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ChevronTexaco Corporation 300-1 December 2003

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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300 Process Control Instrumentation Control Manual

December 2003 300-2 ChevronTexaco Corporation

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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Instrumentation Control Manual 300 Process Control


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

302


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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

306


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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:


K I = Integral mode gain [=] 1/minutes

E(t) = Error [=] %


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:


K C = Controller gain [=] %/% (dimensionless)

E(t) = Error [=] %

τ I = Integral (reset) time [=] minutes



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

309


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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:


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 [=] %



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:



E(t) = Error [=] %


τ D = Derivative time [=] minutes


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


22/98



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:



E(t) = Error [=] %


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+⋅=


23/98



τ D = Derivative time [=] minutes


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

316


24/98



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 – [ ]⋅ ⋅=


25/98



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( )+ +

⎩ ⎭⎨ ⎬⎧ ⎫

⋅=


26/98



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


27/98



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

(%)


28/98



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.


29/98



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


30/98



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


31/98



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 −⋅=+


32/98



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


33/98



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


34/98



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 ( )⋅ ⋅=


35/98



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

325


36/98



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

326


37/98



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 ++ +


38/98



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


39/98



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


40/98



(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


41/98



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


42/98



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


43/98



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


44/98



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

I

C +⎟ ⎠

⎞⎜⎝

⎛ ′+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ′′

′+′= ∫ τ τ


45/98



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

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

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−−=

−+=′


46/98



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

I C +⎟

⎠

⎞⎜⎝

⎛ ′+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ′′

′+′= ∫ τ τ

CU C K K ⋅=′ 6.0 2U I P =′τ 8U D P =′τ

( ) ( ) ( ) ( )

0

0

1CO

dt


t

I

C +⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +′′+= ∫ τ τ


47/98



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

sistema de control 2

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