Armando b corripio ph d p e , michael newell turning of industrial control systems, third edition international society of automation (2015)

The three instrumentation components required for feedback control are: • A sensor/transmitter to measure the process variable and send its value to the controller Measurement • A contro

Third Edition

By Armando B Corripio, Ph.D., P.E.

Chemical Engineering Louisiana State University

and

Michael Newell Automation Designer Polaris Engineering

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Preface to the Third Edition

This third edition of Tuning of Industrial Control Systems has been significantly

simplified from the second edition with the goal of having the discussion more in line with modern control systems and with language that is less aca-demic and more in tune with the vocabulary of the technicians who do the actual tuning of control systems in industry For example, we have eliminated any references to first- and second-order models since these terms are highly mathematical and may discourage some from appreciating the usefulness of the models We have also eliminated the distinction between series and paral-lel PID controllers since most modern installations use the series version and there is not much difference between the tuning of the two versions

We have reduced the tuning strategies to just one; the quarter-decay-ratio (QDR) formulas slightly modified by the Internal Model Control (IMC) rules for certain process characteristics All the tuning strategies are intended for responses to disturbances with a discussion on how to modify these responses

to avoid sudden excessive changes of the controller output on set point changes when such changes are undesirable

Chapter 10 is new and deals with the auto-tuning feature that has become standard on current process control systems We have successfully used the auto-tuning feature in our tuning work on oil refineries as a reference to guide our selection of the final tuning parameters for the controllers

We have kept the previous edition’s discussions on the problems of process nonlinearities and reset windup, and how to compensate for them All of the tuning strategies are demonstrated with computer simulation examples

Contents

Preface to the Third Edition ix

1 – Introduction 1

1-1 The Goal of Tuning 2

1-2 Feedback Control 3

1-3 Other Control Strategies 7

1-4 Organization of the Book 8

1-5 Summary 8

References 8

Review Questions 8

2 – The Feedback Controller 11

2-1 The PID Control Algorithm 12

2-2 Stability of the Feedback Loop 19

2-3 PID Controller Tuning by the Ultimate Gain and Period Method 21

2-4 The Need for Alternatives to Ultimate Gain Tuning 29

2-5 Summary 29

References 30

Review Questions 30

3 – Open-loop Characterization of Process Dynamics 33

3-1 Open-loop Testing—Why and How 34

3-2 Process Parameters from the Open-loop Test 36

3-3 Physical Significance of the Time Constant 41

3-4 Physical Significance of Dead Time 46

3-5 Effect of Process Nonlinearities 50

3-6 Summary 53

References 54

Review Questions 54

4 – How to Tune Feedback Controllers 57

4-1 Tuning from Open-loop Test Parameters 58

4-2 Practical Controller Tuning Tips 67

4-3 Reset Windup 70

4-4 Processes with Inverse Response 71

4-5 Effect of Nonlinearities 74

4-6 Summary 74

References 75

Review Questions 76

5 – Mode Selection and Tuning of Common Feedback Loops 77

5-1 Deciding on the Control Objective 78

5-2 Flow Control 79

5-3 Level and Pressure Control 80

5-4 Temperature Control 87

5-5 Analyzer Control 91

5-6 Summary 92

References 92

Review Questions 92

6 – Tuning Sampled-Data Control Loops 95

6-1 The Discrete PID Control Algorithm 96

6-2 Tuning Sampled-data Feedback Controllers 103

6-3 Selection of the Sampling Frequency 111

6-4 Compensation for Dead Time 113

6-5 Summary 117

References 118

Review Questions 118

7 – Tuning Cascade Control Systems 121

7-1 When to Apply Cascade Control 122

7-2 Selection of Controller Modes for Cascade Control 125

7-3 Tuning of Cascade Control Systems 127

7-4 Reset Windup in Cascade Control Systems 136

7-5 Summary 140

Review Questions 140

8 – Feedforward and Ratio Control 143

8-1 Why Feedforward Control? 144

8-2 Design of Linear Feedforward Controllers 149

8-3 Tuning of Linear Feedforward Controllers 152

8-4 Nonlinear Feedforward Compensation 158

8-5 Summary 165

References 166

Review Questions 166

9 – Multivariable Control Systems 169

9-1 What is Loop Interaction? 170

9-2 Pairing of Controlled and Manipulated Variables 174

9-3 Design and Tuning of Decouplers 186

9-4 Tuning of Multivariable Control Systems 193

9-5 Model Reference Control 196

9-6 Summary 198

References 199

Review Questions 199

10 – The Auto-tuner Application 201

10-1 Operation 202

10-2 Applications 205

10-3 Features and Settings 209

10-4 Summary 212

Review Questions 213

Appendix A – Suggested Reading and Study Materials 215

Appendix B – Answers to Study Questions 217

Index 233

1 Introduction

Automation is essential for the operation of chemical, petrochemical, and refining processes It is required to maintain process variables within safe operating limits while maintaining product purity and optimum operating conditions Because all processes are different in their speed of response and sensitivity to control adjustments and disturbances, the parameters of the automatic controllers must be adjusted to match the process characteristics

This procedure is known as tuning The purpose of this book is to provide

you, the reader, with an understanding of the most commonly used and cessful tuning techniques for the various control strategies used in industry

suc-This first chapter presents a general discussion of the goal of tuning, a tion of feedback control—the most common strategy—and a brief introduc-tion to other common control strategies

descrip-Learning Objectives — When you have completed this chapter, you should be able to

A Define the main goal of tuning a control system.

B Understand the feedback control strategy.

C Identify the various components of a feedback control loop.

1-1 The Goal of Tuning

The goal of tuning is to produce a smoothly operating process One common misconception is that every process variable should be brought to its desired value as quickly as possible and closely maintained at that value When a con-troller is “tightly” tuned to maintain close control of a process variable, it must make large, fast changes in its output, which usually causes disturbances to other variables in the process As the controllers of these other variables take action they, in turn, cause further disturbances that affect other variables Before long the entire process is in a state of continuous change, which is undesirable and may be unsafe in some occasions The situation worsens when the controllers cause oscillatory process responses, because then the process variables will be continuously changing

The following heuristics (“rules of thumb”) may prove helpful to those just starting in the tuning of processes:

• The variability of the controller output should not be excessive; ever, keeping the output variability low must be balanced against the precision with which the process variable is to be controlled

how-• Some variables do not have to be maintained at their desired values The most common example of this is liquid levels, which usually only need to be kept within a safe range

• The controller cannot move the process variable faster than the process can respond, so the controller speed must be matched to the speed of response of the process Some processes respond in a matter of min-utes, while others may take close to an hour or longer to respond Not many processes respond in a matter of seconds

One more item to keep in mind is that there is no such thing as fine-tuning a

controller, particularly a feedback controller In most cases the tuning ters need only be adjusted to one, or at most, two significant digits There are two reasons for this One is that feedback controllers are not that sensitive to variations in the third digit of their tuning parameters The other is that the characteristics of most processes—that is, speed of response and sensitivity to changes in controller output—vary with operating conditions, sometimes slightly and other times not so slightly This means that the controller tuning

parame-parameters are usually compromises selected to work in the range of ing conditions, and so their values are not precise.

operat-Understanding this simplifies the task of tuning because it reduces the ber of values of the tuning parameters to be tried For example, it is a lot easier

num-to decide between gain values of 1.0 or 1.5 than num-to try num-to find out whether the gain should be 1.276 In practice, all three of these values will work about the same

Armed with these heuristics and basic concepts, we are now ready to look at the feedback control strategy

1-2 Feedback Control

Feedback control is the basic strategy for the control of industrial processes It consists of measuring the process variable to be controlled (the controlled variable), comparing the measurement with its desired value or set point, and taking action based on the difference between them to reduce or eliminate the difference—that is, to bring the controlled variable to its desired value The action taken results in the adjustment of a process flow, such as the steam flow

to a heater, which has a direct effect on the controlled variable, such as the outlet process temperature The three instrumentation components required for feedback control are:

• A sensor/transmitter to measure the process variable and send its value to the controller (Measurement)

• A controller to compare the value of the process variable to its desired value, determine the required control action and send it to the final control element (Decision)

• A final control element, usually a control valve or variable speed drive,

to vary the manipulated process flow (Action)

A fourth element of the loop is the process itself, through which the lated flow affects the controlled variable The controlled variable is also

manipu-known as the process variable (PV), its desired value is the set point (SP), and the signal from the controller to the final control element is the controller output (OP)

It is important to realize that a feedback controller does not use a model of the process to compute its output It takes action by trial and error Tuning the controller is the procedure of adjusting the controller parameters to ensure that the controller output converges quickly to its correct value.

In order to better understand the concept of feedback control, consider as an example the process heater sketched in Figure 1-1 The process fluid flows inside the tubes of the heater and is heated by steam condensing on the out-side of the tubes The objective is to control the outlet temperature T of the process fluid in the presence of variations in process fluid flow (throughput or load) F and in its inlet temperature Ti This is accomplished by manipulating

or adjusting the steam flow to the heater Fs and with it the rate at which heat is transferred into the process fluid, thus affecting its outlet temperature

Figure 1-1 Feedback Temperature Control of a Process Heater

TT

TC

SP

PVOP

Condensate

In this example, the outlet temperature T is the (controlled) process variable

PV, the steam flow Fs is the manipulated variable, and changes in the process flow F and inlet temperature Ti are the disturbances that cause the tempera-ture to deviate from its desired value or set point SP The job of the feedback controller is to bring the temperature back to the set point by adjusting the steam flow whenever variations in the process flow or inlet temperature cause the outlet temperature to deviate

In Figure 1-1 the sensor transmitter is shown as a circle with the letters TT in it and the feedback controller is a circle with the letters TC in it This follows the standard ISA instrumentation notation1 in which the first letter denotes the variable being measured or controlled, in this case “T” for temperature, and the second letter is “T” for the transmitter and “C” for the controller The con-trol valve is represented by the symbol shown on the steam line to the heater Its purpose is to adjust the flow of steam (Fs) in response the controller output signal (OP)

The transmitter and the control valve are located in the field while the ler is located in a central control room Today, the signals between the trans-mitter and the controller and between the controller and the control valve are

control-typically digital signals transmitted through a fieldbus or by wireless

transmis-sion The control function is carried out by a computer or distributed control system (DCS) that receives the transmitter signal and transmits the controller output to the control valve The control valve is usually pneumatically oper-ated, requiring that the controller output be converted to an air pressure sig-nal This is done by a current-to-pressure (I/P) transducer

This book uses the instrumentation symbols recommended by the

ISA-5.1-1984 standard for conceptual diagrams, that is, diagrams that convey the basic control concept without regard to the specific implementation hardware In these diagrams the signals are represented as percent of range To facilitate understanding we will deviate slightly from the standard ISA notation for sig-nals and show the signals as arrows to indicate the direction in which the sig-nals travel, as shown in Figure 1-1

Figure 1-2 is a block diagram of the feedback control loop for the process heater It graphically shows the loop around which signals travel: a change in outlet temperature T causes a proportional change in the signal PV to the con-troller; the summer (circle), a part of the controller, calculates the error E or

deviation of the process variable from the set point SP and acts on this error

by changing the signal OP to the control valve; the control valve position changes, causing a change in steam flow Fs to the heater; this in turn causes a change in the outlet temperature T which then starts a new cycle of changes around the loop

The signs in Figure 1-2 represent the action of the various input signals on the output signal; that is, a positive sign means that an increase in input causes an

increase in output—direct action—while a negative sign means that an increase

in input causes a decrease in output—or reverse action For example, the

nega-tive sign by the process flow into the heater means that an increase in flow results in a decrease in outlet temperature Notice that by following the sig-nals around the loop, there is a net reverse action in the loop This property is

known as negative feedback and is a required characteristic of a feedback loop

for the loop to be stable In this example it means that an increase in heater outlet temperature results in a decrease in controller output, which in turn closes the control valve and reduces the steam flow This results in a decrease

in outlet temperature, as desired

To ensure this self-regulating effect the controller must act in the correct tion when the process variable changes In this example the controller action is

direc-reverse, that is, an increase in process variable results in a decrease in

control-Figure 1-2 Block Diagram of the Temperature Control Loop of the Process Heater

Controller Control Valve Heater

Sensor Transmitter

E

+ +

+

+

ler output Other processes may require direct action, for example when a tank

level controller adjusts the flow out of the tank In this case, an increase in uid level in the tank requires that the exit control valve open to increase the

liq-flow out of the tank and decrease the level Consequently, the action (direct or reverse) of the feedback controller is its most important characteristic.

1-3 Other Control Strategies

Although feedback control is by far the most common automatic control egy, there are other strategies that have been known to enhance control per-formance in terms of improving loop stability, preventing initial deviation of the process variable, and allowing tighter control This section will briefly introduce these strategies; their details and tuning procedures will be pre-sented in later chapters

strat-• Cascade Control This strategy consists of cascading feedback controllers

in a hierarchy with each controller adjusting the set point of the ler below it in the hierarchy, the controller at the top of hierarchy, or

control-primary, controls the primary process variable while the output of the

controller at the bottom of the hierarchy adjusts the final control

ele-ment The controllers below the master controller, called secondaries,

control variables that have an effect on the primary controlled variable The basic premise is that the secondary feedback loops improve the sta-bility of the primary controller by speeding up the overall response of the process

• Feedforward and Ratio Control This strategy consists of measuring the

disturbances that affect the controlled variable and adjusting the final control element to prevent deviation from the desired value of the con-trolled variable In general the scheme requires a model of the process

to determine the control adjustment in the final control element back control is combined with the feedforward controller to correct for

Feed-errors in the process model Ratio control is the simplest form of

feedfor-ward control in which the manipulated flow is ratioed to the flow which constitutes the disturbance

• Decoupling This strategy consists of installing decouplers between the

output signals of two or more feedback controllers to reduce the effect

of interaction between the controllers The interaction occurs through

the process when each controller output affects the process variables controlled by the other controllers.

1-4 Organization of the Book

The details of the PID (Proportional-Integral-Derivative) controller are

sented in Chapter 2, and tuning methods for feedback controllers are sented in Chapters 2, 3, and 4 How to select the controller modes for various types of control loops is the subject of Chapter 5 Chapter 6 presents the tun-ing of loops in which the process variable must be sampled, such as composi-tions measured by gas chromatographs and similar analyzers Tuning of cascade control systems is discussed in Chapter 7, design and tuning of feed-forward and ratio controllers in Chapter 8, and design and tuning of decou-plers in Chapter 9 Finally Chapter 10 presents the auto-tuning algorithms available in current computer control systems

pre-1-5 Summary

This first chapter has presented the goals of the tuning procedure and has introduced the feedback control strategy A brief description of other common control strategies has also been presented

References

1 ANSI/ISA-5.1-2009 - Instrumentation Symbols and Identification,

Interna-tional Society of Automation, Research Triangle Park, NC

Review Questions

1-1 What is the main goal of controller tuning?

1-2 Which two process characteristics must be considered when tuning the controller?

1-3 What are the three instrumentation components of a feedback control loop?

1-4 What is the fourth element of the feedback loop?

1-5 What is the most important characteristic of a feedback control loop?

1-6 A controller controls the temperature in an exothermic reactor by ulating the flow of cooling water to the jacket around the reactor What should be the fail position of the cooling water control valve, open or closed? What must be the action of the controller, direct or reverse?1-7 A controller controls the level in a stirred tank reactor by manipulating the flow of the reactants into the reactor Recommend the fail position of the reactants control valve, open or close, and the controller action, direct

manip-or reverse

1-8 A controller controls the composition of a caustic stream by ing the flow of the water that dilutes the concentrated caustic stream entering a mixer The control valve fails closed What must be the con-troller action, direct or reverse?

2 The Feedback

Controller

The basic concept of feedback control was introduced in the preceding ter This chapter presents details of the feedback controller and one of the methods proposed to tune it: the ultimate gain and period method

chap-Learning Objectives—When you have completed this chapter, you should be able to:

A Describe a Proportional-Integral-Derivative (PID) feedback controller.

B Know the functions of each of the three PID control modes.

C Understand how each of the three PID control modes responds.

D Define loop stability.

E Tune PID controllers by the ultimate gain and period method.

2-1 The PID Control Algorithm

The previous chapter showed that the purpose of the feedback controller is to compute its output signal based on the difference between the controlled pro-cess variable and its desired value or set point This section presents the three basic modes used by the controller to compute its output signal

The three basic modes of feedback control are proportional (P), integral (I)—also called reset—and derivative (D)—also called rate The controller can function

in a single mode or in a combination of either two modes or of all three, although today most controllers function in either two or three modes Either

way the device is known as a PID controller, based on the assumption that it

can function in all three modes Each of these modes introduces an adjustable

or tuning parameter into the operation of the feedback controller

Proportional Mode

The purpose of the proportional mode is to cause an instantaneous response

of the controller output to changes in the process variable The adjustable

parameter for the proportional mode is the gain—proportional gain or

con-troller gain—Kc Figure 2-1 illustrates how the proportional mode responds to

the process variable PV assuming that the controller is reverse acting and that the loop is open, that is, that the controller output does not affect the process variable The figure shows that:

• The controller output OP responds instantaneously to the process able PV

vari-• The response is proportional to the gain Kc.

• The proportional mode does not eliminate the sustained deviation set) between the process variable PV and the set point SP

(off-If a controller only has proportional mode there will normally be an offset Since console operators prefer to see all the variables at their set points, not many controllers are proportional only

Integral or Reset Mode

The purpose of the integral or reset mode is to eliminate the deviation

between the process variable and the set point The controller does this by moving its output with time at a rate proportional to the magnitude of the deviation Thus, as long as there is a deviation, the integral mode will keep moving the output The adjustable tuning parameter for the integral mode is the integral time—or reset time—TI, which is inversely proportional to the

rate at which the controller output changes Figure 2-2 illustrates how an

inte-gral reverse-acting controller responds to a sustained deviation between the

PV and the SP with the loop open The figure shows that:

• The output does not change when the deviation is zero

• The output changes continuously as long as there is a deviation

• The response is not instantaneous; that is, the integral mode takes time

to act

• The rate of change is slower the higher the integral time

Figure 2-1 Response of the Proportional Mode with the Loop Open

The step in output shown in the figure is the instantaneous response of the proportional mode It takes the integral mode a period of time equal to TI to duplicate the instantaneous response of the proportional mode.

The integral mode thus forces the process variable to the set point at the expense of slower action than the proportional mode This slow action intro-duces some instability into the response of the loop

Derivative or Rate Mode

The purpose of the derivative or rate mode is to anticipate the movement of the process variable by taking action proportional to its rate of change Just as the slow response of the integral mode decreases the stability of the control loop, the advance response of the derivative mode increases the stability The adjustable tuning parameter of the derivative mode is the derivative or rate time TD Figure 2-3 illustrates the response of the derivative mode to a ramp

Figure 2-2 Response of the Integral Mode to a Step Change in PV with the Loop Open

change in process variable, assuming a direct acting controller and an open loop The figure shows that:

• The derivative mode action is zero when the process variable remains constant

• The derivative response is proportional to the rate of change of the cess variable

pro-• The derivative response is proportional to the derivative time TD

• The derivative mode does not eliminate the sustained deviation between the process variable and its set point

Figure 2-3 Response of the Derivative Mode to a Ramp in the PV with the Loop Open

To better illustrate the anticipation action of the derivative mode, the response

to a ramp in the process variable is shown in Figure 2-4 for a direct-acting troller having both proportional mode (with a gain of 1.0) and derivative mode The initial step in the output is caused by the derivative mode and the continuous change is caused by the proportional mode As a result, the output leads the process variable by a period of time equal to the derivative time Notice that this does not mean the controller can predict the future, since the output cannot change until the process variable starts changing

con-Although the derivative mode increases the stability of the control loop, it has two undesirable characteristics One is that if the transmitter signal (PV) is noisy, the derivative can amplify noise To limit this amplification as the fre-quency of the noise increases, practical controllers have a built-in filter on the derivative mode that limits the amplification factor The other undesirable characteristic is that the derivative mode can cause sudden changes in control-ler output with sudden changes in the process variable This is usually not a problem because very seldom will the process variable change suddenly in practice To prevent sudden changes in set point from causing sudden

changes in output, all practical controllers have the derivative mode work only on the process variable, not on the deviation from the set point

Figure 2-4 Response of a Proportional-Derivative (PD) Controller to a Ramp in PV with the Loop Open

PID Tuning Parameters

The three adjustable tuning parameters of the PID controller are the tional gain Kc, the integral time TI, and the derivative time TD The time parameters are specified in minutes for most controllers, although some brands may require them in seconds Although modern control systems dis-play the process variable in engineering units (°F, lb/hr, barrels/day, psi, etc.), the proportional gain is dimensionless, because it is defined as the change in percent controller output per percent change in the process vari-able’s transmitter output (i.e., the fraction, in percent, that the process variable value is of the calibrated range of the transmitter)

propor-Figure 2-5 illustrates this concept for a temperature controller The left scale shows the process variable PV both in engineering units, °F, and percent of transmitter output The transmitter is calibrated to measure the temperature

in the range of 50°F (0% of the range) to 250°F (100% of the range) The set point SP is assumed to be in the middle of the range, 150°F or 50%

Figure 2-5 Process Variable in Engineering Units (ºF) and Percent of Range Illustration of Controller Proportional Band

0 20 40 60 80 100

50

250

150

0 20 40 60 80

100

SP Kc = 5.0 (20% PB)

Figure 2-5 also illustrates the concept of the controller proportional band (PB)

defined as the fraction of the transmitter output range that causes a 100% change in the controller output OP For the assumed gain of 5.0 the propor-tional band is 20% In some older controllers the gain was specified as the pro-portional band, but that is no longer the practice

Industrial Feedback Controllers

At the time when feedback controllers were individual off-the-shelf ments about 75% of the controllers used in industry were proportional-inte-gral (PI) or two-mode controllers and the balance were proportional-integral-derivative (PID) or three-mode controllers Today control calculations are per-formed by digital control computers and distributed control systems so that all controllers contain all three modes, and to reduce them to two modes one simply sets the derivative time TD to zero As we will see in Chapter 5, there are some control loops in which a single mode would be preferred, either pro-portional or integral, but in most systems it is not possible to specify a single-mode controller

instru-The feedback controllers are displayed for the operators in the control console and provide the following features:

• Process variable display

• Set point display

• Controller output display

• Set point adjustment

• Manual output adjustment

• Auto/Manual switch

• Remote/local set point switch (cascade systems only)

With these the operator can observe the current value of all the variables ciated with the control loop, adjust the set point, and if necessary switch the controller to Manual and adjust the controller output In cascade control sys-tems the operator can switch the slave controller to “local set point” and adjust its set point The controllers are programmed so that the switching from Manual to Auto and from local to remote set point is bump-less; that is,

asso-the controller output does not change, and, optionally, asso-the set point is set to the current value of the process variable when the switch is performed.

When the console is properly authorized under password protection, the instrument person or engineer can access the following features:

• Proportional gain, integral time, and derivative time adjustments

• Direct/reverse action switch

Having introduced the feedback controller in this section, the next section presents the concept of loop stability, that is, the effect of the controller on the process response

2-2 Stability of the Feedback Loop

One of the characteristics of feedback control loops is that they may become unstable The loop is said to be unstable when a small change in a disturbance variable or the set point causes the system to deviate widely from its normal operating point The two possible causes of instability are that the controller has the incorrect action (direct when it should be reverse or vice versa) or that

it is tuned too tightly—that is, the gain is too high, the integral time is too short, the derivative time is too long, or a combination of these Another possi-ble cause is that the process is inherently unstable, but this is rare

When the controller has the incorrect action, instability is manifested by the controller output “running away” to either its upper or its lower limit For example, if the temperature controller on the process heater of Figure 1-1 were set so that an increasing temperature increases its output, a small increase in temperature would result in an opening of the steam valve, which in turn would increase the temperature some more and the cycle would continue until the controller output was at its maximum with the steam valve fully open On the other hand, a small decrease in temperature would result in a closing of the steam valve, which would further reduce the temperature, and the cycle would continue until the controller output was at its minimum point with the steam valve fully closed Thus, the stability of the temperature con-trol loop of Figure 1-1 requires that the controller decrease its output when the process variable increases As we have seen, this is known as reverse action

When the controller is tuned too tightly, instability is recognized by observing that the signals in the loop oscillate, and that the amplitude of the oscillations increases with time, as seen in Figure 2-6 The cause of this instability is that the tightly tuned controller over-corrects for the error and, because of the delays and lags around the loop, the over-corrections are not detected by the controller until sometime later, causing a larger error in the opposite direction and further overcorrection If this process is allowed to continue the controller output will end up oscillating between its upper and lower limits

As pointed out earlier, the oscillatory type of instability is caused by the troller having too high a gain, too short an integral time, or too long a deriva-tive time, or a combination of these This leads into the simplest method for characterizing the process for the purpose of tuning the controller, that of determining the ultimate gain and period of oscillation of the loop

con-Figure 2-6 Response of an Unstable Feedback Control Loop

The first controller tuning method will now be introduced, one that depends

on measuring the characteristics of the control loop by determining the limit

of stability of the closed loop with a proportional controller

2-3 PID Controller Tuning by the Ultimate Gain and Period Method

The earliest published method of characterizing the process for controller ing was proposed by J G Ziegler and N B Nichols.1 The method consists of determining the ultimate gain and period of oscillation of the loop The ulti-mate gain is the gain of a proportional controller at which the loop oscillates with constant amplitude, and the ultimate period is the period of the oscilla-tions The ultimate gain is thus a measure of the controllability of the loop; that is, the higher the ultimate gain, the easier it is to control the loop The ulti-mate period is in turn a measure of the speed of response of the loop, that is, the longer the period, the slower the loop

tun-It follows from the definition of the ultimate gain that it is the gain at which the loop is at the threshold of instability At gains higher than the ultimate gain, the loop signals oscillate with increasing amplitude, as in Figure 2-6 Fig-ure 2-7 shows the response of a loop to a disturbance (for example, an increase

in process flow in the heater of Figure 1-1) with a proportional controller at increasing values of the controller gain As the figure shows, as long as the gain is lower than the ultimate gain, the amplitude of the oscillations

decreases with time When determining the ultimate gain it is very important

to approach it in small gain increments to ensure that it is not exceeded by much, lest the system become violently unstable

The procedure for determining the ultimate gain and period is carried out with the controller in Automatic and with the integral and derivative modes removed It is as follows:

1 Remove the integral mode by setting the integral time to its highest value Alternatively, if the controller model or program allows for switching off the integral mode, switch it off

2 Switch off the derivative mode or set the derivative time to its lowest value, usually zero

3 Carefully increase the proportional gain in steps After each increase, turb the loop by introducing a small step change in set point and observe the response of the controlled and manipulated variables, preferably on a trend recorder The variables should start oscillating as the gain is

dis-increased, as in Figure 2-7

4 When the amplitude of the oscillations remains constant (or mately constant) from one oscillation to the next, the ultimate controller gain has been reached Record it as Kcu

approxi-5 Measure the period of the oscillations from the trend recordings, as in ure 2-8 For better accuracy, time several oscillations and calculate the average period In Figure 2-8, for example, the time required by five oscil-lations is measured and then divided by 5

Fig-6 Stop the oscillations by reducing the gain to about half of the ultimate gain

Figure 2-7 The Response of a Proportional Controller Becomes Oscillatory as the Gain Is Increased

K c = 0.5

K c = 2

K c = 2

K c = 0.5

The procedure just outlined is simple and requires a minimum upset to the process, just enough to be able to observe the oscillations Nevertheless, the prospect of taking a process control loop to the verge of instability is not an attractive one from a process operation standpoint However, it is not abso-lutely necessary in practice to obtain sustained oscillations (see the section on Practical Ultimate Gain Tuning Tips) It is also important to realize that some simple loops cannot be made to oscillate with constant amplitude with just a proportional controller Fortunately, these are usually the simplest loops to control and tune.

Tuning for Quarter-Decay Response

Along with the method just outlined for determining the ultimate gain and period of a feedback control loop, Ziegler and Nichols proposed a set of for-mulas to tune the feedback controller for a specific response, the quarter-decay-ratio response, or QDR for short The QDR response is illustrated in Figure 2-9 for a step change in set point and for a step change in disturbance

Figure 2-8 Determination of the Ultimate Period

K cu = 3.42

5T u

Its characteristic is that each oscillation has an amplitude that is mately one-fourth that of the previous oscillation The formulas proposed by Ziegler and Nichols1 for calculating the QDR tuning parameters of P, PI, and PID controllers from the ultimate gain Kcu and period Tu are summarized in Table 2-1

approxi-Table 2-1 Quarter-Decay Tuning Formulas

Controller Gain Integral Time Derivative Time

P Kc = 0.50 Kcu

PI Kc = 0.45 Kcu TI = Tu/1.2 PID Kc = 0.60 Kcu TI = Tu/2 TD = Tu/8

Figure 2-9 Quarter-Decay-Ratio Responses to Set Point and Disturbance

It is intuitively obvious that for the proportional (P) controller the gain for a QDR response should be one-half of the ultimate gain, as Table 2-1 shows At the ultimate gain, the maximum error in each direction causes an identical maximum error in the opposite direction; at one-half the ultimate gain, the maximum error in each direction is exactly one-half the preceding maximum error in the opposite direction and one-fourth the previous maximum error in the same direction This is the quarter-decay-ratio response.

In Table 2-1 notice that the addition of integral mode results in a reduction of 10% in the QDR gain between the P and the PI controller tuning formulas This is due to the additional lag introduced by the integral mode On the other hand, the addition of the derivative mode allows increasing the controller gain by 20% over the proportional controller Therein lies the justification for the derivative mode, that is, the increase in the controllability of the loop Finally, the derivative and integral times in the PID formulas are in the ratio of 1:4 This is a useful relationship to keep in mind when tuning PID controllers

by trial-and-error (i.e., in those cases when the ultimate gain and period not be determined)

can-Example 2-1 Ultimate Gain Tuning of Process Heater

Determine the ultimate gain and period for the temperature control loop of Figure 1-1, and the quarter-decay tuning parameters for a P, a PI, and a PID controller.

For the temperature control loop, Figure 2-8 shows responses of the process variable PV and the controller output OP with a proportional controller and

a gain of 3.42, which results in sustained oscillations The ultimate gain is then 3.42 A small change in the flow to the heater is used to start the oscilla- tions In the figure, the period of the oscillations is the ultimate period Ultimate gain: K cu = 3.42 (or 100/3.42 = 29%PB)

Ultimate period: 37.5 3.5–

5 - = 6.8 min

Using the formulas of Table 2-1, the QDR tuning parameters are:

P controller: Gain = 0.5(3.42) = 1.7 (or 58%PB)

PI controller: Gain = 0.45(3.42) = 1.5 (or 65% PB)

Figure 2-10 Proportional Controller Responses to a Change in Process Flow

K c = 1.7

K c = 1.0

K c = 1.7

K c = 1.0

Figures 2-11 and 2-12 show the responses of the PI and PID controllers, respectively In each case, the smaller proportional gain results in less oscilla-tory behavior and less initial movement of the controller output, at the expense of a larger initial deviation of the PV and a slower return to the set point This shows that the tuning parameters, particularly the gain, can be varied from the values given by the tuning formulas to obtain the desired response

Notice the offset in Figure 2 10, and the significant improvement that the derivative mode produces in the responses of Figure 2-12 over those of Figure 2-11

Practical Ultimate Gain Tuning Tips

1 In determining the ultimate gain and period, it is not absolutely necessary

to force the loop to oscillate with constant amplitude This is because the ultimate period is not sensitive to the gain as the loop approaches the ulti-mate gain Any oscillation that allows a rough estimate of the ultimate

Figure 2-11 Proportional-Integral (PI) Controller Responses to a Change in Process Flow

period gives good enough values of the integral and derivative times The proportional gain can then be adjusted to obtain an acceptable response For example, notice in Figure 2 7 that, for the case of a gain of 2, the period

of oscillation is 8.0 minutes, which is less than 20% away from the actual ultimate period (6.8 min)

2 The performance of the feedback controller is not usually sensitive to the tuning parameters Thus, when adjusting the parameters from the values given by the formulas one would be wasting time by changing them by less than 50%

3 The recommended parameter adjustment policy is to leave the integral and derivative times fixed at the values calculated from the tuning formu-las, and adjust the gain, up or down, to obtain the desired response

The QDR tuning formulas allow the tuning of controllers for a specific

response when the ultimate gain and period of the loop can be determined

Figure 2-12 Proportional-Integral-Derivative (PID) Responses to a Change in Process Flow

The chapters that follow present alternative methods for characterizing the dynamic response of the loop and for tuning feedback controllers The follow-ing section brings up the need for such alternative methods.

2-4 The Need for Alternatives to Ultimate Gain Tuning

Although the ultimate gain tuning method is simple and fast, other methods

of characterizing the dynamic response of feedback control loops have been developed over the years The need for these alternative methods is based on the fact that it is not always possible to determine the ultimate gain and period

of a loop As pointed out earlier, some simple loops will not exhibit constant amplitude oscillations with a proportional controller

The ultimate gain and period, although sufficient to tune most loops, do not give insight into which process or control system characteristics could be modified to improve feedback controller performance A more fundamental method of characterizing process dynamics is needed to guide such

modifications

There is also the need to develop tuning formulas for responses other than the quarter-decay-ratio response This is because the set of PI and PID tuning parameters that produce a quarter-decay response are not unique It is easy to see that for each setting of the integral and derivative time, there will usually

be a setting of the controller gain that will produce a quarter-decay response This makes for an infinite number of combinations of the tuning parameters that satisfy the quarter-decay-ratio specification

2-5 Summary

This chapter has introduced the three modes of the derivative controller and one method to tune it based on the ultimate gain and period of the closed control loop The next chapter introduces an open loop method for characterizing the dynamic response of the process in the loop; the chapters that follow present tuning formulas based on the parameters of the open-loop model

1 Ziegler, J G and Nichols, N B “Optimum Settings for Automatic

Con-trollers,” Transactions of the ASME, Vol 64, Nov 1942, p 759.

Review Questions

2-1 A controller has a gain of 3 For each of the following cases determine by how much the proportional mode causes the output of the controller to change and in which direction — increase or decrease:

a The PV increases by 10% and the controller is reverse acting

b The PV decreases by 15°C, the transmitter range is 0 to 150°F, and the controller is direct acting

c The PV increases by 250 kg/hr, the transmitter range is 0 to

50,000 kg/hr and the controller is reverse acting

2-2 A direct acting PI controller has a gain of 2 and an integral time of 5 minutes Sketch the response of the controller output when a deviation

of 10% from the set point is instantly applied and sustained By how much has the controller output changed 15 minutes after the deviation was applied?

2-3 A continuous rise in PV of 3% per minute is applied to a reverse acting PID controller with a gain of 1.0 and a derivative time of 0.6 minutes By how much and in which direction is the change in the controller output caused by the derivative mode? Sketch the derivative mode response.2-4 A controller is switched to Automatic and its output starts rising imme-diately and does not stop until it reaches its upper limit What do you think is the cause?

2-5 A controller is switched to Automatic and starts oscillating with ing amplitude of the oscillations What would you do to correct this problem?

increas-2-6 Why do you think that the tuning formulas of Table 2-1 relate the gral and derivative times to the ultimate period of oscillation of the loop?

inte-2-7 After tuning a controller using the formulas of Table 2-1 you find the variation in the controller output when a disturbance upsets the system

is higher than you would like it to be How would you adjust the tuning

to obtain a more reasonable behavior?

3 Open-loop Characterization of Process Dynamics

This chapter shows how to characterize the dynamic response of a process from open-loop step tests, and how to determine the process gain, time con-stant, and dead time from the results of the step tests These parameters of a simple-lag-plus-dead-time (SLPDT) model will be used to tune feedback and feedforward controllers in the chapters to follow

Learning Objectives—When you have completed this chapter, you should be able to:

A Perform an open-loop test and analyze its results.

B Estimate the process gain, time constant, and dead time.

C Understand the physical significance of a simple lag and of dead time.

D Understand process nonlinearity

3-1 Open-loop Testing—Why and How

The preceding chapter showed how to determine the ultimate gain and period

of a feedback control loop by performing a test with the controller on matic output” (i.e., with the loop closed) By contrast, this chapter shows how

“auto-to determine the process parameters—gain, time constant, and dead time—by performing a test with the controller on “manual output,” that is, an open-loop test Such a test presents a more fundamental model of the process than the ultimate gain and period

The purpose of an open-loop test is to determine the relationship between the process variable PV and the controller output OP In the case of a feedback control loop, this relationship is of primary interest However, the relationship between the controlled variable and a disturbance can also be determined, provided that the disturbance variable can be changed and measured This chapter considers only the PV/OP variable pair, since the principles of the testing procedure and analysis are the same for any pair of variables

To better understand the open-loop test concept, consider the temperature feedback control loop of the heater sketched in Figure 3-1 When the controller

is switched to “manual output,” the loop is interrupted at the controller ing direct manipulation of the controller output signal OP The response of the process variable PV to the controller output OP is a combination of the responses of the control valve, the process (the heater) and the sensor/trans-mitter in Figure 3-1 This emphasizes that the two signals of interest in an open-loop test are the controller output variable OP and the transmitter out-put signal PV Notice that in practice, the true process variable, in this case the heater outlet temperature T is not accessible; what is accessible is the measure-ment of that variable (i.e., the transmitter output signal) Similarly, the flow through the control valve, Fs, even if it were measured, is not of as much inter-est as the controller output signal OP, which is the variable directly manipu-lated by the controller

allow-The procedure for performing an open-loop test is simply to cause a step change in controller output OP and record the resulting response of the trans-mitter signal PV The equipment required to cause the change is simply the controller itself, given that its output can be directly manipulated when it is in the Manual state and a means to record the response of the PV Today’s com-

puter and microprocessor-based controllers make the recording of the

response a straightforward task

The simplest type of open-loop test is a step test—that is, a sudden and tained change in the controller output OP Figure 3-2 shows a typical step test and the response of the process variable The S-shaped response is typical of most processes that are self-regulating (i.e., they reach a steady value after the response time is over) What causes this type of response is the fact that the outputs of the different components of the control loop lag their inputs in time (i.e., their outputs do not immediately respond to their inputs) So there are lags in the control valve and the sensor/transmitter, as well as one or more lags in the process Because of this the process variable does not start chang-ing right after the step change is applied As Figure 3-2 shows, the rate of change starts at zero and then increases to a maximum rate that is followed by

sus-a decresus-asing rsus-ate sus-as the vsus-arisus-able sus-approsus-aches its finsus-al stesus-ady vsus-alue

Figure 3-1 Temperature Control of a Process Heater

TT

TC

SP

PV OP

Steam

Steam trap

Condensate

Another reason that generally the process variable does not start changing

immediately is the presence of transportation lag in the loop This is the lag

caused by the time it takes for the process fluid to move through the process However, for most loops real transportation lag, usually of the order of sec-onds, is negligible relative to the lags that are commonly of the order of min-utes (See Section 3-5 for further discussion of transportation lag.)

3-2 Process Parameters from the Open-loop Test

This section shows how to extract the process characteristic parameters from the results of a step test using the step test of Figure 3-2 as an example The three parameters of interest to the tuning of the feedback controller are:

• The process sensitivity or gain, defined as the magnitude of the final

steady change in the process variable for a unit sustained change in the controller output, or

Figure 3-2 Open-Loop Step Test of the Process Heater