System
Time
Y
X
0
XX
0
Y0
> 1 (Overdamped)
= 1 (Critically Damped)
< 1 (Underdamped)
= 0.707 (Butterworth)
K* X
328
ζ
ζ
ζ
Meas
MaxC I
Vol
F K ⋅⋅=
τ ζ
2
1
ζ
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(Eq. 300-34)
where H = Vol Meas
/F Max
[=] minutes
Recall that H is the “residence time” based on the maximum outflow F Max. It is the
time to fill the measurable volume (that is, within the level transmitter range) with
an inflow of F Max and with the outflow valve closed.
Second, Buckley proposed that the level stay within defined bounds for a defined
disturbance. In particular, for an inflow disturbance of half the maximum outflow,
the change in level that results will be half the level transmitter span. In other words,
for this relatively large flow change, level would rise to 100% (assuming it started at
50% level) in order to smooth the outflow.
Figure 300-36 shows how the level and outflow respond to a step change in inflow.
Note that, as expected, there is no oscillation in the response. But the output will
always temporarily exceed (overshoot) the inflow. (With the level process there
always needs to be an imbalance between inflow and outflow to change the level).
The plot shows that at peak level we have the following:
(Eq. 300-35)
Solving for K C gives
(Eq. 300-36)
Fig. 300-36 Level & Outflow Response Plot (Zeta=1)
H K F
Vol
K C Max
Meas
C I ⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛ =⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⎟⎟
⎠
⎞⎜⎜⎝
⎛ =
44τ
1.5
1.0
0.5
0.0
1.5
1.0
0.5
0.0
0 1 2 3 4 5 6 7 8 9 10
(
= 1)
L(t)
H T
F Max
Q In
K C
1.14
0.74
Outflow
Inflow
Level
H Dimensionless Time, t /
Q Out
(t) Q In(t)
329
74.0=⋅⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∆⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∆
∆C
In
Max
T
peak
K Q
F
H
L
( )( )T Peak
Max InC
H L
F Q K
∆∆
∆⋅= 74.0
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In mathematical terms, Buckley’s second criterion specifies that
(Eq. 300-37)
Substitution into the previous equation allows us to solve for controller gain.
(Eq. 300-38)
We can now use this value for controller gain to find the controller integral time.
(Eq. 300-39)
Substituting K C = 0.74 gives
(Eq. 300-40)
In summary, for averaging (flow smoothing) level control (Buckley tuning), use a
controller gain of 0.74 and a controller integral time of 5.4 times the vessel resi-
dence time. For example, for a vessel with a six minute “residence time”, controller
gain would be 0.74 and controller integral time would be 32.4 minutes.
The following plot (Figure 300-37) shows the level and outflow response to an
inflow change equal to half the maximum outflow with Buckley tuning. (The vessel
has a “residence time” of H = 6 minutes).
Notice how the vessel surge volume is used to smooth out the inflow change.Tight Level Control Tuning. Buckley has also solved the response equations for
the general case (that is, for all values of the damping coefficient, ). See
Figure 300-38.
Note that outflow overshoots inflow for any (any controller settings). We will use
these curves to develop tuning guidelines for tight level control
For tight level control, we choose as this will provide the fastest
possible non-oscillatory response. The plot shows that the level peak for is
(Eq. 300-41)
Solving for K C
(Eq. 300-42)
2
1
2
1=
∆
∆⇒=
∆
T
Peak
Max
In
H
L
F
Q
( )( )
( )( )
74.021
2174.074.0 =⋅=
∆∆
∆⋅=
T Peak
Max InC
H L
F Q K
H K F
Vol
K C Max
M
C I ⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⋅=
44τ
H F
Vol
Max
M I ⋅=⎟⎟
⎠ ⎞
⎜⎜⎝ ⎛ ⋅= 4.54.5τ
ζ
ζ
21707.0 ==ζ 707.0=ζ
64.0=⋅⎟⎟
⎠
⎞⎜⎜
⎝
⎛
∆⎟⎟
⎠
⎞⎜⎜
⎝
⎛
∆
∆C
In
Max
T
Peak K Q
F
H
L
) H / L(
) /F Q( . K
T Peak
Max InC ∆∆
∆= 640
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Then we specify a tight level range, e.g. 40% to 60% (starting from 50% level, the
level peak would be one tenth of the level transmitter range) for an inflow distur-
bance of half the maximum outflow. In mathematical terms, we have:
(Eq. 300-43)
Fig. 300-37 Level & Outflow Response — Buckley Tuning
Fig. 300-38 Level & Outflow Peak Plot (Any Zeta)
Outflow, GPM
50.0
0.0 0.0
100.0
150.0
200.0
50.0
100.0
150.0
200.0
Inflow, GPM
100
75
50
25
0
100
75
50
25
0
Level, % Level Controller Setpoint, %
0.0 12.8 25.6 38.4 51.2 64.0
Time, Minutes
0.0 12.8 25.6 38.4 51.2 64.0
Time, Minutes
330
1.0
0.5
0.0
1.4
1.0
0.0
LPeak
H T
F Max
Q In
K C
1.14
0.5 1.0 1.5 2.0
1.1
1.2
1.3
0.707
1.22 0.64
0.74
0.25
0.75
Overdamped Underdamped
Q Out, Peak
Q In
331
10
1
2
1=
∆
∆⇒=
∆
T
Peak
Max
In
H
L
F
Q
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The level controller gain is then
(Eq. 300-44)
We can now use this value for controller gain to find the integral time. Recall that
(Eq. 300-45)
Substituting and solving for τ I gives
(Eq. 300-46)
Substituting K C = 3.2 gives
(Eq. 300-47)
In summary, for tight level control, use a controller gain of 3.2 and a controller inte-
gral time of 0.625 times the vessel “residence time.” For example, a “six minute
vessel” would have a controller gain of 3.2 and controller integral time of 3.75
minutes.
The following plot (Figure 300-39) shows the level and outflow response to an
inflow change equal to half the maximum outflow with “tight” tuning (vessel “resi-
dence time” of H = 6 minutes).
2.3101
21640640 ==
∆∆
∆=
) / (
) / ( .
) H / L(
) /F Q( . K
T Peak
Max InC
H
K
Vol
F K C I
Meas
MaxC I ⋅=⋅⋅
= τ τ
ζ 2
1
2
1
21707.0 ==ζ
H K F
Vol
K C Max
M
C I ⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⋅=
22τ
H F
Vol
Max
M I ⋅=⎟⎟
⎠ ⎞
⎜⎜⎝ ⎛ ⋅= 625.0625.0τ
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Notice how the level controller quickly moves the outflow to keep the level near the
setpoint.
332 Forms of the PID Equation
There are two common forms of the PID equation as implemented in industrialcontrol equipment, that is, distributed control systems (DCS), programmable logic
controllers (PLC), or supervisory control and data acquisition systems (SCADA).
The non-interacting form of the PID algorithm is given below.
(Eq. 300-48)
This is the ISA standard form, and is sometimes called the parallel or ideal form.
The interacting form of the PID algorithm is given below.
(Eq. 300-49)
This is also called the series or factored form.
Fig. 300-39 Level & Outflow Response - Tight Tuning
Outflow, GPM
50.0
0.0 0.0
100.0
150.0
200.0
50.0
100.0
150.0
200.0
Inflow, GPM
100
75
50
25
0
100
75
50
25
0
Level, % Level Controller Setpoint, %
0.0 12.8 25.6 38.4 51.2 64.0
Time, Minutes
0.0 12.8 25.6 38.4 51.2 64.0
Time, Minutes
332
( ) ( ) ( ) ( )
0
0
1CO
dt
t dE t d t E t E K t CO D
t
I C +⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛ +′′+= ∫ τ τ
( ) ( ) ( ) ( )
0
0
11
COdt
t dE t d t E t E K t CO D
t
I
C +⎟ ⎠
⎞⎜⎝
⎛ ′+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ′′
′+′= ∫ τ τ
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In the Honeywell DCS, for example, both the interacting and non-interacting forms
of the PID equation are offered. Yokogawa offers only the non-interacting form.
It is important to note that the tuning parameters are different in the two forms.
Using the same tuning parameters in the two versions will not produce the same
results!
PID Conversion Equations
The equations which follow allow us to convert tuning parameters developed for a
particular PID form to equivalent tuning constants for the other PID form.
For the parallel PID form, we have
(Eq. 300-50)
For the series PID form, we have
Note that, because of the square root term, the equivalent factored version is valid
only for τ D/τ I ≤ 1/4.
( ) ( ) ( ) ( )
0
0
1CO
dt
t dE t d t E t E K t CO D
t
I
C +⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ +′′+= ∫ τ τ
⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛
′′+′= I
DC C K K
τ τ 1
( ) ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ′′
+′=′+′=I
DI DI I
τ
τ τ τ τ τ 1
( ) ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ′′
+′=′+′
′′=
I
DD
DI
I DD
τ
τ τ
τ τ
τ τ τ 1
( ) ( ) ( ) ( )
0
0
11
COdt
t dE t d t E t E K t CO D
t
I C +⎟
⎠
⎞⎜⎝
⎛ ′+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ′′
′+′= ∫ τ τ
( )( )I D
I DC I D
C C K
K K
τ τ
τ τ τ τ
/411
/2/411
2 −−=−+=′
( )( )I D
DI D
I I
τ τ
τ τ τ
τ τ
/411
2/411
2 −−=−+=′
( ) ( )I DI
I D
DD τ τ
τ
τ τ
τ τ /411
2/411
2−−=
−+=′
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Note also that if τ D/τ I ≤ 1/4 (in the non-interacting/ideal form), then (in theinteracting/factored form).
No conversion necessary for P-only or PI control; there is only one equation form.
PID Equation Form Affects Tuning Rules
We will examine how the form of the PID equation affects controller tuning rules.
For example, the Ziegler-Nichols tuning rules are usually stated as follows:
K C = 0.6 ⋅ K CU τ I = P U /2 τ D = P U /8 (Eq. 300-51)
But, what form of the PID equation did they assume? The controllers of the day
were closer to the interacting form than the non-interacting/ideal form. If we assume
that the Z-N tuning rules apply to the interacting form, then the following would be
a complete statement of their rules:
But, suppose the PID equation that was available in our control equipment had the
non-interacting form.
(Eq. 300-52)
We could simply use the conversion equations shown earlier to convert to the
Equivalent Values for the non-interacting form.
K C = (0.6 ⋅ K CU ) ⋅ (1.25) τ I = ( P U /2) ⋅ (1.25) τ I = (P U /2) / (1.25)(Eq. 300-53)
We would then get the same results as if we had used the original values in the inter-
acting PID equation.
However, if the interacting to non-interacting conversions were not made, the effec-
tive proportional gain would be 25% too low (less aggressive), effective integral
time would be 25% shorter (more aggressive), and the effective derivative time
would be 25% longer (more aggressive)
But what if Z-N assumed the non-interacting/ideal formulation? Most textbooks and
many journal articles apply Z-N to the ideal form!
I D τ τ ′=′
( ) ( ) ( ) ( )
0
0
11
COdt
t dE t d t E t E K t CO D
t
I C +⎟
⎠
⎞⎜⎝
⎛ ′+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ′′
′+′= ∫ τ τ
CU C K K ⋅=′ 6.0 2U I P =′τ 8U D P =′τ
( ) ( ) ( ) ( )
0
0
1CO
dt
t dE t d t E t E K t CO D
t
I
C +⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ +′′+= ∫ τ τ
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In this case, the Z-N tuning rules should be stated as follows:
K C = 0.6 ⋅ K CU τ I = P U /2 τ D = P U /8 (Eq. 300-54)
But, suppose the PID Equation we were using had the interacting form.
We could simply convert to the equivalent values for the interacting form.
If we did so, we would then get the same results as if we had used the original
values in the non-interacting PID equation.
However, If the non-interacting to interacting conversions were not made, the effec-
tive proportional gain would be 100% too high (more aggressive), the effective inte-
gral time would be 100% longer (less aggressive), and the effective derivative time
would be 100% shorter (less aggressive).
The following shows Z-N tuning with and without PID form conversion
(Figure 300-40).
The results for the parallel PID and series PID (converted) are very similar but not precisely the same because the conversion equations used didn’t consider the deriv-
ative filter term f
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