Turning of Industrial Control Systems, Third Edition [Third edition] 087664034X, 978-0-87664-034-0, 9781680155556, 1680155555

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Tuning of Industrial Control Systems Third Edition By Armando B. Corripio, Ph.D., P.E. Chemical Engineering Louisiana State University and Michael Newell Automation Designer Polaris Engineering

Notice The information presented in this publication is for the general education of the reader. Because neither the author nor the publisher has any control over the use of the information by the reader, both the author and the publisher disclaim any and all liability of any kind arising out of such use. The reader is expected to exercise sound professional judgment in using any of the information presented in a particular application. Additionally, neither the author nor the publisher has investigated or considered the effect of any patents on the ability of the reader to use any of the information in a particular application. The reader is responsible for reviewing any possible patents that may affect any particular use of the information presented. Any references to commercial products in the work are cited as examples only. Neither the author nor the publisher endorses any referenced commercial product. Any trademarks or tradenames referenced belong to the respective owner of the mark or name. Neither the author nor the publisher makes any representation regarding the availability of any referenced commercial product at any time. The manufacturer’s instructions on the use of any commercial product must be followed at all times, even if in conflict with the information in this publication.

Copyright © 2015 International Society of Automation (ISA) All rights reserved. Printed in the United States of America. 10 9 8 7 6 5 4 3 2 ISBN: 978-0-87664-034-0 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher. ISA 67 Alexander Drive P.O. Box 12277 Research Triangle Park, NC 27709 Library of Congress Cataloging-in-Publication Data in process

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 academic 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 parallel 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.

ix

Contents

Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix 1 – Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1-1. The Goal of Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2. Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3. Other Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4. Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 7 8 8 8 8

2 – The Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2-1. The PID Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2. Stability of the Feedback Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3. PID Controller Tuning by the Ultimate Gain and Period Method . . . . . 2-4. The Need for Alternatives to Ultimate Gain Tuning . . . . . . . . . . . . . . . . 2-5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 19 21 29 29 30 30

3 – Open-loop Characterization of Process Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 33 3-1. Open-loop Testing—Why and How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2. Process Parameters from the Open-loop Test . . . . . . . . . . . . . . . . . . . . . . 3-3. Physical Significance of the Time Constant . . . . . . . . . . . . . . . . . . . . . . . . 3-4. Physical Significance of Dead Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5. Effect of Process Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34 36 41 46 50 53 54 54

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Tuning of Industrial Control Systems, Third Edition

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

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vii

8-3. Tuning of Linear Feedforward Controllers . . . . . . . . . . . . . . . . . . . . . . . 8-4. Nonlinear Feedforward Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152 158 165 166 166

9 – Multivariable Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9-1. What is Loop Interaction?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2. Pairing of Controlled and Manipulated Variables . . . . . . . . . . . . . . . . . 9-3. Design and Tuning of Decouplers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-4. Tuning of Multivariable Control Systems . . . . . . . . . . . . . . . . . . . . . . . . 9-5. Model Reference Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

170 174 186 193 196 198 199 199

10 – The Auto-tuner Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10-1. Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3. Features and Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

202 205 209 212 213

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 successful tuning techniques for the various control strategies used in industry. This first chapter presents a general discussion of the goal of tuning, a description of feedback control—the most common strategy—and a brief introduction to other common control strategies.

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

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Tuning of Industrial Control Systems, Third Edition

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 controller 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; however, keeping the output variability low must be balanced against the precision with which the process variable is to be controlled. • 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 minutes, 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 parameters 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

Introduction

3

parameters are usually compromises selected to work in the range of operating conditions, and so their values are not precise. Understanding this simplifies the task of tuning because it reduces the number of values of the tuning parameters to be tried. For example, it is a lot easier to decide between gain values of 1.0 or 1.5 than to try 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 manipulated flow affects the controlled variable. The controlled variable is also 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).

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Tuning of Industrial Control Systems, Third Edition

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 outside 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

SP

Steam

OP

TC

Fs

PV

F Ti

TT

Process fluid

T

Steam trap

Condensate

Introduction

5

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 temperature 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 control 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 controller is located in a central control room. Today, the signals between the transmitter and the controller and between the controller and the control valve are typically digital signals transmitted through a fieldbus or by wireless transmission. 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 operated, requiring that the controller output be converted to an air pressure signal. This is done by a current-to-pressure (I/P) transducer. This book uses the instrumentation symbols recommended by the ISA-5.11984 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 signals and show the signals as arrows to indicate the direction in which the signals 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 controller; the summer (circle), a part of the controller, calculates the error E or

6

Tuning of Industrial Control Systems, Third Edition

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.

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

Ti

F

-

SP

E

+

Controller

T

Fs +

OP+

+

Control Valve

+

Heater

-

PV

Sensor Transmitter

+

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 negative 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 signals 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 direction when the process variable changes. In this example the controller action is reverse, that is, an increase in process variable results in a decrease in control-

Introduction

7

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 liquid level in the tank requires that the exit control valve open to increase the 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 strategy, there are other strategies that have been known to enhance control performance 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 presented in later chapters. • Cascade Control. This strategy consists of cascading feedback controllers in a hierarchy with each controller adjusting the set point of the controller below it in the hierarchy, the controller at the top of hierarchy, or primary, controls the primary process variable while the output of the controller at the bottom of the hierarchy adjusts the final control element. 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 stability 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 controlled variable. In general the scheme requires a model of the process to determine the control adjustment in the final control element. Feedback control is combined with the feedforward controller to correct for errors in the process model. Ratio control is the simplest form of feedforward 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

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Tuning of Industrial Control Systems, Third Edition

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 presented in Chapter 2, and tuning methods for feedback controllers are presented 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 tuning of loops in which the process variable must be sampled, such as compositions measured by gas chromatographs and similar analyzers. Tuning of cascade control systems is discussed in Chapter 7, design and tuning of feedforward and ratio controllers in Chapter 8, and design and tuning of decouplers in Chapter 9. Finally Chapter 10 presents the auto-tuning algorithms available in current computer control systems.

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, International 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?

Introduction

9

1-6. A controller controls the temperature in an exothermic reactor by manipulating 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 or reverse. 1-8. A controller controls the composition of a caustic stream by manipulating the flow of the water that dilutes the concentrated caustic stream entering a mixer. The control valve fails closed. What must be the controller action, direct or reverse?

2 The Feedback Controller

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

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.

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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 process 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 controller 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 variable PV. • The response is proportional to the gain Kc. • The proportional mode does not eliminate the sustained deviation (offset) between the process variable PV and the set point SP. 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.

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13

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

PV SP

OP

Kc = 1.0 Kc = 2.0

time 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 integral 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.

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

PV SP

Kc = 1.0 OP

TI = 6 TI = 12 time

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 introduces 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

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15

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 process variable. • 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

PV SP

OP

TD = 6.0 TD = 3.0

0 time

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Tuning of Industrial Control Systems, Third Edition

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 controller 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.

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

Kc = 1.0

OP

10

PV

TD = 10 SP time

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 frequency 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 controller 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.

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17

PID Tuning Parameters The three adjustable tuning parameters of the PID controller are the proportional 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 display 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 variable’s transmitter output (i.e., the fraction, in percent, that the process variable value is of the calibrated range of the transmitter). 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

T, °F 250

150

50

PV, %

OP, %

100

100

80

80

60

60

SP

Kc = 5.0 (20% PB)

40

40

20

20

0

0

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Tuning of Industrial Control Systems, Third Edition

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 proportional band is 20%. In some older controllers the gain was specified as the proportional band, but that is no longer the practice.

Industrial Feedback Controllers At the time when feedback controllers were individual off-the-shelf instruments about 75% of the controllers used in industry were proportional-integral (PI) or two-mode controllers and the balance were proportional-integralderivative (PID) or three-mode controllers. Today control calculations are performed 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 proportional or integral, but in most systems it is not possible to specify a singlemode controller. 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 associated with the control loop, adjust the set point, and if necessary switch the controller to Manual and adjust the controller output. In cascade control systems 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,

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19

the controller output does not change, and, optionally, 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 possible 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 control 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.

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Tuning of Industrial Control Systems, Third Edition

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.

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

As pointed out earlier, the oscillatory type of instability is caused by the controller having too high a gain, too short an integral time, or too long a derivative 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.

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21

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 tuning 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 ultimate 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 oscillations. 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 ultimate period is in turn a measure of the speed of response of the loop, that is, the longer the period, the slower the loop. 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. Figure 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.

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Tuning of Industrial Control Systems, Third Edition

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

Kc = 2

Kc = 0.5

Kc = 2

Kc = 0.5

3. Carefully increase the proportional gain in steps. After each increase, disturb 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 increased, as in Figure 2-7. 4. When the amplitude of the oscillations remains constant (or approximately constant) from one oscillation to the next, the ultimate controller gain has been reached. Record it as Kcu. 5. Measure the period of the oscillations from the trend recordings, as in Figure 2-8. For better accuracy, time several oscillations and calculate the average period. In Figure 2-8, for example, the time required by five oscillations is measured and then divided by 5. 6. Stop the oscillations by reducing the gain to about half of the ultimate gain.

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23

Figure 2-8. Determination of the Ultimate Period

Kcu = 3.42

5Tu

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 absolutely 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 formulas to tune the feedback controller for a specific response, the quarterdecay-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.

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Tuning of Industrial Control Systems, Third Edition

Its characteristic is that each oscillation has an amplitude that is approximately 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.

Table 2-1. Quarter-Decay Tuning Formulas Controller

Gain

Integral Time

P

Kc = 0.50 Kcu

PI

Kc = 0.45 Kcu

TI = Tu/1.2

PID

Kc = 0.60 Kcu

TI = Tu/2

Derivative Time

TD = Tu/8

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

A A/4 SP

Set point change

SP

A/4 Disturbance A

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25

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 cannot be determined).

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 oscillations. In the figure, the period of the oscillations is the ultimate period. Ultimate gain: Ultimate period:

Kcu = 3.42 (or 100/3.42 = 29%PB) 37.5 – 3.5 ------------------------ = 6.8 min 5

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Tuning of Industrial Control Systems, Third Edition

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) TI = 6.8/1.2 = 5.7 min

PID controller:

Gain = 0.6(3.42) = 2.0 (50%PB) TI = 6.8/2 = 3.4 min TD = 6.8/8 = 0.85 min

Figure 2-10 shows the response of the controller output and the outlet process temperature to an increase in process flow for the proportional controller with a QDR gain of 1.7 and with a lower gain of 1.0. The figure shows that the lower gain results in a larger initial deviation of the PV and a larger offset, but the oscillations are smaller and the required variation in controller output is less.

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

Kc = 1.7

Kc = 1.0

Kc = 1.7

Kc = 1.0

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27

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

Kc = 1.5

Kc = 1.0 TI = 5.7 min

Kc = 1.5 Kc = 1.0

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 ultimate gain. Any oscillation that allows a rough estimate of the ultimate

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Tuning of Industrial Control Systems, Third Edition

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

Kc = 2.0

Kc = 1.0

TI = 3.4 min TD = 0.85 min

Kc = 2.0

Kc = 1.0

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 formulas, 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.

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29

The chapters that follow present alternative methods for characterizing the dynamic response of the loop and for tuning feedback controllers. The following 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 proportional-integralderivative 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.

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References 1. Ziegler, J. G. and Nichols, N. B. “Optimum Settings for Automatic Controllers,” 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 immediately 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 increasing amplitude of the oscillations. What would you do to correct this problem?

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2-6.

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

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 constant, 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.

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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 “automatic output” (i.e., with the loop closed). By contrast, this chapter shows how to determine the process parameters—gain, time constant, and dead time—by performing a test with the controller on “manual output,” that is, an openloop 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 allowing 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/transmitter 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 output 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 measurement 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 interest as the controller output signal OP, which is the variable directly manipulated by the controller. 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 transmitter 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-

Open-loop Characterization of Process Dynamics

35

Figure 3-1. Temperature Control of a Process Heater SP

Steam

OP

TC

Fs

PV

F Ti

TT

Process fluid

T

Steam trap

Condensate

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 sustained 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 changing 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 a decreasing rate as the variable approaches its final steady value.

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Tuning of Industrial Control Systems, Third Edition

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

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 seconds, is negligible relative to the lags that are commonly of the order of minutes. (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

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37

Final steady change in PV Gain = -------------------------------------------------------------------Change in OP • The process time constant, a measure of how long it takes for the process to respond. • The process dead time, a measure of how long it takes for the process variable to start changing after the step change in controller output is applied. These three parameters are discussed in detail below. Together they constitute a simple-lag-plus-dead-time (SLPDT) model of the loop response that will be used to tune the feedback controller. These parameters are estimated from the step response of the loop and are then used to tune the feedback controller.

Gain The gain determined from the step response is the product of the gains of the control valve (or other final control element), the process, and the sensor/ transmitter. It must be expressed as percent change in transmitter output per percent change in controller output, not in engineering units. Figure 3-3 shows the determination of the gain from the step response of Figure 3-2. As in most control computer-generated plots, the PV is displayed in engineering units, °F, and the final steady change in the PV is (209.5 – 190) = 19.5°F. To convert to percent of the transmitter output, we need the range of the transmitter. Let us say it is 50 to 250°F; the change in the PV is then: 19.5°F -------------------------------- 100% = 9.75% ( 250 – 50 )°F Since this change is caused by a 5% change in controller output, the gain is: 9.75% --------------- = 1.95 5%

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Tuning of Industrial Control Systems, Third Edition

Figure 3-3. Determination of Loop Parameters from Step Response

t2 = 15 – 5 = 10 min 190 + 0.632(19.5) = 202.3°F

209.5 – 190 = 19.5°F

190 + 0.283(19.5) = 195.5°F

t1 = 10 – 5 = 5 min

Dead Time and Time Constant The dynamic response is characterized by two parameters: the dead time and the time constant. Although Ziegler and Nichols2 proposed two different parameters that must be determined by graphical construction, Cecil Smith1 proposed to fit a model consisting of dead time and a single lag to the response. Smith’s method requires only reading two points from the response and avoids the cumbersome graphical construction. The procedure consists of reading the times in the response at which the PV reaches 28.3% and 63.2% of its total change. These times correspond, respectively, to one-third of the time constant and one time constant in the response of a single lag, so the difference between them is two-thirds of the time constant. They are picked in the area of high rate of change of the response, which results in more accurate determination of the two time values. The dynamic parameters are then: Time constant = 1.5(t2 – t1)

Open-loop Characterization of Process Dynamics

39

Dead time = t2 – (Time constant) where t1 and t2 are, respectively, the times at which the response reaches 28.3% and 63.2% of its total change. From Figure 3-3, the results for our example are: At PV = 190 + 0.283(19.5) = 195.5°F

t1 = 10 – 5 = 5 min

At PV = 190 + 0.632(19.5) = 202.3°F

t2 = 15 – 5 = 10 min

The time constant is then:

1.5(10 – 5) = 7.5 min

And the dead time is:

10 – 7.5 = 2.5 min

Note that we subtracted 5 minutes from the times read in the response to determine the times from the application of the step change in controller output.

Example 3-1. Step Test on a Hot Oil Temperature Controller Figure 3-4 shows the step response of a hot oil temperature controller in a refinery. The coil outlet temperature is controlled by manipulating the flow of hot oil through the coil, which is in the convection section of a furnace. A step increase of 3% in the controller output at time zero results in the response shown in the figure. The response data are read off the trend plot and entered into a spreadsheet for plotting and analysis. The range of the temperature transmitter is 0 to 1200°F. The change in the process variable is:

( 635.8 – 638.2 )°F -------------------------------------------- 100% = – 0.8% ( 1200 – 0 )°F

The gain is then –0.8%/3% = –0.267.

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Figure 3-4. Determination of Loop Parameters from Step Test of a Hot Oil Temperature Controller 638.5

638.2°F 638.0

638.2 + (0.283)(-2.4) = 637.5°F

Temperature, °F

637.5

637.0

t1 = 54 sec

638.2 + (0.632)(-2.4) = 636.7°F

636.5

635.8 - 638.2 = -2.4°F 636.0

t2 = 87 sec

635.8°F

635.5 0

50

100

150

200 time, sec

Measurement

250

300

350

400

Calculated

From the figure, t1 = 54 seconds and t2 = 87 seconds. The data were manually obtained from the trend recorder on the control system in a refinery and entered into a spreadsheet. Interpolation was then used on the spreadsheet to obtain the model parameters. The time constant and dead time are then: Time constant:

1.5(87 – 54) = 50 sec (0.82 min)

Dead time:

87 – 50 = 37 sec (0.62 min)

The dashed line in Figure 3-4 is a plot of the response of a process with a gain of –0.267, a transportation lag of 37 sec, and a time constant of 50 sec. The negative sign of the gain means that an increase in controller output results in a decrease in temperature because it causes an increase in the process flow through the coil. The controller must then have direct action.

Open-loop Characterization of Process Dynamics

41

This example illustrates how modern computer control systems allow the precise determination of the step response parameters using a small step change in controller output that results in a very small change in the process variable (less than 1%). This step response method of characterizing the process dynamics applies only when the process is self-regulating; that is, when it is one that reaches a new steady state when driven by a sustained change in controller output. There are two types of processes that are not self-regulating: imbalanced or integrating processes, and open-loop unstable processes. A typical example of an imbalanced process is the liquid level in a tank, and an example of an unstable process is an exothermic chemical reactor. It is obviously impractical to perform step tests on processes that are not self-regulating. Fortunately, most processes are self-regulating.

3-3. Physical Significance of the Time Constant Although the process time constant and dead time can be estimated from an open-loop step test (as described in the previous section), it is important to examine the physical significance of these two dynamic parameters of the process. Doing this will allow estimation of the process time constant and dead time from physical process characteristics (e.g., volumes, flow rates, valve sizes) when it is not convenient to perform the step test. This section concerns the time constant, and the next section explores the dead time. To understand the physical significance of the time constant, consider some of the physical systems whose dynamic response can be characterized by a single time constant and no dead time. Such systems consist of a single capacitance to store mass, energy, momentum or electricity, and a conductance to the flow of these quantities. We will call such single capacitance/conductance systems “simple lags” or just “lags.” Figure 3-5 presents several examples of systems that constitute simple lags.

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Figure 3-5. Physical Systems with Simple Lag Dynamics: (a) Electrical Circuit; (b) Liquid Storage Tank; (c) Gas Surge Tank; (d) Blending Tank

R

Inlet flow

ein

C

eout

Level

Outlet flow (b)

(a) Outlet flow

Inlet flow

Kv

F1 C1

F C V

F2 C2

P V T Kv (c)

(d)

The time constant of a lag is defined as the ratio of its capacitance to its conductance or the product of the capacitance times the resistance (the resistance is the reciprocal of the conductance): Capacitance τ = ---------------------------------- = Capacitance × Resistance Conductance where τ is the time constant. The concepts of capacitance, resistance, and conductance are best understood by analyzing the physical systems of Figure 3-5. In each of them there is a physical quantity which is conserved, a rate of flow of that quantity, and a potential that drives the flow. The capacitance is defined by the amount of quantity conserved per unit of potential: Amount of quantity conserved Capacitance = --------------------------------------------------------------------------------Potential

Open-loop Characterization of Process Dynamics

43

The conductance is the ratio of the flow to the potential that drives it: Flow of quantity conserved Conductance = ------------------------------------------------------------------------Potential To better understand the physical meanings to the terms just presented, consider each of the physical systems of Figure 3-5. Electrical Circuit (Figure 3-5a). For this system the quantity conserved is electric charge, the potential is electric voltage, and the flow is electric current. The capacitance is provided by the ability of the capacitor to store electric charge and the conductance is the reciprocal of the resistance of the electrical resistor. The time constant is then given by: τ = RC where: R =

the resistance of the electrical resistor, ohms

C =

the capacitance of the electrical capacitor, farads

and the time constant τ is in seconds. Liquid Storage Tank (Figure 3-5b). In this common process system, the quantity conserved is the volume of liquid (assuming constant density), the capacitance is provided by the ability of the tank to store liquid, and the potential for flow through the valve is provided by the level of liquid in the tank. The capacitance is volume of liquid per unit level (i.e., the cross-sectional area of the tank), and the conductance is the change in flow through the valve per unit change in level. The time constant can then be estimated by: τ = A/Kv where: A =

the cross sectional area of the tank, ft2

Kv =

the conductance of the valve, (ft3/min)/ft

(3-1)

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Tuning of Industrial Control Systems, Third Edition

The conductance of the valve depends on the valve size, and is usually known in terms of flow per unit pressure drop. Note that the change in pressure drop across the valve per unit change in level can be calculated by multiplying the density of the liquid times the local acceleration of gravity. Gas Surge Tank (Figure 3-5c). This system is analogous to the liquid storage tank. The quantity conserved is the mass of gas, the potential that drives the flow through the valve is the pressure in the tank, and the capacitance is provided by the ability of the tank to store gas as it is compressed. The capacitance can be calculated by the formula MV/zRT lb/psi, where V is the volume of the tank (ft3), R is the ideal gas constant (10.73 psi – ft3/lbmole – °R), z is the compressibility factor of the gas, M is its molecular weight (lb/lbmole), and T is its absolute temperature (°R). The conductance of the valve is expressed in change of mass flow per unit change in pressure drop across the valve. The time constant of the tank can be estimated by the formula: τ = (MV/zRT)/Kv

(3-2)

where: Kv =

the conductance of the valve, (lb/min)/psi

Blending Tank (Figure 3-5d). The change of temperature and composition in a blending tank is governed by the phenomena of convection transfer of energy and mass, respectively. Assuming that the material in the tank is perfectly mixed, the capacitance is provided by the ability of the material in the tank (usually a liquid) to store energy and mass of the various components of the mixture entering the tank, and the conductance is the total flow through the tank. The potential for energy transfer is the temperature, and for mass transfer is the concentration of each component. In the absence of chemical reactions and heat transfer through the walls of the blender, the time constant for both temperature and composition is given by: τ = V/F where: V =

the volume of the tank, ft3

F

the total flow through the tank, ft3/min

=

(3-3)

Open-loop Characterization of Process Dynamics

45

If there is a chemical reaction, the time constant for the concentration of reactants is decreased because the conductance is increased to the sum (F + kV) where k is the reaction coefficient, defined here as the change in reaction rate divided by the change in the reactant concentration. The conductances are added because the processes of reaction and convection occur in parallel. Similarly, if there is heat transfer to the surroundings or to a coil or jacket, the time constant for temperature changes is reduced because the conductance is increased to the sum (F + [UA/ρCp]) where U is the coefficient of heat transfer (Btu/min-ft2-°F), A is the heat transfer area (ft2), ρ is the density of the fluid (lb/ft3), and Cp is the heat capacity of the fluid (Btu/lb°F). In this case the conductances are additive because the processes of conduction and convection occur in parallel. For the preceding examples of simple lags the time constant may be estimated from process parameters and thus a dynamic test on the process is not needed. For more complex processes such as distillation columns and heat exchangers, the time constant cannot be estimated because it is made up of many resistance-capacitance combinations in series and in parallel. For these systems the only recourse is to perform a dynamic test such as the one presented earlier in this chapter.

Example 3-3. Estimation of the Time Constant of a Surge Tank The surge tank of Figure 3-5c is for an air compressor. It runs at a temperature of 150°F, and has a volume of 10 ft3. The valve can pass a flow of 100 lb/hr at a pressure drop of 5 psi when the pressure in the tank is 30 psig. Estimate the time constant of the response of the pressure in the tank to variations in inlet pressure. The capacitance of the tank is its ability to store air as the density of air changes with pressure, which is the potential for flow. Assuming that air at 30 psig behaves as an ideal gas (z = 1), and using the fact that its molecular weight M is 29, the capacitance is: Vρ/P = VM/RT = (10)(29)/(10.73)(150+460) = 0.0443 lb/psi

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Tuning of Industrial Control Systems, Third Edition

The conductance of the valve can be estimated from the formulas given by valve manufacturers to size the valves. Since the pressure drop through the valve is small compared to the pressure in the tank, the flow is “subcritical” and the conductance is given by the following formula: Kv = W (1 + ΔPv/P)/(2ΔPv) = (100/60)[1 + 5/(30+14.7)]/[(2)(5)] = 0.1853 (lb/min)/psi The time constant is then: τ = 0.0443/0.1853 = 0.24 min (14.3 sec) The conductance calculated for the valve is the change in gas flow per unit change in tank pressure, P. It takes into account the variation in gas density with pressure, and the variation in flow with the square root of the product of density times the pressure drop across the valve, ΔPv. For critical flow, when the pressure drop across the valve is more than one-half the upstream absolute pressure, the conductance can be calculated by the formula Kv = W/P.

3-4. Physical Significance of Dead Time Pure dead time, also called transportation lag or time delay, occurs when the process variable is transported from one point to another, hence the term “transportation lag.” At any point in time, the variable downstream is what the variable upstream was one dead time earlier, hence the term “time delay.” This is all illustrated in Figure 3-6. When the upstream variable C1 first starts changing at the upstream point, it takes one dead time before the downstream variable C2 starts changing, hence the term “dead time.” The dead time can be estimated from the following formula: L Distance t 0 = ---------------------- = --v Velocity

(3-4)

Different physical variables travel at different velocities, as follows: • Electric voltage and current travel at the velocity of light: 300,000 km/s or 984,000,000 ft/s.

Open-loop Characterization of Process Dynamics

47

Figure 3-6. Physical Occurrence of Dead Time (Transportation Lag or Time Delay) and Response

L C1

v

C2

C1 Dead time L/v C2

Time • Pressure and flow travel at the velocity of sound in the fluid; for example, 340 m/s or 1,100 ft/s for air at ambient temperature. • Temperature, composition, and other fluid properties travel at the velocity of the fluid, up to about 5 m/s (15 ft/s) for liquids, and up to about 60 m/s (200 ft/s) for gases. • Solid properties vary at the velocity of the solid (e.g., paper in a paper machine or coal in a conveyor). These numbers show that for the reasonable distances, which are typical of process control systems, pure dead time is only significant for temperature, composition, and other fluid and solid properties. The velocity of a fluid in a pipe can be calculated by the following formula: v = F/Ap

(3-5)

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Tuning of Industrial Control Systems, Third Edition

where v

=

the average velocity, ft/s

F

=

the volumetric flow, ft3/s

Ap =

the cross sectional area of the pipe, ft2

Given that (as shall be seen shortly) the dead time makes a feedback loop less controllable, most process control loops are designed to reduce the dead time as much as possible. Dead time can be reduced by installing the sensor as close to the equipment as possible or in the equipment itself. Although pure dead time is usually not significant for most processes, the process dead time estimated from the response to the step test arises from phenomena which are not necessarily transportation lag, but consist of the presence of two or more lags in series (e.g., the trays in a distillation column). When these processes are modeled with a simple lag, the dead time is needed to represent the delay caused by the multiple lags in series. Figure 3-7 shows the response of composition in a blending train when it consists of one, three, and five tanks in series, assuming that the total blending volume is the same. For example, each of the five tanks has one-fifth the volume of the single tank. As the figure shows, the higher the number of tanks in series, the longer it takes for the process to start changing and the shorter the total response time. This behavior makes the model dead-time-to-time-constant ratio higher, making the loop less controllable since it takes the feedback controller longer to see the change in PV relative to the time it takes the PV to respond. At the limit, an infinite number of infinitesimal tanks in series results in a pure dead time equal to the time constant of the single tank—that is, the total volume divided by the volumetric flow. Most real processes fall somewhere between the two extremes of single perfectly mixed processes, and transportation (unmixed) processes. The simplelag-plus-dead-time (SLPDT) model used to model such processes is the simplest model that can be used for characterizing them. It is the model commonly used to tune the controllers by practitioners in industry and by autotune software.

Open-loop Characterization of Process Dynamics

49

Figure 3-7. Composition Response of a Series of Blending Tanks in Series

1

PV

F

V

F

V/3

V/3

V V/3 /

F

V/5

V/5

V/5

PV PV V/5

V/5

3

5

Ft/V

Example 3-4. Estimation of Dead Time Estimate the dead time of temperature of a liquid flowing through a 1-inch standard pipe at 10 gpm (gallons per minute). The distance that the liquid must travel is 100 ft. A pipe manual or engineering handbook gives the cross-sectional area of the 1-inch standard pipe: Ap = 0.00600 ft2. The velocity of the liquid in the pipe is then: v = (10 gpm)/[(7.48 gal/ft3)(60 s/min)(0.00600 ft2)] = 3.71 ft/s Dead time: t0 = (100 ft)/(3.71 ft/s) = 26.9 s (0.45 min)

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Tuning of Industrial Control Systems, Third Edition

3-5. Effect of Process Nonlinearities A characteristic of most refining and petrochemical processes is that they are nonlinear. In general, there are two types of nonlinearities: those that arise from the variation of dynamic parameters with operating conditions, and those that result from saturation of the final control elements (e.g., control valves driven against their upper or lower operating limits). The variation of the process gain, time constant, and dead time with process operating conditions causes the controller performance to vary as process conditions change. Because of this, a controller is usually tuned so that its performance is best at the design operating point and is acceptable over the expected range of operating conditions. The formulas in the preceding section show that for concentration and temperature (Equation 3-3), the time constant is inversely proportional to the flow and thus to the throughput. For liquid level and gas pressure (Equations 3-1 and 3-2), the time constant varies with the valve conductance, Kv, which varies with the valve characteristics and the pressure drop across the valve, and the dead time is inversely proportional to the velocity (Equation 3-4), which is in turn proportional to the flow (Equation 3-5). Control valve characteristics such as equal percentage are usually selected to maintain the process gain constant, which for liquid level and gas pressure is equivalent to keeping the valve conductance constant (the valve gain is the reciprocal of the valve conductance). Equal-percentage valves. Of the three parameters of a process, the gain is the one with the greatest influence on the performance of the control system. As pointed out in the preceding paragraph, such devices as equal-percentage characteristic control valve are used to maintain the loop gain as constant as possible. The equal-percentage characteristic, shown in Figure 3-8, is particularly useful for maintaining a constant gain because the gain of most rate processes (e.g., fluid flow, heat transfer, mass transfer) decreases as the flow increases, that is, as the valve opens. As Figure 3-8 shows, the gain of an equal-percentage valve increases as the valve is opened, compensating for the decrease in the process gain. In fact, the equal-percentage valve is designed to produce equal percentage increments in the valve capacity for equal increments in controller output. This makes the gain of the valve proportional to

Open-loop Characterization of Process Dynamics

51

Figure 3-8. Control Valve Equal-Percentage Characteristics 100% 90% 80%

Valve capacity

70% 60% 50% 40% 30% 20% 10% 0% 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Valve position

the controller output, provided that the pressure drop across the valve remains approximately constant. Reset Windup. The other type of process nonlinearity is caused by saturation of the controller output and of the final control element, not necessarily at the same points. Saturation gives rise to various degrees of the problem known as reset windup, which happens when the reset or integral mode drives the controller output against one of its limits. Reset windup is worse when the controller output limit is different from the corresponding limit of its destination (e.g., the position of the control valve). Reset windup is more common in batch processes and during the startup and shutdown of continuous processes, but the possibility of windup must always be kept in mind when tuning controllers. Some apparent tuning problems are really caused by unexpected reset windup. Chapter 4 looks at reset windup in more detail.

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Tuning of Industrial Control Systems, Third Edition

The following example illustrates the variation of the process gain in a process heater. It takes advantage of the fact that for the heater, the gain can be calculated from a simple steady-state energy balance on the heater.

Example 3-5. Variation in Process Heater Gain with Process Flow At design conditions, the process flow through the heater of Figure 3-1 is F=16 kg/s, process fluid inlet temperature is Ti=50°C, and it is desired to heat the process fluid to T = 90°C. The process fluid has a specific heat of Cp=3.75 kJ/kg–°C, and the steam supplies Hv = 2250 kJ/kg upon condensing. Heat losses to the surroundings can be neglected. The temperature transmitter range is 50 to 150°C, the control valve has linear characteristic with constant pressure drop, and delivers 2.0 kg/s of steam when fully opened. Calculate the gain of the heater in terms of the sensitivity of the outlet temperature to changes in steam flow. An energy balance on the heater, neglecting heat losses, yields the following formula: FCp(T – Ti) = FsHv where Fs is the steam flow and the other terms have been defined in the statement of the problem. The desired gain is the steady state change in outlet temperature per unit change in steam flow: Hv Change outlet temperature K = ------------------------------------------------------------------------- = ---------Change steam flow FC p Notice that the gain is inversely proportional to the process flow F. At the design flow of 16 kg/s the process gain is: °C 2250 kJ/kg ---------------------------------------------- = 37.5 -------------kg ⁄ s 3.75kJ  16kg -------------  --------------------  s   kg * °C

Open-loop Characterization of Process Dynamics

53

For the linear valve with constant pressure drop, the gain of the valve is equal to its capacity, (2 kg/s)/100% = 0.02 kg/s/%, and the gain of the transmitter is (100%)/(150°C – 50°C) = 1.0%/°C. The dimensionless gain is then: % °C 0.020kg ⁄ s K = 37.5 -------------- ---------------------------1.0 ------- = 0.75 °C % kg ⁄ s Now, if the process were to be run at one-half its full capacity, 8 kg/s, the gain at this capacity would be: % 2250kJ ⁄ kg 0.020kg ⁄ s K = ------------------------------------------- ---------------------------1.0 ------- = 1.50 % °C 3.75kJ  8kg ----------  --------------------  s   kg * °C This doubling of the process gain could cause the loop to become unstable if the controller was tuned at the design process flow. One way to compensate for variable process gain is to use an equal-percentage control valve instead of a linear valve. As discussed above, the equalpercentage valve is designed to have its gain proportional to the valve position, so as the process flow decreases and the control valve closes, the gain of the valve decreases proportional to the flow. This example shows the variation of the process gain, indicating that the process heater is nonlinear. As mentioned earlier, this decrease in process gain with an increase in flow is characteristic of many process control systems, hence the popularity of equal-percentage control valves, which exactly compensate for this gain variation.

3-6. Summary This chapter showed how to perform and analyze a process step test to determine the parameters of a simple-lag-plus-dead-time (SLPDT) model of the process. These parameters are the gain, the time constant, and the dead time. It also presented the physical significance of these parameters and showed how to estimate them from process design parameters for some simple process loops. The chapters to follow will use the estimated dynamic parameters to design and tune feedback, feedforward, and multivariable controllers.

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Tuning of Industrial Control Systems, Third Edition

Regardless of the method used to measure the dynamic parameters of a process, it is important to realize that even a rough estimate of the process dynamic parameters can be quite helpful in tuning and troubleshooting process control systems.

References 1. Smith, C. L., Digital Computer Process Control, Scranton: International Textbook Co., 1972. 2. Ziegler, J. G., and Nichols, N. B., “Optimum Settings for Automatic Controllers,” Transactions ASME, V. 64, Nov. 1942, p. 759.

Review Questions 3-1. Summarize the procedure for performing an open-loop step test on a process. 3-2. What are the parameters of a single-lag-plus-dead-time (SLPDT) model of the process? Give a brief description of each one. 3-3. Figure 3-9 shows the response of the composition out of a reactor to a step change in the controller output at time 1.0 min. The composition controller has a range of 0 to 1.5 lb/gal. Estimate the parameters of a simple-lag-plus-dead-time model of the response. 3-4. A passive low-pass filter can be built with a resistor and capacitor. For use in printed circuit boards, the maximum magnitudes of these components are, respectively, 10 megohms (million ohms) and 100 microfarads (millionth of farad). What would be the maximum time constant of a filter built with these components? 3-5. The liquid surge tank of Figure 3-5b has an area of 20 ft2 and the valve has a conductance of 50 gpm/ft of level change (1 ft3 = 7.48 gallons). Estimate the time constant of the response of the level to a step change in inlet flow. 3-6. The blending tank of Figure 3-5d has a volume of 2000 gallons. Calculate the time constant of the composition response for product flows of (a) 50 gpm, (b) 500 gpm, and (c) 5000 gpm.

Open-loop Characterization of Process Dynamics

55

Figure 3-9. Open-loop Step Response for Review Question 3-3

3-7. The blending tank of Figure 3-5d mixes 100 gpm of concentrated solution at 20 lb/gallon with 400 gpm of dilute solution at 2 lb/gallon. Calculate the steady-state product concentration in lb/gallon. How much would the outlet concentration change if the concentrated solution rate were to change to 110 gpm, with all other conditions remaining the same? Calculate the process gain. 3-8. Repeat the previous question assuming that the initial rates are 10 gpm of concentrated solution and 40 gpm of dilute solution, and the concentrated solution is changed to 11 gpm to do the test. Also estimate the time constant of the tank for both questions if the tank has a volume of 5,000 gal.

4 How to Tune Feedback Controllers

This chapter presents formulas for tuning controllers based on the three parameters obtained from the open-loop step test presented in the previous chapter: gain, time constant, and dead time.

Learning Objectives — When you have completed this chapter, you should be able to: A. Tune feedback controllers from estimates of the process gain, time constant, and dead time. B.

Compare controller tuning methods.

C. Identify factors that affect controller performance. D. Recognize reset windup and know how to avoid it.

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4-1. Tuning from Open-loop Test Parameters Quarter-Decay-Ratio Tuning Formulas Besides the formulas for quarter-decay ratio (QDR) response tuning based on the ultimate gain and period of the loop (see Table 2-1 in Chapter 2), Ziegler and Nichols3 also developed formulas for tuning feedback controllers for QDR response which are based on process gain K, time constant τ, and dead time to. These formulas are given in Table 4-1. Table 4-1. Tuning Formulas for Quarter-Decay Ratio (QDR) Response Controller

Gain

Integral Time

P

Kc = τ/Kto

PI

Kc = 0.9τ/Kto

TI = 3.33to

PID

Kc = 1.2τ/Kto

TI = 2.0 to

Derivative Time

TD = 0.5to

The formulas of Table 4-1 are very similar to those of Table 2-1. Notice, for example, that in both sets of formulas the proportional gain of the PI controller is 10% lower and the PID gain is 20% higher than that of the P controller, and that the derivative or rate time is one-fourth of the integral or reset time for the PID controller. The ratio of the integral time of the PID controller to that of the PI controller is 1.7, which is also the same as in Table 2-1; that is, the derivative mode allows the integral mode to be 1.7 times faster. The formulas of Table 4-1, however, provide important insight into how the parameters of the process affect the tuning of the controller and thus the performance of the loop, in particular: • The controller gain is inversely proportional to the process gain K. Since the process gain represents the product of the gains of all the elements in the loop other than the controller (control valve, process equipment, and sensor/transmitter), this means that the loop response depends on the loop gain; that is, it depends on the product of all the elements in the loop. It also means that if the gain of any of the elements should change because of recalibration, resizing, or nonlinearity (see Section 3-5), the response of the feedback loop will change unless the controller gain is readjusted.

How to Tune Feedback Controllers

59

• The controller gain must be reduced when the ratio of the loop dead time to its time constant increases. This means that the controllability of the loop decreases when the ratio of the process dead time to its time constant increases, and leads us to define the ratio of dead time to time constant as the uncontrollability parameter of the loop: t P u = ---0τ where: to = the loop dead time τ = the loop time constant Note that it is the ratio of the dead time to the time constant that determines the degree of uncontrollability of the loop. In other words, a process with a long dead time is not uncontrollable if its time constant is much longer than its dead time. • The speed of response of the controller, which is determined by the integral and derivative times, must match the speed of response of the process. The QDR formulas match these response speeds by relating the controller time parameters to the process dead time. These three conclusions can be helpful in guiding the tuning of feedback controllers, even in cases when the tuning formulas cannot be used directly because the process parameters cannot be accurately estimated. For example, if the performance of a well-tuned controller should deteriorate during operation, look for a change in the process gain, its uncontrollability parameter, or its speed of response. At other times the controller performance may be poor because the integral time is much shorter than the process response time, because in such a case the process cannot respond as fast as the controller wants it to respond. The point here is that the speed of response of the process must be considered when setting the integral time. This is what the tuning formulas of Tables 2-1 and 4-1 do. The conclusions just drawn from the tuning formulas, coupled with the methods for estimating time constants and dead times given in Sections 3-4 and 35, can also guide the design of the process and its instrumentation. For exam-

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ple, loop controllability can be improved by reducing the dead time between the manipulated variable and the sensor, or by increasing the process time constant. It is also possible to quantitatively estimate the effect of process, control valve, and sensor nonlinearities on the variability of the loop gain and thus determine the need for readjusting the controller gain when process conditions change.

Internal Model Control Tuning Although a number of sets of tuning formulas have been developed since the Ziegler-Nichols3 formulas, all of them give controller parameters in the same ballpark and in tuning, ballpark values are all that is needed. The ZieglerNichols formulas of Table 4-1 are thus sufficient to get the tuning procedure started. However, these formulas are empirical, and can be expected to be valid for a common range of loop controllability, with say uncontrollability values of 0.05 to 0.5. A set of tuning formulas based on more fundamental theory was developed by Martin1, and independently by Rivera and Morari2. These formulas, now commonly known as the Internal Model Control (IMC) formulas, give results similar to the QDR formulas except for the integral time, which is set equal to the loop time constant: TI = τ As expected, this results in more conservative (slower) tuning than with the QDR formulas when the loop uncontrollability parameter is less than 0.5 and faster responses for loops with higher dead-time-to-time-constant ratios. Therefore, the following strategy is indicated: Use the QDR formulas but limit the integral time to a maximum value equal to the model time constant. The IMC tuning method does not result in a specific value for the controller gain since the formula for it contains an adjustable parameter, so it is just as good to use the gain from the QDR formulas. In addition, the formula for the derivative time for the IMC method is exactly the same as the one for QDR. The following examples illustrate controller tuning using the open-loop model parameters.

How to Tune Feedback Controllers

61

Example 4-1. Comparison of PI and PID tuning In Section 3-2, it was determined that the open-loop test parameters for the heat exchanger of Figure 3-1 were: Gain = 1.95 Time constant = 7.5 min Dead time = 2.5 min. The formulas of Table 4-1 produce the following QDR tuning parameters for a PI controller: Kc = (0.9/1.95) (7.5/2.5) = 1.4

TI = 3.33(2.5) = 8.3 min

The QDR parameters for a PID controller, also from Table 4-1, are: Kc = (1.2/1.95) (7.5/2.5) = 1.8

TI = 2(2.5) = 5.0 min TD = 2.5/2 = 1.2 min

Figure 4-1 compares the PI and PID controller responses of the temperature transmitter output PV and of the controller output OP using these tuning parameters for a step increase in process flow to the heater. For this loop with an uncontrollability parameter of 2.5/7.5 = 0.33, the advantage of adding the derivative mode is obvious: it produces a smaller initial deviation and maintains the temperature closer to the set point for the entire response, with fewer oscillations. In addition, the initial change in controller output is not much greater for the PID response than it is for the PI response. The next two examples address the question of whether the QDR tuning parameters will always perform this well, regardless of the degree of loop controllability.

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Figure 4-1. Responses of PI and PID Controllers to a Disturbance Input on Process Heater tuned by QDR Formulas

PID PI

PID PI

Example 4-2. Control of a Very Controllable Loop An open-loop test on a loop results in a time constant of 11.6 min and a dead time of 0.9 min. Since the uncontrollability parameter is small (0.9/11.6 = 0.08), a PI controller should perform well for this loop. Using Table 4-1, the QDR tuning parameters are: KKc = 0.9(11.6/0.9) = 11.6 By contrast, the IMC integral time is:

TI = 3.33(0.9) = 3.0 min TI = τ = 11.6 min.

Figure 4-2 shows the responses of the transmitter output PV and the controller output OP to a step change in the disturbance variable. As expected, the shorter integral time recommended by the QDR formula in comparison with the IMC rule results in a faster return to the set point with about the same oscillatory behavior. This is at the expense of a slightly larger initial change in the controller output.

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Figure 4-2. Responses of PI Controllers for a Very Controllable Loop Tuned with QDR and IMC Formulas

TI = 3 min TI = W = 11.6 min

KKc = 11.6

TI = 3 min

TI = W = 11.6 min

Example 4-3. Tuning for a Low Controllability Loop An open-loop test on a loop results in a time constant of 3.5 min and a dead time of 4.5 min. For this low-controllability loop, a PID controller is indicated. The QDR tuning parameters from Table 4-1 are: KKc = 1.2(3.5/4.5) = 0.93

TI = 2(4.5) = 9.0 min TD = 4.5/2 = 2.2 min

By contrast, the IMC integral time is TI = τ = 3.5 min. Figure 4-3 shows the responses of the transmitter output PV and the controller output OP to a step change in the disturbance variable. As expected the shorter integral time proposed by IMC returns the PV to the set point faster, and in this case without oscillation. Besides, this is accomplished here without the expense of a higher overshoot in the initial change in controller output.

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Figure 4-3. Responses of PID Controllers For Low Controllability Loop Tuned with QDR and IMC Formulas

TI = W = 3.5 min

TI = 9 min KKc = 0.93 TD = 2.2 min

TI = W = 3.5 min TI = 9 min

The preceding two examples show excellent performance by the controller when tuned using the QDR formulas with the integral time limited to the value of the time constant. Notice that the loop gain was less than one-tenth of the gain for the uncontrollable process of Example 4-3 than for the highly controllable process of Example 4-2. Because of the smaller gain, the initial deviation of the PV from the set point is larger in Example 4-3 than it is in Example 4-2. This is the cost of the long dead time in the loop.

Response to a Change in Set Point In the preceding examples, the responses are to a step change in the disturbance variable, such as that of the process flow to an exchanger. In continuous processes, the objective of the control system is to maintain the process variable at or near the set point in the presence of disturbances. The set point is seldom changed. However, what happens when there is a need for the operator to change the set point? When the controller gain is high, a sudden change

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in set point can cause a large change in the controller output, as shown in Figure 4-4. The response shown in Figure 4-4 is to a 3°F change in set point for the process of Example 4-2. Since the loop gain is 11.6, the figure shows that the controller output initially changes by over 30%. Note that this is for a process gain K = 1; if the process gain is higher the change in output will be smaller, but if the process gain is lower than usual the controller output will have a larger change. This sudden large change in controller output is bound to cause a disturbance to other loops. For example, if the controller manipulates the steam flow as in Figure 3-1, there may be a drop in the steam header pressure that will affect other systems on that steam line.

Figure 4-4. Responses of a Loop with High Proportional Gain to a Step Change in Set Point (Set Point Shown as a Dashed Line)

SP

KKc = 11.6 TI = 3 min

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There are several ways to prevent sudden large changes in controller output when the set point is changed: 1. Decrease the controller gain so as to prevent the large change in controller output. This, however, may reduce the tightness of control on disturbance inputs. 2. Configure the controller so that the proportional mode acts on the process variable and not on the deviation from the set point. When the set point is changed, the output will ramp to the new value at a rate controlled by the integral time. 3. Configure the controller so that the set point is always ramped at an adjustable rate when the set point is changed. 4. Have the operator slowly change the set point in small steps. Keep in mind that a large change in controller output happens only when the controller gain is high, as in Example 4-2. This is not a problem with less controllable loops such as the one in Example 4-3. A similar problem could occur with the derivative mode except that the default configuration for the derivative mode is to act on the process variable and not on the deviation from set point. There are cases in which set point changes are common, such as with batch processes and on-line optimization. One recent development in industrial operations is the incorporation of on-line optimization programs that automatically change controller set points as the optimum conditions change. Most of these programs contain limits on the size of the set point changes they make. At any rate, as mentioned above, one sure way to prevent large changes in controller output with set point changes is to have the proportional mode act on the process variable instead of on the deviation from set point. As long as there is integral mode, this option will not affect the performance of the controller on the disturbance variables. It is important to recognize that good controller performance on maintaining the process variable at or near the set point must be balanced against too much action on the controller output. The reason is that the controller output usually causes disturbances to other control loops and in some cases manipu-

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lates safety-sensitive variables. For example, in a furnace temperature controller, the controller output could be manipulating the fuel flow to the furnace. A large drop in fuel flow could cause a loss of the flame in the firing box. The examples in this section have illustrated the performance of the controller when tuned with the parameters of the open-loop test. To summarize our findings: • Derivative mode provides superior performance for processes with a high dead-time-to-time-constant ratio—that is, those processes with high uncontrollability parameter. • Except for controllers that must constantly respond to set point changes (e.g., slaves in cascade loops; see Chapter 7), the controller should be tuned for good performance on disturbance inputs, and sudden set point changes should be limited in magnitude. • For very controllable processes, the tuning formulas call for high loop gains. • Very uncontrollable processes require low loop gains that result in large initial deviations of the process variable from its set point. Better performance is possible with techniques such as dead time compensation (Section 6-4) or feedforward control (Chapter 8).

4-2. Practical Controller Tuning Tips The following is a collection of tips the authors hope will be useful in making the controller-tuning task more efficient and satisfying. 1. Tune coarse, not fine. The realization that the performance of a feedback controller is not sensitive to the precise adjustment of its tuning parameters significantly simplifies the tuning task. Faced with the infinite possible combinations of precise tuning parameter values, you might give up the task of tuning before you even get started; however, once you realize that controller performance does not require precise setting of the tuning parameters, the number of significantly different combinations is reduced to a workable number. You will also find satisfaction in the large improvements in performance achievable by coarse tuning and frustration in the little incre-

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mental improvement achievable by fine tuning. How coarse? When tuning a controller, you should seldom change a parameter by less than one-half of its current value. 2. Tune with confidence. One of the reasons that controller performance is not sensitive to precise tuning parameter settings is that any of the parameters may be adjusted to make up for non-optimal values of the other parameters. A successful approach is to select the integral time first, set the derivative time to about one-fourth of the integral time or, if the dead time is known, to one-half of the dead time, then adjust the proportional gain to obtain tight control of the controlled variable without undue variations in the manipulated variable. If the response is still too oscillatory, double the integral and derivative times; if it is too slow in approaching the set point, halve the integral and derivative times, then readjust the gain. When you obtain satisfactory performance, LEAVE IT ALONE. DO NOT TRY TO FINE TUNE IT FURTHER. If you try to fine tune it, you will be disappointed by the insignificant incremental improvement. 3. Use all of the available information. You may be able to gather enough information about the process equipment to estimate the gain, time constant, and dead time of the process without having to resort to the open-loop step test (see Sections 3-3 and 3-4). You can also gather information during trial-and-error tuning that allows you to estimate the integral and derivative times from the period of oscillation of the loop or the total delay around the loop (dead time plus time constant). The latter can be estimated by the time difference between peaks in the controller output and the corresponding peaks in the transmitter signal. 4. Try a longer integral time. Many times, poor loop response can be traced to trying to bring the controlled variable back to its set point faster than the process can respond. In such cases, increasing the integral time allows you to increase the process gain and improve the response. 5. Tuning very controllable processes. Processes with uncontrollability parameters less than 0.1 have very large ultimate gains, which are difficult to determine by the closed-loop method of Chapter 2. When the uncontrollability parameter is less than 0.1, most tuning formulas result in very high gains and very short integral times.

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When the formulas result in higher gains and shorter integral times than seem reasonable, let your judgment override the formulas. 6. Tuning very uncontrollable processes. For processes with uncontrollability parameters of 1 and higher, it is important to recognize that even an optimally tuned feedback controller will display poor performance; that is, show large initial deviations on disturbance inputs and slow return to set point. In such cases, improved performance can be achieved through feedforward control (see Chapter 8) or dead time compensation in the feedback controller (see Section 6-4). 7. Beware of problems that are not related to tuning. The following problems interfere with the normal operation of a controller and although they may appear to be tuning problems, they are not: • Reset windup, caused by saturation of the controller output (see Section 4-3). • Interaction between loops (see Chapter 9). • Processes with inverse or overshoot response, caused by parallel effects of opposite direction between a process input and the controlled variable (see Section 4-4). • Changes in process parameters because of nonlinearities, which must be handled by adaptive control methods. • Control valve hysteresis—that is, the valve stops at a different position than the one desired, and the difference changes directions depending on the direction of motion of the valve. Hysteresis is due to dry friction on the valve packing. It causes the controller output to oscillate around the desired position of the valve. • Limit cycles due to nonlinear behavior. All of these problems cause poor feedback controller performance that must be handled by means other than controller tuning; for example decoupling, feedforward control, adaptive control, or the use of valve positioners. The chapters that follow present these techniques.

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4-3. Reset Windup The problem of reset windup or saturation of the controller output is one that may often be considered a tuning problem, when in reality it cannot be resolved by tuning the controller. It is therefore important to be able to recognize the symptoms of reset windup and to know how to resolve them. A properly tuned controller will behave well as long as its output remains in a range where it can change the manipulated variable, but it will behave poorly if, for some reason or other, the effect of the controller output on the manipulated flow is lost. A gap between the limit on the controller output and the operational limit of the control valve is the most common cause of reset windup. The symptom is a large overshoot of the controlled variable while the integral mode in the controller is crossing the gap. Reset windup occurs most commonly during startup and shutdown, but it can also occur during product grade switches and large disturbances during continuous operation. Momentary loss of a pump may also cause reset windup. To illustrate a typical occurrence of reset windup, consider the temperature control of a large reactor by manipulation of coolant flow to the jacket, as shown in Figure 4-5. The figure shows the start-up of the reactor with the controller in Automatic. Because initially the temperature is so much lower than the set point, the integral mode closes the coolant valve fully (100% controller output since the control valve fails open). A large overshoot is caused because the control valve does not start opening until the temperature reaches the set point, point “a” in the figure. This is because the integral mode keeps driving the valve to the closed position as long as the temperature is below the set point. At point “a” the integral mode starts to reduce the controller output to open the coolant valve, but because it takes too long to reach the desired flow (about 20% “CO” in the figure), the temperature continues to increase and the control valve goes to the fully open position (0% “CO”). The temperature eventually drops to the set point and the controller brings it under control. The large temperature overshoot (or undershoot in other cases) is the symptom of reset windup. One way to prevent the overshoot of Figure 4-5 would be to start up with the controller in Manual state. The console operator must then watch the temperature and as it approaches the set point, set the controller output to, for exam-

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Figure 4-5. Response of Reactor Startup Shows Overshoot Caused by Reset Windup

Overshoot due to reset windup SP a

Coolant valve closed

Valve opened

ple, 50% or if known, to a value near the required output (20% in this example) before switching the controller to Automatic. The problem of reset windup can also occur during normal operation when there is a gap between the limits of the controller output and the operating limits of the valve position or other manipulated variable. For example, if in case of Figure 4-5 the controller output were limited to -10% to 110% while the valve operates between 0 and 100%, windup would occur when a large disturbance causes the controller output to enter the gap. This is because the controller has no effect on the process variable while its output is in the gap and the integral mode keeps the controller output in the gap until the process variable crosses the set point, resulting in an overshoot.

4-4. Processes with Inverse Response Some processes exhibit what is known as inverse response; that is, an initial move in a direction opposite to the final steady-state change when the input is

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a step change. A typical example of a process with inverse response is an exothermic reactor with the feed colder than the reactor. An increase in the feed rate to the reactor causes the temperature to initially drop due to the larger rate in cold feed. However, the increase in the inlet flow of the reactants eventually increases the rate of the reaction, and with it the rate of heat generation by the reaction. This causes the temperature in the reactor to end up higher than it was initially. Another typical example of inverse response is the level in the steam drum of a water tube boiler when the steam demand changes. The inverse response is caused by the phenomena of “swell” and “shrink” of the steam bubbles in the boiler tubes. Figure 4-6 shows the response of the temperature control of an exothermic reactor to a step increase in reactant flow followed by a step increase in the inlet temperature. The temperature is controlled by a PID controller tuned by the QDR formulas. Since the reactants enter at a lower temperature than the reactor, the reactor temperature initially decreases, but as the reactant concentration increases, the reaction rate and corresponding rate of heat generation increase, causing the reactor temperature to increase. The initial drop in temperature fools the controller into increasing its output to decrease the coolant flow (the control valve fails open), but eventually a lower controller output— higher coolant flow—is required because of the higher rate of heat generation. As the figure shows, the result is a very oscillatory response, particularly in the controller output. By comparison, the step decrease in inlet temperature results in lower amplitude of the oscillations because there is no inverse response to the inlet temperature. Note that the temperature control loop is very controllable because the reactor has a high capacitance and hardly any dead time; this is why the deviations in the temperature are so small. Feedforward control (Chapter 8) can compensate for the effect of inverse response to improve the performance of the feedback controller when necessary. The feedforward model considers the long-term effect of the disturbance and takes action that cancels out the initial change in the feedback controller output in the wrong direction. This is basically how the “swell” and “shrink” problems in boiler levels are handled by two- and three-element boiler level control systems.

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Figure 4-6. The Inverse Response to a Change in Reactant Flow Causes Oscillations because the Controller Output Moves in the Wrong Direction Initially

Step increase in

Step decrease in

reactants flow

inlet temperature

When the inverse response is to a change in the controller output, the loop becomes very uncontrollable. For example, if in the reactor of Figure 4-6 the temperature were controlled by manipulating the reactant flow, every action by the controller would be followed by an immediate change in the temperature in the wrong direction. This would be worse than if the process had a dead time equal to the duration of the inverse response. Fortunately, such a situation is extremely rare, but it should be kept in mind when troubleshooting difficult tuning problems.

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4-5. Effect of Nonlinearities As discussed in Section 3-5, most processes exhibit nonlinear behavior; that is, their sensitivity to changes in controller output and dynamic behavior change with operating conditions. This means that although the controller can perform well at the set of operating conditions at which it is tuned, the performance can deteriorate at some other operating conditions. One characteristic of many processes is that they become more sensitive at lower throughput rates. Figure 4-7 shows the temperature response of the process heater of Figure 3-1 at full production rate and at half production rate. The controller is PID tuned for QDR response at full production rate. The figure shows that at half production rate, the response becomes highly oscillatory. This is because there is half as much process fluid to absorb the heat provided by the steam, making the temperature twice as sensitive to the action of the controller. As discussed in Section 3-5, an equal-percentage control valve will provide a lower loop gain at lower controller outputs to compensate for the higher process sensitivity.

4-6. Summary This chapter presented a simple set of tuning formulas for feedback controllers based on the parameters of the open-loop test: the gain, the time constant, and the dead time. The set of formulas originally proposed by Ziegler and Nichols3 for quarter-decay-ratio response was proposed with a limitation on the integral time when the process is very uncontrollable. The limit on the integral time is the time constant, which is the value proposed by the IMC formulas. The effectiveness of the tuning formulas was demonstrated for normal processes as well as for very controllable and very uncontrollable processes. The advantage of the PID over the PI controller was also demonstrated. Helpful tuning hints were presented for those instances when the open-loop test cannot be performed. Several process characteristics that reduce the performance of the controller were presented, namely reset windup, inverse response, and process nonlin-

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Figure 4-7. The Response Is More Oscillatory at Half Production Rate because the Loop Gain is Twice as High than at Full Rate

Full production rate

Half production rate

earity. The problem of large initial changes in the controller output when the set point is changed and when the controller gain is high was also discussed. The next chapter presents the selection of controller modes and tuning for a number of common control loops.

References 1. Martin, J. Jr., Corripio, A. B. and Smith, C. L. “How to Select Controller Modes and Tuning Parameters from Simple Process Models,” ISA Transactions, V. 15 (Apr. 1976), pp. 314-319. 2. Rivera, D. E., Morari, M. and Skogestad, S. “Internal Model Control, 4. PID Controller Design,” Industrial and Engineering Chemistry Process Design and Development, V. 25 (1986), p. 252.

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3. Ziegler, J. G. and Nichols, N. B. “Optimum Settings for Automatic Controllers,” Transactions ASME, V. 64, (Nov. 1942), p. 759.

Review Questions 4-1. Based on the tuning formulas given in this chapter, how must you change the controller gain if, after the controller is tuned, the process gain were to double because of the nonlinear behavior of the process? 4-2. How is the uncontrollability of a feedback loop measured? 4-3. Assuming that the quarter-decay-ratio formulas of Table 4-1 give the same tuning parameters as those of Table 2-1, what relationship can be established between the controller ultimate gain and the gain with the uncontrollability parameter of the process in the loop? What is the relationship between the ultimate period and the process dead time? 4-4. Compare the following processes in regards to sensitivity, speed of response, and controllability: Process A

Process B

Process C

Gain

0.5

2.0

4.0

Time constant, min

2.0

30

5.0

Dead time, min

0.2

3.0

3.0

4-5. Estimate the tuning parameters of a PID controller for the three processes of question 4-4. 4-6. Why would one want to configure the controller so that the proportional mode acts on the process variable and not the deviation from set point? What would the response of the controller output be when the controller is configured as such and the set point is changed? 4-7. What is the typical symptom of reset windup? What causes it? How can it be prevented? 4-8. What is known as inverse response? What effect does it have on the performance of a feedback controller and why?

5 Mode Selection and Tuning of Common Feedback Loops

The preceding chapters dealt with the tuning of feedback controllers for general processes that can be represented by a single-lag-plus-dead–time (SLPDT) model. This chapter presents tuning guidelines for the most typical process control loops, specifically flow, level, pressure, temperature, and composition control loops.

Learning Objectives—When you have completed this chapter, you should be able to: A. Decide on the appropriate control objective for a loop. B.

Select proportional, integral, and derivative modes for specific control loops.

C. Design and tune simple feedback controllers for flow, level, pressure, temperature, and composition.

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5-1. Deciding on the Control Objective Although the most common objective for feedback control is to maintain the controlled variable at its set point, there are some control situations, often in the control of level or pressure, when it is acceptable to just maintain the controlled variable within an acceptable range. The difference between these two objectives is important because, as Chapter 2 showed, the purpose of the integral mode is to eliminate the offset or steady-state deviation of the process variable from the set point. Consequently, integral mode is not required when it is acceptable to allow the controlled variable to vary over a range. One advantage of eliminating the integral mode is that it permits higher proportional gain, thus reducing the initial deviation of the controlled variable caused by disturbances. There are two situations when integral mode is not required: • When the process is so controllable—a time lag with insignificant dead time—that the proportional gain can be set high to maintain the controlled variable in a very narrow range. • When it is desirable to allow the controlled variable to vary over a wide range so that the control loop attenuates the oscillations caused by recurring disturbances. The first of these situations calls for proportional (P) or proportional-derivative (PD) controllers with very high gains, or for on-off controllers. These may be found in the control of level in evaporators and reboilers, and in the control of temperature in refrigeration systems, ovens, constant-temperature baths, and air conditioning/heating systems. On-off controllers can be used when the time constant is long enough that the cycling it necessarily causes is of a very low frequency; otherwise, P or PD controllers are used to modulate the operation of the manipulated variable. In either case, the dead band of the onoff controller or the proportional band of the P or PD controllers can be set very narrow. Derivative mode can be added to compensate for the lag in the sensor or final control element and thus improve stability. The second situation calls for proportional controllers with as wide a proportional band as possible. These are found in the control of level in intermediate storage tanks and condenser accumulators, and in the control of pressure in

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79

gas surge tanks, because in these cases the purpose of the tank is to attenuate variations in process flow.

5-2. Flow Control Flow control is the simplest and most common of the feedback control loops. The schematic diagram of a flow control loop in Figure 5-1 shows that there are no lags between the control valve that causes the flow to change and the flow sensor/transmitter (FT) that measures the flow. Since most types of flow sensors (orifice, venturi, flow tubes, magnetic flowmeters, turbine meters, coriolis, etc.) respond very fast, the only significant lag in the flow loop is the control valve actuator, and most actuators have time constants of the order of a few seconds.

Figure 5-1. Schematic of a Flow Control Loop

SP

FC

FT

Several controller design theories (Internal Model Control2 controller synthesis1, etc.) suggest that the controller for a very fast loop should contain only integral mode. In practice, flow controllers have been traditionally PI controllers tuned with low proportional gains and very short integral times, of the order of seconds, which are essentially pure integral controllers. Such an approach is acceptable when flow is controlled to maintain a constant rate, with rare changes in the flow set point by the operator.

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However, when the flow controller is the slave in a cascade control scheme (see Chapter 7), it is important for the flow to respond quickly to set point changes. This requires a proportional-integral controller with a gain near unity, which to maintain stability may require an increase in the integral time from the few seconds normally used in flow controllers. The IMC2 tuning rules (see Section 4-1) suggest that the integral time be set equal to the time constant of the loop, usually that of the control valve actuator. In cascade situations, tight flow control is indicated. The proportional gain should also be increased when hysteresis of the control valve causes variations in the flow around its set point. As mentioned in Section 4-2, hysteresis is caused by static friction in the valve packing that creates a difference between the actual valve position and the corresponding controller output. The error changes direction according to the direction in which the valve stem must move, and this causes a dead band around the desired valve position; that is, a band within which the valve does not respond to changes in the controller output. Increasing the flow controller gain reduces the amplitude of the flow variations caused by hysteresis. A valve positioner also reduces hysteresis and speeds up the valve response, but positioners are usually difficult to cost-justify for flow control loops.

Example 5-1. Flow Control with Valve Hysteresis Figure 5-2 shows responses of a flow control loop with valve hysteresis for two different tunings of the controller. The top curve is for the traditional tuning of low gain and fast integral (that is, a short integral time), while the bottom curve is for a more aggressive tuning of a gain of 1.5 and the same integral time. As the figure shows, the more aggressive tuning reduces the variations in flow caused by hysteresis in the valve.

5-3. Level and Pressure Control There are two reasons for controlling level and pressure: • To keep them constant because of their effect on process or equipment operation.

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Figure 5-2. Oscillations on a Flow Control Loop with Valve Hysteresis are Reduced in Amplitude with a Higher Controller Gain

Kc = 0.25

TI = 0.2 min

Kc = 1.5

TI = 0.2 min

• To smooth out variations in flow while satisfying the material balance. Keeping level and pressure constant calls for “tight” control, while smoothing out variations in control usually calls for “averaging” control. Pressure is to gas systems what level is to most liquid systems, although liquid pressure is sometimes controlled.

Tight Control One example of tight liquid level control and one example of tight pressure control are shown in Figure 5-3. The control of level in natural-circulation evaporators and reboilers is important because too low a level causes deposits on the bare hot tubes and overheating of the tubes at the top. Conversely, too high a level causes elevation of the boiling point, reducing the heat transfer rate and preventing the formation of bubbles, which enhances heat transfer by promoting turbulence. The example of tight pressure control or pressure regu-

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lation is the control of the pressure in a liquid or gas supply header. It is important to keep the pressure in the supply header constant to prevent disturbances to the using processes when there is a sudden change in the demand of one or more of the using processes. The design of tight level and pressure control systems requires a fast-acting control valve, with a positioner if necessary, to prevent secondary time lags that would cause oscillatory behavior at high controller gains. If the level or pressure controller is cascaded to a flow controller, the latter must be tuned as tight as possible, as discussed in the preceding section. Normally, only proportional mode is needed for tight level or pressure control. The proportional gain must be set high, from 10 to over 100 (proportional band of 1 to 10% of range). If the lag of the level or pressure sensor is significant, derivative mode could be added to compensate for it and to afford a higher gain. The derivative time should be set approximately equal to the time constant of the sensor (see the next section). Integral mode should not be used, since it would require a reduction of the proportional gain. Many modern controllers cannot be configured to be proportional only or proportional-derivative. When the integral mode cannot be removed, a long integral time should be used for tight level control to permit higher proportional gains. The reason the integral mode can be slow is that the high proportional gain keeps the process variable near the set point all the time.

Averaging Level Control Two examples of averaging level control are shown in Figure 5-4: the control of level in a surge tank (a) and in a condenser accumulator drum (b). Both the surge tank and the accumulator drum are intermediate process storage tanks. The liquid level in these tanks has absolutely no effect on the operation of the process. It is important to realize that the purpose of an averaging level controller is to smooth out flow variations while keeping the tank from overflowing or running empty. If the level were to be controlled tightly in such a situation, the outlet flow would vary just as much as the inlet flow(s), and it would be as if the tank (or accumulator) were not there. The averaging level controller should be proportional-only with a set point of 50% of range, a gain of 1.0 (proportional band of 100%), and an output bias of 50%. Note that the “bias” is the controller output when the process variable is

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Figure 5-3. Examples of Tight Control: (a) Evaporator Level; (b) Supply Header Pressure

Vapors

Feed Steam

LC

T

LT

Condensate (a)

Product

PC PT

(b)

Loads

Supply

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Figure 5-4. Examples of Averaging Level Control: (a) Surge Tank; (b) Condenser Accumulator

Feeds SP Surge tank

LT

LC

Outlet flow

(a) Vapors

Condenser

Accumulator

Column

LT

LC

FC FT

(b)

Reflux

Distillate

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85

at the set point, a term that is only important when the controller has no integral mode. This will cause the outlet valve to be fully open when the level is at 100% of range and fully closed when the level is at 0% of range, thus using the full capacity of the valve and of the tank. A proportional gain higher than unity would reduce the effective capacity of the tank for smoothing variations in flow, while a gain lower than unity would reduce the effective capacity of the control valve and create the possibility of the tank overflowing or running dry. With the proposed design the tank behaves as a low-pass filter to flow variations; a low-pass filter allows low-frequency input through while it attenuates high frequency variations. The time constant of such a filter is: A ( h max – h min ) τ = ----------------------------------------K c F max

(5-1)

where: A

= the cross-sectional area of the tank, ft2

hmin and hmax = the low and high points of the range of the level transmitter, respectively, ft Fmax = the maximum flow through the control valve when opened fully (100% controller output), ft3/min Kc

= the controller gain

The controller gain is assumed to be 1.0 in this design. When the level controller is cascaded to a flow controller, Fmax is the upper limit of the range of the flow transmitter in the flow control loop. Note that a proportional gain greater than unity results in a reduction of the filter time constant and therefore less smoothing of the variations in flow. A good way to see it is to note that doubling the gain would be equivalent to reducing either the tank area or the transmitter range by a factor of two, thus reducing the effective capacity of the tank. On the other hand, reducing the controller gain to half would be equivalent to reducing the capacity of the valve by half, thus increasing the possibility of the tank overflowing. Although averaging level control can be accomplished by a simple proportional controller, most level control applications use PI controllers. This is because control room operators have an aversion to variables that are not at

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their set points. The process in a level control loop is unlike most other loops in that it does not self-regulate; that is, the level tends to continuously rise or fall when the feedback controller is not in Automatic. This usually means that for level control loops, a time constant cannot be determined by an open-loop test. Even when there is some degree of self-regulation, the process time constant is very long, on the order of one hour or longer. Because of this, PI controllers in level control loops have the following characteristics: • The level, and the flow that is manipulated to control it, oscillate for a long period. Sometimes the period is so long that the oscillation is imperceptible, unless it is trended over a very long time. • The shorter the integral time, the shorter the period of oscillation. • The level control loop is unstable when the integral time is equal to or shorter than the time constant of the control valve. • Unlike most other loops, there is a range of controller gains over which the oscillations increase as the controller gain is decreased. This leads to the following general rules for tuning PI controllers for averaging level control: • Set the integral time to 60 minutes or longer. • Set the proportional gain to at least 1.0. Averaging pressure control is not as common as averaging level control because in the case of gas systems, a simple fixed resistance on the outlet of the surge tank is usually all that is required to smooth out variations in flow.

Intermediate Level Control There are intermediate situations that do not require a very tight level control but where it is important not to allow the level to swing over the full range of the transmitter, as in averaging level control. A typical example is a blending tank, where the level controls the tank volume, and therefore the residence time for blending. If a ±5% variation in residence time is acceptable, a proportional controller with a gain of 5 to 10 or even lower could be used, since the flow would not be expected to vary over the full range of the control valve capacity.

Mode Selection and Tuning of Common Feedback Loops

87

Example 5-2. Tight and Averaging Level Control Figure 5-5 shows responses of the control of the level in a tank such as the one in Figure 5-4(a) with the level controller tuned for averaging and for tight level control. The inlet flow into the tank increases by increments of 200 gpm as several batch processes dump their contents into the tank. The tank has a total capacity of 10,000 gallons, while the valve has a flow capacity of 1,000 gpm when fully open. Figure 5-5 also illustrates that averaging level control (the continuous lines) averages out the variation of the inlet flow, resulting in a smooth variation of the outlet flow. On the other hand, tight level control (the dashed lines) maintains the level nearly constant, but this requires that the outlet flow essentially follow the variations in the inlet flow, just as if the tank were not there. In this example the averaging level controller has a gain of 1.0 and the tight controller has a gain of 20; both have integral times of 20 minutes.

5-4. Temperature Control Temperature controllers are usually proportional-integral-derivative (PID), the derivative mode being required to compensate for the lag time of the temperature sensor, which is usually significant. The sensor time constant can often be estimated by the following formula: MC τ s = ------------phA where: M

= the mass of the sensor, including the thermowell, kg

Cp

= the average specific heat of the sensor, kJ/kg-°C

h

= the film coefficient of heat transfer, kW/m2-°C

A

= the area of contact of the thermowell, m2

(5-2)

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Figure 5-5. Responses of Averaging Level Control (Continuous Lines) and Tight Level Control (Dashed Lines) on a Surge Tank

Level Kc = 1 Kc = 20 TI = 20 min

Outlet flow

Kc = 20

Kc = 1

When these units are used, the time constant is calculated in seconds. Temperature is the variable most often controlled in chemical reactors, furnaces, and heat exchangers. When the temperature controller manipulates the flow of steam (Figure 3-1) or fuel to a heater or furnace (Figure 5-6), the rate of heat supplied to the process fluid is proportional to the flow of steam or fuel. This is because the heat of condensation of the steam and the heating value of the fuel remain approximately constant with load. However, when the manipulated variable is cooling water or hot oil, the rate of heat removed or supplied to the process fluid is very nonlinear with water or oil flow because the heat transfer rate requires that the outlet utility temperature moves closer to its inlet temperature as the heat transfer rate increases. This means that it requires higher increments in flow for equal increments in heat rate as the load increases. To reduce the nonlinear nature of the loop, the temperature controller TC is sometimes cascaded to a heat rate controller (QC), as in Figure 5-7. The process variable for the heat rate controller is the rate of heat transfer

Mode Selection and Tuning of Common Feedback Loops

89

in the exchanger, which is proportional to the flow and to the change in temperature of the hot oil: Q = FoilCp(Toin - Toout) where: Q

= rate of heat transfer

Foil

= flow rate of the hot oil

Cp

= specific heat of the hot oil

Toin = inlet temperature of the hot oil Toout = outlet temperature of the hot oil This calculation is carried out in the heat rate controller QC in Figure 5-7 to determine the process variable of the controller. The process outlet temperature controller TC sets the set point of the heat rate controller QC.

Example 5-3. Estimate of Temperature Sensor Time Constant Estimate the time constant of an RTD (resistance temperature device) weighing 0.23 kg and having a specific heat of 0.15 kJ/kg-°C. The thermowell is cylindrical with an outside diameter of 12.5 mm and a length of 125 mm. The film coefficient of heat transfer between the fluid and the thermowell is 0.5 kW/m2-°C. The area of the thermowell is: A = πDL = 3.1416(0.0125)(0.125) = 0.0049 m2. The time constant, from Equation (5-1), is estimated as: τs = (0.22)(0.15)/(0.5)(0.0049) = 13.5 s (0.22 min).

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Figure 5-6. Temperature Control of a Process Furnace

SP Process flow

TC TT

Air

Fuel

Most industrial temperature controllers can usually be tuned by the methods outlined in Chapters 2, 3, and 4. There are a few exceptions: • The control of the outlet temperature from reformer furnaces by manipulation of the fluid flow involves very fast loops similar to flow control loops. (Reformer furnaces are used to carry out highly endothermic catalytic reactions. They differ from regular furnaces in that the tubes are packed with catalyst.) The controllers can be tuned as flow controllers (see Section 5-2). • The control of laboratory constant temperature baths by manipulation of power to electric heaters is usually done with on-off controllers or high-gain proportional controllers.

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Figure 5-7. Temperature Control of Process Heater by Manipulation of the Heat Rate Temperature Control of Process Heater by Manipulation of the Heat Rate

Hot oil Toin

TT

Foil

FT

SP TC SP QC

Process in

TT

Toout

TT

Process out

5-5. Analyzer Control The major problem with the control of composition by analyzer is usually associated with the sensor/transmitter. Sampling of process streams introduces significant dead time into the loop because sensors are often slow, plus some measurement noise occurs if the sample is not representative due to poor mixing. In addition, sensor measurements are sensitive to temperature and other process variables. Analyses of hydrocarbon mixtures are done by chromatographic separation, which is discontinuous in time; they also have a time delay of about the same magnitude as the period of the analysis cycle, compounding the control problem. In spite of all the sources for time delays in sampling and analysis, since it is the ratio of the dead time to the process time constant that determines the uncontrollability of the loop (see Chapter 4), if the combination of the analysis sampling period and time delay is less than the process time constant, a proportional-integral-derivative (PID) controller is indicated. The tuning strategy of Chapters 2 and 4 can be used. On the other hand, if the total dead time is on

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the order of several process time constants, theories such as IMC2 and controller synthesis1 call for a pure integral controller. This is because the process responds fast relative to the time frame in which the analysis is done. Chapter 6 discusses the tuning of controllers that make use of sampled, rather than continuous, measurements.

5-6. Summary This chapter presented some guidelines for selecting and tuning feedback controllers for several common process variables. While flow control calls for fast PI controllers with low gains, level and pressure control can be achieved with simple proportional controllers with high or low gains, depending on whether the objective is tight control or smoothing of flow disturbances. When PI controllers are used for level control, the integral time should be long, on the order of one hour or longer. PID controllers are commonly used for temperature and analyzer control.

References 1. Martin, J. Jr., Corripio, A. B. and Smith, C. L. “How to Select Controller Modes and Tuning Parameters from Simple Process Models,” ISA Transactions, V. 15 (Apr. 1976), pp. 314-319. 2. Rivera, D. E., Morari, M. and Skogestad, S. “Internal Model Control, 4. PID Controller Design,” Industrial and Engineering Chemistry Process Design and Development, V. 25 (1986), p. 252.

Review Questions 5-1. Briefly state the difference between tight level control and averaging level control. In which of the two is it important to maintain the level at the set point? Give an example of each. 5-2. What type of controller is recommended for flow control loops? Indicate typical values for the gain and integral times. 5-3. What type of controller is indicated for tight level control? Indicate typical gains for the controller.

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5-4. What type of controller is indicated for averaging level control? Indicate typical gains for the controller. 5-5. When a PI controller is used for averaging level control, what should the integral time be? Would an increase in gain increase oscillations or decrease oscillations? 5-6. Estimate the time constant of a temperature sensor weighing 0.03 kg, with a specific heat of 23 kJ/kg-°C. The thermowell has a contact area of 0.012 m2 and the heat transfer coefficient is 0.6 kW/m2-°C. 5-7. Why are PID controllers commonly used for controlling temperature? 5-8. What is the major problem with the control of composition?

6 Tuning Sampled-Data Control Loops

This chapter deals with tuning methods for loops—such as analyzer control loops—in which the process variable cannot be measured continuously. In such loops the process variable must be sampled at discrete intervals of time, at which the control calculations are carried out and the controller output is updated, to be kept constant until the next update.

Learning Objectives–When you have completed this chapter you should be able to: A. Understand the effect of sampling on control loop performance. B.

Tune sampled-data control systems.

C. Apply feedback controllers to loops with dead time compensation.

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6-1. The Discrete PID Control Algorithm Most of the process industries today use computers and microprocessors to carry out the basic feedback control calculations. Microprocessors carry out the control calculations in distributed control systems (DCS), programmable logic controllers (PLC), and single-loop controllers, while larger computers carry out higher-level control functions, many of which include supervisory feedback control. Unlike analog instruments, digital devices must sample the controlled variable and compute and update the controller output at discrete intervals of time. The formulas that are programmed to calculate the controller output are discrete versions of the feedback controllers presented in Chapter 2. A particular way to arrange a formula for the calculation is called an “algorithm.” When the process variable is measured continuously, the control system samples it fast enough relative to the response time of the process for the sampling to be of no consequence to the performance of the controller. In such cases, the algorithm used to do the PID calculations is as transparent to the control engineer as the pneumatic and electronic circuits that are used to determine the controller output in analog controllers. However, when the sampling interval is of the order of the process response time, the control algorithm becomes of interest. This section introduces the PID (proportional-integral-derivative) algorithm. Since there is no extra cost for programming all three modes of control, most algorithms contain all three and then use flags and logic to allow the control engineer to specify any single mode or combination of two modes or all three modes. Since the feedback control calculation is made at regular intervals of time, the process variable (PV) is sampled only when the controller output (OP) is calculated and updated. The controller output is updated at the sampling instants and is held constant for one sampling interval (the period of time between output updates) of duration T. Formulas for carrying out the PID algorithm calculations are given in Table 61. In this algorithm, the process variable (PVk) is first used to calculate Yk, the output of a proportional-derivative (PD) calculation. This is done to avoid undesirable pulses in the controller output on set point changes by having the

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97

derivative mode work on the process variable PVk instead of on the deviation from set point. The PD calculation also contains a filter with time constant αTD, which is intended to limit the magnitude of pulses in the controller output upon sudden changes in the process variable. It is seldom desirable for the derivative mode of the controller to respond to set point changes, because on a set point change there would be a large change in the controller output lasting for just one sample—that is, a large undesirable output pulse known as a “derivative kick.” Such pulses are completely avoided by the algorithm of Table 6-1 since the derivative mode, acting on the process variable, does not “see” changes in set point. Table 6-1. Discrete PID Control Algorithm Proportional-Derivative (P+D) unit:

( α + 1 )T D αT D T Y k = -------------------- Y k – 1 + -------------------- PV k + ------------------------- ( PV k – PV k – 1 ) T + αT D T + αT D T + αT D Deviation from set point: Ek = SPk - Yk Increment in controller output:

T ΔM k = K c  E k – E k – 1 + ------ E k  T  I

Controller output: Mk = Mk-1 + ΔMk where: SPk = set point PVk = process variable (measurement) Mk = controller output Ek = error or set point deviation α = derivative filter parameter T = sampling interval, min Kc = proportional gain TI = integral time TD = derivative time

The deviation from the set point or error (E) in Table 6-1 is for a reverse acting controller. For a direct-acting controller the terms Yk and SPk are reversed in the formula or, alternatively, the controller gain is set to a negative value. When either the derivative time TD is set to zero (PI controller) or the process

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variable reaches a steady value, Yk = PVk, the algorithm still drives the process variable to the set point. The filter parameter α of Table 6-1 has a special meaning; its reciprocal, 1/α, is the amplification factor on the change of the PV at each sampling instant and is also called the “dynamic gain limit.” Note that, if α were set to zero, the amplification factor on the change in PV would have no limit. For example, if the sampling interval is one second (1/60 min) and the derivative time is one minute, the change in PV at each sample with α = 0 would be multiplied by a factor of 60 (TD/T = 60). By setting the nonadjustable parameter α to a reasonable value, say 0.1, the change in PV cannot be amplified by a factor greater than 10, independent of the sampling interval and the derivative time. The dynamic limit permits setting the derivative time to any desired value without the danger of introducing large undesirable pulses in the controller output.

Example 6-1. Response of the Proportional-Derivative (PD) Calculation to a Ramp Calculate the output of the derivative term on the PD unit (the equation that calculates Yk in Table 6-10) of the PID control algorithm to a ramp that starts at zero and increases by 1% with each sample. Use a sampling interval of 1 s and a derivative time of 0.5 min. The derivative filter parameter is α = 0.1. Substitution of the values given and of the process variable at each sample into the series controller of Table 6-1 produces the results summarized in Table 6-2. The results for the “ideal” derivative unit are calculated using the filter parameter α = 0. Table 6-2. Results of PD Calculation on a Ramp Input for Example 6-1 Sample, s

0.

1.

2.

3.

4.

5.

10.

20.

40.

PVk

0.

1.

2.

3.

4.

5.

10.

20.

40.

Yk

0.

8.5

15.2

20.3

24.5

27.9

38.3

49.9

70.0

Ideal

30.

31.0

32.0

33.0

34.0

35.0

40.0

50.0

70.0

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99

For example, for the value of Y1, at the first sample Y0 = 0, PV1 = 1%, PV0 = 0%, TD = 0.5(60) = 30 s, so: ( 0.1 + 1 )30 1 0.1 ( 30 ) Y 1 = ---------------------------- 0 + ---------------------------- 1 + ---------------------------- ( 1 – 0 ) = 0 + 0.25 + 8.25 = 8.50 1 + 0.1 ( 30 ) 1 + 0.1 ( 30 ) 1 + 0.1 ( 30 ) Notice that the unfiltered (ideal) derivative unit jumps to 30 at time 0 with increments of 1 at each sample. Both of these responses are shown graphically in Figure 6-1. The unfiltered derivative unit is leading the PV by one derivative time (30 s), while the derivative unit with the filter, after a brief lag, also leads the PV by one derivative time. In practice, the lag is too small to significantly affect the performance of the controller.

Figure 6-1. Response of Proportional-Derivative (P+D) Unit to a Ramp Input, with and without the Filter 80

70

60

Yk 50

40

D=0

30

PVk

D = 0.1

20

10

TD = 30 T

0 0

5

10

15

20

25

t/T

30

35

40

45

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Eliminating Proportional Kick on Set Point Changes Similar to the derivative kick, the sudden change in controller output caused by the proportional mode right after a change in set point is known as “proportional kick,” although it is not a pulse. It too can be eliminated by replacing the deviation Ek with the output of the derivative unit Yk in the calculation of the increment in controller output in the control algorithm of Table 6-1. Modern computer- and microprocessor-based controllers offer the option of having the proportional mode act on either the deviation or the process variable. This option, known as proportional-on-PV, must be selected on the following basis: • If the controller is a main controller, with infrequent changes in set point, the proportional mode should act on the process variable. This allows the tuning of the controller for disturbance inputs (higher gain) without the danger of large overshoots on sudden set point changes (see Section 4-3). • If the controller is the slave of a cascade control scheme (see Chapter 7), the proportional mode must act on the deviation from set point. Otherwise, when the main controller changes the set point of the slave, the slave will not respond immediately, as it must for the cascade scheme to work. It is important to realize that the reason the proportional-on-PV option is selected is to allow the operator to make changes in set point without the fear of causing a sudden change in the controller output. As would be expected, the resulting approach to the new set point will be slower than when the proportional mode acts on the deviation from set point. The rate of approach to set point is controlled by the integral time when the proportional-on-PV option is selected. As in the case of the derivative-on-PV option, the performance of the controller on disturbance inputs is the same whether the proportional mode acts on the deviation or on the PV, because in both cases the set point does not change.

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Nonlinear Proportional Gain Practically all modern computer- and microprocessor-based controllers offer the option of a nonlinear gain parameter. The purpose is to have the proportional gain increase as the error or deviation from the set point increases: Kc = KL(1 + KNL|Ek|)

(6-1)

where: KL

= the gain at zero error

KNL = the increase in gain per unit increase in error The bars around the error indicate the absolute value or magnitude of the error. By using the absolute value of the error, the gain increases when the error increases in either the positive or the negative direction. The nonlinear gain option is normally used with averaging level controllers (see Section 5-3) because it allows a wider variation of the level near the set point while still preventing the tank from overflowing or running dry, as illustrated in Figure 6-2. In the figure, the gain at the middle of the range is 0.25 (the dashed line) and unity is at the two extremes of the range. The nonlinear gain allows greater smoothing of flow variations with a given tank; that is, it makes the tank look as if it has a larger capacity than it does, as long as the flow varies near the middle of its range. Some computer controllers provide the option of having a zero gain at zero error, a feature which is desirable in some pH control schemes.

Example 6-2. Adjusting the Nonlinear Gain An averaging level controller is proportional-only with a gain of 1, a set point of 50%, and an output bias of 50%. Determine the value of the nonlinear gain required to be able to reduce the gain at zero error to 0.25 while still keeping the tank from overflowing or running dry.

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Figure 6-2. Controller Output OP versus Process Variable PV for a Proportional Controller with Nonlinear Gain 100%

80%

OP

60%

40%

20%

0% 0%

20%

40%

60%

80%

100%

PV

Prevention of the tank’s overflowing or running dry requires that the valve be fully open when the level is at 100% of the range and fully closed when the level is at 0%. Since the set point is 50%, either requirement takes effect when the magnitude of the error is ±50%. With the output bias of 50%, using the upper limit requirement in Equation 6-1: 100% = 50% + Kc(100% - 50%) = 50% + 0.25[1 + KNL(50%)](50%) KNL = [(100% - 50%)/(0.25)(50%) - 1]/50% = 0.06 The controller gain at each end of the range is: Kc = 0.25[1 + 0.06(50%)] = 1.0

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Recall from Equation 5-1 that the time constant of the tank is inversely proportional to the controller gain, thus the effective capacity of the tank, as used for smoothing flow variations, can be increased from its real value at full and zero flow to four times that value at one-half of full flow.

Section Summary This section presented the computer- and microprocessor-based PID control algorithm and the options that are made available by its configurable nature. The next section concerns the tuning of sampled-data controllers.

6-2. Tuning Sampled-data Feedback Controllers The tuning formulas of Chapters 2 and 4, although intended for continuous controllers, can be applied to sampled-data (discrete) controllers as long as the effect of sampling is taken into consideration. This section presents a simple correction of the tuning formulas for the effect of sampling, and formulas which are specifically applicable to discrete controllers. Keep in mind, however, that for most loops the sampling frequency is high relative to the dynamics of the rest of the loop, and the effect of sampling is therefore nonexistent.

Tuning by Ultimate Gain and Period The formulas for quarter-decay-ratio response presented in Chapter 2, based on the ultimate gain and period of the loop, can be applied directly to sampled-data controllers because the effect of sampling is accounted for in the experimentally determined ultimate gain and period. Increasing the sampling interval decreases the ultimate gain and increases the ultimate period, because slower (that is, less frequent) sampling makes the feedback control loop less controllable and slower to respond.

Tuning by Open-loop Test Parameters When the controller is tuned using the process parameters of gain, time constant, and dead time, estimated as shown in Chapter 3, the effect of sampling is not included in the process model. This is because the process model is obtained from a step test in controller output, as you can recall from Chapter

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3, and such a step will always take place at a sampling instant and will remain constant after that. Moore and his coworkers3 developed a simple correction of the controller tuning parameters for the effect of sampling. They point out that when a continuous signal is sampled at regular intervals of time and then reconstructed by holding the sampled values constant for each sampling period, the reconstructed signal is effectively delayed by approximately one-half the sampling interval, as shown by the dashed line in Figure 6-3. In sampled-data systems the controller output is held constant between updates, thus adding one-half the sampling interval to the dead time of the rest of the loop components. The correction for sampling is then simply to add one-half the sampling interval to the dead time obtained from the step response. The uncontrollability parameter is then given by: T t 0 + --2 P u = --------------τ

(6-2)

where: Pu

= the uncontrollability parameter

t0

= the model dead time, min

τ

= the model time constant, min

T

= the sampling interval, min

Tuning Formulas for Sampled-data (Discrete) Controllers Dahlin1 introduced a procedure for synthesizing sampled-data controllers. This synthesis procedure can be used to develop a set of controller tuning formulas. The tuning formulas, shown in Table 6-3, have the advantage of accounting exactly for the effect of sampling, so that they apply over any set of values of the loop parameters and the sampling time. For details of the derivation of these formulas, see Smith and Corripio4. The formulas of Table 6-3 contain an adjustable parameter q that affects only the controller gain. This parameter is adjusted in the range of 0 to 1 to shape the tightness of the closed-loop response. If the model parameters were an

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105

Figure 6-3. Effective Delay of Sampling and Holding a Signal is One Half the Sampling Interval

Sample interval = 2 min

Table 6-3. Tuning Formulas for Discrete PI Controller Given the loop parameters: K = gain τ = time constant, min t0 = dead time, min T = sampling interval, min q = an adjustable parameter, in the range of 0 to 1.

Let

t N = -----oT

a = e

T – --τ

Proportional gain:

( 1 – q )a K c = ------------------------------------------------------------------------------K( 1 – a)[1 + N(1 – q)]

Integral time:

aT T I = ---------------1–a

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exact fit to the process response, the value of q would be the fraction of the error at any one sample that would remain after one dead time plus one sampling interval. For example, setting q = 0 specifies the process variable to match the set point after N + 1 samples, where N is the number of samples of dead time—that is, the sample time divided by the sampling interval. This would result in the highest gain, and therefore in the tightest control. However, for any value of q, the tightness of the closed-loop response depends on the ratio of the sampling interval to the time constant, T/τ. A more fundamental adjustable parameter is the closed-loop time constant τc, which can be related to the time parameters of the loop—short for fast processes and long for slow processes. If τc is specified, the value of q can be computed by:

q = e

T – ---τc

Setting q = 0 results in an upper limit for the controller gain. This value can be used as a guide for the initial tuning of the controller. As is the case with the tuning formulas of Chapter 4, the upper limit of the controller gain decreases with increasing the process dead time, parameter N, in number of samples. The formulas of Table 6-3 are intended to tune only a PI controller. Two time constants plus a dead time would be required to tune a PID controller by this procedure, but it is difficult to accurately determine more than one time constant and a dead time from a simple open-loop step test. As mentioned earlier, the formulas of Table 6-3 are applicable to any value of the loop parameters and the sampling interval; moreover, the controller gain can be adjusted to obtain fast response with reasonable variation of the controller output. They are highly recommended because they relate the integral and derivative times to the process time constants, thus reducing the tuning procedure to the adjustment of the controller gain.

Example 6-3. Discrete Control of Process Heater Temperature Use the tuning formulas of Table 6-3 to tune the temperature controller for the heater of Section 3-1. Use a PI controller with sampling intervals of 1 s, 10 s, 30 s, 1 min, 2 min, and 4 min.

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107

The process parameters for the heater were determined in Section 3-1: K = 1.95

τ = 7.5 min

t0 = 2.5 min

The tuning parameters, for a sampling interval of 0.5 min and q = 0, are:

N = 2.5 ⁄ 0.5 = 5

a = e

0.5 – ------7.5

= 0.936 0.5 ( 0.936 ) T I = -------------------------- = 7.3 min 1 – 0.936

( 1 – 0 )0.936 K c = ------------------------------------------------------------------------------------------ = 1.2 1.95 ( 1 – 0.936 ) ( 1 – 0 ) [ 1 + 5 ( 1 – 0 ) ] TD = 0

For the other sampling times the tuning parameters are given in Table 6-4.

Table 6-4. Tuning Parameters for Example 6-3 Sampling interval, min

1/60

1/6

0.5

1.0

2.0

4.0

Dead time, N

150

15

5

2

1

0

Maximum Kc (q = 0)

1.5

1.4

1.2

1.2

0.84

0.72

Integral time, min

7.5

7.4

7.3

7.0

6.5

5.7

Notice that the gain is lower and the integral time is shorter as the sampling interval is increased. This means that the loop is less controllable at the longer sampling interval. Also notice that there is a very small change in the tuning parameters when the sampling frequency is increased from 6 times per minute to 60 times per minute. Since most control systems sample at the rate of more than once per second, the effect of sampling is negligible for most control loops. Figure 6-4 shows the responses of the heater temperature controller to a step change in disturbance with the tuning parameters given in Table 6-4 and sampling intervals of 0.5, 1.0, and 2.0 min. The figure shows that there is little difference in controller performance for sampling interval times of 0.5 and 1 min, but the performance deteriorates with a sampling interval of 2 min.

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Figure 6-4. Responses of Heater Temperature Control with Sampling Intervals of 0.5, 1.0 and 2.0 min

T = 1 min T = 0.5 min

T = 2 min

Fast Process/Slow Sampling When the sampling interval is more than three or four times the dominant process time constant, the process reaches steady state in responding to each controller output move before it is sampled again. This may happen because the process is very fast or because the sensor is an analyzer with a long cycle time. For such situations the formulas of Table 6-2 result in a pure integral controller: Mk = Mk-1 + KIEk where: 1–q K 1 = ----------------------------------------K[1 + N(1 – q)]

(6-3)

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Notice that for the case N = 0 and q = 0, the controller gain is the reciprocal of the process gain. This result makes sense, since a loop gain of 1.0 is what is needed to reduce the error to zero in one sampling interval if the process reaches steady state during that interval. An interesting application of this case is a chromatographic analyzer sampling a fast process. The nature of such an analyzer is that the composition is not available to the controller until the end of the analysis cycle, because it takes a full analyzer cycle to separate the mixture and analyze it. This means that the process dead time is approximately one sampling interval, or N = 1. For q = 0, Equation 6-3 gives a gain of KI = 1/K(1 + 1) = 1/2K, or one-half the reciprocal of the process gain. This also makes sense, because when action is taken by the controller, it takes two sampling intervals for the controller to see the result of that action, so the formula says to spread the corrective action equally over two samples.

Example 6-4. Slow Sampling of Process Heater Outlet Temperature For the process heater of Section 3-1, calculate the maximum gain for the PI controller using the formulas of Table 6-2 and sampling times of 5, 10, and 20 min. Also calculate the gain of the pure integral controller, given by KcT/ TI (this is the same as the KI of Equation 6-3). This problem is just a continuation of the progression of the sampling interval in Example 6-3. The results are summarized in Table 6-5. Table 6-5. Tuning Parameters for Example 6-4 Sampling interval, min

5

10

20

Dead time, N

0

0

0

Maximum gain (q = 0)

0.54

0.18

0.038

Integral time, min

5.2

3.6

1.5

Integral gain, KcT/TI

0.51

0.51

0.51

As the sampling interval is increased, the proportional term disappears while the gain of the pure integral controller remains constant and equal to the reciprocal of the model gain, 1/1.95 = 0.51. Figure 6-5 compares the temperature control responses to a step change in flow to the heater for a PI controller with sampling intervals of 5, 10, and 20 min. The tuning parame-

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ters are those shown in Table 6-5 except that the gain was reduced by half (q = 0.5) with the sampling interval of 5 min, and to one-fourth (q = 0.75) with the sampling intervals of 10 and 20 min. As expected, controller performance is a lot slower than it is in the responses of Figure 6-4, but still acceptable when the sampling interval is forced to be that long. This shows that the tuning formulas of Table 6-2 can be applied to a wide range of sampling-interval-to-time-constant ratios.

Figure 6-5. Responses of Heater Temperature Control with Sampling Intervals of 5, 10, and 20 min

T = 5 min q = 0.5

T = 10 min q = 0.75

T = 20 min q = 0.75

To summarize, the formulas presented in Table 6-2 can be used with the openloop test model, resulting in a PI controller. They are applicable over a wide range of sampling intervals and dead-time-to-time-constant ratios. Although formulas used to tune PID controllers have not been developed by this method, this does not present a problem because the derivative mode should not be effective when the process variable is sampled slowly.

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6-3. Selection of the Sampling Frequency Most microprocessor-based controllers (e.g., DCS) have a fixed processing frequency of about one to ten output updates per second. For most feedback control loops, such a short sampling interval has no effect on controller performance and the controller can be considered to be continuous. On the other hand, computer control systems and higher-level DCS functions, allow the control engineer to select the sampling interval of each controller. Although in theory the minimum sampling interval results in maximum loop performance, there is a point of diminishing returns where further reduction in the loop sampling interval results in minor improvement in loop performance at the expense of overloading the process control system and limiting the number of loops it can process. However, computers have become faster and with today’s hardware, the limit on the number of loops they can process is no longer a consideration in selecting the processing frequency. The relationship between sampling interval—the period of time between output updates—and controller performance is a function of the time constant and dead time of the process. In fact, a good way to select the sampling interval is to look at the ratio of sampling interval to process time constant versus the ratio of process dead time to time constant or to process uncontrollability parameter. It makes sense to ratio the sampling interval to the process time constant because the relative change in the process output from one sample to the next depends only on this ratio; that is, the relative change will be the same for a process with a 1-minute time constant sampled once every 5 seconds as for a process with a 10-minute time constant sampled every 50 seconds. It also makes sense to relate the sampling interval to the uncontrollability parameter, because the dead time imposes a limit on controller performance, and this limit is met at higher sample-time-to-time-constant ratios the higher the dead-time-to-time-constant ratio for the process. What is the problem with too long a sampling interval? The problem is that for a very slow process the increments in controller output at each update become very small and may be lost in the precision with which the controller output is calculated. The reason the controller output becomes small is because:

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• The increment in output contributed by the proportional and derivative modes is very small because the process variable changes very little from one sample to the next. • The increment in output contributed by the integral mode is very small because the ratio of the sampling interval to the integral time, T/TI, is very small. For example, the pressure control loop of an ammonia synthesis loop, where the pressure is controlled with the purge flow, has a time constant of about 40 minutes (2,400 s). If the controller is updated 10 times per second and the integral time is of the order of magnitude of the time constant, the term T/TI in Table 6-1 is 0.1/2,400 = 0.00004. This makes the controller output, when the error is 1% and the output 50%, equal to 50.00004%, so that there is an imperceptible change on the position of the valve after each update of the controller output. The reason this is not commonly a problem is that the controller output is usually computed with double precision (twice as many digits as normal in the representation of the output result) and the output can be accurately computed so that after many samples (about 10,000 in this case), the small increments are not lost. In our experience, the control loop performance is acceptable when the controller update frequency is as high as one-tenth the loop time constant. At one time, Fisher Controls marketed a very successful computer control package for ammonia plants that used a sampling interval of 5 minutes to control the synthesis loop pressure with the purge flow.

Optimizing Feedback Loops Many modern computer control installations use feedback controllers to minimize the consumption of energy and to maximize the production rate. A common example of such control loops is the technique of “valve position control” in which a controller looks at the output of another controller or the equivalent, the valve position, and keeps it close to fully open or fully closed. Such controllers are designed to drive the process toward its constraints over a very long time period, and their sampling intervals should be much longer than the sampling interval of the controller whose output they control, maybe 30 times as long or longer. This is to prevent the valve position controller from continuously introducing disturbances and interactions into the control system.

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Sometimes the valve position controller is designed with a “gap” or dead band around its set point so that it only takes action when the controlled valve position is outside that dead band. Once again, the purpose of the gap is to prevent the valve position controller from introducing disturbances and interactions into the control system.

6-4. Compensation for Dead Time It has been clearly established in Chapter 4 and Section 6-1 that feedback controllers cannot perform well when the process has a high ratio of dead time to time constant. The total loop gain must be low for such processes, so the deviations of the controlled variable from its set point cannot be kept low in the presence of disturbances. One way to improve the performance of the feedback controller for low controllability loops is to design a controller that compensates specifically for the process dead time. This section presents two controllers that have been proposed to compensate for dead time: the Smith Predictor and the Dahlin Dead Time Compensation Controller (Dahlin Controller). Dead time compensation requires that past values of the controller output be stored and played back. Not until the advent of computer- and microprocessor-based controllers was the storage and playback of control signals possible. The computer memory provides the ability to store and retrieve past sampled values.

The Smith Predictor Smith5 proposed a dead time compensator which consisted of an internal model of the process to be driven on-line by the controller output and continuously compared with the controlled variable to correct for model errors and disturbances. A block diagram of the scheme, known as the “Smith Predictor,” is shown in Figure 6-6. Notice that in the process model, the dead time term is separated from the rest of the model transfer function so that the model output, after being corrected for model error and disturbance effects, can be fed to the feedback controller in such a manner that the process dead time is bypassed, thereby compensating for the dead time. A disadvantage of the Smith Predictor is that although it requires a model of the process, it does not use the model to design or tune the feedback control-

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ler, so that it ends up with too many adjustable parameters: the model parameters plus the controller tuning parameters. Because there are so many parameters to adjust, there is no convenient way to adjust the closed-loop response when the model does not properly fit the process. Given the nonlinear nature of process dynamics, any technique that depends heavily on exact process modeling is doomed to fail.

Figure 6-6. Block Diagram of the Smith Predictor for Dead Time Compensation Disturbances

SP Controller

Process with Dead Time

OP

+

PV

+ Process Model Corrected model output

Model Dead Time + +

-

Model error

The Dahlin Controller The controller synthesis procedure introduced by Dahlin1 produces a feedback controller that is exactly equivalent to the Smith Predictor in compensating for the dead time, but with the advantage that the controller tuning parameters are obtained directly from the model parameters. Those interested in the details of the derivation can see Smith and Corripio4. The Dahlin dead time compensation controller can be reduced to a PID controller with an extra term. The only modification to the controllers of Table 6-1 is in the calculation of the controller output: Mk = Mk-1 + ΔMk + (1 - q)(Mk-N-1 - Mk-1)

(6-4)

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where ΔMk can be computed by the control algorithm of Table 6-1. The last term in the calculation of the output provides the dead time compensation. Note that the term vanishes when there is no dead time, N = 0. The actual controller is tuned with the formulas of Table 6-2 except for the controller gain, which is given by: ( 1 – q )a K c = --------------------K(1 – a)

(6-5)

Comparison of these formulas with the corresponding ones in Table 6-2 shows that these lack the term [1 + N(1 - q)] in the denominator. Recall that this term decreases the controller gain to account for dead time. Since the controller of Equation 6-4 explicitly compensates for dead time, its gain can be higher. The Dahlin Controller is used extensively to control processes with long dead times. A common application is the control of paper machines, where the properties of the paper can only be measured after it has gone through the drying process, which introduces significant dead time. One characteristic of this application is that the dead time is relatively constant and can be determined precisely. Dead time compensation presents problems in other processes in which the dead time depends on flow and other process variables (see Section 3-4).

Example 6-5. Dead Time Compensation Control of Process Heater Compare the response of the temperature controller for the process heater of Figure 3-1 with and without dead time compensation. Use a PI controller with a sampling interval of 1 min, which is approximately one-tenth of the time constant (7.5 min). The open-loop parameters for the steam heater are a gain of 1.95, a time constant of 7.5 min, and a dead time of 2.5 min. The dead time compensation term requires two sampling intervals of dead time: N = int(t0/T) = int(2.5/1) = 2

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Using the formulas of Table 6-2, the tuning parameters are: a = e(-1/7.5) = 0.875 Without dead time compensation, with q = 0.67: Kc = (1-0.67)(0.875/0.125)/(1.95)[1+2(1-0)] = 0.4 TI = (0.875/0.125)*1 = 7.0 min With dead time compensation and q = 0.67: Kc = (1-0.67)(0.875/0.125)/1.95 = 1.2 TI = 7.0 min

Figure 6-7 compares the steam heater responses of the controllers to a step increase in process flow to the heater. The dead time compensation controller results in a smaller deviation from set point and less oscillation than the regular PI controller. The improvement in performance is not spectacular, probably because this is a relatively controllable process. The value of q = 0.67 was selected to prevent excessive oscillation in the controller output. The proportional gains are then one-third, (1 – 0.67) = 0.33, of the maximum gains. More sophisticated dynamic compensation controllers have been proposed in the last few years, for example the Vogel-Edgar controller6 and Internal Model Control2. These controllers can incorporate a more precise compensator than the Dahlin Controller, provided that a precise model of the process is available. Nevertheless, the Dahlin Controller has been applied successfully to the control of paper machines and other processes with high dead-time-totime-constant ratios.

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Figure 6-7. Response of Heater Temperature Control with and without Dead Time Compensation

PI controller with dead time compensation Standard PI controller

6-5. Summary This chapter presented various sampled-data feedback controllers, how to tune them, and how to select the sampling interval for them. The control algorithm of Table 6-1 and the tuning formulas of Table 6-2 are strongly recommended, in addition to being the most commonly used in sampled-data control applications. For processes with high dead-time-to-time-constant ratios, the Dahlin Controller, Equation 6-4, is commonly used in industry and is also recommended here.

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References 1. Dahlin, E. B. “Designing and Tuning Digital Controllers,” Instruments and Control Systems, V. 41, June 1968, p. 77. 2. Garcia, C. E. and Morari, M. “Internal Model Control, 1. A Unifying Review and Some Results,” Industrial and Engineering Chemistry Process Design and Development, V. 21, 1982, pp. 308-323. 3. Moore, C. F., Smith, C. L. and Murrill, P. W. “Simplifying Digital Control Dynamics for Controller Tuning and Hardware Lag Effects,” Instrument Practice, V. 23, Jan. 1969, p. 45. 4. Smith, C. A. and Corripio, A. B. Principles and Practice of Automatic Process Control 2nd ed., New York: Wiley, 1997, Chapter 15. 5. Smith, O. J. M. “Closer Control of Loops with Dead Time,” Chemical Engineering Progress, V. 53, May 1957, pp. 217-219. 6. Vogel, E. F. and Edgar, T. F. “A New Dead Time Compensator for Digital Control,” Proceedings ISA/80, Research Triangle Park: ISA, 1980.

Review Questions 6-1. How are computer- and microprocessor-based controllers different from analog controllers? 6-2. What is “derivative kick”? How is it prevented? Why is a “dynamic gain limit” needed in the derivative term of the PID controller? 6-3. How and why would you eliminate “proportional kick” on set point changes? Will the process variable approach its set point faster or slower when proportional kick is avoided? When must proportional kick be allowed? 6-4. What is the advantage of a nonlinear proportional gain in averaging level control situations? In such a case, what must the nonlinear gain be for the gain to be 0.2 at zero error and still have the controller output reach its limits when the level reaches its limits (0 and 100%)? Assume a level set point of 50% and an output bias of 50%.

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6-5. A process has a gain of 1.6%, a time constant of 20 min and a dead time of 5 min. Calculate the tuning parameters for a discrete controller if the sampling interval is (a) 4 s, (b) 1 min, (c) 5 min and (d) 50 min. 6-6. Repeat question 6-5, but for a controller with dead time compensation. Specify also how many samples of dead time compensation, N, must be used in each case. 6-7. What is the basic idea behind the Smith Predictor? What is its major disadvantage? How does the Dahlin Controller with dead time compensation overcome the disadvantage of the Smith Predictor?

7 Tuning Cascade Control Systems

Cascade control is a common strategy for improving the performance of process control loops. In its simplest form it consists of closing a feedback loop inside the primary control loop by measuring an intermediate process variable. This chapter presents an overview of cascade control and the tuning of cascade control systems.

Learning Objectives—When you have completed this chapter, you should be able to: A. Know when to apply cascade control and why. B.

Select the control modes and tune the controllers in a cascade control system.

C. Recognize reset windup in cascade control systems and know how to prevent it.

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7-1. When to Apply Cascade Control Figure 7-1 shows a typical cascade control system for controlling the temperature in a jacketed exothermic chemical reactor. (Note that the steam valve shown in the figure is used at startup to preheat the reactor and is not relevant to the following discussion.) The control objective is to control the temperature in the reactor, but instead of having reactor temperature controller TC 1 directly manipulate the jacket coolant valve, the jacket temperature is measured and controlled by a different controller, TC 2, which is the one that manipulates the valve. The output of reactor temperature controller TC 1, the “primary” controller, is connected or cascaded to the set point of jacket temperature controller TC 2, the “secondary” controller. Notice that only the reactor temperature set point is maintained at the operator-set value; the jacket temperature set point varies to whatever value is required to maintain the reactor temperature at its set point. A block diagram of the reactor cascade control strategy, shown in Figure 7-2, clearly shows that the secondary control loop is inside the primary control loop. There are three major advantages to using cascade control: 1. Any disturbances that affect the secondary variable are detected and compensated by the secondary controller before they have time to affect the primary control variable. Examples of such disturbances for the reactor of Figure 7-1 are the coolant inlet temperature and pressure. 2. The controllability of the outside loop is improved because the inside loop speeds up the response of the process dynamic elements between the control valve and the secondary variable. In the reactor example, the speed of response of the jacket temperature is increased, resulting in a more controllable loop for the reactor temperature. 3. Nonlinearities of the process in the inner loop are handled by that loop and are removed from the more important outer loop. In the reactor example, the nonlinear relationship between temperature and coolant flow is made a part of the inner loop by the cascade arrangement, while the outer loop enjoys the linear relationship between reactor and jacket temperatures. Since the secondary loop should be more controllable than the primary loop, variations in the process gain are less likely to cause instability when isolated in the secondary loop.

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Figure 7-1. Cascade Temperature Control on a Jacketed Exothermic Chemical Reactor

SP TC 1 Reactants TC 2

TT 2

TT 1

Water out

Coolant Products Steam

A side benefit of cascade control, what we in Louisiana call lagniappe, or a “free gift”, is that when the primary loop must be switched off because of maintenance or another issue, the operator can directly adjust the set point of the secondary controller instead of having to adjust the control valve position directly. This is particularly beneficial when the secondary loop is a flow control loop. Because cascade control requires investment on an additional sensor (TT) and controller (TC 2) over simple feedback control, it is important that these three advantages result in significant improvement in control performance. Such improvement depends on the inner loop responding faster than the outer loop, because all three advantages depend on it. If the inner loop is not faster to respond than the outer loop:

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Figure 7-2. Block Diagram of Cascade Temperature Control of Exothermic Chemical Reactor Disturbances

SP

+ -

Master Controller TC-1

+ PV2

PV1

Slave Controller TC-2

TJ Jacket

TR Reactor

Jacket TT 2

Reactor TT 1

• Disturbances into the inner loop will not be eliminated fast enough to avoid their affecting the primary control variable. • Speeding up of the inner loop would result in a decrease in the controllability of the overall loop because its dead-time-to-time-constant ratio would increase. • Nonlinearities would become a part of the slower and possibly less controllable inner loop, thus affecting the stability of the control system. Besides the inner loop having to be faster to respond than the outer loop, the success of cascade control also requires that the sensor of the inner loop be fast and reliable. One would not consider, for example, cascading a temperature controller to a chromatographic analyzer controller. On the other hand, the sensor for the inner loop does not have to be accurate, only repeatable, because the integral mode in the primary controller compensates for errors in the measurement of the secondary variable. In other words, it is acceptable for the inner loop sensor to be wrong as long as it is consistently wrong, and to the same degree. Finally, cascade control would not be able to improve the performance of loops that are already very controllable; for example, this occurs with liquid

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level and gas pressure control loops, or when the controlled variable does not have to be maintained tightly around its set point, as in averaging level control. When a level controller is cascaded to a flow controller, it is usually justified by the greater flexibility in the operation of the process, not by improved control performance. The above sections have examined the reasons and requirements for using cascade control. The following sections will look at how to select the controller modes for cascade control systems and how to tune them.

7-2. Selection of Controller Modes for Cascade Control In a cascade control system, the primary controller has the same function as the controller in a single feedback control loop: to maintain the primary process variable at its set point. It follows that the selection of controller modes for the primary controller should follow the same design guidelines presented for a single controller in Chapter 5. On the other hand, the function of the secondary controller is not the same as that of the primary or single controller; it therefore requires different design guidelines. Unlike the primary or single feedback controller, the secondary controller is constantly responding to changes in set point, which it must follow as quickly as possible with a small overshoot and decay ratio, as defined in Chapter 2. It is also desirable that the secondary controller transmit changes in its set point to its output as quickly as possible and if possible, amplify them, because the output of the secondary controller is the one that manipulates the final control element. If the secondary controller is to speed up the response of the primary controller, it must transmit changes in the primary controller output (secondary set point) to the final control element at least as fast as if it were not there. It is evident, then, that the secondary controller: • Must have proportional mode. • Proportional mode must act on the deviation from set point. • Should have a proportional gain of 1.0 or greater if stability permits it. If the gain of the secondary controller is greater than unity, changes in the primary controller output result in higher immediate changes in the final control

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element than when a single feedback loop is used. This amplification results in a faster response of the primary loop.

Integral Mode in the Secondary Controller The use of integral and derivative modes in the secondary controller depends on the application. Recall from previous chapters that adding integral mode results in a reduction of the proportional gain, while adding derivative mode results in an increase in the proportional gain. This may suggest that all secondary controllers should be proportional-derivative (PD) controllers, but this is not the general case. As mentioned earlier, integral mode is not needed in the secondary controller to eliminate the offset because the integral mode of the primary controller can adjust the set point of the secondary controller to compensate for the offset. However, if the secondary loop is fast responding and is subject to large disturbances—for example, a flow loop—the offset in the secondary controller would require corrective action by the primary controller and therefore a deviation of the primary process variable from its set point. A fast-acting integral mode on the secondary controller would eliminate the need for corrective action from the primary controller and the deviation in the primary process variable. The integral mode should not be used in those secondary loops in which the gain is limited by stability, and those in which the disturbances to the inner loop do not cause large offsets in the secondary controller. The jacket temperature controller of the reactor in Figure 7-1 is a typical example of a secondary loop that does not require integral mode.

Derivative Mode in the Secondary Controller A common rule states that derivative mode should not be used in both the secondary and primary controller, and since derivative would do the most good on the least controllable loop, which is the primary loop, the rule essentially reduces to never having derivative mode in the secondary controller. This rule is based on the following reasoning: 1. Having all three modes in both the primary and secondary controller results in requiring six tuning parameters, which without the proper guidelines, makes the tuning task more difficult.

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2. It is undesirable to put two derivative units in series in the loop. Both of these reasons can be argued away as follows: • Guidelines, such those presented in previous chapters, simplify the task of tuning. For example, keeping the derivative time to about onefourth the integral time, or to about one-half the dead time when known, reduces the number of parameters in the cascade loop to four: two gains and two integral times. • By having the derivative of the secondary controller act on the process variable instead of on the deviation from set point, the derivative mode will not be in series with the derivative unit in the primary controller. The purpose of the derivative unit in the secondary controller is to compensate for sensor lag or loop dead time and to allow for a higher secondary controller gain, with less overshoot and a low decay ratio. When the secondary process variable is temperature, as in Figure 7-1, derivative mode may improve overall performance. When the inner loop is fast and very controllable (e.g., flow loops), the secondary controller does not require derivative mode.

7-3. Tuning of Cascade Control Systems The tuning of the controllers in a cascade control system must be carried out from the inside out; that is, the innermost loop must be tuned first, then the loop around it, and so on. The block diagram of Figure 7-2 shows why this is so: each inner loop is part of the process of the next outer loop. Each loop in a cascade system must be tuned tighter and faster than the loop around it; otherwise, the set point of the secondary loop would vary more than its measured variable, resulting in poorer control of the primary process variable. Ideally, the secondary variable should follow its set point as quickly as possible, but with little overshoot and oscillations. If quarter-decay-ratio (QDR) response is used for the secondary controller, the gain must be adjusted to prevent excessive overshoot on set point changes. The ideal overshoot for the secondary process variable to a set point change is 5 to 10%.

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After each inner loop has been tuned in succession, the primary loop can be tuned to follow any desired performance criteria by any of the methods of Chapters 2, 4, 5 and 6. Given that what is special in cascade systems is the tuning of the secondary loop(s), some typical secondary loops, namely flow, temperature, and pressure, are briefly discussed next. Keep in mind, however, that any variable—including composition—can be used as a secondary variable provided it can be measured quickly and reliably (e.g., when a simple continuous thermal conductivity detector is used to measure the hydrogen composition in the ammonia synthesis process - see Example 7-2).

Secondary Flow Loop In modern computer control systems, flow is the innermost loop in most cascade control schemes because it allows the operator to intervene in the control scheme by taking direct control of the manipulated flow. Figure 7-3 shows a typical temperature-to-flow control scheme. The flow transmitter compensates for variations in the pressure drop across the control valve and absorbs any nonlinearities of the valve. If the square root of the differential pressure is extracted, the secondary measured variable and thus the output of the primary controller, becomes linear with the flow. The flow controller in a cascade scheme must be tuned tight. A proportionalintegral (PI) controller can be used with the integral time set equal to the time constant of the control valve (see Section 5-2), and a gain of unity or slightly higher. If hysteresis or dead band in the valve position is a problem, the higher gain of the flow controller will help reduce the variations in flow required to overcome the hysteresis.

Secondary Temperature Loop There are two difficulties with using temperature as the secondary measured variable: the sensor lag and the possibility of reset windup. Fortunately, both problems can be handled. The next section deals with the windup problem. The sensor lag can be compensated for by using derivative mode in the secondary controller with the derivative time set equal to the sensor time constant. As mentioned above, the derivative mode must act on the secondary process variable only, not on the deviation from set point, to prevent having two derivative units in series.

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Figure 7-3. Temperature to Flow Cascade Control of Reflux in a Distillation Column

Vapors to condenser SP TC

TT

Column SP

FC FT

Reflux The reactor temperature control scheme of Figure 7-1 is a typical example of a secondary temperature controller. In this application, temperature has the advantage over coolant flow as a secondary variable in that it compensates for changes in both coolant header pressure and inlet temperature, while coolant flow only compensates for variations in coolant header pressure. The temperature controller also closes a loop around the jacket, reducing its effective time constant and thus making the reactor temperature control loop more controllable.

Secondary Pressure Loop Pressure is a good secondary variable to use because it can be measured easily, quickly, and reliably. Figure 7-4 shows a temperature-to-pressure cascade

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system. The pressure in the steam chest in the reboiler directly determines the heat transfer rate because it controls the steam condensing temperature and therefore the difference in temperature across the heat transfer area. Like temperature, pressure presents the difficulty of reset windup, discussed in the next section. Another difficulty with pressure as a secondary variable is that it can move out of the transmitter range and thus get out of control. For example, in the scheme of Figure 7-4, if at a low production rate the reboiler temperature drops below 100°C (212°F), the pressure in the steam chest will drop below atmospheric pressure, getting out of the transmitter range, unless the pressure transmitter is calibrated to read negative pressures (vacuum).

Figure 7-4. Temperature to Pressure Cascade Control of a Distillation Column

Reboiler

Sampled-data Cascade Control When both the primary and the secondary controllers are carried out by a computer or microprocessor, the secondary (inner) loop is usually processed

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at a higher frequency than the outer loop, so that the secondary controller has time to respond to a set point change from the primary controller before the next set point change takes place. Recall that the inner loop should respond faster than the outer loop. If the sampling frequency is low and the same for both the primary and secondary loops, the secondary loop must be processed after the primary loop; otherwise, the change in set point will be delayed by one sample before the secondary loop can take action. One important consideration when digital feedback algorithms are cascaded is bumpless transfer from Manual to Automatic. This is done by initializing the output of the primary controller to the process variable of the secondary controller when the loops are switched to automatic control, making for a smooth transition to Automatic.

Example 7-1. Cascade Control of Jacketed Chemical Reactor This example shows how to tune the cascade control system for the jacketed chemical reactor of Figure 7-1. For comparison, the response of a single reactor temperature controller is compared to the response of the cascade control system. The single reactor temperature controller, TC 1, manipulates the coolant valve directly, while in the cascade scheme the reactor temperature controller, TC 1, sets the set point of a jacket temperature controller, TC 2, which in turn manipulates the coolant valve, as in Figure 7-1. For the purposes of this example, the manual steam valve is always closed. To obtain the process parameters, a step test in the controller output connected to the coolant valve is performed with the controllers in Manual, and both the reactor temperature and the jacket temperature are recorded. The following results are obtained: From the response of the reactor temperature: K = 2.2

τ = 7.5 min

t0 = 1.5 min.

From the response of the jacket temperature: K = 1.9

τ = 1.5 min

t0 = 0 min.

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Although an increase in coolant flow results in a decrease in both the reactor and jacket temperatures, the signs on the process gains are positive because, for safety, the coolant valve fails to open. As a result, an increase in controller output results in a decrease in coolant flow and consequently an increase in the temperatures, hence the positive gains. Use the Ziegler-Nichols QDR tuning formulas of Table 4-1 to tune the single reactor temperature PID controller: Kc = 1.2(7.5/1.5)/2.2 = 2.8 TI = 2.0(1.5) = 3.0 min

TD = 0.5(1.5) = 0.75 min.

The parameters from the response of the jacket temperature are used to tune jacket temperature controller TC 2 in the cascade scheme. Since the dead time is zero, a PI controller is indicated, the IMC rule of Section 4-1 is used for the integral time and the gain can be as high as desired. To keep the overshoot to set point changes reasonable: Kc = 1.5

TI = τ = 1.5 min

TD = 0.

Once jacket temperature controller TC 2 is tuned, it is switched to Automatic and a step test in its set point is applied with the reactor temperature TC 1 set in Manual. The response of the reactor temperature is recorded with the following results: K = 1.1

τ = 4.5 min

t0 = 0.75 min.

Comparison with the results of the response to the step in coolant flow shows that the reactor temperature loop has both a shorter time constant and a shorter dead time when the jacket temperature controller is used. Recall, however, that these parameters depend on the tuning of the jacket temperature controller. For example, if a higher gain were used for TC 2, the time parameters would be shorter still.

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Reactor temperature controller TC 1 is now tuned for QDR response from the above parameters. Kc = 1.2(4.5/0.75)/1.1 = 6.6 TI = 2(0.75) = 1.5 min

TD = 0.5(0.75) = 0.37 min.

These values result in a response that is too oscillatory, so the proportional gain must be reduced to 3.3.

Figure 7-5 compares the responses of the single reactor temperature controller and the cascade control scheme to a 5°C step increase in coolant inlet temperature followed by a step decrease of 20% in the flow of the reactants. Since the response of the outlet coolant temperature to the change in inlet coolant temperature is immediately detected and corrected for by the secondary controller, the reactor temperature in the cascade scheme hardly deviates from its set point. The cascade scheme immediately increases the coolant flow to compensate for the increase in inlet coolant temperature. The figure shows that the cascade control scheme also improves the response of the reactor temperature to a step decrease in reactant feed to the reactor. However, the improvement in performance is not as dramatic because the feed flow has a direct effect on the reactor temperature and cannot be corrected in time by the jacket temperature controller. The improvement in control is due to the faster response of the reactor temperature to controller output in the cascade scheme. Another reason that the performance improvement is not as dramatic for the feed flow disturbance is the inverse response of the temperature to the feed flow. This is because the reactants are colder than the reactor and the decrease in the flow of reactants causes an immediate rise in temperature, but the decrease in flow of reactants also causes a decrease in reactant concentration that eventually results in a decrease in reaction rate and consequently in temperature. The following is an example of a successful industrial application of cascade control. It is an example of composition-to-composition cascade, which is not very common. It also shows a three-level cascade control system, with the flow controller being the lowest level.

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Figure 7-5. Responses of Temperature Control on an Exothermic Reactor with Simple Feedback (Continuous Line) and Cascade (Dashed Line) for a Step Increase in Inlet Coolant Temperature Followed by a Step Decrease in Reactants Flow

Example 7-2. Control of Hydrogen/Nitrogen Ratio in an Ammonia Synthesis Loop Figure 7-6 shows a simplified diagram of the synthetic ammonia process. Air, natural gas (CH4), and steam are mixed in the reforming furnace and after the carbon dioxide (CO2) is removed, a mixture of hydrogen and nitrogen is obtained and is fed to the synthesis loop compressor. The flow in the synthesis loop is about six to seven times the flow of fresh feed, since the

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synthesis reactor converts only about 15% of the hydrogen-nitrogen mixture to ammonia (NH3) in each pass. This high recycle-to–fresh-feed ratio makes for a long time constant for the synthesis loop, compared to a short time constant for the reforming process, a situation ideal for cascade control.

Figure 7-6. Cascade Control of Reactor Inlet Composition and Synthesis Loop Pressure in the Ammonia Process SP

SP AC 1

RC 2

Vent

SP

Air Compressor

FC 2

FT

AT 1

SP AC 2 Synthesis Gas Compressor

Air AT 2

SP

Ammonia Synthesis Reactor

FC 1

FT Reforming Process

Natural Gas

RC 3 FC 3

Steam

FT

SP PC 4

PT SP SP

Ammonia Product

SP

CO2 FC 4

FT Purge

The objective is to control the hydrogen to nitrogen ratio (H/N) of the mixture entering the synthesis reactor at its optimum value (about 2.85 for a slight excess of nitrogen). The primary controller (AC 1) receives the measurement of the composition at the reactor inlet from a very accurate analyzer (AT 1). The output of the primary controller adjusts the set point on the secondary controller (AC 2). The secondary controller receives the measurement of the composition of the fresh feed from a fast and inexpensive analyzer (AT 2), usually a simple thermal conductivity detector, and its output adjusts the ratio of air to natural gas through ratio controller (RC 2). The ratio controller, in turn, adjusts the set point of the process air flow controller (FC 2). To pre-

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vent from throttling the air compressor, flow controller (FC 2) adjusts a valve on a vent on the compressor discharge. This example illustrates the point made earlier about the secondary measurement not having to be accurate but having to be fast and consistent. Inaccuracy in the secondary measurement is corrected by the integral mode of the primary controller. On the other hand, the measurement of the primary controller can be slow, but it must be accurate. Disturbances in the reforming process are handled quickly by the secondary controller, before they have a chance to affect the primary controlled variable. Figure 7-6 also shows a pressure-to-flow cascade loop for the control of the pressure in the synthesis loop. In this cascade the primary controller is the pressure controller (PC) and the secondary controller is the purge flow controller (FC 4). The purge is a small stream removed from the loop to avoid the accumulation of inert gases (e.g., argon and methane) and the excess nitrogen. Although both cascade control loops of Figure 7-6 could be carried out with analog controllers, computer control offers an unexpected virtue to this scheme: patience. For example, in one installation where the pressure control scheme was carried out with analog controllers, the primary controller was operated on Manual because it was swinging the purge flow all over its range. This was because the time constant of this loop is about one hour. A digital controller with a sampling interval of 5 minutes and an integral time of 45 minutes was able to maintain the pressure at its optimum set point on the same installation.

7-4. Reset Windup in Cascade Control Systems Chapter 4 showed that a discrepancy between the operating range of a single feedback controller output and a control valve causes undesirable overshoot of the controlled variable after a period of saturation of the control valve. Such range discrepancies are more common in cascade control systems because the range of the transmitter on the secondary loop is usually wider than the operating range of the secondary process variable, particularly when the secondary process variable is temperature or pressure. To illustrate the problem of cascade reset windup, consider the start-up of the jacketed reactor of Figure 7-1. Both controllers are initially in Manual, with the

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cooling water valve closed and the steam valve manually opened to bring the reactor up to the operating temperature, 104°C (see Figure 7-7). The jacket temperature transmitter, TT 2, has a range of 40 to 115°C, and the steam condenses at 110°C, which is the value of the jacket temperature when the steam valve is closed and the cascade control system is initialized and switched to Automatic. This is done before the reactor temperature reaches its 104°C set point, say when it reaches 100°C. Following the bumpless transfer procedure of the control system, the output of the primary controller is initialized to the measured temperature of the secondary controller, 110°C. At this time the jacket temperature begins to drop because the steam has been turned off and the reactor is at the lower temperature of 100°C, while the reactor temperature is rising because of the heat of the reaction. For the time that the reactor temperature is between 100 and 104°C (its set point), the control situation is as follows: • The secondary controller sees a jacket temperature below its set point (110°C) and calls for the cooling water valve to remain closed. • The primary controller also sees its temperature below set point and calls for an increase in the jacket temperature set point above the current 110°C value. Most computer and DCS controllers detect that the secondary controller output is limited or “clamped” at the closed position and prevent the primary controller from increasing its output since this would only call for the closing of the coolant valve, which is already closed. Does this logic prevent the cascade control system from winding up? Let us see what happens next. Notice that a gap has been created between the set point of the secondary controller, clamped at 110°C, and its measured temperature, the jacket temperature that drops to the reactor temperature as soon as the steam is turned off. As the reactor temperature crosses its set point of 104°C, the primary controller starts decreasing the set point of the secondary controller to bring the temperature down, but the coolant valve will not open until the set point of the secondary controller drops below its measured temperature; that is, until the gap mentioned earlier is overcome. Since the set point of the secondary controller will change at a rate controlled by the integral time of the primary controller, it takes a long time for the coolant valve to start to open and the reactor temperature overshoots its set point badly, the common symptom of reset

windup. The situation continues as the coolant valve is driven from closed to open and back again, as the oscillations of the dashed lines in Figure 7-7 show. As you can see, the saturation or “clamp limit” detection system could not avoid reset windup in this case.

Figure 7-7. Oscillatory Behavior Caused by Reset Windup in the Cascade Control of an Exothermic Reactor Startup (Dashed Lines) and Solution Using Reset Feedback on the Master Controller (Continuous Lines)

Oscillations caused by reset wind-up

Coolant valve closed

Coolant valve opened

Reset Feedback An elegant and effective way to protect against cascade reset windup is the use of the “reset feedback” feature on the primary controller. In the cascade

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scheme the measured variable of the secondary loop, expressed in percent of transmitter range, is fed back to the primary controller and used in the calculation of the controller output by the control algorithm of Table 6-1, known as a “velocity” algorithm, and appears as follows: Mk = FBk + ΔMk where: Mk

= the output of the primary controller and set point of the secondary controller

FBk

= the reset feedback variable, in this case the measured variable of the secondary loop

ΔMk = the incremental output of the primary controller, which is calculated as in Table 6-1. By using this formula to update the set point of the secondary controller at every processing of the primary controller, there is no possibility of windup because the primary controller will call for an increase or decrease of the secondary variable from its current measured value, not from its set point. This eliminates the gap between the secondary process variable and its set point when the controller output is saturated. The result of the use of the reset feedback feature on the cascade temperature control of the jacketed exothermic reactor is shown by the continuous lines in Figure 7-7. Although the initial overshoot of the temperature cannot be eliminated because the coolant valve starts in the closed position, further oscillations are eliminated. The use of the reset feedback approach requires that the secondary loop be sampled more frequently than the primary loop, and that the secondary controller have integral mode. Otherwise, any offset in the secondary controller will cause an offset in the primary controller, even if the primary controller has integral mode.

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7-5. Summary This chapter presented the cascade control scheme; that is, the cascading of a primary controller to a secondary controller to improve control performance. The discussion included the reasons for using cascade control, the selection of modes for the secondary controller, and the procedure for tuning cascade control systems. It also looked at cascade reset windup and ways to protect against it. Cascade control has proliferated in computer control installations because there is essentially no cost for the additional secondary controllers. The only additional cost in a computer control system is the cost of one transmitter and one multiplexer input channel for each secondary loop.

Review Questions 7-1. What are the three major advantages of cascade control? 7-2. What is the main requirement for a cascade control system to result in improved control performance? What is required of the sensor for the secondary loop? 7-3. Are the tuning and selection of modes different for the primary controller in a cascade control system than for the controller in a simple feedback control loop? Explain. 7-4. What is different about the secondary controller in a cascade control system? When should it not have integral mode? If the secondary is to have derivative mode, should it operate on the process variable or on the error (deviation)? 7-5. In what order must the controllers in a cascade control system be tuned? Why? 7-6. What are the two major difficulties with using temperature as the process variable of the secondary controller in a cascade control system? How can they be handled? 7-7. Why is pressure a good variable to use as the secondary variable in cascade control? What are the two major difficulties with using pressure as the secondary variable?

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7-8. What is the relationship between the processing frequencies of the primary and secondary controllers in a computer cascade control system? 7-9. How can reset windup occur in a cascade control system? How can it be avoided?

8 Feedforward and Ratio Control

This chapter presents the design and tuning methods of feedforward and ratio control strategies. Along with cascade control, these strategies can be classified as multiple input, single output (MISO) because they require more than one process measurement but only one final control element (usually a control valve) because there is only one control objective.

Learning Objectives—When you have completed this chapter you should be able to: A. Understand when to apply feedforward and ratio control. B.

Know when to use and how to tune a static feedforward compensator.

C. Tune dynamic feedforward compensators.

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8-1. Why Feedforward Control? Chapter 4 showed that some feedback loops are more controllable than others and that the uncontrollability measure of a feedback loop is the ratio of the dead time to the time constant. When this ratio is high, on the order of unity or greater, feedback control cannot prevent disturbances from causing large deviations of the process variable from its set point. It is then that the strategies of feedforward and ratio control—ratio control being the simplest form of feedforward control—can improve control performance the most. The strategy of feedforward control consists of measuring the major disturbances to the controlled process variable—the control objective—and calculating the change in output variable required to compensate for them. The following are characteristics of feedforward control: • It is in theory possible to have perfect control, that is, zero deviation from set point at all times. (This is not so for feedback control, which must operate on the deviation.) • An accurate model of the process is needed to design the feedforward controller. The model must include the effects of both the disturbances and the output variable on the process variable. • All disturbances must be measured and compensated for. Alternately, feedback trim can be added to compensate for disturbances that have a minor effect on the process variable, or that vary too slowly to merit measurement (e.g., ambient conditions and heat exchanger scaling). Feedforward compensation can be a simple proportionality between two signals, or more complex material and energy balance calculations involving the measured disturbances and the output variable. No matter how simple or complex the steady state compensation, compensation for process dynamics is usually accomplished with a simple linear lead-lag unit, which will be introduced later in this chapter. The motivation for feedforward control is best presented by comparing it to feedback control. Figure 8-1 shows a block diagram of the typical feedback control loop. The characteristics of feedback control that make it so convenient are:

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• The controller is a standard off-the-shelf instrument or software algorithm. • The feedback controller can be tuned on-line, by trial and error, so that a model of the process is not needed to implement it. • The integral mode of the controller computes the value of the controller output OP required to keep the process variable PV at its set point SP. Opposite these very desirable characteristics there are two undesirable ones: • Disturbances cause the process variable to deviate from its set point before the controller can take action. • Overcorrections occur because of delays in the process and sensor that can cause the process variable to oscillate around its set point. These problems are significant in process systems because of the long time delays involved, sometimes of the order of hours. One remedy to these problems is feedforward control.

Figure 8-1. Block Diagram of Simple Feedback Loop

U G2 SP Feedback Controller

+

OP G1

-

Sensor

+

- PV

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Simple Feedforward Control Figure 8-2 shows a block diagram for feedforward control. The technique consists of measuring the disturbance, U, instead of the process variable being controlled, PV. Corrective action begins as soon as the disturbance enters the system and in theory, can prevent any deviation of the process variable from its set point. However, this requires an exact model of the process and its dynamics, plus exact compensation for all possible disturbances. The “set point element” 1/G1 of Figure 8-2 provides for calibrated adjustment of the set point and seldom includes any dynamic compensation. The “feedforward element” G2/G1 of Figure 8-2 simulates the effect of the disturbance on the process variable (block G2) and compensates for the lags and delays on the output variable (block G1). Notice that the signals always travel forward; that is, there is no loop in the diagram, so the feedforward controller cannot introduce or prevent instability in the process response.

Figure 8-2. Block Diagram of Feedforward Controller

U G2/G1 G2 -

SP 1/G1

+

OP G1

+

+

PV

Feedforward-Feedback Control It is seldom practical to measure all the disturbances that affect the process variable. A more reasonable approach is to measure only those disturbances that are expected to cause the greatest deviations in the process variable and handle the so-called “minor disturbances” by adding “feedback trim” to the feedforward controller. Figure 8-3 shows a block diagram for a feedforwardfeedback control system. Note that the feedback controller takes the place of

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the set point element of Figure 8-2, and only the feedforward element is necessary in the combined control scheme. A feedforward element is required for each disturbance measured.

Figure 8-3. Block Diagram of Feedforward Controller with Feedback Trim

U G2/G1 G2 Feedback Controller

SP

+

OP G1

+

+

PV

+ -

Sensor

When the outputs of the feedforward and feedback controllers are summed, as in Figure 8-3, the presence of the feedforward controller does not affect the response of the loop to inputs other than the measured disturbance, thus the feedback controller tuning does not have to be adjusted because of the installation of the feedforward controller. Economics dictates that only those disturbances that are frequent enough and important enough—in regards to their effect on product quality or safety, or for similar considerations—should be measured and compensated for with a feedforward controller. The advantages of the feedforward-feedback scheme are: • The feedback controller takes care of those disturbances that are not important enough to be measured and compensated for.

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• The feedforward controller does not have to compensate exactly for the measured disturbances since any minor errors in the model are trimmed off by the feedback loop, hence the term “feedback trim.” Because of these advantages, feedback trim is a part of almost every feedforward control scheme.

Ratio Control The simplest form of feedforward control is ratio control. It simply consists of the establishment of a ratio between two flows. Figure 8-4 shows an example of ratio control between the steam and process flows of a steam heater. In this example, the process flow is the disturbance or “wild” flow, and the steam is the manipulated flow. The steam flow controller takes care of both variations in the pressure drop across the control valve and its nonlinearity. By maintaining a constant ratio when the process flow is changed by the operator or by another controller, the outlet process temperature is kept constant as long as the steam latent heat and process inlet temperature remain constant; in other words, the ratio controller compensates only for variations in process flow to the heater. The temperature feedback controller in the figure, TC, adjusts the ratio to provide the feedback trim in this example to compensate for variations in process inlet temperature and steam heat of condensation. The diagram of the control of the ammonia process in Figure 7-6 shows two ratio controllers to compensate for variations in the natural gas flow to the process. Some control engineers prefer to calculate the ratio by dividing the manipulated flow by the wild flow and then controlling the ratio with a feedback controller, as in RC in Figure 8-5. This alternative has the disadvantage of creating a very nonlinear feedback control loop; note that the gain of the feedback loop in Figure 8-5 is inversely proportional to the wild flow, which is the major disturbance. The ratio controllers in some computer and distributed control systems display the calculated ratio, but do not use it for control. Instead, the output is calculated by multiplying the input or wild flow by the ratio set point, as in Figure 8-4.

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Figure 8-4. Ratio Control of Process Heater with Feedback Trim SP

RC SP

Steam

FT

SP

FC

TC OP

Fs

PV

FT

TT

F Process fluid

T

Steam trap

Condensate

8-2. Design of Linear Feedforward Controllers Based on the block diagram of Figure 8-2, the feedforward controller and the process constitute two parallel paths between the disturbance U and the process variable PV. A simple linear feedforward model assumes that the process variable response is the sum of its separate responses to the output variable OP and to the disturbance U: PV = G1(OP) + G2(U) In this equation, OP is the output variable, U is the disturbance, and G1 and G2 represent the effects of the manipulated variable and the disturbance, respectively, on the process variable PV.

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Figure 8-5. Feedback Control of Calculated Ratio

Wild flow FT

(B/A)set

A

B/A

Manipulated flow

B

FY %

SP RC

FT

The value of the output OP required to keep PV = SP is given by: G 1 OP = ------- SP – ------2- U G1 G1

(8-1)

This is the design equation for the feedforward controller having set point SP and disturbance U as inputs and output variable OP as output. Equation 8-1 provides the design formulas for both the set point and feedforward elements of Figure 8-2: Set point element: 1 G s = ------G1 Feedforward element: G G F = ------2G1

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When feedback trim is used, as in Figure 8-3, only the feedforward element is needed, since the feedback controller takes the place of the set point element.

Simple Linear Models for Feedforward Control When the process elements G1 and G2 are modeled with simple single-lagplus-dead-time (SLPDT) models, the feedforward controller can be built out of standard algorithms available in most commercial process control programs. The feedforward controller then consists of three elements: GF = (Gain)(Lead-Lag)(Dead Time Compensator)

(8-2)

with: K Gain = – ------2K1 Lead of τ Lead – Lag = --------------------------1Lag of τ 2 Dead Time Compensator = t02 - t01

(8-3)

where: K1

= the gain of the manipulated variable on the process variable (gain of G1)

K2

= the gain of the disturbance on the process variable (gain of G2)

τ1, τ2 = the time constants of G1 and G2, respectively, min t01, t02= the dead times of G1 and G2, respectively, min Although the feedforward controller of Equation 8-2 results from simple single-lag process models, there is no incentive to use more complex dynamic compensation terms. For example, use of process models with more than one lag would call for a compensator with additional parameters than the lead-lag unit, making it harder to tune while offering little improvement in performance over a well-tuned lead-lag unit.

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The dead time compensator of Equation 8-3 can only be realized when the dead time between the disturbance and the process variable is longer than the dead time between the manipulated variable and the process variable. Otherwise, it would call for the feedforward correction to start before the disturbance takes place, which is obviously not possible. The dead time compensator requires the memory of digital devices (computers and microprocessors) for its implementation. The dead time compensator can often be left out because the lead-lag unit can be tuned to provide all of the required dynamic compensation, thus simplifying the tuning task. In general, the dead time compensator should only be used when the lead-lag unit cannot do the job by itself.

8-3. Tuning of Linear Feedforward Controllers Of the three terms of the feedforward controller of Equation 8-2, the gain is always required and the dynamic compensators are optional. When only the gain is used, the feedforward controller is called a “static” compensator.

Gain Adjustment The adjustment of the feedforward gain can be carried out with the feedback controller in Manual or Automatic. If it is done with the feedback controller in Manual, when the gain is not correct, the process variable will deviate from its set point after a sustained disturbance input. The gain can then be adjusted until the process variable is at the set point again. Because of process nonlinearities, the required feedforward gain may change with operating conditions, thus exact compensation may not be possible with a simple linear controller. If the feedforward gain is adjusted with the feedback controller in Automatic, the variable to observe is the output of the feedback controller. If the feedback controller has integral mode, the process variable will always return to its set point after a disturbance, but if the feedforward gain is incorrect, the output of the feedback controller will be changed to compensate for the error in the feedforward controller. The feedforward gain must then be adjusted until the feedback controller output returns to its initial value. As before, process nonlinearities will prevent a single value of the gain from working for all process conditions.

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The one thing to remember when tuning the feedforward gain is that it is necessary to wait until the system reaches steady state before making the next adjustment.

Tuning the Lead-Lag Unit The most commonly used feedforward dynamic compensator is the lead-lag unit, which is available either as an analog off-the-shelf device or as a control block in computer and DCS control systems. To understand how to tune a lead-lag unit it is important to know how it responds to step and ramp signals. Keep in mind that both the lead and the lag time constants are adjustable and that either one can be longer than the other. Figure 8-6 shows the response of the lead-lag unit to a step change in its input for both the lead being longer than the lag and for the lag being longer than the lead, assuming in each case that the gain is unity. The initial change in the output of the lead-lag unit is always equal to the ratio of the lead to the lag, so that there is an initial overcorrection when the lead is longer than the lag and a partial correction when the lag is longer than the lead. In either case, the output approaches the steady state correction asymptotically, at a rate determined by the lag time constant. Figure 8-7 shows the response of the lead-lag unit to a ramp input, both for the lead longer than the lag and the lag longer than the lead, assuming unity gain. The figure shows where the terms “lead” and “lag” come from: the output of the lead-lag unit, after a transient period, either leads the input ramp by the difference between the lead and the lag or lags it by the difference between the lag and the lead. The ramp response is more typical than the step response to the type of inputs provided by the disturbances in a real process. The ramp response is similar to the response to the rising and dropping portions of slow-oscillating disturbances.

Tuning Procedure for a Lead-Lag Unit With the responses to step and ramp inputs in mind, tuning the lead-lag unit becomes a simple procedure, as follows: 1. Decide by how much to lead or lag the feedforward correction to the disturbance; this fixes the difference between the lead and the lag.

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Figure 8-6. Response of Lead-Lag Units to an Input Step Change 3.5

3

Output Lead = 2xLag

2.5

Input

2

Output Lead = 0.5xLag

1.5

1

0.5

0 0

5

10

15

20

25

30

35

40

45

2. Select the ratio of the lead to the lag based on how much to amplify or attenuate sudden changes in the disturbance inputs. For example, suppose it is desired to lead the disturbance by one minute; a lead of 1.1 minutes and a lag of 0.1 minutes give an amplification factor of 1.1/0.1 = 11, while a lead of 3 minutes and a lag of 2 minutes give an amplification factor of only 3/2 = 1.5. If the disturbance is noisy; for example, in the case of a flow, the second choice is preferred since it results in less amplification of the noise. Although it is possible to have a lag with zero lead, it is not possible to have a lead without a lag. The ratio of the lead to the lag should not be greater than 10. When a net lag is required, the lead can usually be set to zero, simplifying the tuning task.

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Figure 8-7. Response of Lead-Lag Units to an Input Ramp 120

100

Net lead

Output Lead > Lag

80

Net lag

60

Input

40

Output Lead < Lag

20

0 0

5

10

15

20

25

30

35

40

45

Computer Lead-Lag Algorithm A common computer formula to implement a lead-lag unit is given by: τ LD Y k = Y k – 1 + ( 1 – a ) ( X k – 1 – Y k – 1 ) + --------- ( X k – X k – 1 ) τ LG

where: Xk

= the input at the kth sample

Yk

= the output at the kth sample

τLD , τLG = the lead and lag constants, respectively, min a

= τLG/(T + τLG) = filter parameter

T

= the sampling interval, min.

(8-4)

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The actual algorithms used in commercial computer control programs use various forms of approximations for the filter parameter “a”, but it is always a function of the sampling interval and the lag time constant. Note that the effect of the lead is just to multiply the change in input at each sample by the ratio of the lead to the lag. In other words, for the computer lead-lag algorithm, the input change at each sample is a step change. Equation 8-4 is for a unity gain. If the gain is different from unity, it can be applied to the signal before or after the lead-lag calculation.

Tuning the Dead Time Compensation Term Besides lead-lag dynamic compensation, the ability of computers and microprocessors to store information in its memory allows compensation for dead time. When should dead time compensation be used? The following guidelines are suggested to simplify the overall tuning procedure: • Dead time compensation is not possible when the dead time is negative because it would require taking action ahead of the disturbance, so when the dead time is negative it should be added to the lead term in the lead-lag unit and only the corrected lead-lag unit should be used. • Instead of using a lead in the lead-lag unit and a dead time compensation term, it is simpler to just subtract the dead time from the lead and use only the corrected lead-lag unit. • When the dead time in the dead time compensation term is longer than the lead in the lead-lag unit, it is simpler to subtract the lead from the dead time and use a lag without lead and a corrected dead time compensation term. In other words, dead time compensation should be used only when a lag without a lead would cause the feedforward correction to take place too soon. Dead time compensation is accomplished by storing the feedforward corrective action at each control update in a memory stack and then retrieving it several sampling intervals later for output to the process. The output of the dead time compensator is equal to its input N samples earlier: Yk = Xk-N

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where N is the number of samples of dead time and unity gain is assumed. Figure 8-8 shows a plot of the responses of dead time compensation to an input signal. Notice that the dead time compensator does not start responding until one dead time after the change in the input; the output then reproduces the input exactly. The dead time compensator is easy to tune, since it only has one dynamic parameter, the number of samples of dead time N. Before applying dead time compensation, it is important to ensure that the dead time does not delay the action in a feedback control loop. Recall that dead time always makes a feedback control loop less controllable. The reason it can be used in feedforward control is that the corrective action always goes forward; that is, no loop is involved.

Figure 8-8. Response of Dead-Time Compensator to an Input Signal 7

6

5

Input

4

Dead Time = 20 min

3

Dead Time = 10 min

2

1

0 0

5

10

15

20

25

30

35

40

45

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8-4. Nonlinear Feedforward Compensation Although linear feedforward compensation can significantly improve control performance, process nonlinearities cause the performance of the linear feedforward controller to deteriorate when process conditions change. Simple nonlinear models, based on knowledge of the process, can be used to design feedforward compensators that perform well over a wide range of operating conditions. The idea is to use the basic principles of physics to replace the steady-state gain of the linear feedforward controller with more precise calculations reflecting the full nonlinear interaction between the process variables. The control calculations are kept simple by designing the controller from steady-state relationships and then using lead-lag and dead time compensators to compensate for process dynamics. The outline of the design procedure is as follows: 1. State the control objective; that is, define which process variable needs to be controlled and what its set point is. It is useful to write the objective in the form: Process variable = set point The set point should be adjustable by the operator and should not be a constant. 2. Enumerate the possible measured disturbances. Which disturbances can be easily measured? How much and how fast is each expected to vary? How much would it cost to measure each of them? It is not really necessary to make a precise cost estimate or get a price bid from a vendor, but just to be aware that, for example, a composition sensor may be more expensive to buy and maintain than a flow or temperature sensor. 3. Select the output variable, that is, the variable to be adjusted by the feedforward controller. When the feedforward controller is cascaded to a secondary controller, the output variable should be defined as the set point of the secondary controller; for example, the flow of the manipulated stream instead of the valve position. 4. From basic principles, usually material and energy balances, write the formulas relating all the variables defined in the first three steps. Keep them

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as simple as possible. Solve for the output variable so that it can be calculated from the measured disturbances and the set point of the process variable. The resulting formula or formulas constitute the design equation(s) to be programmed into the computer for on-line execution. Caution: the formula must use the set point of the process variable and not its measured value. 5. Re-evaluate the list of measured disturbances. The effect of the expected variation of each disturbance on the process variable can be calculated from the basic design formulas; if the effect of a disturbance is small the disturbance need not be measured. On the other hand, there may be a disturbance that was not on the original list which may be found from the formulas to have a significant effect on the process variable. The decision to measure or not to measure must weigh the effect of the disturbance, its expected magnitude, speed and frequency of variation, and the cost of measuring it. Unmeasured disturbances are treated as constants in the design equation, equal to their design or average expected values. Alternatively, if they are difficult to measure but are still expected to vary, they may be adjusted by feedback trim. 6. Introduce the feedback trim, if any, into the design equation. This is done by grouping unknown terms and unmeasured disturbances as much as possible and letting the output of the feedback controller adjust the group of terms that is expected to vary the most. A simple and effective approach is to have the output of the feedback controller adjust the set point of the feedforward controller. 7. Decide whether dynamic compensation is needed and how it is to be introduced into the design. Simple lead-lag or dead time compensators are commonly used. A separate dynamic compensator should be installed on each measured disturbance. It is not good practice to install the dynamic compensator in such a way that it becomes part of the feedback trim loop, especially if it contains dead-time compensation. 8. Draw the instrumentation diagram for the feedforward controller. This is a diagram showing the various computations and relationships between the signals. It is good practice to draw it so that all the input signals enter from the top (or left) and the output signals exit at the bottom (or right). It is at this point that implementation details, largely dependent on the equipment used, must be decided upon. A good design should be able to

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continue to operate safely when some of its input measurements fail, a characteristic of the design known as “graceful degradation.” The feedforward controller can then be programmed on the control computer or configured on the distributed control system. The following example illustrates this design procedure. For other good examples, see the texts by Luyben1 and by Smith and Corripio3.

Example 8-1. Feedforward Temperature Control of a Process Heater An example of a nonlinear model for feedforward control is given by the process heater application described by Shinskey2. Figure 8-9 shows a sketch of the steam heater and feedforward controller. The design procedure is as follows: 1. Control objective: To = Toset

(8-5)

2. Measured disturbances: W, the flow through the exchanger, kg/h Ti, the inlet temperature, °C 3. Manipulated variable: F, steam flow controller set point, kg/h 4. A steady-state energy balance on the exchanger yields the equation for the static feedforward controller: FHv = WCp(To - Ti) + QL where: Cp

= the specific heat of the fluid, kJ/kg-°C

Hv

= the heat of vaporization of the steam, kJ/kg

QL

= the heat loss rate, kJ/h.

(8-6)

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Figure 8-9. Feedforward Control of a Process Heater with Feedback Trim

SP Feedforward Controller

Fset Toset

SP

Steam

FT

SP

FC

TC OP

F Ti

PV

W

TT

FT

TT

Process fluid

To

Steam trap

Condensate

5. At this point it is possible to evaluate the quantitative effect of the possible disturbances on the outlet temperature; such analysis may determine that the heat loss rate is as important as the two measured disturbances but is difficult to measure and is thus a candidate for feedback trim adjustment. Conversely, the inlet temperature may not have enough effect to merit the cost of measuring it, in which case the feedforward controller becomes a simple steam-to-process-flow ratio controller. 6. The need for feedback trim is determined by considering how much the unknown terms in the design formula are expected to vary. The three unknown terms are the physical properties, Cp and Hv, and the heat loss rate, QL. The three can be lumped together by assuming that the heat loss rate is proportional to the heat transfer rate: QL = (1 - η)FHv

(8-7)

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where η is a heater efficiency, or a fraction of the energy input which is transferred to the process fluid. Substitution of Equation 8-7 into Equation 8-6 and solving for the manipulated variable yields the design formula:

F

set

Cp set = ----------( T o – T i )W Hv η

(8-8)

Notice that the outlet temperature in the formula has been replaced by its set point; that is, the control objective, Equation 8-5, has been substituted into the design formula to ensure that it is enforced by the feedforward controller. In modern computer control systems it is possible to retrieve the set point from the feedback controller to use in the feedforward calculation, so that only one set point has to be entered by the operator. This is an important design requirement. All the unknowns of the model have been lumped into a single coefficient, Cp/Hvη, and it would seem natural for the feedback trim controller to adjust this coefficient to correct for variations in the specific heat Cp, the steam latent heat of condensation Hv, and the heater efficiency η. However, these parameters are not expected to vary much, thus it would be undesirable for the feedback trim controller to control by adjusting a term that is not expected to vary. A better control system structure results if the feedback controller output is made to adjust the set point of the feedforward controller or equivalently, the product of the unknown coefficient and the set point, as follows:

F

set

Cp   =  OP – -----------T W H v η i 

where: OP = CpToset/Hvη = output of feedback controller The coefficient Cp/Hvη becomes the tunable gain of the inlet temperature correction. This term can be calculated from measured values of the temperatures and flows, averaged over long enough periods of time. From Equation 8-8:

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163

Cp F = ---------------------------------------W ( To – Ti ) Hv η 7. The feedforward formula is derived from an energy balance on the heater at steady state. Dynamic compensation will probably be required because changes in steam flow, which is the output variable, are delayed by the lags of the control valve and steam chest, while the process flow will have a faster effect on the outlet temperature. On the other hand, the effect of changes in inlet temperature will be delayed by the transportation lag in the heater. To compensate for these dynamic imbalances, lead-lag units can be applied to the two measured disturbances before they are used in the computation. 8. Figure 8-10 shows the instrumentation diagram for the feedforward controller. In some computer control systems, the multiplier may be carried out as a ratio controller, with the ratio being set by the adder, which combines the feedback controller output and the inlet temperature correction.

Example 8-2. Tuning of Lead-Lag Units for the Process Heater Tune the lead-lag units for the steam heater feedforward controller of the preceding example. Figure 8-11 compares the responses of the outlet temperature to a decrease in process flow followed by an increase in process inlet temperature with (a) a well-tuned feedback controller, (b) a static feedforward controller and (c) a feedforward controller with lead-lag compensation. Notice that with static compensation the process outlet temperature rises even though the steam flow is immediately decreased in proportion to the process flow. It is evident from the graph that the steam flow needs to lead the process flow, because the immediate action still allows the outlet temperature to deviate in the same direction as when feedforward control is not used. Curve (c) in Figure 8-11 uses a lead of 2.8 min and a lag of 2.0 min for a net lead of 0.8 min. Since the process flow is expected to be a noisy signal, these values limit the amplification of the noise to a factor of 1.4. With this tuning the

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Figure 8-10. Instrumentation Diagram of Feedforward Controller for Process Heater

TT

Toset

TT

Ti

To

Lead-Lag 1

FT

W

SP

TC

Lead-Lag 2

+ -

Adder

Toset - Ti Multiplier

Fset lead-lag unit reduces the deviation of the outlet temperature to about onethird the deviation of the simple feedback controller and about two-thirds that of the static feedforward controller. In terms of the response of the outlet temperature to a 10°C increase in inlet temperature, the action of the static feedforward controller is hindered by the lag of the inlet temperature sensor, causing the correction to be too slow to prevent the outlet temperature from deviating as much as if feedforward compensation were not used. Once again this can be corrected by inserting a net lead to dynamically correct for the disturbance. Curve (c) of Figure 811 shows the response with a lead of 2.1 min and a lag of 0.4 min for a net lead of 1.7 min, considerably reducing the initial deviation in outlet temperature through both simple feedback control and static compensation.

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165

Figure 8-11. Response of Temperature Control of Process Heater: (a) Feedback control; (b) Static Feedforward Control; (c) Feedforward Control with Dynamic Compensation

(a) (b) (c)

Decrease in process flow

Increase in process inlet temperature

To better demonstrate the performance of feedforward control, the feedforward responses of Figure 8-11 do not include feedback trim. Although in this case the feedforward model is accurate, feedback trim is almost always needed in practice to correct for inaccuracies in the feedforward model. The preceding example has a characteristic typical of many successful feedforward control applications: the formulas used in the compensation are simple steady-state relationships. If dynamic compensation is needed, lead-lag and dead time compensation may be added to the nonlinear steady-state compensator. The moral is: keep your design super simple.

8-5. Summary In summary, ratio and feedforward control complement feedback control, reducing the magnitude of the deviations of the process variable caused by disturbances. The feedforward controller is free of stability concerns but its

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application requires a model of the process. The best approach is a combination of feedforward and feedback control. Ratio control is the simplest form of feedforward control; it establishes a simple proportionality between two flows.

References 1. Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers, 2nd ed. New York: McGraw-Hill, 1990, Sections 8.7 and 11.2. 2. Shinskey, F. G. “Feedforward Control Applied,” ISA Journal, Nov. 1963, p. 61. 3. Smith, C. A. and Corripio, A. B. Principles and Practice of Automatic Process Control, 3rd ed. New York: Wiley, 2006, Chapter 11.

Review Questions 8-1. Why isn’t it possible to have perfect control; that is, the process variable always being equal to the set point, with feedback control alone? Is perfect control possible with feedforward control? 8-2. What are the main requirements of feedforward control? What are the advantages of feedforward control with feedback trim over pure feedforward control? 8-3. What is ratio control? What is the control objective of the air-to-natural gas ratio controller in the control system sketched in Figure 7-6 for the ammonia process? Which are the measured disturbance and the manipulated variable for that ratio controller? 8-4. What is a lead-lag compensator? How is it used in a feedforward control scheme? Describe the step and ramp responses of a lead-lag unit. 8-5. It is desired to lead a disturbance in a feedforward controller by 1.5 minutes. If the amplification factor for the noise in the disturbance measurement must not exceed 2, what must the lead and lag be? 8-6. What is dead time compensation in a feedforward controller? When can it be used? When should it be used?

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167

8-7. Design a feedforward controller to compensate for changes in process flow, inlet temperature, and supplementary fuel flow in the outlet temperature control of the furnace shown in Figure 8-12. Specifically discuss each of the eight steps of the procedure outlined in the text.

Figure 8-12. Process Furnace for Study Question 8-7

Flue gas

W Ti FT TT Process Stream Fset SP F FC FT Main Fuel

To TT

Fs FT Air

Auxiliary Fuel

9 Multivariable Control Systems

Previous chapters have looked at the tuning of feedback controllers from a single loop point of view; that is, a single control objective and a single controller output were considered at a time. This chapter concerns the effect of interaction between multiple control objectives and the tuning of multivariable control systems.

Learning Objectives—When you have completed this chapter, you should be able to: A. Understand how interaction with other loops affects the performance of a feedback control loop. B.

Estimate the extent of interaction between loops.

C. Pair process variables and controller outputs so that the effects of interaction are minimized. D. Adjust the tuning of feedback controllers to account for interaction. E.

Design decouplers for multivariable control systems.

F.

Recognize advanced multivariable control systems.

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9-1. What is Loop Interaction? When two or more feedback loops are installed on a process or unit operation (e.g., distillation column, evaporator, etc.), the possibility arises of interaction between the loops. This means that each process variable is affected by more than one controller output as shown in Figure 9-1, where in controlling the total flow and concentration out of a catalyst blender, both process variables are affected by each of the two controller outputs, which are the flows of the concentrated and dilute inlet streams. The problem that arises is known as loop interaction. Since multiple control objectives are involved, the problem can also be viewed as the design of a multivariable control system.

Figure 9-1. Multivariable Control of a Catalyst Blender

SP AC OP1

SP FC

x F1x1

F OP2

AT FT

F2x2

Effect of Loop Interaction Consider the representation of the 2x2 multivariable control system of Figure 9-2. The terms G11 and G21 represent the effect of controller output OP1 on the two process variables PV1 and PV2, while G12 and G22 are the corresponding effects of controller output OP2. The two controllers, GC1 and GC2, act on their respective deviations from set point, SP1 – PV1 and

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171

SP2 – PV2, to produce the two controller outputs. Signals SP1 and SP2 represent the set points of the loops. In the diagram of Figure 9-2 each of the four process blocks, G11, G12, G21 and G22, includes the gains and dynamics of the final control elements (valves), the process and the sensor/transmitters. For simplicity the disturbances are not shown.

Figure 9-2. Block Diagram of a 2x2 Control System

-

SP1 +

Controller 1

OP1

+

PV1

G11 + G12

G21 SP2 +

-

Controller 2

OP2

+ G22

+

PV2

To look at the effect of interaction, assume that the gains of all four process blocks are positive; that is, an increase in each controller output results in an increase in each of the process variables. Let us follow the sequence of events plotted in Figure 9-3: 1. Suppose that at a point in time a step change in controller output OP1 takes place with both loops in Manual (opened). Figure 9-3 shows the responses of both process variables, PV1 and PV2, where the time of the step change is marked as point “a”. 2. Now suppose that at time “b” control loop 2 is closed (switched to Automatic) and that it has integral or reset mode. Controller output OP2 will decrease until process variable PV2 comes back down to its original value, assumed to be its set point.

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3. The decrease in OP2 also causes, through interaction block G12, a decrease in process variable PV1, so that the net change in PV1 is smaller than the initial change. Note that this initial change is the only change that would take place if there were no interaction, or if controller 2 were kept in Manual. The difference between the initial change and the net change in PV1 is the effect of interaction. It depends on the effect that OP1 has on PV2 (G21), the effect that OP2 has on PV2 (G22)—which determines the necessary corrective action on OP2—and the effect that OP2 has on PV1 (G12). Note also that the steady-state effect of interaction depends only on the process gains, not on the controller tuning, provided that controller 2 has integral mode.

Figure 9-3. Effect of Interaction for a 2x2 Control System with all Four Gains Positive

b

OP1

b SP1

PV1 a

a

OP2

a

b

b

PV2 SP2 a

Time

The authors invite you to verify that a step in OP2, followed by closing control loop 1, has the same effect on PV2—at least qualitatively—as the effect just observed on PV1. It will be shown shortly that the relative effect of interaction is quantitatively the same for control loop 2 as it is for control loop 1.

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In the case just analyzed, all four process gains were assumed positive (direct action). The effect of interaction was in the opposite direction as the direct (initial) effect of the step change, resulting in a net change that was smaller than the initial change. This type of situation, in which the two loops “fight each other,” is known as “negative” interaction. Note that it is possible for the effect of interaction to be greater than the initial effect, in which case the net change will be in the opposite direction as the initial change. Here we could say that “the wrong loop wins the fight,” a situation that results from incorrect pairing of the loops, as shall be shown shortly. You can easily verify that if any two of the process gains were positive, and the other two were negative, the interaction would also be negative. If one of the four process gains has a sign opposite to that of the other three, the effect of interaction would be in the same direction as the direct action and the net change would be larger than the initial change, as you can also verify. This is the case of “positive” interaction, in which the two loops “help each other.” Positive interaction is usually easier to handle than negative interaction, because the possibility of inverse response (i.e., the process variable moving in the wrong direction right after a change) or of open-loop overshoot exists only in the case of negative interaction. It is evident that both positive and negative interaction can be detrimental to the performance of the control system. This is because the response of each loop is affected when the other loop is switched into and out of Automatic, or when its output saturates. In summary, the following are characteristics of loop interaction: • For interaction to affect the performance of the control system, it must work both ways; that is, each controller output must affect both process variables through the process. Notice that if either G12 or G21 is absent from the diagram of Figure 9-2, there is no interaction effect. • Because of interaction, a set point change to either loop produces at least a transitory change in both process variables. • The interaction effect on one loop can be eliminated by interrupting the other loop. That is, if one of the two controllers is switched to Manual the remaining loop is no longer affected by interaction.

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The following sections look at two ways to approach the problem of loop interaction: • Pair the process variables and controller outputs so as to minimize the effect of interaction between the loops. • Combine the controller output signals through decouplers to eliminate the interaction between the loops. More advanced multivariable control design techniques will be discussed in a later section.

9-2. Pairing of Controlled and Manipulated Variables The first step in the design of a control system for a process is usually the selection of the control loops; that is, the selection of those variables that must be controlled and of those variables that are to be manipulated to control them. This pairing task has traditionally been performed by the process engineer and based mostly on intuition and knowledge of the process. Fortunately, for a good number of loops, intuition is all that is necessary. However, when the interactions involved in a system are not clearly understood and the “intuitive” approach produces the wrong pairing, control performance is poor. The expedient solution is then to switch the troublesome controllers to Manual which, as pointed out in the preceding section, eliminates the effect of interaction. The many controllers operating in Manual in control rooms throughout industry are a testimony of failures to correctly pair the variables in the system. Each one is a failure of an attempt to apply automatic control. A method to quantitatively determine the correct pairing of process variables and controller outputs in a multivariable system was developed by Bristol1. It is popularly known as the Relative Gain Matrix or Interaction Measure, and it requires only steady-state information that is easy to obtain off-line. The fact that dynamic information is not included is, on the other hand, the one objection that has kept the method from being accepted more widely than it has been.

Open-loop Gains Consider the 2x2 system of Figure 9-2. If a change is applied to controller output OP1, while keeping the other controller output constant, and the

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175

changes in process variables PV1 and PV2 are measured, the open-loop gains can be calculated: Change in PV K 11 = ---------------------------------------1Change in OP 1 Change in PV K 21 = ---------------------------------------2Change in OP 1

(9-1)

Similarly, when a change is applied to OP2, keeping OP1 constant, the other two open-loop gains can be calculated: Change in PV K 12 = ---------------------------------------1Change in OP 2 Change in PV K 22 = ---------------------------------------2Change in OP 2

(9-2)

The open-loop gains can be determined from the steady-state equations or the computer simulation programs used to design the plant. There is a natural tendency to try to use the open-loop gains in the pairing of the variables. However, it is immediately apparent that PV1 and PV2, and OP1 and OP2 do not necessarily have the same dimensions. Thus, attempting to compare open-loop gains would be like trying to decide between buying a new sofa or a new house. To overcome this problem, Bristol1 proposes to compute relative gains that are independent of dimensions.

Closed-loop Gains Because of interaction, the effect of OP1 on PV1 is different when the other loop is closed than when it is opened, as discussed in the previous section. This requires the definition of the closed-loop gains K11', K21', K12' and K22'. These are defined exactly in Equations 9-1 and 9-2, but with the changes in PV1 determined with PV2 kept constant, and the changes in PV2 determined with PV1 kept constant. For example, to determine K11', a change is made in OP1 and the change in PV1 is measured while a feedback controller with integral mode controls PV2 by adjusting OP2.

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However, closed-loop tests are not needed because the closed-loop gains can be computed from the open-loop gains previously defined. For example, when both OP1 and OP2 change, the total change in PV1 can be estimated by the sum of the two changes: Change in PV1 = K11(change in OP1) + K12(change in OP2) and similarly for the total change in PV2. Now, if PV2 is kept constant, its change is zero: Change in PV2 = K21(change in OP1) + K22(change in OP2) = 0. Solving for the change in OP2 required for PV2 to remain constant: K 21 Change in OP 2 = – -------- ( Change in OP 1 ) K 22 Substitute to obtain the total change in PV1: K 12 K 21  Change in PV 1 =  K 11 – ------------------ ( Change in OP 1 ) K 22   The bracketed expression is then the closed-loop gain K11'. The closed-loop gains for each of the other three pairings can be similarly derived.

Relative Gains (Interaction Measure) Bristol’s1 relative gains or measures of interaction are obtained by dividing each open-loop gain by the corresponding closed-loop gain: K µ ij = -------ijK ij' where µij is the relative gain for the pairing of process variable PVi with controller output OPj.

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The formulas of Equations 9-3 can be used to compute the relative gains for any 2x2 system: K 11 K 22 µ 11 = µ 22 = ------------------------------------------K 11 K 22 – K 12 K 21 K 12 K 21 µ 12 = µ 21 = ------------------------------------------K 12 K 21 – K 11 K 22

(9-3)

It makes sense that the interaction measure for the PV1-OP1 pair is the same as for the PV2-OP2 pair since they represent one option in the 2x2 system, the other option being PV1-OP2 and PV2-OP1. The relative gains are dimensionless and can therefore be compared to one another. To minimize the effect of interaction, the process variables and controller outputs are paired so that the relative gain for the pair is closest to unity. This results in the least change in gain when the other loop of the pair is closed. Note that for the case of no interaction, the open-loop gain is equal to the closed-loop gain, and the relative gains are 1.0 for one pairing and 0.0 for the other.

Example 9-1. Calculation of Relative Gains of a Blender In the blender of Figure 9-1, a change of 5 kg/h in F1, the dilute inlet stream, results in a steady-state increase of 5 kg/h in F, the outlet flow, and a decrease of 0.5% in x, the outlet mass% of solute. A change of 2 kg/h in F2, the concentrated inlet stream, results in a steady-state increase of 2 kg/h in F and an increase of 0.8% in x. Determine the relative gains and pair the flow controller, FC, and mass fraction controller, AC, so as to minimize interaction. From the change in F1, the open-loop gains are: KF1 = (5 kg/h)/(5 kg/h) = 1.0 Kx1 = (-0.5%)/(5 kg/h) = -0.1%/(kg/h)

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From the change in F2: KF2 = (2 kg/h)/(2 kg/h) = 1.0 Kx2 = (0.8%)/(2 kg/h) = 0.4%/(kg/h) From Equation 9-3, the relative gains are: µF1 = µx2 = (1.0)(0.4)/[(1.0)(0.4) - (-0.1)(1.0)] = 0.8 µF2 = µx1 = (-0.1)(1.0)/[(-0.1)(1.0) - (1.0)(0.4)] = 0.2 This means that for the pair F1 with F and F2 with x, the steady-state gain of each loop increases to 1/0.8 = 1.25 (a 25% change) when the other loop is closed, while for the pair F1 with x and F2 with F, the gain of each loop increases by a factor of 1/0.2 = 5 (a 400% change) when the other loop is closed! Obviously the first pairing is significantly less sensitive to interaction than the second.

Extension to Systems with More than Two Control Objectives Equation 9-3 can be used to compute the relative gains for any 2x2 control system. For systems with more than two process and controller outputs, the open loop gain of each loop is determined with all the other loops opened, and the closed loop gain implies that all the other loops are closed. The relative gain for each process variable, controller output pair is still defined as the ratio of the open-loop gain to the closed-loop gain for that pair. The calculation of the relative gains involves the inversion of the matrix of open-loop gains, which is K in Equation 9-4. It is therefore helpful to use a computer and canned programs to perform the following matrix operations: 1. Compute the inverse of the matrix of open-loop gains. 2. Transpose the inverse matrix. 3. Multiply each term of the open-loop gain matrix by the corresponding term of the transposed inverse matrix to obtain the corresponding term of the relative gain matrix.

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Properties of the Relative Gains 1. The relative gains are not only non-dimensional, but are also normalized in the sense that the sum of the gains of any row or column of the matrix is unity. You can verify this fact for the 2x2 by adding the relative gain formulas for each pairing, that is, µ11 + µ12 = 1. This property also applies to systems with more than two process and controller outputs. 2. For the 2x2 system, when the two loops help each other (positive interaction), the relative gains are between 0 and 1; conversely, when the two loops fight each other (negative interaction), one set of relative gains is greater than unity and the other set is negative. Notice that a negative relative gain means that the net action of the loop reverses when the other loop is opened or closed, a very undesirable situation. 3. For a system with more than two control objectives, the concept of positive and negative interaction must be applied on a pair by pair basis. If the relative gain for a process variable and controller output pair is positive and less than unity, the interaction is positive; that is, that pair is “helped” by the interaction of all the other loops. On the other hand, if the relative gain for a pair is greater than unity or negative, the interaction is negative. In other words, the combined action of all other loops causes a change in the process variable that is in the opposite direction as the direct change caused by the controller output in the pair.

Example 9-2. Control of Composition and Flow in a Catalyst Blender Consider the blender of Figure 9-1, where the objectives are to control the composition x and flow F of the product stream by adjusting the positions of the control valves on the two feed streams. Which of the two controllers should be paired to which valve to minimize the effect of interaction? The relative gains can be used to determine this. (Note: Although ratio control should be used here, this still leaves the question of which flow should be ratioed to which, and the answer to our original question will also answer this one. In fact, the ratio controller is really a form of decoupling here.)

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Unlike the specific numerical solution that was developed in Example 9-1, a general solution for the blender will be developed here. To do this, the conservation of mass and the conservation of solute are used to develop formulas for the open-loop gains. Conservation of mass: F = F1 + F2 F1 x1 + F2 x2 Conservation of solute: x = ----------------------------F1 + F2 Using differential calculus, the steady-state gains are: KF1 = Kv1 F2 ( x1 – x2 ) K x1 = ---------------------------K 2 v1 ( F1 + F2 )

KF2 = Kv2 F1 ( x2 – x1 ) K x2 = ---------------------------K 2 v2 ( F1 + F2 )

where Kv1 and Kv2 are the valve gains in (kg/h)/fraction valve position. Next substitute the open-loop gains into the formulas for the relative gains, Equations 9-3. A little algebraic manipulation produces the following general expressions for the relative gains: F1 µ F1 = µ x2 = ----------------F1 + F2

F2 µ F2 = µ x1 = ----------------F1 + F2

In words, the pairing that minimizes interaction has the flow controller adjusting the larger of the two flows and the composition controller adjusting the smaller of the two flows. If a ratio controller were to be used, the smaller flow would be ratioed to the larger flow, with the flow controller adjusting the larger flow and the composition controller adjusting the ratio. It could easily be shown that the ratio controller decouples the two loops so that a change in the product stream flow does not affect the composition. Note that the valve gains Kv1 and Kv2 do not affect the relative gains. This is why they were not considered in Example 9-1.

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For most processes, the relative gains tell all that needs to be known about interaction. They are determined from the open-loop steady-state gains, which are easy to determine by either on-line or off-line methods. However, in systems with negative interaction, the pairing recommended by relative gain analysis may not result in the best control performance because it does not consider the dynamic response. This is illustrated in the following example.

Example 9-3. Two-point Composition Control of a Distillation Column Figure 9-4 shows a sketch of a distillation column with five controller outputs and five process variables. The column separates a 50% mixture of benzene and toluene into a distillate product with 95% benzene and a bottoms product with 5% benzene. It is desired to maintain the compositions of the distillate and bottoms products at their set points. In a distillation column, temperature can provide an indirect measurement of composition, so the two temperature controllers (TC 1 and TC 2) control the composition of the two products by inference. Three secondary objectives are to maintain the vapor balance by controlling the column pressure (PC), and the liquid balances by controlling the levels in the accumulator drum (LC 1) and column bottom (LC 2). The five controller outputs adjust the flow rates of the two products, the reflux flow, the steam flow to the reboiler, and the cooling rate of the condenser. The two-level variables, LC 1 and LC 2, do not affect the operation of the column directly thus they cannot be made a part of the interaction analysis. However, the decision about which streams control the levels has an effect on the interaction between the other control loops. Two schemes are considered below. To reduce the problem to a 2x2, assume that the column pressure controller (PC) manipulates the condenser cooling rate.

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Figure 9-4. Multivariable Control of a Distillation Column, a 5x5 Control System

Condenser

SP

PC

SP

PT LC 1

LT TT TC 1 SP

Feed

TT SP LC 2

Reflux

Distillate

SP TC 2

LT Steam Reboiler

Bottoms

Scheme 1. Level Control by Product Stream Manipulation In this scheme, commonly known as “Energy Balance Control,” the distillate rate is adjusted to control the level in the condenser accumulator (LC 1), and the bottoms rate is adjusted to control the bottom level (LC 2), as in Figure 9-5. This leaves two unpaired control loops: the two temperature controllers to adjust the steam and reflux rates.

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Figure 9-5. Energy Balance Control of Distillation Column

Condenser

SP

PC

SP

PT LC 1

LT TT TC 1 SP

Feed

TT SP LC 2

Reflux

l Distillate

SP TC 2

LT Steam Reboiler

Bottoms

Sensitivity tests performed on a simulation of the column yield the following open-loop gains:

TC 1 TC 2

Reflux -2.85 -0.438

Steam 1.16 2.53

The relative gains are: ( – 2.85 ) ( 2.53 ) µ 11 = µ 22 = ----------------------------------------------------------------------------- = 1.08 ( – 2.85 ) ( 2.53 ) – ( – 0.438 ) ( 1.16 )

TC 1 TC 2

Reflux 1.08 -0.08

Steam -0.08 1.08

The recommended pairing is the same as the intuitive one: control the overhead composition with the reflux flow and the bottoms composition with the

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bottoms flow. The interaction is negative, with the gain of each loop decreasing by 8% when the other loop is closed.

Scheme 2. Bottom Level by Steam Manipulation In this scheme, known as “Direct Material Balance Control,” the bottom level controller manipulates the steam rate, and the bottom temperature controller manipulates the bottoms product flow, as in Figure 9-6. The top of the column remains the same as before. The loops to be paired involve the two temperature controllers TC 1 and TC 2 with the reflux flow and the bottoms product flow, respectively. Figure 9-6. Direct Material Balance Control of Distillation Column: Bottoms Temperature is Controlled by Adjusting the Bottoms Product Flow

Condenser

SP

PC

SP

PT LC 1

LT TT TC 1 SP

Feed

Reflux

l Distillate

SP TT SP LC 2

TC 2

LT Steam Reboiler

Bottoms

The sensitivity study on the simulated column gives the following open-loop gains:

TC 1 TC 2

Reflux -0.35 0.07

Bottoms -1.05 -1.93

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The relative gains are: ( – 0.35 ) ( – 1.93 ) µ 11 = µ 22 = ----------------------------------------------------------------------------- = 0.90 ( – 0.35 ) ( – 1.93 ) – ( 0.07 ) ( – 1.05 )

TC 1 TC 2

Reflux 0.90 0.10

Bottoms 0.1 0.90

The pairing for this scheme is also the obvious one: top temperature with reflux and bottom temperature with bottoms product flow, and the relative gains show about 10% positive interaction; that is, the two loops help each other, indicated by the relative gains being positive and less than unity. It would appear, then, from steady-state relative gain analysis that the Direct Material Balance Control results in positive interaction, which is preferred to the negative interaction resulting from Energy Balance Control. Unfortunately, the Energy Balance Control scheme was found by simulation to perform better in this particular example than the Direct Material Balance Control scheme. The reason is dynamic interaction, which goes undetected by the relative gain matrix. For the first scheme the open-loop responses are monotonic; that is, the temperature stays between its initial value and its final value during the entire response. On the other hand, for the second scheme the openloop responses exhibit inverse response; that is, the temperature moves in one direction at the beginning of the response and then moves back to a final value on the opposite side of its initial value. This causes the feedback controller to initially take action in the wrong direction, degrading the performance of the control system. Although in this particular example relative gain analysis fails to properly predict which of the two control schemes performs better, it is still useful in verifying that the intuitive pairing is the correct one for each scheme. It would also have evaluated the interaction for each scheme correctly had all the responses been monotonic. This example also shows that the arrangement of the level controllers affects the interaction between the other loops in the column.

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9-3. Design and Tuning of Decouplers Although relative gain analysis usually results in the pairing of variables that minimizes the effect of loop interaction, it does not eliminate it. When the relative gains approach 0.5, the effect of interaction is the same regardless of the pairing. In the case of negative interaction, when one set of relative gains is negative and the other is much greater than unity, the proper pairing still produces significant interaction. The only solution to this problem is to compensate for interaction by designing a decoupler. A decoupler is a signal processor that combines the controller outputs to produce the signals to the control valves, other final control devices, or secondary controller set points. Its operation can best be understood by considering the of a decoupled 2x2 system shown in Figure 9-7.

Figure 9-7. Block Diagram of Decoupled 2x2 Control System

-

SP1 +

+

Controller OP1 1

+

G12

G21

D2

+

-

Controller 2

+

+ D1

SP2

PV1

G11

+

OP2 +

+ G22

+

PV2

Each of the two decoupler terms, D1 and D2, can be considered to be feedforward controllers for which the “disturbances” are the controller output signals OP1 and OP2. The design of the decouplers is therefore identical to the design of a feedforward controller presented in Chapter 8.

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Decoupler Design Formulas The objective of decoupler term D2 is to compensate for the effect of OP2 on PV1, or rather, to prevent changes in the output of the second controller from affecting the process variable of the first loop. Decoupler design is to make the total change in PV1, which is the sum of the changes caused by the two paths from OP2 to PV1, equal to zero: Change in PV1 = D1G11(Change in OP2) + G12(Change in OP2) = 0 Solving for the decoupler term D1: G 12 D 1 = – -------G 11

(9-4)

Similarly, decoupler term D2 is designed to compensate for the effect of OP1 on PV2, and from the block diagram of Figure 9-7: G 21 D 2 = – -------G 22

(9-5)

Decoupling, like feedforward, can be designed to varied degrees of complexity. The simplest is given by linear static compensation (i.e., forfeiting the dynamic compensation), which can be accomplished in practice by a simple summer (adder) with adjustable gains. The next degree of complexity is to add dynamic compensation in the form of lead-lag units (see Chapter 8). Ultimately, nonlinear models of the process could be used to design nonlinear decouplers, following the procedure outlined in Chapter 8. Equations 9-4 and 9-5 assume linear models.

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Decoupling and Control Loop Performance Unlike the feedforward controller, the decoupler forms a part of the feedback loop and can thus introduce instability into the system. Consider the total effects that OP1 has on PV1 and that OP2 has on PV2: Change in PV1 = [G11 + D2G12](Change in OP1)

(9-6)

Change in PV2 = [G22 + D1G21](Change in OP2)

(9-7)

It is possible for dynamic compensation to call for unstable terms in D1 and D2. These terms must obviously be left out of the decouplers to maintain stability. Another aspect of decoupling is that, as Equations 9-6 and 9-7 show, two parallel paths exist between each controller output and its process variable. For processes with negative interaction these two parallel paths have opposite signs, creating either an inverse response or an overshoot in the open-loop step response of each decoupled loop. It is important to realize, however, that the parallel paths are not created by the decouplers since they were already present in the un-decoupled system (in the interaction and direct effects). It is evident from the design of the decoupler that the steady-state effect of the decoupler on any one loop is the same as what the integral mode of the other loops would have if the decoupler were not used. What, then, does the decoupler achieve? Basically, through decoupling the effect of interaction is made independent of whether the other loops are opened or closed. However, problems may still arise in one loop if the controller output of another loop is driven to the limits of its range, because the decoupling action is then blocked by the saturation of the final control element. It is therefore important to select the correct pairing of controller outputs and process variables even when decoupling is used, so that saturation of one of the controller outputs in the multivariable system will not drastically affect the performance of the other loops.

Half Decoupling As discussed earlier, the interaction effect depends on both controller outputs affecting both process variables. Thus, interaction can be eliminated by decou-

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pling one loop and letting the other loop be affected; this can be achieved by implementing either D1 or D2 but not both. This is referred to as “half decoupling.” In deciding which decoupler to select, the first consideration may be which of the process variables is more important to keep at its set point. A secondary consideration may be the ease with which the dynamic terms of the decouplers can be implemented.

Example 9-4. Design a Decoupler for the Catalyst Blender The two objectives of the control system for the catalyst blender of Figure 9-1 are the control of the product composition and the control of product flow. Since the blender is full of liquid, the response of the total flow to changes in each of the input flows is instantaneous, thus the decoupler for the total flow should not require dynamic compensation. The response of the product composition should be that of a simple lag with a time constant equal to the residence time of the tank—the tank volume divided by the total flow. Since this time constant is the same for the composition response to either input flow, no dynamic compensation should be required for the composition decoupler either. The application of the linear decoupler design formulas, Equations 9-4 and 9-5, results in the following formulas for the signals to the control valves, assuming that F1 is the largest of the two flows and, for minimum interaction, it is used to control the total flow. This is the pairing determined by relative gain analysis in Example 9-2: K v2 - ( OP 2 – OP 20 ) M 1 = OP 1 – --------K v1

F 2 K v1 - ( OP 1 – OP 10 ) M 2 = OP 2 – --------------F K 1

v2

where OP10 and OP20 are the controller outputs at initialization. The coefficients correct for the sizes of the two valves and, in the second formula, for the ratio between the two inlet flows that is required to maintain the composition constant. This ratio is a function of the two inlet

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stream compositions and the product composition set point. If any of these compositions were to vary, the gain of the decoupler would have to be readjusted. There is, however, another way to design the decoupler which does not require readjustment of the parameters when process conditions change. It consists of using simple process models to set up the structure of the control system, as discussed next.

Decoupler Design from Process Models The conservation of total mass balance and of mass of solute from Example 9-2 provides the models needed to design the decouplers. Conservation of total mass indicates that the output of the product flow controller should manipulate the sum of the two inlet flows, so the output of the flow controller is assumed to be the total inlet flow, and the smaller flow is subtracted from it to determine the larger flow: F1set = OP1 - F2

(9-8)

This formula requires the measurement of the smaller flow and flow control of the larger flow. The conservation of solute mass shows that the product composition depends on the ratio of the flows rather than on any one of the inlet flows. It is then assumed that the output of the composition controller is the ratio of the smaller flow to the larger flow, and the smaller flow is calculated as follows: F2set = OP2F1

(9-9)

This formula requires that the smaller flow also be controlled. Figure 9-8 shows the diagram of the resulting control system. In this scheme the ratio controller keeps the product composition from changing when the total flow is changed, and the adder keeps the total flow from changing when the composition controller takes action. The multivariable control system is therefore fully decoupled. The last two design formulas, Equations (9-8) and (9-9), do not show the scale factors that may be necessary to convert the flow signals into percent of the

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Figure 9-8. Decoupler Control of Catalyst Blender from Basic Model

-

Adder

OP2 = (F2/F1)set

F1set

FC

SP

SP RC

FT

FC AC

SP F1

SP

OP1 = (F1 + F2)set

+

x

F2set

F1x1

SP FC

F AT FT

F2 FT F2x2

scales of the flow controllers. The scale factors depend on the spans of the two flow transmitters rather than on the sizes of the control valves. The flow controllers allow the signals to be linear with flow, and they also take care of changes in pressure drop across the control valves.

Example 9-5. Decoupler Control of Catalyst Blender The catalyst blender control system of Figure 9-1 consists of an analyzer controller AC adjusting the dilute stream F2 and a continuous flow controller FC adjusting the concentrated stream F1. At design conditions each inlet flow is 20 kg/hr, with the concentrated stream containing 50% catalyst and the dilute stream being pure solvent. The resulting product stream has a composition of 25% catalyst and the tank contains 5 kg of solution for a time constant of 7.5 min.

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The analyzer controller is a PI controller tuned as follows: Kc = 3

TI = 6 min

A PI controller for the flow controller is tuned thus: Kc = 0.9

TI = 0.1 min

Figure 9-9 shows the responses of the product composition and flow, as well as the inlet flows, for a step increase of 10 kg/hr in product flow set point followed by a 2% increase in product composition set point. The continuous curves plot the responses for simple feedback controllers and the dashed lines are for the decoupled system. Notice that for the change in product flow the decoupler immediately changes both flows, keeping the ratio between the two flows constant and the product composition constant, while the un-decoupled system must correct the dilute flow to bring the composition back to set point. Similarly, for the change in product composition, the decoupler keeps the total flow constant while the un-decoupled system requires a small delay in correcting the flows, causing the total flow to dip temporarily. The reason that the improvement in control afforded by the decoupler is not more dramatic is that the blender is a highly controllable system that allows the composition controller to be very tightly tuned. Nevertheless, this example shows that the decoupler can fulfill its objective of maintaining each process variable constant when the other one is changed.

In summary, decoupling is a viable strategy for multivariable control systems. Its design is similar to feedforward control, although simpler in that it does not require additional measurements of process variables. Unlike feedforward, the decoupler forms part of the loop response and affects its stability. Applications of decoupling are usually restricted to 2x2 systems. More sophisticated control strategies are used with systems involving more than two control objectives.

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Figure 9-9. Product Flow and Composition Control of Blender with Decoupler (Dashed Lines) and with Simple Feedback Controllers (Solid Lines)

9-4. Tuning of Multivariable Control Systems It is obvious from the preceding analysis of interacting loops that the interaction is going to affect the response of each loop; that is, the tuning parameters and manual/automatic state of each loop affects how the other loops respond. This section shows how to account for the effect of interaction when tuning each loop in a multivariable control system. The first thing to do when tuning interacting loops is to prioritize the control objectives; in other words, to rank the process variables in the order in which is important to maintain them at their set points. The second thing to do is to check the relative gain for the most important variable and decide whether it is necessary to detune the other loops. The principle behind this approach is that a loosely tuned feedback control loop with low gain and slow integral behaves as if it were opened; that is, it will make slow enough changes in its

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controller output to allow the controller of the important variable to correct for the effect of interaction. The decision on how loosely to tune the less important loops is based on how different from unity the relative gain is for the most important loop. It is understood that the controller output for the most important variable has been selected to make the relative gain for that loop as close to unity as possible. When there are more than two interacting loops, the tightness of tuning for each loop will decrease with its rank of diminishing importance. An alternate approach to detuning less important loops is to install decouplers that compensate for the effect of the action of the less important loops on the most important loop. Decouplers that compensate for the effect of the action of the most important loop on the other loops should not be installed, especially if the relative gain for that loop is greater than unity. This is because, as discussed in the preceding section with the decoupled block diagram of Figure 9-7, the action of the decoupler affects the loop whose action is compensated for. If the relative gain for a loop is greater than unity or is negative (negative interaction), the decoupler action will be in the opposite direction to the direct action of the controller output, causing inverse response or overshoot, which makes the loop less controllable. Note that for loops with negative interaction, detuning the other loops slows down the parallel effect in the opposite direction if the decoupler is not used. Thus, for example, if the top loop in Figure 9-7 were the more important of the two, decoupler D1 should be used, but not decoupler D2. If at least two of the control objectives in a multivariable control system are of equal importance, it is necessary to tune them as tightly as possible. In such a case they should be tuned in the order of decreasing speed of the loop response. If one of the important control loops can be tuned to respond much faster than the others, it should be tuned first and kept in Automatic while the other loops are tuned. By this means, the response used for tuning the slower loops will include the interaction effect of the faster loop. For example, in the control system of the blender in Figure 9-1, the flow controller should be faster than the composition controller because the flow responds almost instantaneously while the composition lags by the time constant of the tank. Therefore, the flow controller must be tuned first and kept in Automatic while the composition controller is tuned.

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If all the loops are of equal importance and speed of response, each must be tuned while the other loops are in Manual. The gain of each loop must then be adjusted by multiplying the gain obtained when all other loops were opened by the relative gain for the loop: Kcij' = Kcijµij

(9-10)

where: Kcij' = the adjusted controller gain Kcij

= the controller gain tuned with all the other loops opened

µij

= the relative gain for the loop.

This adjustment accounts for the change in steady-state gain when the other loops are closed, but it does not account for dynamic effects. If some of the loops are slower than the others or can be detuned, the relative gains for the remaining loops must be recalculated as if those were the only interacting loops; that is, as if the slower or detuned loops were always opened. The gain adjustment suggested by Equation 9-10 should be sufficient for those loops with positive interaction since their response remains monotonic when the other loops are closed. However, the loops with negative interaction may require retuning after the other loops are closed because the other loops will cause either inverse or overshoot responses that normally require lower gains and slower integral than monotonic loops. Note that the formula results in a gain reduction for the loops with positive interaction and a gain increase for the loops with negative interaction, assuming that the pairing with the positive relative gain is always used, as it should be. When decouplers are used, they must be tuned first and kept active while the feedback controllers are tuned. Recall that perfect decoupling has the same effect on a loop as if the other loops were very tightly tuned. For example, in the blender control system of Figure 9-8, the ratio and mass balance controllers must be tuned first and then kept active while the flow and composition controllers are tuned.

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Example 9-6. Tuning of a Catalyst Blender Compare the responses of the composition control of the blender in Figure 9-1 with and without product flow control. Assume the same conditions as in Example 9-5, but also assume that the analyzer AT samples the composition at intervals of one minute and that it takes one minute of dead time to carry out the analysis. Because of the additional dead time of the analysis and the sampling, the analyzer controller AC tuning changes as follows: Gain = 2

Integral time = 3 min

The tuning of the flow controller is the same as in Example 9-5. The responses to a flow increase of 10 kg/hr are compared in Figure 9-10 with the continuous line corresponding to the case when both the product flow and the composition are controlled (when the product flow controller is in Automatic) and the dashed line when only the product composition is controlled (when the product flow controller is in Manual). The oscillatory behavior of the response when both the flow and composition are controlled shows that the gain of both loops is increased by interaction. In this case, the increase is by a factor of two since the relative gain is 0.5 (the loops help each other). Were the gain of the composition controller to be reduced by one-half, the response would be the same as when the flow is not controlled.

9-5. Model Reference Control A number of multivariable control schemes that are currently in widespread use in the process industries can be classified under the general umbrella of Model Reference Control. Many such schemes for multivariable control and optimization are commercially available. Some, like Dynamic Matrix Control2 (DMC) use multivariable linear models, while others use artificial neural network-based nonlinear models. Although the technical aspects of these advanced schemes are outside the scope of this book, this section briefly dis-

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Figure 9-10. Control of Product Composition with the Product Flow Controller in Automatic (Solid Lines) and in Manual (Dashed Lines)

cusses Model Reference Control and presents an example. For a simple introduction to the mathematics of the DMC scheme see Smith and Corripio3. A model reference controller uses an on-line process model and feedback from a process measurement to correct for unmeasured disturbances and model error. The more successful schemes are not restricted to specific model structures, such as the single-lag–plus-dead-time model of Chapter 3. Instead, the models are developed from process data. For example, DMC models consist of unit step responses of each process variable PV to each controller output OP and measured disturbance D. One common characteristic of all the successful model reference controllers is that they use the models to predict the future response of the process or dependent variables. Then, by comparing current process measurements with the values predicted by the model for the current time, the predicted values are

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corrected. The corrected predicted values from the model are then used to determine the changes in the manipulated or independent variables that minimize the deviations of the dependent variables from their set points. Because the different process variables have different units of measure (e.g., temperatures, flows, compositions, etc.), their deviations must be weighted in the function to be minimized. One way this is done is by defining an “equalconcern error” for each variable. For example, equal-concern errors in a given application may be 5°C, 200 kg/h, 2 weight%, etc. Weighing the deviations by the reciprocals of the equal-concern errors normalizes them into deviations of equivalent magnitude. Another common characteristic of model reference controllers is the presence of penalties in the function to be minimized for excessive movements in the controller outputs. In fact, the penalty factors for the controller output moves, known as “move-suppression parameters,” are some of the tuning parameters of the multivariable control system. Another characteristic is provision for optimization of the set points. In a linear scheme like DMC, a Linear Program is used to do the optimization, which means that the system is driven to its constraints, since linear systems cannot have optimums inside the range of operating conditions. Since there are constraints in both the set points and the controller outputs, and the number of degrees of freedom is equal to the number of manipulated (independent) variables, the optimum operating conditions occur when the sum of the number of variables constrained is equal to the number of controller outputs. Finally, model reference control systems are designed to handle constraints in both the dependent and independent variables. The main concern addressed by these techniques is that when one or more variables are driven against a constraint, the optimum values of the remaining variables are not the same as when all the variables can be set to their optimum values.

9-6. Summary This chapter dealt with multivariable control systems and their tuning. It showed the effect that loop interaction has on the response of feedback control systems and presented two methods to deal with it: Bristol’s relative gains for quantitatively determining the amount of interaction and for selecting the

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pairing of process and controller outputs that minimize the effect of interaction, and loop decoupling. One example, the distillation column, showed that dynamic interaction, which cannot be detected by the relative gains, must also be considered when selecting pairing process variables and controller outputs.

References 1. Bristol, E. H. “On a Measure of Interaction for Multivariable Process Control,” IEEE Transactions on Automatic Control, V. AC-11, Jan. 1966, pp.133134. 2. Cutler, C. R. and Ramaker, B. L. “DMC - A Computer Control Algorithm,” AIChE 1979 Houston Meeting, Paper #516; New York: AIChE, 1979. 3. Smith, C. A. and Corripio, A. B. Principles and Practice of Automatic Process Control, 2nd ed., New York: Wiley, 1997.

Review Questions 9-1. Under what conditions does loop interaction take place? What are its effects? What two things can be done about it? 9-2. For any given loop in a multivariable (interacting) system, define the open-loop gain, the closed-loop gain, and the relative gain (interaction measure). 9-3. How are the relative gains used to pair process variables and controller outputs in an interacting control system? What makes it easy to determine the relative gains? What is the major shortcoming of the relative gain approach? 9-4. In a 2x2 control system the four relative gains are 0.5. Is there a best way to pair the variables to minimize the effect of interaction? By how much does the gain of a loop change when the other loop is closed? Is the interaction positive or negative? 9-5. Define positive and negative interaction. What is the range of values of the relative gain for each type of interaction?

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9-6. The open-loop gains for the top and bottom compositions of a distillation column are the following: Distillate Composition Bottoms Composition

Reflux 0.05 -0.02

Steam -0.02 0.05

Calculate the relative gains and pair the compositions of the distillate and bottoms to the reflux and steam rates so that the effect of interaction is minimized. 9-7. The automated showers in the house of the future will adjust the hot and cold water flows to maintain constant water temperature and flow. In a typical design the system is to deliver 2.5 gallons per minute (gpm) of water at 110°F by mixing water at 170°F with water at 80°F. Determine the open-loop gains, the relative gains and the preferred pairing for the two control loops. Hint: the solution to this problem is identical to that of Example 9-2. 9-8. Design a decoupler to maintain the temperature constant when the flow is changed in the shower control system of the preceding exercise. Dynamic effects can be neglected.

10 The Auto-tuner Application

This chapter will describe the functionality and operation of the auto-tuner application that is built into most process control software packages on the market at this time. Knowing how to properly set up and use the application is essential to obtaining acceptable results from the auto-tuner.

Learning Objectives–When you have completed this chapter, you should be able to: A. Know how to properly set up the auto-tuner application. B.

Know how to use the auto-tuner application.

C. Know what to do after the auto-tuner has finished performing its calculations.

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10-1. Operation The auto-tuner application is an advanced software application that automatically handles all the conventional functions of tuning for the engineer or technician. Everything that you have learned in this book thus far, the auto-tuner can do for you. Does this mean that you can forget all of the preceding chapters? Definitely not! You must have a working knowledge of the principles that underlie those tuning procedures to be able to use the auto-tuner reliably and effectively. The auto-tuner is simply a tool that allows the engineer or technician to perform the tuning procedures discussed in this book in a timely, more efficient manner. However, the application must be set up for the task at hand. This requires knowing how to configure the application for each loop that needs to be tuned. Once the application has been configured properly, it will perform satisfactorily. The auto-tuner will perform the step test, make the necessary measurements and solve the calculations depending on the selected algorithm. There are many algorithms to choose from, including Modified Ziegler-Nichols, Internal Model Control, and Lambda tuning among others. Once the autotune test has been completed, the user can also switch between different algorithms instantaneously to compare the different outcomes based on the different algorithms. The auto-tuner can, in most cases, perform the step tests, make the measurements and solve the calculations in less time than that which is required to manually perform a step test, thereby saving a considerable amount of time during the start-up of a process or during the general tuning of process loops.

Local Override In use, the auto-tuner will take control of the loop being tuned and put the loop into LOCAL OVERRIDE. In Figure 10-1, LOCAL OVERRIDE is displayed on the faceplate as “LO”. This will let the board operator know that another program is controlling the output. During this period, the auto-tuner has control of the control valve (or another final control element) and it will make adjustments according to how it has been configured. This means that the auto-tuner will move the final control element as it sees fit for the purpose of measuring the process parameters to obtain the desired tuning results, instead of maintaining process control during normal operations.

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Because the auto-tuner’s main function is to generate oscillations and to thereby measure process dynamics generated from those oscillations, the process is not being controlled optimally during this procedure. Instead, the autotuner is actually causing fluctuations in the process control loop being tuned, which may cause even larger disturbances downstream from the loop being tuned. It is thus important to stay in constant communication with the board operator while the loop is in LOCAL OVERRIDE. Should the process start to become unstable, including the loops that are downstream from the loop being tuned, the operator will need to retake control of the loop immediately to mitigate any developing problems. Most control systems allow the operator to retake control of the process immediately by simply changing the set point on the faceplate.

Figure 10-1. Faceplate Showing the Control Loop in LOCAL OVERRIDE (LO)

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Step Size The step size can be selected depending on several variables of the loop being tuned, the most important of which is process gain. Be aware that by performing a step test, you are causing a disturbance in a running process, and the goal is to keep the disturbance to a minimum. For loops with a high process gain, a small step size should generally be used. A step size of 3% of scale or less can be chosen if there is very little process noise in the loop. On some loops, a step size as small as 0.5% of scale can produce good results. For noisier loops or loops with a low process gain, a larger step size should be selected. If you are unsure of the process gain, start with a smaller step size and retest with a larger step size if necessary. Figure 10-2 shows the autotuner application tuning page. You can see that the step size parameter (in the lower left corner) is selectable by dropping down the step size selection box.

Figure 10-2. The Auto-tuner Application Step Size Selection Box

One method that will aid in selecting an initial step size is to examine the plant historian. Look at the historical data that was recorded during recent process upsets and see whether small output responses were needed to rectify a large process upset. If so, you can assume that the process gain is large and you should start with a small step size. On the other hand, if you notice that

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large output responses were necessary to bring a small disturbance under control, then you can assume that the loop has a small process gain and you can begin with a larger step size. In either case, you may increase the step size in subsequent tuning tests should you find it necessary. Use caution when selecting a step size for integrating (non-self-regulating) loops since integrating loops (usually product level measurements in tanks or vessels) will not reach steady state and will at some point reach an undesirable level if left uncontrolled long enough. Remember that while the autotuner is active, it puts the control loop in LOCAL OVERRIDE and its main objective is to inject an error in the process to measure the process response, as opposed to maintaining control of the process. The larger the chosen step size, the faster the integrating loop will approach an intolerable limit. Always choose the smallest possible step size for integrating loops.

10-2. Applications The first requirement in tuning any process is that the loop must be stable. If the loop is in automatic mode and the process variable is stable, you should have no problem with auto-tuning the process while it is in automatic mode. If the historical trend shows that the process variable is stable overall but has some significant process noise, you may still be able to tune the loop using a larger step size. If stability cannot be achieved in automatic mode, you should switch the loop to manual mode. Remember to monitor the process variable any time that the loop is not under automatic control. If after switching the loop to manual mode you still observe variability in the loop, then there is another loop acting on the loop under test. You must find the loop that is interacting with the loop under test and stabilize that loop first to obtain a steady process variable in preparation for the tuning procedures. Keep in mind that other process problems, such as feed composition changes or equipment malfunctions, may also cause interactions with the loop under test. Interacting loops may be easy to locate if they are cyclic or recurring, but they may be harder to find if they are intermittent and interact with the loop under test less frequently. The facility process engineer may be able to offer helpful advice when it comes to tracking down interacting loops.

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Not all loops can or should be tuned using the auto-tuner. In some cases, the auto-tuner may not be able to achieve a sufficient response from the process variable to provide accurate results. On some slower loops, the auto-tuner may seem to “hang up” and not move the final control element as required to generate oscillations in the process. When this happens, you have the option to either wait and see whether the auto-tuner will switch states or to abort the procedure and tune the loop using an alternate method. In all cases, you should closely monitor the reaction of any loop in which the tuning parameters have been modified. If the response of the loop is not acceptable, you can retune the loop, or if necessary, revert to the original tuning values until the process becomes stable again. We will discuss the application of the auto-tuner later in this chapter. The auto-tuner is just one tool that the engineer or technician has at his or her disposal to help maintain the most efficient control of the process. A control loop with a tight control response (when required) can save the facility unnecessary expense compared to a control loop that has high variability. For instance, if you take the example of a process heater in which the fuel gas flow is controlled by the heater outlet temperature, you can see where a steady control loop would cost less in expensive fuel gas when compared to a poorly tuned loop in which the outlet temperature falls a few degrees and the control valve opens a significant amount allowing a surge of fuel gas into the heater to recover the few degrees of lost heat. Such a loop can be tuned using any method, but if the time it takes to complete the tuning can be shortened by using the auto-tuner, then the technician or engineer can move on to the next loop sooner, thereby increasing profitability for the company.

Problematic Loops and Dead Time Before attempting tuning or auto-tuning, you should ensure that all parts of the process loop are working properly. Tuning will not rectify inherent problems within a loop, such as a valve that has hysteresis or stiction and is not transitioning properly from opening to closing or from closing to opening. Examine the location of the sensing element (sensor) in relation to the final control element. If there is a large amount of piping in between the two, then you might expect a long dead time in the process loop. Likewise, if the controlled variable is on one product stream and the manipulated variable is on another product stream, then it may take longer for the response to be noticed.

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The auto-tuner has special settings for slow time constant loops. These slow time constant loops will usually take longer to auto-tune than standard loops. They should be watched closely and the tuning should be aborted if the process does not respond within a time period that will not interfere with normal process operations. Loops that are classified as dead-time-dominant present a special challenge to the auto-tuner. Some auto-tuners have a dead-time-dominant setting that can be used to help with these loops. Be sure to take advantage of these advanced settings when working with more challenging loops such as dead–time-dominant and other more challenging loops. The auto-tuner works well with fast loops. A typical flow loop usually has a small time constant. Look back at Figure 10-2 and notice how quickly the step test takes place. When this is the case, the auto-tuner can almost always finish the testing and report the results faster than the process control engineer can manually perform the step test. A loop that has a small time constant will also settle back to normal operating conditions very quickly after the step test disturbances have been injected. Most pressure loops are very fast by nature and are easily tunable using the auto-tuner application. Some fast-acting level loops are also good candidates for auto-tuning. Tight level control is easily achievable using the auto-tuner on fast level loops. Loops with a small time constant provide the most benefit from using the auto-tuner application because of the relatively short amount of time it takes to complete the tuning process using the auto-tuner. There may be some loops in the process that would respond better if they were not tightly tuned. For instance, the liquid level in a large tower may be tuned less tightly to allow the level to fluctuate within reasonable limits if this fluctuation does not cause disturbances in other loops (see Figure 10-3). If the level control loop in the tower has a control valve that controls the level, and the product leaving the tower is the feed to another unit, it may be more desirable to have the feed flow rate to the downstream unit remain more uniform while allowing the level to fluctuate a bit more. In this case, the technician could use the auto-tuner to achieve tight level control and then modify the suggested tuning parameters to deaden the process response time a small amount, or simply choose a more conservative tuning method, such as Lambda tuning.

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Figure 10-3. Level in Unit 1 Tower Allowed to Fluctuate Within Acceptable Limits to Maintain Steady Feed Rate to Unit 2 Tower

Loops with a very small process gain should be monitored closely during the auto-tune process. See Figure 10-4 for an example of a loop with a small process gain. Notice that a step size of 12 percent of span was chosen for tuning this particular loop. Also notice on the graph that the process variable only moved a very small amount given the very large step size that was chosen.

Figure 10-4. Control Loop with a Small Process Gain

The Auto-tuner Application

209

Process Noise Loops with a high level of process noise should also be monitored closely during the tuning procedure. At times, the auto-tuner may actually mistake process noise for a movement of the process variable PV in an incorrect direction. In these cases, the auto-tuner may abort the tuning procedure, forcing you to start over again. If possible, PV filtering may be used to limit the amount of process noise seen by the auto-tuner. Remember that the autotuner cannot distinguish between process noise and an actual process reading. Use the minimum amount of filtering required to limit the process noise and don’t forget to set the filter back to normal (if required) after the tuning process is completed. In all tuning procedures the loop (or loops) should be evaluated closely after any change in tuning parameters is made. Watch for the loop’s response to load changes, upsets, and set point adjustments. If the loop does not perform satisfactorily, you may try retuning the loop, or you may want to reinstall the original tuning parameters. The auto-tuner has a button that will allow you to revert to the original tuning parameters easily, should it be required.

10-3. Features and Settings The auto-tuner package has many features and settings that can make the job of tuning an easy one for the process control technician or engineer. Many control system manufacturers include auto-tune software with their control system. Differences in available features may vary greatly between different manufacturers. You must read the manufacturer’s documentation and become familiar with the features of the software that you are using before attempting tuning using the auto-tune software. If an auto-tune software package is not included with the control system software, it may have to be purchased separately, or a license may have to be purchased separately from the control system license to allow the software to operate properly.

Desired Response Most auto-tuner software has a feature that will allow the user to choose certain options in the tuning such as Normal, Fast, or Slow to obtain a particular desired response from the tuning. The Fast option allows the user to try a

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more aggressive approach for loops that need to return to set point quickly, with overshoot being of a minimal concern. The Slow option reduces the amount of overshoot experienced at the cost of PV being away from set point longer. The Normal option is a trade-off between a quick return to set point and a minimal amount of overshoot. Figure 10-5 illustrates the DESIRED RESPONSE parameter in the highlighted drop-down box. As you can see, in this case a NORMAL response has been selected.

Figure 10-5. Auto-tuner Desired Response Set to NORMAL

Tuning Methods Some expanded features of the software will allow you to choose which tuning method is employed to enhance the performance of the loop. You may choose Ziegler-Nichols, Lambda, or Internal Model Control for more desirable tuning results. One of the best things about the auto-tuner is that once you have performed the step test and recorded the process dynamics of the loop, you do not have to retest each time you wish to switch between tuning methods. By simply selecting the new tuning method, you will instantly have the new calculation results and will be able to quickly compare them to one another with just a few clicks of the mouse.

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211

Simulation of Tuning Response Another feature of some auto-tuners currently on the market allows you to simulate loop response for a variety of tuning settings. As you select different phase margin and gain margin settings by clicking the mouse in the shaded area, the simulated response is updated in the form of a chart that displays the projected amount of overshoot versus the time required to reach set point again. See Figure 10-6 for an example of the Simulate feature that is built into the auto-tuner. This tool can come in handy for the user who wants to see the process response plotted in graphical form. If the facility has pre-published overshoot criteria, this tool will be invaluable to the user.

Figure 10-6. The Simulate Feature Built into the Auto-tuner

As you can see, the recommended setting is represented by a dot with RECOMMENDED beside it on the “Tuning for Robustness” graph. The second dot depicts the simulated tuning value. It can be placed anywhere in the “Tuning for Robustness” graph by simply clicking the mouse in the shaded area. The auto-tuner will instantly recalculate the predicted response to the simulated tuning values and plot the results in graphical form in the “Simulation Response” area of the auto-tuner’s Simulate tab.

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Using the Auto-Tuner To use the auto-tuner to perform an automatic tuning procedure on a control loop, the user should perform the following steps: 1. Select the control loop to be tuned. 2. Observe the current reaction of the control loop. 3. Select the DESIRED RESPONSE. 4. Communicate your intentions to the board operator. 5. Press the TEST button on the ON-DEMAND TUNING tab. 6. Monitor the loop response to the final control element movements. 7. When the auto-tune procedure is finished, determine whether a comparison between different tuning methods is needed. 8. Determine whether simulation would enhance the tuning values. 9. Press the UPDATE button to install the new tuning parameters. 10. Monitor the newly installed tuning parameters and control loop response.

10-4. Summary The auto-tuner application does not replace the knowledge necessary to perform tuning procedures. The auto-tuner can drastically reduce the amount of time required for manual tuning step testing and the required mathematical calculations, but it must be properly configured before beginning any tuning procedures. Proper setup is crucial to obtaining accurate results. By doing the necessary preparation work before running the auto-tuner application, the engineer or technician can ensure a better outcome. The application will take control of the final control element and cause it to move to generate a measurable response in the process variable. Constant communication with the process operators is a must during the period that the control loop is in LOCAL OVERRIDE.

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The user can select a certain desired response before testing. Once the process dynamics have been measured, the user can switch between several of the most common tuning methods available. The auto-tuner application also has the ability to project expected responses to process disturbances based on input from the user using the simulation feature that is included with the auto-tuner.

Review Questions 10-1. Briefly describe LOCAL OVERRIDE and explain its effects on the control loop. 10-2. While selecting a step size for auto-tuning a control loop, you notice that the process historian shows that the last upset took large movements of the final control element to bring the process variable back to set point. What can you assume about this loop and what step size should you begin with? 10-3. What is the first requirement in tuning any process? 10-4. What types of loops does the auto-tuner work best on? 10-5. Explain why the user must maintain constant contact with the board operator while performing tuning procedures. 10-6. Describe the Simulate feature and how it outputs the results of simulated tuning.

Appendix A Suggested Reading and Study Materials

Blevins, T. L., McMillan, G. K., Wojsznis, W. K., and Brown, M. W., Advanced Control Unleashed: Plant Performance Management for Optimum Benefit, ISA, Research Triangle Park, NC. McMillan, G. K. and Cameron, R. A., Advanced pH Measurement and Control, 3rd edition, ISA, Research Triangle Park, NC. McMillan, G. K., and Toarmina, C. M., Advanced Temperature Control, 2nd edition, ISA, Research Triangle Park, NC. McMillan, G. K., Good Tuning, A Pocket Guide, 3rd edition, ISA, Research Triangle Park, NC Murrill, P. W., Fundamentals of Process Control Theory, 3rd edition, ISA, Research Triangle Park, NC. Shinskey, F. G., Process Control Systems, 3rd ed., McGraw-Hill, New York, NY, 1989. Smith, C. A., and Corripio, A. B., Principles and Practice of Automatic Process Control, 3rd edition, Wiley, New York, NY, 2006. (Note: The material on Dynamic Matrix Control is in the 2nd edition, 1997).

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Appendix B Answers to Study Questions

Chapter 1 1-1.

The main goal of controller tuning is to produce a smoothly operating process (Section 1-1); that is, to reduce the variability of the process and the manipulated variables.

1-2.

The two process characteristics to be considered when tuning the controller are the process sensitivity or gain and its rate of response (Section 1-1).

1-3.

The three instrumentation components in a feedback control loop are the sensor transmitter (measurement), the controller (decision), and the control valve (action) (Section 1-2).

1-4.

The fourth element of the feedback loop is the process (Section 1-2).

1-5.

The most important characteristic of a feedback control loop is its action, direct or reverse, chosen so that there will be a net reverse action around the loop; that is, when the process variable changes the controller will take action to move it in the opposite direction (Section 1-2).

1-6.

The fail position of the cooling water valve must be open so that coolant is not lost on loss of power; the controller signal then closes the valve, so the controller action must be reverse. In other words, when the tempera-

217

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ture increases the controller output decreases, opening the valve to supply a higher-cooling water flow. 1-7.

The reactant’s control valve must fail closed so that the reactor does not overflow on loss of power. The controller action must then be reverse; that is, increasing level decreases the controller output to close the valve and reduce the flow of reactants into the reactor.

1-8.

The controller must have direct action; that is, increasing caustic composition must increase the controller output to increase the flow of dilution water and decrease the composition of the caustic product.

Chapter 2 2-1.

Proportional gain of 3. a. The controller output decreases by 3(10%) = 30%. b. The change in PV is a decrease of 15°C/(150°C – 0)100% = 10%. Controller output decreases by 3(10%) = 30%. c. The PV increases by: (250 kg/hr)/(50,000 kg/hr – 0)100% = 0.5%. Controller output decreases by 3(0.5%) = 1.5%.

2-2.

The proportional mode causes an immediate increase in output of 2(10%) = 20%. The integral mode increases the controller output at the rate of 2(10%)/(5 min) = 4%/min. In 15 minutes the output increase is 20% + (15 min)(4%/min) = 80%. The sketch is a step up of 20% followed by a continuous rise of 4%/min.

2-3.

The derivative mode causes an immediate decrease in controller output of (3%/min)(0.6 min) = 1.8%. The sketch is just a step down of 1.8%.

2-4.

The cause is that the controller has the incorrect action. To correct, change the controller action.

2-5.

The problem is that the controller gain is too high (or the integral time is too low). To correct, decrease the controller gain by at least half. If this does not work, adjust the integral time using the period of the oscillations as a guide; that is, set the integral time to be of the order of magnitude of the period of the oscillations.

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219

2-6.

The integral and derivative times in Table 2-1 are related to the period of oscillation of the loop because the period is an indication of the speed of response of the loop.

2-7.

To reduce the variability of the controller output, reduce the controller gain.

Chapter 3 3-1.

The procedure for the open-loop step test is as follows (Section 3-1): a. Switch the controller to manual output. b. Apply a small step change, 1 to 3%, in the controller output. c. Record on the same trend the controller output and the process variable until the PV reaches a new steady value. d. Analyze the PV response to obtain the gain, dead time, and time constant.

3-2.

The parameters of the SLPDT model are the gain, the dead time, and the time constant (Section 3-2). The gain is an indication of the sensitivity of the process variable to the controller output; the dead time is a measure of how long it takes the PV to start responding to the change in controller output; and the time constant is a measure of how long it takes the process to respond to the action of the controller.

3-3.

From the response, the change in controller output is: 45.7% – 40.7% = 5.0% The steady change in outlet concentration is: 0.74 lb/gal – 0.50 lb/gal = 0.24 lb/gal For a transmitter range of 0 to 1.5 lb/gal, the change in PV is: (0.24 lb/gal)(100%)/(1.5 – 0)lb/gal = 16.0% The gain is: 16.0%/5% = 3.2 The 28.3% point is: 0.5 + 0.283(0.24) = 0.568 lb/gal From the graph, t1 = 7.5 – 1 = 6.5 min

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The 63.2% point is: 0.5 + 0.632(0.24) = 0.652 lb/gal From the graph, t2 = 15 – 1 = 14.0 min The time constant is: 1.5(14.0 – 6.5) = 11.2 min The dead time is: 14.0 – 11.2 = 2.8 min Note that 1 min is subtracted from the time readings because the step change in controller output is applied at 1 min. 3-4.

For a maximum resistor of 10 megohms and capacitor of 100 µfarads, the maximum filter time constant is (Section 3-3): (10x106 ohms)(100x10-6 farads) = 1,000 sec

3-5.

The time constant of the surge tank is (Section 3-3): (20 ft2)(7.48 gal/ft3)/(50 gpm/ft) = 3.0 min

3-6.

The time constants at each flow are (Section 3-3): a. (2000 gal)/(50 gpm) = 40 min b. (2000 gal)/(500 gpm) = 4.0 min c. (2000 gal)/(5000 gpm) = 0.4 min

3-7.

At the base conditions the total flow is (Section 3-2): 100 + 400 = 500 gpm The outlet concentration is: [(100)(20) + (400)(2)]/500 = 5.60 lb/gal At 110 gpm of the concentrated solution the total flow is: 110 + 400 = 510 gpm The outlet concentration is: [(110)(20) + (400)(2)]/510 = 5.88 lb/gal The process gain is: (5.88 – 5.60)/(110 – 100) = 0.028 (lb/gal)/gpm

3-8.

At the base conditions the total flow is (Sections 3-2, 3-3): 10 + 40 = 50 gpm

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The outlet concentration is: [(10)(20) + (40)(2)]/50 = 5.60 lb/gal At 11 gpm of the concentrated solution the total flow is: 11 + 40 = 51 gpm The outlet concentration is: [(11)(20) + (40)(2)]/51 = 5.88 lb/gal The process gain is: (5.88 – 5.60)/(11 – 10) = 0.28 (lb/gal)/gpm For a tank volume of 5,000 gal, the time constants are (Section 3-3, Equation 3): For question 3-7: (5,000 gal)/(500 gpm)= 10 min For this question: (5,000 gal)/(50 gpm) = 100 min After the change in flow, they are 9.8 and 98 min, respectively. These answers illustrate the effect of the process flow on the gain and time constants of the process model due to a simple and common nonlinearity.

Chapter 4 4-1.

If the process gain doubles due to nonlinear behavior, the controller gain must be halved to maintain the same loop gain and controller performance (Section 4-1).

4-2.

The uncontrollability is the ratio of the dead time to the time constant. The actual magnitude of the time constant and the dead time determine how fast the loop can respond, but not its controllability (Section 4-1).

4-3.

The proportional gain from the ultimate gain is Kcu/2 and from the uncontrollability parameter is 1/KPu. For them to be the same, Kcu = 2/ KPu. The PID integral time from the ultimate period is Tu/2 and from the dead time is 2t0, so for them to be the same, Tu = 4t0 (Section 4-1).

4-4.

The sensitivity is the measure of how much the process variable changes when the controller output changes; that is, the gain, so Process A is the least sensitive and Process C the most. The speed of response is deter-

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mined by the process time constant, so Process B is the slowest and Process A the fastest. The uncontrollability is the ratio of the dead time to the time constant, so Process C is the least controllable and Processes A and B are equally controllable (Section 4-1). 4-5.

Using the tuning strategy proposed in the chapter, the PID tuning parameters are (Section 4-1): Process A:

Process B:

Process C:

Kc = (1.2/0.5)(2.0/0.2) = 24 TI = 2(0.2) = 0.4 min TD = 0.2/2 = 0.1 min Kc = (1.2/2.0)(30/3.0) = 6.0 TI = 2(3.0) = 6.0 min TD = 3.0/2 = 1.5 min Kc = (1.2/4.0)(5.0/3.0) = 0.5 TI = 5.0 min (use IMC rule, 2(3) > 5) TD = 3.0/2 = 1.5 min

4-6.

The controller would be configured to act on the process variable to prevent large changes in controller output when the set point is changed and the controller gain is high. When configured in this manner, the output ramps to the new required value at the rate controlled by the integral time when the set point is changed (Section 4-1).

4-7.

Reset windup shows up as a large overshoot in the process variable caused by the controller output being driven to one of its limits (for example, control valve fully opened or closed) by the integral mode. It usually happens on start-up or after a large disturbance and can be prevented by keeping the controller in Manual until the process variable and the controller output are near their design values. In addition, gaps between the controller output limits and the control valve (or other final control element) limits must be avoided (Section 4-3).

4-8.

Inverse response is the case where the process variable initially moves in the opposite direction of its eventual direction of change. It is detrimental to the performance of a feedback controller because it causes the controller to initially move the output in the wrong direction (Section 4-4).

Answers to Study Questions

223

Chapter 5 5-1.

Tight level control is indicated when the level has significant effect on the process operation, as in a natural-circulation evaporator or reboiler. Averaging level control is to be used when it is necessary to smooth out sudden variations in flow, such as in a surge tank receiving discharge from batch operations to feed a continuous process. Tight level control is the one that requires the level to be kept at or very near its set point (Section 5-3).

5-2.

For flow control loops a proportional-integral (PI) controller is recommended with a gain near but less than 1.0. The integral time is usually small, on the order of 0.05 to 0.1 minutes (Section 5-2).

5-3.

For tight level control a proportional controller with a high gain, usually greater than 10, should be used. When the lag of the control valve is significant, a proportional-derivative controller could be used. If a proportional-integral controller is used, the integral time should be long, on the order of one hour or longer (Section 5-3).

5-4.

For averaging level control a proportional controller with a gain of 1.0 should be used, because this provides maximum smoothing of variations in flow while still preventing the level from overflowing or running dry (Section 5-3).

5-5.

When a PI controller is used for averaging level control, the integral time should be long, on the order of one hour or longer. At some values of the gain, an increase in gain would decrease oscillations in the flow and the level (Section 5-3).

5-6.

Time constant, from Equation 5-2: t = (0.03 kg)(23 kJ/kg-°C)/[(0.012 m2)(0.6 kW/m2-°C)] = 96 s (1.6 min)

5-7.

PID controllers are commonly used for temperature control so that the derivative mode compensates for the lag of the temperature sensor, which is usually significant (Section 5-4).

5-8.

The major difficulty with the control of composition is the dead time introduced by sampling and by the analysis (Section 5-5).

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Chapter 6 6-1.

Computer- and microprocessor-based controllers perform the control calculations at discrete intervals of time, with the process variable being sampled and the controller output updated only at the sampling instants, while analog controllers calculate their outputs continuously with time (Section 6-1).

6-2.

The “derivative kick” is a pulse on the controller output that takes place at the next sample after the set point is changed and lasts for one sample. It can be prevented by having the derivative term act on the process variable instead of on the error (deviation) (Section 6-1). The derivative filter or “dynamic gain limit” is needed to prevent large amplification of changes in the process variable when the derivative time is much longer than the algorithm sampling interval.

6-3.

“Proportional kick” is a step change in controller output right after a set point change; it can be eliminated by having the proportional term act on the process variable instead of on the error, so that the operator can apply large changes in set point without danger of upsetting the process. When proportional kick is avoided, the process variable approaches the set point slowly after it is changed, at a rate determined by the integral time. The proportional kick must not be avoided whenever it is necessary to have the process variable follow set point changes quickly, as is the case in the secondary controller of a cascade system (Section 6-1).

6-4.

The nonlinear gain allows the proportional controller gain to be smaller than unity when the error is near zero, which is equivalent to having a larger tank in an averaging level control situation. To have a gain of 0.2 (400% PB) at zero error, the nonlinear gain must be (Equation 6-1): KNL = [(1/0.2) 1]/50 = 0.08 This calculation assumes a proportional-only controller with an output bias term of 50% and a set point of 50% (Section 6-1).

6-5.

Using the formulas of Table 6-2, with q = 0 (for maximum gain) and the following parameters (Section 6-2):

Answers to Study Questions

K = 1.6

6-6.

225

t = 20 min

Sample time, min

0.067

1

5

50

a = exp(T/t)

.9934

0.905

0.368

0.0067

N = t0/T

75

5

1

0

Gain

2.0

1.7

0.90

0.046

Integral time, min

20.0

19.5

17.6

4.47

If the algorithm has dead time compensation, the gain can be higher because it does not have to be adjusted for dead time. This does not affect case (d) because the dead time is less than one sample, and therefore, no dead time compensation is necessary. From Equation 6-5 and Table 6-3 (Section 6-4): Sample time, min

6-7.

t0 = 5 min

0.067

1

5

N = t0/T

75

5

1

Gain

153

5.9

1.8

Integral time, min

20

19.5

17.6

The basic idea of the Smith Predictor is to bypass the process dead time to make the loop more controllable. This is accomplished with an internal model of the process responding to the manipulated variable in parallel with the process. The basic disadvantage is that a complete process model is required but it is not used to tune the controller, creating too many adjustable parameters (Section 6-4). The Dahlin Algorithm produces the same dead time compensation as the Smith Predictor but it uses the model to tune the controller, reducing the number of adjustable parameters to one: q.

Chapter 7 7-1.

Cascade control (1) takes care of disturbances into the secondary loop reducing their effect on the controlled variable, (2) makes the primary loop more controllable by speeding up the secondary loop and (3) handles the nonlinearities in the inner loop, where they have less effect on controllability (Section 7-1).

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7-2.

For cascade control to improve the control performance, the secondary loop must be faster than the primary loop. The sensor of the secondary loop must be reliable and fast, although it does not have to be accurate (Section 7-1).

7-3.

The primary controller in a cascade control system has the same requirements as the controller in a simple feedback control loop, thus the tuning and mode selection of the primary controller are no different from those for a single controller (Section 7-2).

7-4.

The tuning of the secondary controller is different because it has to respond to set point changes, which it must follow quickly but without excessive overshoot. The secondary controller should not have integral mode when it can be tuned with a high enough proportional gain to maintain the offset small. If the secondary is to have derivative mode, it must act on the process variable so that it is not in series with the derivative mode of the primary controller (Section 7-2).

7-5.

The controllers in a cascade system must be tuned from the inside out, because each secondary controller forms part of the process controlled by the primary around it (Section 7-3).

7-6.

Temperature as the secondary variable (1) introduces a lag because of sensor lag and (2) may cause integral windup because its range of operation is narrower than the transmitter range. These difficulties can be handled by (1) using derivative on the process variable to compensate for the sensor lag and (2) having the secondary measurement fed to the primary controller as its reset feedback variable (Section 7-3).

7-7.

Pressure is a good secondary variable because its measurement is fast and reliable. The major difficulties are (1) that the operating range may be narrower than the transmitter range and (2) that part of the operating range may be outside the transmitter range—for example, vacuum when the transmitter range includes only positive gage pressures (Section 7-3).

7-8.

In a computer or DCS cascade control system the secondary controller must be processed more frequently than the primary controller (Section 7-3).

7-9.

Integral windup can occur in cascade control when the operating range of the secondary variable is narrower than the transmitter range. To pre-

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vent it, the secondary measurement can be passed to the integral feedback of the primary; in such a scheme the primary always takes action based on the current measurement, not on its set point (Section 7-4).

Chapter 8 8-1.

A feedback controller acts on the deviation of the process variable from the set point. Thus, if there is no deviation, there is no control action. In theory, perfect control is possible with feedforward control, but it requires perfect process modeling and compensation (Section 8-1).

8-2.

To be used by itself, feedforward control requires that all the disturbances be measured and accurate models be developed of how the disturbances and the manipulated variable affect the controlled variable (Section 8-1). Feedforward with feedback trim has the advantage that only the major disturbances have to be measured and compensation does not have to be exact, because the integral action of the feedback controller takes care of the minor disturbances and the model inaccuracies.

8-3.

Ratio control consists of maintaining constant the ratio of two process flows by manipulating one of them. It is the simplest form of feedforward control (Section 8-1). For the air-to-natural-gas ratio controller of Figure 7-5: • Control objective: Maintain constant the nitrogen-to-hydrogen ratio of the fresh synthesis gas. • Measured disturbance: Natural gas flow (production rate). • Manipulated variable: The set point of the air flow controller.

8-4.

A lead-lag unit is a linear dynamic compensator consisting of a lead (a proportional plus derivative term) and a lag (a low-pass filter), each having an adjustable time constant. It is used in feedforward control to advance or delay the compensation so as to dynamically match the effect of the disturbance (Section 8-3). The step response of a lead-lag unit is an immediate step of amplitude proportional to the lead-to-lag ratio, followed by an exponential

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approach to the steady-state compensation at a rate controlled by the lag time constant. The response of a lead-lag unit to a ramp is a ramp that leads the input ramp by the difference between the lead and the lag time constants, or lags it by the difference between the lag and the lead time constants. 8-5.

To lead by 1.5 minutes with amplification of 2: 1.5 min = lead – lag = 2(lag) – lag = lag Therefore, a lag of 1.5 minutes and a lead of 3.0 minutes (Section 8-3).

8-6.

Dead time compensation consists of storing the feedforward compensation and playing it back some time later. The time delay is the adjustable dead time parameter (Section 8-3). Dead time compensation can be used only when the feedforward action is to be delayed and a computer or microprocessor device is available to implement it. It should be used only when a simple lag cannot effectively delay the feedforward compensation.

8-7.

Design of feedforward controller for the process furnace (Section 8-4): 1. Control objective: To = Toset 2. Measured disturbances: W, process flow, kg/h Fs, supplementary fuel flow, m3/min Ti, inlet process temperature, °C 3. Manipulated variable: Fset, main fuel flow, m3/min 4. Steady-state energy balance on furnace: (FΔHm + FsΔHs)η = WCp(To - Ti) where ΔHm is the heating value of the main fuel in kJ/m3, ΔHs is that of the supplementary fuel gas in kJ/m3, η is the efficiency of the furnace and Cp is the specific heat of the process fluid in kJ/kg-°C. Solve for the manipulated variable and substitute the control objective: Fset = (Cp/ηΔHm)(Toset - Ti)W - (ΔHs/ΔHm)Fs

Answers to Study Questions

229

5. Numerical values are needed to evaluate the importance of each disturbance. The change in each disturbance required to cause a given change in main fuel flow would be calculated (Section 8-4). 6. Feedback trim can be added as in Example 8-1: Feedback output: OP = Toset Design formula: Fset = (Cp/ηΔHm)[OP - Ti]W - (ΔHs/ΔHm)Fs 7. Lead-lag units must be installed on the process flow and inlet temperatures, but not on the supplementary fuel gas flow, because its dynamic effect should match that of the main fuel gas flow (Section 8-4). 8. Instrumentation diagram (Figure B-8-1):

Figure B-8-1.

TT

Toset

TT

Ti Lead-Lag 1

Lead-Lag 2

TC OP -

W

SP

To

FT

FT

+

Fs

Adder

Toset - Ti Multiplier

+ Adder

Fset

-

230

Tuning of Industrial Control Systems, Third Edition

Chapter 9 9-1.

Loop interaction takes place when the controller output of each loop affects the process variable of the other loop. The effect is that the gain and the dynamic response of each loop change when the auto/manual state or tuning of the other loops changes (Section 9-1). When loop interaction is present, we can (1) pair the loops in the way that minimizes the effect of interaction and (2) design a control scheme that decouples the loops.

9-2.

Open-loop gain of a loop is the change in its process variable divided by the change in its controller output when all other loops are opened (in Manual) (Section 9-2). Closed-loop gain is the gain of a loop when all other loops are closed (in Automatic) and have integral mode. Relative gain (interaction measure) for a loop is the ratio of its open-loop gain to its closed–loop gain.

9-3.

To minimize interaction for a loop, the relative gain for that loop must be as close to unity as possible. Thus, the loops must be paired to keep the relative gains close to unity, which in a system with more than two control objectives may require ranking the objectives (Section 9-2). The relative gains are easy to determine because they involve only a steady-state model of the process, which is usually available at design time. The main shortcoming of the relative gain is that it does not take into account the dynamic response of the loops.

9-4.

When all four relative gains are 0.5, the effect of interaction is the same for both pairing options. The gain of each loop will double when the other loop is switched to Automatic. The interaction is positive; that is, the loops help each other (Section 9-2).

9-5.

When the effect of interaction with other loops is in the same direction as the direct effect for that loop, the interaction is positive; if the interaction and direct effects are in opposite direction, the interaction is negative. For positive interaction, the relative gain is positive and less than

Answers to Study Questions

231

unity, while for negative interaction the relative gain is either negative or greater than unity (Section 9-2). 9-6.

Interaction for top composition to reflux and bottoms composition to steam (Section 9-2): (0.05)(0.05)/[(0.05)(0.05) - (-0.02)(-0.02)] = 1.19 Relative gains:

| Reflux Yd | 1.19 Xb | 0.19

Steam -0.19 1.19

The top composition must be paired to the reflux and the bottoms composition to the steam to minimize the effect of interaction. 9-7.

Let H be the flow of hot water in gpm, let C be the flow of cold water in gpm, let F be the total flow in gpm and let T be the shower temperature in °F. The mass and energy balances on the shower, neglecting variations in density and specific heat, give the following formulas: F=H+C

T = (170H + 80C)/(H + C)

These are the same formulas as for the blender of Example 9-2. The relative gains are therefore: | Hot F | H/F T | C/F

Cold C/F H/F

For the numbers in the problem: H = (2.5 gpm)(110 - 80)/(170 - 80) = 0.83 gpm C = 2.5 – 0.83 = 1.67 gpm Since the cold water flow is the higher flow, use it to control the flow and use the hot water flow to control the temperature. The relative gain for this pairing is (Section 9-2): C/F = 1.67/2.5 = 0.67 The gain of each loop increases by a factor of 1/0.67 = 1.5. A 50% increase in gain when the other loop is closed (Section 9-4).

232

Tuning of Industrial Control Systems, Third Edition

9-8.

As in the second part of Example 9-4, we can use a ratio controller to maintain a constant temperature when the flow changes. We would then ratio the hot water flow (smaller) to the cold water flow (larger) and manipulate the cold water flow to control the total flow. The design ratio is 0.5 gpm of hot water per gpm of cold water (Section 9-3).

Chapter 10 10-1. LOCAL OVERRIDE is when another program takes control of the output and moves it to measure process parameters to obtain tuning results. 10-2. The loop has a small process gain and you should start with a larger step size. 10-3. The process must be stable. 10-4. Loops with a fast (that is, small) time constant. 10-5. If the process starts to become unstable, the board operator will need to retake control of the loop immediately. 10-6. The Simulate feature allows the user to simulate loop response for different tuning settings. The simulated response is presented in the form of a chart that displays the projected amount of overshoot versus the time required to reach set point again.

INDEX

Index Terms

Links

A action (direct or reverse)

7

adaptive control

69

algorithm

12

96–98

100

103

115

131

139

145

151

155–156

202

224–225

112

128

134

96

136

153

ammonia synthesis analog

224 analysis cycle

91

109

analyzer control

92

95

191–192

196

arrow

124

5

automatic output auto-tuner application auto-tuning

34 202 8

205–207

213

82

84–87

101

125

223–224

batch process

51

66

87

bias

82

101–102

118

44

49

averaging level control

B

224 blending tank

42 86

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms block diagram

Links 5–6

113–114

122

124

127

144–147

149

C capacitance cascade control

cascade dontrol

41–45

72

7–8

18

80

100

121–125

127–129

131

133

135–138

140

143

225–226

130

135

characteristics control loop control valve

21 50–51

derivative mode

16

feedback control

144

feedback control loops

19

feedforward control

144

loop interaction

173

PI controllers process

86 1–2

29

74

217

clamp limit

138

clamped

137

closed-loop gain closed-loop time constant coarse tuning compensation for dead time composition control

computer cascade control

175–178

199

41

230

106 67 113

156

54

77

190

193–196

180

141

conductance

41–46

conductance, valve

43–44

46

54 This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

50

Index Terms

Links

control objective

77–78

122

143–144

158

160

162

169–170

178–179

192–194

227–228

230

3

5–6

34–35

37

50–53

58

60

69–70

72

74

79–80

82

85–86

122–123

128

136

143

148

163

179

186

189

191

202

206–207

217–218

222–223

21

25

59–61

63–64

76

113

122

124

221

control valve

controllability

225 controllable process

64

116

controlled variable

3

7

34

68–70

78

96

113

125

136

206

225

227

18

27

80

128

223

8

11

18

28–29

87

91

101

224

9

217–218

controller proportional-integral

proportional-integral-derivative

96 proportional-only

82

single-mode

18

three-mode

18

two-mode

18

controller action

6

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms controller gain

controller output controller synthesis

Links 12

21–22

29

58–60

64

66

75–76

80–81

85–86

97

102

104

106

109

115

127

195

218

221–222

224

3 79

92

104

80

113

33–34

38–41

46–50

53

57–60

62–64

68

72–74

78

91

103–106

111

119

127

132

144

156–157

196

206

219–223

67

69

95

113–117

119

156–157

165–166

225

228

dead time compensator

113

151–152

157–159

decoupler

7–8

169

174

186–194

200

114 current-to-pressure transducer

5

D Dahlin controller dead band

113–117 78 128

dead time

dead time compensation

decoupling

7

dependent variables derivative

filter

197 12

16

126–127

226

97–98

224

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

98

Index Terms

Links

derivative (Cont.) kick

97

100

118

14–16

21

25

27

58

61

66–67

77–78

82

97

110

112

126

128

140

218

223

226

15–20

24

28–30

58–60

68

82

97–99

106

127–128

224 mode

time

224 unit

98–100

derivative-on-PV

127–128

100

desired response

209–210

digital controller

136

direct action

6–7

40

194

218

direct material balance control distillation column

distributed control systems

disturbance

173

184–185 45

48

129–130

170

181–184

199

5

18

96

148

160

1–2

5

7

19

21

23–24

31

34

62–67

69–72

78

82

92

100

107

112–113

122

124

126

133

136

144–147

149–154

156

161

164–165

171

186

203–205

207

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

disturbance (Cont.) 213

222

227

229

116

146

151–152

156

159

163

165

187–189

dynamic gain limit

98

118

dynamic interaction

185

199

162

228

dynamic compensation

225

224

E efficiency electrical circuit

43

electronic circuits

96

energy balance control equal percentage equal-concern error equal-percentage valves error

estimation of time

182

185

50 198 50 5–7

20

25

80

97

101–102

106

109

112–113

118

124

140

148

152

197

205

224

41

45

106

108–109

1

3–4

7–8

11–12

78

96

123

144

150

164–166

198

5

8

19–20

23

29

34

49

F fast process feedback control

feedback control loop

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

feedback control loop (Cont.)

feedback controller

79

103

111

125

140

144

148

157

169

193

217

226

2

4–5

7

11–12

19

23

28–29

36–37

48

67

69

72

76

86

113–114

125

136

145–148

151–152

159

162–164

175

185

222

144

146–149

151

159

161–162

165–166

227

229

7

67

69

72

144–146

148

151

157

160–161

163

165–166

192

227 feedback trim

feedforward control

227 feedforward element

146

feedforward-feedback control

146

fieldbus

5

filter parameter fine tuning flow control

frequency

97–98

155–156

2

68

79–81

85

90

92

123

190

196

223

16

78

85

112

131

159

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

G gain

3

11–12

16

18

20

22

24–26

30

34

36–37

39–40

50

53

57–58

60–61

64

74

76

80

82

87

92

101

105

107

110

113

126

128

132

148

151–153

156–157

162

177

179

181

184–185

190

194–196

199

217

219

223

225

230 closed-loop

175–178

199

230

nonlinear

101–102

118

224

open-loop

174–178

180

183–184

199–200

230

175–180

183

185–186

189

193–196

198–200

178

180–181

113

137

44

79

relative

230–231 steady-state

158 195

variation

53

gap

70–71 139

gas surge tank graceful degradation

42 160

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

H half decoupling

188–189

heat exchanger

45

61

88

44–45

50

81

87

89

93

130

161

3–6

19

21

25

34–36

52

61–62

74

88

90–91

107

115

149

160–161

163–165

144 heat rate controller (QC) heat transfer

heater

88

206 efficiency

162

feedforward controller

163

temperature hydrogen/nitrogen ratio hysteresis

106–110

117

134 69

80–81

128

60

62–64

74

80

92

132

92

108–109

13–14

21

25

51

58

66

70–71

78–79

82

85

112

124

126

136

139–140

206

I IMC

222 independent variables integral controller

198 79–80 223

integral mode

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

integral mode (Cont.)

integral time

integrating process interaction

interaction measure

145

152

172

175

188

218

222

226

230

13

17

19–21

24–25

30

58–60

62–64

66

68–69

74

79–80

82

86–87

92

97

100

105

107

109

112

127–128

132

136–137

196

218

221–225

41 7

69

112–113

158

169

171–173

175

178–181

184–185

189

193–194

196

199–200

205

174

176–177

199

230 intermediate level control

86

internal model control (IMC)

60

62–64

74

80

92

132

71–74

76

133

173

185

188

194

222

222 intuitive

174

inverse response

J jacketed reactor

136

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

L lag

41

153

lagniappe

123

lead

153

lead-lag compensation

163

lead-lag unit

144

151–153

155–156

163–164

166

187

72

81–82

84–87

92

96

118

125

182

207

227–228 level control

223–224 linear feedforward controllers liquid storage tank

149 42–44

local override

202–203

local set point

18

loop interaction

152

205

212–213

170

173–174

186

198–199

230

5

22

60

68

70–71

78

88

149

151–152

160

162

166

174

206

217

225

227–228

M manipulated variable

manual output master controller material balance control measured disturbance

34 7

138

184–185 144

147–148

158–161

163

166

197

227–228

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms microprocessor

microprocessor-based controllers

Links 96

130

156

228

35

100–101

111

113

118

224

96

205

microprocessor-based PID control algorithm

103

minor disturbances

146

mode

12

152

automatic

205

derivative

14–16

21

25

27

30

58

61

66–67

78

82

87

97

110

126–128

218

223

226

13–14

21

25

51

58

70–71

78–79

82

112

124

126

136

139–140

145

152

171–172

175

188

218

222

226

12

14

16

30

66

76

82

100

125

integral

230 manual

205

proportional

218 rate

14

reset

13

171

selection

77

226

model reference control

196

move-suppression parameters

198

multiple input, single output (MISO)

143

multiplexer

140 This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms multivariable control

Links 169–170

174

182

190

192–194

196

173

179

181

185–186

188

194–195

199

231

198

N negative feedback negative interaction

6

noise

209

nonlinear feedforward compensation

158

nonlinear gain nonlinearity

101–102

118

224

58

148

221

174

181

12

26–27

78

126

139

226

78

90

174–178

180

199–200

230

33–36

58

61–63

67

74

86

103

110

O off-line offset

on-off controllers open-loop gain

open-loop test

optimizing feedback loops

112

oscillations

203

output

184

2

4–7

12–14

16

18–20

25–27

30

34–37

39–41

50–51

54

61–63

65–76

80

82

85

95–98

100–104

106

108

111–114

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

output (Cont.)

overshoot

116

118

122

125

128

131

133

135–137

139

144

146

148–150

152–153

155–159

162–163

169–171

173–174

176–179

181

186

188

190

194

197–199

202

204

218–219

221–222

224

229–230

63

69–71

100

125

127

132

136–137

139

173

188

194–195

210–211

222

226

232

173–180

183

185–186

188–189

195

199–200

69

194

P pairing

230 parallel

45 225

parallel paths

149

PD controller

78

percent controller output

17

perfect control performance

188

144

166

227

7

28–29

50

57–59

64

66–69

72

74

76

95–96

99–100

107

110–113

116

121

123–125

127

133

140

144

151

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

performance (Cont.) 158

165

169

173–174

181

185

188

210

221–222

25–26

30

58

61–63

74

79

85–86

92

97

105–106

109–110

115–116

132

192

223

226 pH control

101

PI controller

PID

8

PID algorithm

96

PID controller

11–12

17

21

24–27

30

58

61–64

72

76

92–93

106

110

114

132

223

173

179

185

195

230

80–81

86

92

112

125

136

pneumatic

96

positive interaction

pressure control

primary controller

122

problematic loops

206

process dead time

37

48

59

76

106

109

111

113

225

33

50

52–53

55

57–59

65

68

109

122

204

208

220–221

process gain

232 process noise

209

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

process nonlinearity

33

51

74

process time constant

37

41

60

86

91

106

108

111

222

1–3

5–8

12–19

25

34–36

41

46

64

66–67

71

76

78

82

88–89

91–92

95–98

100

102

106

110

112

115

118

121

125–128

131

136

139–140

144–145

149

151–152

158–159

166

169–174

176

178–179

181

187–188

192–193

197–199

205–206

208–209

212–213

217

219

221–222

224

226–227

process variable

230 processing frequency programmable logic controllers (PLC)

111 96

proportional band

17–18

78

82

proportional controller

21–23

25–26

29

78

85–86

90

92

102

223

12

17

19

22

27–28

58

65

68

78

80

82

85–86

97

101

118

125–126

133

218

221

226

proportional gain

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms proportional kick proportional mode

proportional-integral controller

proportional-integral-derivative controller

Links 100

118

224

12–14

16

66

76

82

100

125

218

18

27

128

223

8

11

18

28–29

87

91

101

224

96–98

100

224

3

5

12–17

25–27

30

34

37–39

48

61–64

96–99

102

145–146

149–150

170–172

175–177

187–188

209–210

218–219

23

25

58

74

133

24

26

61–63

132

80

96 proportional-only controller proportional-on-PV pulse PV

82 100

Q QDR response

QDR tuning

quarter-decay ratio (QDR) response

28

58

R rate time ratio control

reactor

14

58

7–8

143–144

148–149

166

179

227

9

41

54

70–73

122–123

126

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

reactor (Cont.) 129

131–133

135

137

139

218

175–180

183

185–186

189

193–196

198–200

174

178

185

138–139

226

reset time

13

58

reset windup

51

57

69–71

74

76

121

128

130

136

138

140–141

222

42–43

45

86

19

217

relative gain

230–231 matrix reset feedback

resistance resistance temperature device (RTD)

89

reverse

6

reverse action

6

running away

19

S sample time

106

225

sampling frequency

103

107

111

131 sampling period saturation

secondary controller self-regulating

91

104

50–51

69–70

138

188

136

122 6

35

41

48

60

78

82

87

108

123–124

127–128

145

158

164

206

223

226

205 sensor

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms sensor time constant sensor/transmitter

Links 87

89

128

3

34–35

37

58

79

91

45

48–49

98

127–128

226

3

5–7

12–19

22–24

27

30

61–71

75–76

78–80

82

86

89

92

96–98

100–102

106

113

116

118

122–123

125–128

131–133

135–137

139

144–148

150–152

158–159

162

166

170–171

173

181

186

189–190

192–193

198

203

209–210

213

222–224

226–227

232

80

100

171 series

set point

set point element

146

shrink

72

simple lags

41

simulation of tuning response

211

simulation response

211

single-mode controller

18

slave

67

slave controller

18

slow sampling

108–109

Smith Predictor

113–114

119

225

7

11

14

16

19

21

78

80

124–126

stability

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

stability (Cont.) 165

188

192

205 static compensation static compensator

163–164

187

152

static friction

80

steady-state

52

55

71

78

158

160

165

172

174–175

177–178

228

158

178

180–181

148

160

steady-state gain

195 steam heater

115–116

step size

204

step test

33

35–36

39–41

48

53–54

57

68

103

106

131–132

202

204

207

210

212

4

6

19

25

35

70

72

87

90–91

108–110

117

123–124

129

139

160

165

167

223

219 stiction

206

subcritical

46

swell

72

T temperature control

temperature-to-flow control scheme three-mode controller

128 18

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms tight control

time constant

time delay

Links 68

81

83

87

92

206

33–34

37–46

48

50

53–55

57–60

62–64

68

74

76

78–80

82

85–86

89

91

93

97

103–106

108

111–113

115

119

128–129

132

135–136

144

151

153

156

189

191

194

207

219–223

227–228

232

46–47

91

145

40

46–48

228 transducer

5

transfer function

113

transportation lag

36 163

tuning

1

tuning for robustness

211

tuning methods

210

tuning parameter

2–3

12–14

17

24–29

61–63

67–68

76

104

107

109

114

116

119

126

193

198

206–207

209

212

222

two-mode controller two-point method

18 181

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

U ultimate gain

ultimate period

uncontrollability

uncontrollable process

21

24–25

27

29

34

58

76

103

21

23

25

27

31

76

103

221

59–62

67–69

76

91

104

111

144

221–222

64

67

69

19–21

41

53

86

188

203

74 unstable

232

V vacuum

130

226

valve characteristics

50

conductance

50

gain

50

hysteresis

69

valve position control

112

velocity

139

80–81

W windup

51

57

69–71

74

76

121

128

130

136

138–141

222

226

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Z Ziegler and Nichols

23–24

38

74

This p a g e ha s b e e n re fo rma tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

58