Handbook of Pi And Pid Controller Tuning Rules 2nd edition

Handbook of

Pi and P I D Controller Tuning Rules 2nd Edition

Aidan O*Dwyer Imperial College Press

Handbook of

PI and PID Controller Tuning Rules 2nd Edition

This page is intentionally left blank

Handbook of

PI

and P I D

Controller Tuning Rules 2nd Edition

Aidan O'Dwyer Dublin Institute of Technology, Ireland

ICP

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HANDBOOK OF PI AND PID CONTROLLER TUNING RULES (2nd Edition) Copyright © 2006 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 1-86094-622-4

Printed in Singapore by B & JO Enterprise

Dedication

Once again, this book is dedicated with love to Angela, Catherine and Fiona and to my parents, Sean and Lillian.

This page is intentionally left blank

Preface

Proportional Integral (PI) and Proportional Integral Derivative (PID) controllers have been at the heart of control engineering practice for seven decades. However, in spite of this, the PID controller has not received much attention from the academic research community until the past fifteen years, when work by K.J. Astrom, T. Hagglund and F.G. Shinskey, among others, has sparked a revival of interest in the use of this "workhorse" of controller implementation. There is strong evidence that PI and PID controllers remain poorly understood and, in particular, poorly tuned in many applications. It is clear that the many controller tuning rules proposed in the literature are not having an impact on industrial practice. One reason is that the tuning rules are not very accessible, being scattered throughout the control literature; in addition, the notation used is not unified. The purpose of this book is to bring together and summarise, using a unified notation, tuning rules for PI and PID controllers. The author restricts the work to tuning rules that may be applied to the control of processes with time delays (dead times); in practice, this is not a significant restriction, as most process models have a time delay term. It is the author's belief that this book will be useful to control and instrument engineering practitioners and will be a useful reference for students and educators in universities and technical colleges. I would like to thank the School of Control Systems and Electrical Engineering, Dublin Institute of Technology, for providing the facilities needed to complete the book. Finally, I am deeply grateful to my mother, Lillian and my father, Sean, for their inspiration and support over many

VII

viii

Handbook of PI and PID Controller Tuning Rules

years and to my family Angela, Catherine and Fiona, for their love and understanding. Aidan O'Dwyer

Contents

Preface

vii

1. Introduction 1.1 Preliminary Remarks 1.2 Structure of the Book

1 1 2

2. Controller Architecture 2.1 Introduction 2.2 PI Controller Structures 2.3 PID Controller Structures 2.3.1 Ideal PID controller structure and its variations 2.3.2 Classical PID controller structure and its variations 2.3.3 Non-interacting PID controller structure and its variations 2.3.4 Other PID controller structures 2.3.5 Comments on the PID controller structures 2.4 Process Modelling 2.5 Organisation of the Tuning Rules 3. Tuning Rules for PI Controllers 3.1 FOLPD Model 3.1.1 Ideal controller - Table 3 3.1.2 Ideal controller in series with a first order lag -Table 4 3.1.3 Ideal controller in series with a second order filter - Table 5 3.1.4 Controller with set-point weighting - Table 6 3.1.5 Controller with proportional term acting on the output 1 - Table 7 3.1.6 Controller with proportional term acting on the output 2 - Table 8 ix

5 5 6 7 7 11 12 17 19 20 24 26 26 26 61 62 63 65 66

x

Handbook of PI and P1D Controller Tuning Rules 3.2 FOLPD Model with a Positive Zero 3.2.1 Ideal controller - Table 9 3.2.2 Ideal controller in series with a first order lag - Table 10 3.3 FOLPD Model with a Negative Zero 3.3.1 Ideal controller in series with a first order lag -Table 11 3.4 Non-Model Specific 3.4.1 Ideal controller - Table 12 3.4.2 Controller with set-point weighting - Table 13 3.5 IPD Model 3.5.1 Ideal controller - Table 14 3.5.2 Ideal controller in series with a first order lag -Table 15 3.5.3 Controller with set-point weighting - Table 16 3.5.4 Controller with proportional term acting on the output 1 - Table 17 3.5.5 Controller with proportional term acting on the output 2 -Table 18 3.5.6 Controller with a double integral term - Table 19 3.6 FOLIPD Model 3.6.1 Ideal controller - Table 20 3.6.2 Controller with set-point weighting - Table 21 3.6.3 Controller with proportional term acting on the output 1 - Table 22 3.6.4 Controller with proportional term acting on the output 2 - Table 23 3.7 SOSPD Model 3.7.1 Ideal controller - Table 24 3.7.2 Controller with set-point weighting - Table 25 3.8 SOSIPD Model - Repeated Pole 3.8.1 Controller with set-point weighting - Table 26 3.9 SOSPD Model with a Positive Zero 3.9.1 Ideal controller - Table 27 3.10 SOSPD Model (repeated pole) with a Negative Zero 3.10.1 Ideal controller in series with a first order lag - Table 28 3.11 Third Order System plus Time Delay Model 3.11.1 Ideal controller - Table 29 3.11.2 Controller with set-point weighting - Table 30 3.12 Unstable FOLPD Model 3.12.1 Ideal controller-Table 31

67 67 68 69 69 70 70 75 76 76 83 84 85 87 88 89 89 92 94 101 102 102 120 122 122 123 123 125 125 126 126 128 130 130

Contents

3.12.2 Controller with set-point weighting - Table 32 3.12.3 Controller with proportional term acting on the output 1 - Table 33 3.12.4 Controller with proportional term acting on the output 2 - Table 34 3.13 Unstable FOLPD Model with a Positive Zero 3.13.1 Ideal controller - Table 35 3.13.2 Ideal controller in series with a first order lag -Table 36 3.13.3 Controller with set-point weighting - Table 37 3.14 Unstable SOSPD Model (one unstable pole) 3.14.1 Ideal controller-Table 38 3.15 Unstable SOSPD Model with a Positive Zero 3.15.1 Ideal controller - Table 39 3.15.2 Controller with set-point weighting - Table 40 3.16 Delay Model 3.16.1 Ideal controller - Table 41 3.17 General Model with a Repeated Pole 3.17.1 Ideal controller - Table 42 3.18 General Model with Integrator 3.18.1 Ideal controller - Table 43 4. Tuning Rules for PID Controllers 4.1 FOLPD Model 4.1.1 Ideal controller - Table 44 4.1.2 Ideal controller in series with a first order lag -Table 45 4.1.3 Ideal controller in series with a second order filter -Table 46 4.1.4 Ideal controller with weighted proportional term -Table 47 4.1.5 Ideal controller with first order filter and set-point weighting 1 - Table 48 4.1.6 Controller with filtered derivative - Table 49 4.1.7 Blending controller - Table 50 4.1.8 Classical controller 1 - Table 51 4.1.9 Classical controller 2 - Table 52 4.1.10 Series controller (classical controller 3) - Table 53 4.1.11 Classical controller 4 - Table 54 4.1.12 Non-interacting controller 1 - Table 55 4.1.13 Non-interacting controller 2a - Table 56 4.1.14 Non-interacting controller 2b - Table 57

XI

135 137 140 141 141 142 143 145 145 147 147 148 149 149 152 152 153 153 154 154 154 181 183 185 186 187 193 194 208 209 211 212 213 215

Handbook of PI and PID Controller Tuning Rules

4.1.15 Non-interacting controller based on the two degree of freedom structure 1 - Table 58 4.1.16 Non-interacting controller based on the two degree of freedom structure 2 - Table 59 4.1.17 Non-interacting controller based on the two degree of freedom structure 3 - Table 60 4.1.18 Non-interacting controller 4 - Table 61 4.1.19 Non-interacting controller 5 - Table 62 4.1.20 Non-interacting controller 6 (I-PD controller) - Table 63 4.1.21 Non-interacting controller 7 -Table 64 4.1.22 Non-interacting controller 11 - Table 65 4.1.23 Non-interacting controller 12 - Table 66 4.1.24 Industrial controller - Table 67 Non-Model Specific 4.2.1 Ideal controller - Table 68 4.2.2 Ideal controller in series with a first order lag - Table 69 4.2.3 Ideal controller in series with a second order filter -Table 70 4.2.4 Ideal controller with weighted proportional term - Table 71 4.2.5 Controller with filtered derivative - Table 72 4.2.6 Ideal controller with set-point weighting 1 - Table 73 . . . . 4.2.7 Classical controller 1 - Table 74 4.2.8 Series controller (classical controller 3) - Table 75 4.2.9 Classical controller 4 - Table 76 4.2.10 Non-interacting controller based on the two degree of freedom structure 1 - Table 77 4.2.11 Non-interacting controller 4 - Table 78 4.2.12 Non-interacting controller 9 - Table 79 BPD Model 4.3.1 Ideal controller - Table 80 4.3.2 Ideal controller in series with a first order lag -Table 81 4.3.3 Ideal controller with first order filter and set-point weighting 2 - Table 82 4.3.4 Controller with filtered derivative - Table 83 4.3.5 Controller with filtered derivative and dynamics on the controlled variable - Table 84 4.3.6 Classical controller 1 - Table 85 4.3.7 Classical controller 2 - Table 86

217 222 223 224 226 227 230 231 232 233 235 235 241 243 245 246 252 253 254 255 256 257 258 259 259 262 263 264 265 266 268

Contents

4.3.8 Classical controller 4 - Table 87 4.3.9 Non-interacting controller based on the two degree of freedom structure 1 - Table 88 4.3.10 Non-interacting controller based on the two degree of freedom structure 3 - Table 89 4.3.11 Non-interacting controller 4 - Table 90 4.3.12 Non-interacting controller 6 (I-PD controller) - Table 91 4.3.13 Non-interacting controller 8 -Table 92 4.3.14 Non-interacting controller 10 -Table 93 4.3.15 Non-interacting controller 12 -Table 94 4.4 FOLIPD Model 4.4.1 Ideal controller - Table 95 4.4.2 Ideal controller in series with a first order lag - Table 96 4.4.3 Ideal controller with weighted proportional term -Table 97 4.4.4 Controller with filtered derivative - Table 98 4.4.5 Controller with filtered derivative with set-point weighting 1 - Table 99 4.4.6 Controller with filtered derivative with set-point weighting 3 - Table 100 4.4.7 Ideal controller with set-point weighting 1 - Table 101 . . . 4.4.8 Classical controller 1 - Table 102 4.4.9 Classical controller 2 - Table 103 4.4.10 Series controller (classical controller 3) - Table 104 4.4.11 Classical controller 4 - Table 105 4.4.12 Non-interacting controller based on the two degree of freedom structure 1 - Table 106 4.4.13 Non-interacting controller 4 - Table 107 4.4.14 Non-interacting controller 6 (I-PD controller) - Table 108 4.4.15 Non-interacting controller 8 - Table 109 4.4.16 Non-interacting controller 11 - Table 110 4.4.17 Non-interacting controller 12 - Table 111 4.4.18 Industrial controller - Table 112 4.4.19 Alternative controller 1 - Table 113 4.4.20 Alternative controller 2 - Table 114 4.5 SOSPD Model 4.5.1 Ideal controller - Table 115 4.5.2 Ideal controller in series with a first order lag -Table 116

xiii

269 270 272 273 274 276 277 278 279 279 282 284 285 286 287 289 290 292 293 294 295 297 298 303 304 305 306 307 308 309 309 332

Handbook of PI and PID Controller Tuning Rules 4.5.3 Ideal controller in series with a first order filter -Table 117 4.5.4 Ideal controller in series with a second order filter -Table 118 4.5.5 Controller with filtered derivative - Table 119 4.5.6 Controller with filtered derivative in series with a second order filter - Table 120 4.5.7 Ideal controller with set-point weighting 1 - Table 121 . . . 4.5.8 Ideal controller with set-point weighting 2 - Table 122 .. . 4.5.9 Classical controller 1 - Table 123 4.5.10 Classical controller 2 - Table 124 4.5.11 Series controller (classical controller 3) - Table 125 4.5.12 Non-interacting controller 1 - Table 126 4.5.13 Non-interacting controller based on the two degree of freedom structure 1 - Table 127 4.5.14 Non-interacting controller 4 - Table 128 4.5.15 Non-interacting controller 5 - Table 129 4.5.16 Non-interacting controller 6 - Table 130 4.5.17 Alternative controller 4 - Table 131 I2PD Model 4.6.1 Controller with filtered derivative with set-point weighting 2 - Table 132 4.6.2 Controller with filtered derivative with set-point weighting 4 - Table 133 4.6.3 Series controller (classical controller 3) - Table 134 4.6.4 Non-interacting controller based on the two degree of freedom structure 1 - Table 135 4.6.5 Industrial controller - Table 136 SOSIPD Model (repeated pole) 4.7.1 Non-interacting controller based on the two degree of freedom structure 1 - Table 137 SOSPD Model with a Positive Zero 4.8.1 Ideal controller - Table 138 4.8.2 Ideal controller in series with a first order lag -Table 139 4.8.3 Controller with filtered derivative - Table 140 4.8.4 Classical controller 1 - Table 141 4.8.5 Series controller (classical controller 3) - Table 142 4.8.6 Classical controller 4 - Table 143 4.8.7 Non-interacting controller 1 -Table 144 4.8.8 Non-interacting controller based on the two degree of freedom structure 1 - Table 145

336 337 338 339 340 341 342 351 352 354 363 368 370 371 372 374 374 376 378 379 380 381 381 383 383 385 386 387 388 389 390 391

Contents

4.9 SOSPD Model with a Negative Zero 4.9.1 Ideal controller - Table 146 4.9.2 Controller with filtered derivative - Table 147 4.9.3 Classical controller 1 - Table 148 4.9.4 Classical controller 4 - Table 149 4.9.5 Non-interacting controller 1 - Table 150 4.9.6 Non-interacting controller based on the two degree of freedom structure 1 - Table 151 4.10 Third Order System plus Time Delay Model 4.10.1 Ideal controller - Table 152 4.10.2 Ideal controller in series with a first order lag - Table 153 4.10.3 Controller with filtered derivative - Table 154 4.10.4 Non-interacting controller based on the two degree of freedom structure 1 - Table 155 4.11 Unstable FOLPD Model 4.11.1 Ideal controller - Table 156 4.11.2 Ideal controller in series with a first order lag - Table 157 4.11.3 Ideal controller with set-point weighting 1 - Table 158 . . . 4.11.4 Classical controller 1 -Table 159 4.11.5 Series controller (classical controller 3) - Table 160 4.11.6 Non-interacting controller based on the two degree of freedom structure 1 - Table 161 4.11.7 Non-interacting controller 3 - Table 162 4.11.8 Non-interacting controller 8 - Table 163 4.11.9 Non-interacting controller 10 - Table 164 4.11.10 Non-interacting controller 12 - Table 165 4.12 Unstable SOSPD Model (one unstable pole) 4.12.1 Ideal controller - Table 166 4.12.2 Ideal controller in series with a first order lag - Table 167 4.12.3 Ideal controller with set-point weighting 1 - Table 168 . . . 4.12.4 Classical controller 1 - Table 169 4.12.5 Series controller (classical controller 3) -Table 170 4.12.6 Non-interacting controller 3 - Table 171 4.12.7 Non-interacting controller 8 - Table 172 4.13 Unstable SOSPD Model (two unstable poles) 4.13.1 Ideal controller - Table 173 4.13.2 Ideal controller with set-point weighting 1 - Table 174 .. . 4.14 Unstable SOSPD Model with a Positive Zero

xv

392 392 393 394 395 396 397 398 398 399 400 401 403 403 408 410 414 415 416 420 421 423 425 427 427 430 431 432 433 434 439 442 442 444 445

xvi

Handbook of PI and PID Controller Tuning Rules

4.14.1 Ideal controller in series with a first order lag -Table 175 4.14.2 Non-interacting controller based on the two degree of freedom structure 1 - Table 176 4.15 Delay Model 4.15.1 Ideal controller - Table 177 4.15.2 Ideal controller in series with a first order lag -Table 178 4.15.3 Classical controller 1 -Table 179 4.16 General Model with a Repeated Pole 4.16.1 Ideal controller - Table 180 4.16.2 Ideal controller in series with a first order lag -Table 181 4.17 General Stable Non-Oscillating Model with a Time Delay 4.17.1 Ideal controller - Table 182 4.18 Fifth Order System plus Delay Model 4.18.1 Ideal controller - Table 183 4.18.2 Controller with filtered derivative - Table 184 4.18.3 Non-interacting controller 7 - Table 185 5. Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers 5.1 Introduction 5.2 The Analytical Determination of Gain and Phase Margin 5.2.1 PI tuning formulae 5.2.2 PID tuning formulae 5.3 The Analytical Determination of Maximum Sensitivity 5.4 Simulation Results 5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for all Values of Delay 5.5.1 PI controller design 5.5.1.1 Processes modelled in FOLPD form 5.5.1.2 Processes modelled in IPD form 5.5.2 PID controller design 5.5.2.1 Processes modelled in FOLPD form - classical controller 1 5.5.2.2 Processes modelled in SOSPD form - series controller 5.5.2.3 Processes modelled in SOSPD form with a negative zero - classical controller 1 5.5.3 PD controller design 5.6 Conclusions

445 447 449 449 450 451 452 452 453 454 454 455 455 457 460

462 462 463 463 466 469 470 475 475 475 477 480 480 482 482 483 484

Contents

xvii

Appendix 1 Glossary of Symbols Used in the Book

485

Appendix 2 Some Further Details on Process Modelling

493

Bibliography

507

Index

535

Chapter 1

Introduction

1.1 Preliminary Remarks The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to compensate most practical industrial processes has led to their wide acceptance in industrial applications. Koivo and Tanttu (1991), for example, suggest that there are perhaps 510% of control loops that cannot be controlled by single input, single output (SISO) PI or PID controllers; in particular, these controllers perform well for processes with benign dynamics and modest performance requirements (Hwang, 1993; Astrom and Hagglund, 1995). It has been stated that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers (Bialkowski, 1996) and that, in process control applications, more than 95% of the controllers are of PID type (Astrom and Hagglund, 1995). The PI or PID controller implementation has been recommended for the control of processes of low to medium order, with small time delays, when parameter setting must be done using tuning rules and when controller synthesis is performed either once or more often (Isermann, 1989). However, despite decades of development work, surveys indicating the state of the art of control industrial practice report sobering results. For example, Ender (1993) states that, in his testing of thousands of control loops in hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the automatic controllers are poorly tuned). The situation l

2

Handbook of PI and PID Controller Tuning Rules

does not appear to have improved in recent years as Van Overschee and De Moor (2000) report that 80% of PID controllers are badly tuned; 30% of PID controllers operate in manual with another 30% of the controlled loops increasing the short term variability of the process to be controlled (typically due to too strong integral action). The authors state that 25% of all PID controller loops use default factory settings, implying that they have not been tuned at all. These and other surveys (well summarised by Yu, 1999, pages 1-2) show that the determination of PI and PID controller tuning parameters is a vexing problem in many applications. The most direct way to set up controller parameters is the use of tuning rules; obviously, the wealth of information on this topic available in the literature has been poorly communicated to the industrial community. One reason is that this information is scattered in a variety of media, including journal papers, conference papers, websites and books over a period of seventy years. The author has recorded 408 separate sources of tuning rules since the first such rule was published by Callender et al. (1935/6). In a striking statistic, 293 sources of tuning rules have been recorded since 1992, reflecting the upsurge of interest in the use of the PID controller recently. The purpose of this book is to bring together, in summary form, the tuning rules for PI and PID controllers that have been developed to compensate SISO processes with time delay. Tuning rules for the variations that have been proposed in the 'ideal' PI and PID controller structure are included. Considerable variations in the ideal PID controller structure, in particular, are encountered; these variations are explored in more detail in Chapter 2. 1.2 Structure of the Book Tuning rules are set out in the book in tabular form. This form allows the rules to be represented compactly. The tables have four or five columns, according to whether the controller considered is of PI or PID form, respectively. The first column in all cases details the author of the rule and other pertinent information. The final column in all cases is labelled "Comment"; this facilitates the inclusion of information about the tuning

Chapter 1: Introduction

3

rule that may be useful in its application. The remaining columns detail the formulae for the controller parameters. Chapter 2 explores the range of PI and PID controller structures proposed in the literature. It is often forgotton that different manufacturers implement different versions of the PID controller algorithm (in particular); therefore, controller tuning rules that work well in one PID architecture may work poorly on another. This chapter also details the process models used to define the controller tuning rules. Chapters 3 and 4 of the book detail, in tabular form, tuning rules for setting up, respectively, PI controllers and PID controllers (and their variations), for a wide variety of process models. P controller and I controller tuning rules are also defined (as subsets of PI controller tuning rules), and PD controller tuning rules are defined (as a subset of PID controller tuning rules), for processes whose model includes an integrator. One hundred and eighty-three such tables are provided altogether. To allow the reader to access data readily, the author has arranged that each table start on its own page of the book; each table is preceded by the controller used, together with a block diagram showing the unity feedback closed loop arrangement of the controller and process model. In Chapter 5 of the book, analytical calculations of the gain and phase margins of a large sample of PI and PID controller tuning rules are determined, when the process is modelled in first order lag plus time delay (FOLPD) form, at a range of ratios of time delay to time constant of the process model. Results are given in graphical form. An important feature of the book is the unified notation that is used for the tuning rules; a glossary of the symbols used is provided in Appendix 1. Appendix 2 outlines the range of methods that are used to determine process model parameters; this information is presented in summary form, as this topic could provide data for a book in itself. However, sufficient information, together with references, is provided for the interested reader. Finally, a comprehensive reference list is provided. In particular, the author would like to recommend the contributions by McMillan (1994), Astrom and Hagglund (1995), Shinskey (1994), (1996), Tan et al. (1999a), Yu (1999), Lelic and Gajic (2000) and Ang et al. (2005) to the

4

Handbook of PI and PID Controller Tuning Rules

interested reader, which treat comprehensively the wider perspective of PID controller design and application.

Chapter 2

Controller Architecture

2.1 Introduction The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain as follows: U(s) = G c (s)E(s)

(2.1)

G c (s) = K c (l + -^- + Tds)

(2.2)

with

and with Kc = proportional gain, Tj = integral time constant and Td = derivative time constant. If T; = oo and Td - 0 (i.e. P control), then it is clear that the closed loop measured value, y, will always be less than the desired value, r (for processes without an integrator term, as a positive error is necessary to keep the measured value constant, and less than the desired value). The introduction of integral action facilitates the achievement of equality between the measured value and the desired value, as a constant error produces an increasing controller output. The introduction of derivative action means that changes in the desired value may be anticipated, and thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID controller allows contributions from present controller inputs, past controller inputs and future controller inputs. Many variations of the PI and, in particular, the PID controller structure have been proposed. As Tan et al. (1999a) suggest, one

5

6

Handbook of PI and PID Controller Tuning Rules

important reason for the non-standard structures is due to the transition of the controllers from pneumatic implementation through electronic implementation to the present microprocessor implementation. The variations in the controller structures are detailed below. 2.2 PI Controller Structures Tuning rules have been detailed for seven PI controller structures: 1. Ideal controller

G c (s) = K(

i + -L

(2.3)

T s

V

i 7

2. Ideal controller in series with a first order lag: G c (s) = Kc 1 +

T:S

1

1 + Tfs

(2.4)

3. Ideal controller in series with a second order filter (also labelled the 'generalised PID' controller (Lee and Shi, 2002)): 1 1 + b fl s + b f2 s2 ^ G c (s) = Kc 1 + v T>sy l + a fl s + a f2 s j

(2.5)

4. Controller with set-point weighting: U(s) = Kc 'i-L^ E(s)-aK c R(s) V

T

(2.6)

is/

5. Controller with proportional term acting on the output 1: U(s) = ^ E ( s ) - K c Y ( s )

(2.7)

Ts 6. Controller with proportional term acting on the output 2: r 1\ U(s) = K 1+ E(s)-K,Y(s) V Tfsy 7. Controller with a double integral term:

(2.8)

Chapter 2: Controller Architecture

G c (s) = Kc

'

1 1+

V

1 ^

(2.9)

+ ^il

S

7

T i2 S j

2.3 PID Controller Structures Forty-six such structures have been considered. The labelling of these structures has not been consistent in the literature; for example, the first PID controller structure considered (labelled the ideal controller below) has also been labelled the 'non-interacting' controller (McMillan, 1994), the 'ISA' algorithm (Gerry and Hansen, 1987) or the 'parallel noninteracting' controller (Visioli, 2001). The controller structures have been divided into four types, as described below. 2.3.1 Ideal PID controller structure and its variations f

1. Ideal controller:

G c (s) = Kc

A 1 1 + — + Tds

V

T s

.

(2.10)

j

A variation of the controller is labelled the 'parallel' controller structure (McMillan, 1994). This variation has also been labelled the 'ideal parallel', 'noninteracting', 'independent' or 'gain independent' algorithm: G c (s) = K c + - i - + Tds

(2.11)

TiS

The controller structure is used in the following products: (a) Allen Bradley PLC5 product (McMillan, 1994) (b) Bailey FC19 PID algorithm (EZYtune, 2003) (c) Fanuc Series 90-30 and 90-70 independent form PID algorithm (EZYtune, 2003) (d) Intellution FIX products (McMillan, 1994) (e) Honeywell TDC3000 Process Manager Type A, non-interactive mode product (ISMC, 1999) (f) Leeds and Northrup Electromax 5 product (Astrom and Hagglund, 1988)

Handbook of PI and PID Controller Tuning Rules

(g) Yokogawa Field (EZYtune, 2003).

Control

Station

(FCS) PID algorithm

2. Ideal controller in series with a first order lag: f

\

G c (s) = Kc l + ^ v

T

+ Tds

is

;

1

Tfs + l

(2.12)

3. Ideal controller in series with a first order filter: / b s +l 1 G c (s) = Kt 1 + — + Tds fl V Tis J a fl s + l

(2.13)

4. Ideal controller in series with a second order filter: / \ l + b fl s 1 G c (s) = Kt 1 + — + Tds T s v i J l + a n s + a f2 s

(2.14)

5. Ideal controller with weighted proportional term: (

1

A

G c (s) = Kt b + — + THs T S V i J

(2.15)

6. Ideal controller with first order filter and setpoint weighting 1: r > 1 1 (2.16) U(s) = Kc i + — + T d s R(S)1±^_Y(S) 1 + sT V Tis J Tfs + 1 7. Ideal controller with first order filter and setpoint weighting 2: / \ 1 1 U(s) = Kc R<.)l±2^i-Y(., K0Y(s) Tfs + 1 V Tis l + sTr (2.17) 8. Ideal controller with first order filter and dynamics on the controlled variable: / A_ J ^ T V ( l K ) 1 _JC_ t ± U(s) = Kt 1 + — + Tds Tfs + 1 • Ts + 1 Ts + 1 9. Controller with filtered derivative:

Chapter 2: Controller Architecture

9

G c (s) = Kc 1 + — + - d (2.19) T s * l+ s ^ N This structure is used in the following products: (a) Bailey Net 90 PID error input product with N = 10 (McMillan, 1994) and FC156 Independent Form PID algorithm (EZYtune, 2003) (b) Concept PIDP1 and PID1 PID algorithms (EZYtune, 2003) (c) Fischer and Porter DCU 3200 CON PID algorithm with N = 8 (EZYtune, 2003) (d) Foxboro EXACT I/A series PIDA product (in which it is an option labelled ideal PID) (Foxboro, 1994). (e) Hartmann and Braun Freelance 2000 PID algorithm (EZYtune, 2003) (f) Modicon 984 product with 2 < N < 30 (McMillan, 1994; EZYtune, 2003) (g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with N = 10 (ISMC, 1999) and the S7 FB41 CONTC PID product (EZYtune, 2003). 10. Controller with filtered derivative in series with a second order filter: (

\

l + b fl s + b f2 s

G c (s) = Kt T s

>

V

l + a fl s + a f2 s 2

1 + T< N

(2.20)

11. Controller with filtered derivative with setpoint weighting 1:

f

\ ,

1

G c (s) = K( 1 + — + T:S V

l + b fl s R(s)-Y(s) l + a fl s

T

dS %r -l + ' Ld N

s j

12. Controller with filtered derivative with setpoint weighting 2:

(2.21)

Handbook of PI and PID Controller Tuning Rules

10

f

l + b f l s + b f 2 s"

G c (s) = Kc T s

.

l + afls + af2s2

l+ ^ s N

-R(s)-Y(s)

(2.22)

13. Controller with filtered derivative with setpoint weighting 3:

r

\

G c (s) = K ( 1 + — + — h Xs . X 1+ ^ s v N ;

1+ b

»S+

b

"S22+b»S33R(s)-Y(s)A

(2.23)

14. Controller with filtered derivative with setpoint weighting 4: G c (s) = Kc

1+ —

Xs

+•

d

, . Td 1+ ^ s N

'1

+ b

"S

+ b -

^ + b " S 3 + b " S 4 4 R ( s ) - Y(s)'

1 + Sf-jS H S-f-^S

~H S ^ T S

+3.^48

j

(2.24) 15. Controller with filtered derivative and dynamics on the controlled variable: T s 1 d G c (s) = K t 1 E(s)-K0Y(s) 1+-Xs--^ , . d. Xd 1 + sN

(2.25)

16. Ideal controller with setpoint weighting 1: U(s) = K c (F p R(s) - Y ( s ) ) + ^ ( F i R ( s ) - Y(s))+ K c X d s(F d R(s) - Y(s)) (2.26) 17. Ideal controller with setpoint weighting 2: \ ( 1 1 U(s) = K, 1 + — + T d s R(s)-Y(s) T s V i j TiV+TjS +1

(2.27)

Chapter 2: Controller Architecture

11

18. Blending controller: f

G c (s) = Kc

1 Ts

>

(2.28) s This structure is used in the Yokogawa EFCS/EFCD Field Control Station product (Chen et al., 2001). 2.3.2 Classical PID controller structure and its variations 19. Classical controller 1: This controller is also labelled the 'cascade' controller (Witt and Waggoner, 1990), the 'interacting' or 'series' controller (Poulin and Pomerleau, 1996), the 'interactive' controller (Tsang and Rad, 1995), the 'rate-before-reset' controller (Smith and Corripio, 1997), the 'analog' controller (St. Clair, 2000) or the 'commercial' controller (Luyben, 2001). G c (s) = Kt ' l +

1 J'

l + STd

(2.29) T N The structure is used in the following products: (a) Honeywell TDC Basic/Extended/Multifunction Types A and B products with N = 8 (McMillan, 1994) (b) Toshiba TOSDIC 200 product with 3.33
Ts

12

Handbook of PI and PID Controller Tuning Rules

l + NTds T,Sy 1 + Tds j

20. Classical controller 2: G c (s) = Kc 1 + -

(2.30)

21. Series controller (Classical controller 3): This controller is also labelled the 'interacting' controller, the 'analog algorithm' (McMillan, 1994) or the 'dependent' controller (EZYtune, 2003). G c (s) = Kt 1 + T.Sy

(l + sTd)

(2.31)

The structure is used in the following products: (a) Turnbull TCS6000 series product (McMillan, 1994) (b) Alfa-Laval Automation ECA400 product (Astrom and Hagglund, 1995) (c) Foxboro EXACT 760/761 product (Astrom and Hagglund, 1995). 22. Classical controller 4: This controller is also labelled the 'interacting' controller (Fertik, 1975). 1

(2.32) l + - TdS 1+ s^ NJ The structure is used in the following products: (a) Bailey FC156 Classical Form PID product (EZYtune, 2003) (b) Fischer and Porter DCI 4000 PID algorithm (EZYtune, 2003)

G e (s) = Kt

2.3.3 Non-interacting PID controller structure and its variations 23. Non-interacting controller 1: This controller is also labelled the 'reset-feedback' controller (Huang et al., 1996). ]_

%-Y(s) 1+N This structure is used in the following products: U(s) = Kt 1 +

TiS,

E(s)

(2.33)

Chapter 2: Controller Architecture

13

(a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products with N = 0 (ISMC, 1999). (b) Moore Model 352 Single-Loop Controller product (Wade, 1994). 24. Non-interacting controller based on the parallel controller structure: 24a. Non-interacting controller 2a: 1 A

f

U(s) = Kt 1 +

E(s)-

Tds

(2.34)

Y(s)

N 24b. Non-interacting controller 2b: /

U(s) =

K

c+

v

1A E(s)T jSy

Tds

Y(s)

(2.35)

N

25. Non-interacting controller based on the two degree of freedom structure 1: This controller is also labelled the 'm-PID' controller (Huang et al, 2000) the 'ISA-PID' controller (Leva and Colombo, 2001) and the 'P-I-PD (only P is DOF) incomplete 2DOF algorithm' by Mizutani and Hiroi (1991). U(s) = Kt ,

1

T

dS

E(s)-K c ct +

PT d s 1+

R(s)

(2.36)

^

N N v This structure is used in the following products: (a) Bailey Net 90 PID PV and SP product (McMillan, 1994) (b) Yokogawa SLPC products with <x = - l , (B = - 1 , N=10 (McMillan, 1994) (c) Omron E5CK digital controller with p = 1 and N=3 (ISMC, 1999). 26. Non-interacting controller based on the two degree of freedom structure 2:

Handbook of PI and PID Controller Tuning Rules

14

f

1 + - U - ^E(s)-K (

U(s) = K(

T 8

«

l+ ^ s N ,

I

\ PTds

a+ V

X

R(s)

1+ T s

l+ ^ s N

i

(2.37)

27. Non-interacting controller based on the two degree of freedom structure 3: U(s) = K j ( b - l ) + (c-l)T d s]R(s) ( 1 + K, 1 + — + Tds R(s) T s v .

(l + sT f ) :

-Y(s)

(2.38)

28. PD-I-PD (only PD is DOF) incomplete 2DOF algorithm (Mizutani andHiroi, 1991): f

U(s) = Kc a-

_1_

V %Tds

1

Td_

1+-5-S

^

1 + Xs

R(s)

N , 1 1+— + T s

T dS —-

Y(s) (2.39) 1+ ^ s N However, no tuning rules are available for this controller structure. •K,

'

29. PI-PID (only PI is DOF) incomplete 2DOF algorithm (Mizutani and Hiroi, 1991): U(s) = K( a +

1 T;s

P TiS(l + TiS).

R(s)-K £ 1 +

T;s

+

=7-

Y(s)

1 + ^-s N

(2.40) However, no tuning rules are available for this controller structure.

Chapter 2; Controller Architecture

15

30. Super (or complete) 2DOF PID algorithm (Mizutani and Hiroi, 1991): U(s) = Kc a1 — J _ + _£?>_ Ts 1 + Ts I+3L N

R(s)

Y(s) (2.41)

-K, v

N

Mizutani and Hiroi (1991) state that 0 < a < 1, p > 1 and % ^ 1, and suggest starting values of a = 0.4, P = 1.35 and % = 1.25. However, no tuning rules are available for this controller structure. 31. Non-interacting controller 3: KTs U(s) = Kt 1 + — E(s)- c d Y(s) (2.42) . T iSy 1 + ^L N This structure is used in the following products: (a) Allen Bradley SLC5/02, SLC5/03, SLC5/04, PLC5 and Logix5550 products (EZYtune, 2003). (b) Modcomp product with N = 10 (McMillan, 1994). 32. Non-interacting controller 4: This controller is also labelled the 'PI+D' controller (Chen, 1996). U(s) = Kt 1 +

E(s)-K c T d sY(s)

(2.43)

This structure is used in the following products: (a) ABB 53SL6000 Product (ABB, 2001). (b) Genesis product (McMillan, 1994) (c) Honeywell TDC3000 Process Manager Type B, non-interactive mode product (ISMC, 1999) (d) Square D PIDR PID product (EZYtune, 2003)

16

Handbook of PI and PID Controller Tuning Rules

33. Non-interacting controller 5: U(s) = Kc b + E(s)-(c + Tds)Y(s) T,s

(2.44)

34. Non-interacting controller 6 (I-PD controller): (2.45)

U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s) T:S

This structure is used in the Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al, 1987) and the Honeywell TDC3000 Process Manager Type C non-interactive mode product (ISMC, 1999). 35. Non-interacting controller 7: K„

n

T s

'

Y(s)

(2.46)

U(s) = K c | l + - j - + T d s|E(s)-K i (l + Tdis)Y(s)

(2.47)

U(s) = - ^ E ( s ) - K , T:S

T

1- " N

36. Non-interacting controller 8: V

T,s

37. Non-interacting controller 9: \ / 1 U(s) = Kt 1 + — + Tds E(s)--^sY(s) N Ts

.

v

(2.48)

;

38. Non-interacting controller 10: U(s) = Kc

E(s)-K f -^-s + 1 Y(s) \

T s

39. Non-interacting controller 11:

i y

VTm

J

(2.49)

Chapter 2: Controller Architecture

U(s) = Kt

T=s

17

(l+T d s)E(s)-K i (l + Tdis)Y(s)

(2.50)

40. Non-interacting controller 12: f

1

A

is

j

U(s) = Kc T

V

E(s)-K,Y(s) 1 + Tds

(2.51)

41. Non-interacting controller 13: U(s) = Kc v

T s

Tds

E(s)-

Y(s)

1 + I l s + I^ s 2

;y

(2.52)

N 2N This structure is used in the Foxboro IA PID product, with 10 < N <50(EZYtune, 2003); however, no tuning rules are available for this controller structure. 2.3.4 Other PID controller structures 42. Industrial controller (Kaya and Scheib, 1988): (

U(s) = Kc 1+ — . T,Sy

^

1 +T s

R(s)-i±M Y (s) V

(2.53)

N

This structure is used in the following products: (a) Fisher-Rosemount Provox product with N = 8 (ISMC, 1999; McMillan, 1994) (b) Foxboro Model 761 product with N=10 (McMillan, 1994) (c) Fischer-Porter Micro DCI product with N = 0 (McMillan, 1994) (d) Moore Products Type 352 controller with 1 < N < 30 (McMillan, 1994) (e) SATT Instruments EAC400 product with N=8.33 (McMillan, 1994) (f) Taylor Mod 30 ESPO product with N=16.7 (McMillan, 1994)

18

Handbook of PI and PID Controller Tuning Rules

(g) Honeywell TDC3000 Process Manager Type B, interactive mode product with N = 10 (ISMC, 1999) f

V

1 + Ts

43. Alternative controller 1: G c (s) = Kc

1+ v

1 + Tds

(2.54)

^ N y\

N j

44. Alternative controller 2: 2

G c (s) = Kc

l + -^s N

„ 2

l + 0.5xms + 0.0833xm s

1 + TjS T

(2.55)

N

j \

45. Alternative controller 3: / K„

U(s) = - ^ R ( s ) - K ( 1+ T:S . T,sy

l + sTd

Y(s)

(2.56)

N ; This structure is used in the following products: (a) Honeywell TDC3000 Process Manager Type C, interactive mode product with N = 10 (ISMC, 1999) (b) Honeywell TDC3000 Universal, Multifunction and Advanced Multifunction products with N = 8 (ISMC, 1999) However, no tuning rules are available for this controller structure. 46. Alternative controller 4: U(s) = Kt

E(s) + N

R(s)

(2.57)

Chapter 2: Controller Architecture

19

2.3.5 Comments on the PID controller structures In some cases, one controller structure may be transformed into another; clearly, the ideal and parallel controller structures (equations (2.10) and (2.11)) are very closely related. It is shown by McMillan (1994), among others, that the parameters of the ideal PID controller may be worked out from the parameters of the series PID controller, and vice versa. The ideal PID controller is given in Equation (2.58) and the series PID controller is given in Equation (2.59). f

1

^

+ Tdps

(2.58)

T~I (l + sTds)

(2.59)

G cp (s) = Kcp 1 + G cs (s) = Kc 1 + Then, it may be shown that ( K

cp =

l + _d!L

X.

^•cs ' *ip ~~ V i s

+

*ds/'

*dp

_

X. X.S+X*

Similarly, it may be shown that, provided X- > 4T(d p • l

K„5 = 0.5K„

dP

1 +Jl-4-X.

Xdp X. = 0.5X, 1 + 1 1 - 4 L

Xds=0.5Xdp 1 - 1 - 4 -

dP

Astrom and Hagglund (1996) point out that the ideal controller admits complex zeroes and is thus a more flexible controller structure than the series controller, which has real zeroes; however, in the frequency domain, the series controller has the interesting interpretation that the zeroes of the closed loop transfer function are the inverse values of Tis and Tds. O'Dwyer (2001b) developed a comprehensive set of tuning rules for the series PID controller, based on the ideal PID controller; as these are not strictly original tuning rules, only representative examples are included in the relevant tables. In a similar manner, the parameters of an ideal controller in series with a first order filter may be determined from the parameters of the classical controller 1 (Witt and Waggoner, 1990) and the parameters of

20

Handbook of PI and PID Controller Tuning Rules

the non-interacting controller 10 structure may be determined from the parameters of the ideal PID controller (Kaya et al, 2003). 2.4 Process Modelling Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables (see Chapters 3 and 4). These modelling strategies are outlined in Appendix 2. The following models are defined: K e~STm 1. First order lag plus time delay (FOLPD) model ( G m (s) = — ) 2. FOLPD model with a positive zero ( G m ( s ) = -

mV

m3J

l + sT m l

K m ( l + sT m 3 )e3. FOLPD model with a negative zero ( G m (s) = — ^ ^ 1 + sT ml 4. Non-model specific K e"SXm 5. Integral plus time delay (IPD) model ( G m (s) = — 2 ) s 6. First order lag plus integral plus time delay (FOLIPD) model

)

(Gm(S)= v r v s(l + s T j

7. Second order system plus time delay (SOSPD) model K e~SXm K e~SXm (0 ( )= r0 w - ' T. 1 v;^ T .,. + ,° - -( 1 + T jxuT. I .)> 8. Integral squared plus time delay ( I 2 P D ) model ( G m ( s ) = —m-^ ) s 9. Second order system (repeated pole) plus integral plus time delay (SOSIPD) model ( G m (s) = fm& "" ) s(l + sT m ) 2

Chapter 2: Controller Architecture

10. SOSPD model with a positive zero (s)- K j l - s T ^ y " (G ( m ( ) (l + sTml)(l + s T m 2 ) ° r G m ( S ) 11. SOSPD model with a negative zero

21

K m (l-sT m3 )eTmlV+2^Tinls + l )

( G m ( s ) = K - ( l + S > > - " ; o r G m ( s ) , Km(l + STm3)e m (l + sTml)(l + sTm2) T m l 2 s 2 + 2^ m T m l s + l -ST„

12. Third order system plus time delay model 3 ST + bb,2Ss22 +b ,^ , -. K " mm\(l- +• b J,S + b3.sS )e" ,. " 1s + 3 (G m (s)= m A(l + a,s ' + a22 s,2 + 3a 3 sA or ) K e""m Gm(S)= ( l + sTml)(l+msTm2)(l + S T m 3 ) ) 13. Fifth order system plus delay model K m (l + bIs + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 ) e - ^ (G m (s) = r j 5 4 n ' \1 + a t s + a2sSTm + a3s + a 4 s + a5s J K e~ 14. Unstable FOLPD model (G m (s) = —* ) T m s-1 15. Unstable FOLPD model with a positive zero K m (l-sT m 3 )e(G m (s)= m \ mi ) Tms-1

16. Unstable SOSPD model (one unstable pole) K e"STm (Gm(s)=7™ ) ( T mlS-lX 1 + sTm2) 17. Unstable SOSPD model with a positive zero

(Gm(s)= J Y ^ T m 3 i e .) '~TmlV-2SmTmls + l 18. Unstable SOSPD model (two unstable poles) K e~STm (G (

19. Delay model (G m (s) = K m e _s ' m )

(s)= m ( )

-^-

)

(Tmls-lXTm2s-l))

K e"s'tm 20. General model with a repeated pole (G m (s) = — ) (l + s T J n

Handbook of PI and PID Controller Tuning Rules

22

21. General model with integrator K

n( T m ,i s + i )n( T m / s 2 + 2 ^„i T m 2 i s + i ) j

i

22. General stable non-oscillating model with a time delay Tables 1 and 2 show the number of tuning rules defined for each PI or PID controller structure and each process model. The following data is key to the process model type in Tables 1 and 2: Model 1: Stable FOLPD model Model 2: Non-model specific Model 3: IPD model Model 4: FOLIPD model Model 5: SOSPD model Model 6: Other models Table 1: PI controller structure and tuning rules - a summary Process model Controller Equation (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) Total

1

2

3

4

5

6

Total

172 3 2 7 1 2 0 187

42 0 0 2 0 0 0 44

46 2 0 4 9 1 1 63

10 0 0 3 14 1 0 28

32 0 0 3 0 0 0 35

62 2 0 13 7 2 0 86

364 7 2 32 31 6 1 443

Chapter 2: Controller Architecture

23

Table 2: PID controller structure and tuning rules - a summary 1

2

(2.10,2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) Subtotal

100 7 0 2 2 1 0 1 16 0 0 0 0 0 0 0 0 1 130

51 2 0 2 2 0 0 0 11 0 0 0 0 0 0 1 0 0 69

(2.29) (2.30) (2.31) (2.32) Subtotal

54 1 6 2 63

(2.33) (2.34) (2.35) (2.36) (2.37) (2.38) (2.42) (2.43) (2.44) (2.45) (2.46) (2.47) (2.48)

1 5 6 12 4 2 0 10 2 10 1 1 0

Process model Controller Equation

3

4

5

6

Ideal structure + variations 22 12 55 36 1 5 13 9 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 2 3 5 0 0 1 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0 0 2 0 1 0 0 1 0 1 9 0 0 2 0 0 0 0 0 24 27 77 62 Classical structure + variations 5 5 8 20 9 1 0 2 2 0 1 5 0 7 6 1 1 1 0 2 8 11 11 29 17 Non-interacting structure + variations 0 0 0 5 2 0 0 0 0 0 0 0 0 0 0 6 1 5 7 13 0 0 0 0 0 0 0 1 0 0 4 0 0 0 0 2 4 3 0 3 0 1 0 0 0 0 8 11 1 0 0 1 0 0 0 2 4 0 1 0 0 1 0 0 0

Total

276 37 1 5 5 1 1 1 39 1 1 1 2 2 1 12 2 1 389 101 6 25 7 139 8 5 6 44 4 3 4 22 3 30 2 8 1

Handbook of PI and PID Controller Tuning Rules

24 Process model Controller Equation (2.49) (2.50) (2.51) Subtotal

0 0 1 55

(2.53) (2.54) (2.55) (2.57) Subtotal

8 0 0 0 8

Total

2

1

|

256

3

4

6

5

Total

Non-interacting structure + variations (continued) 2 0 1 0 0 0 0 1 0 0 1 0 1 1 0 27 J 21 23 17 Other controller structures 1 0 0 1 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 2 1 0 4 0 2 |

85

|

59

|

59

|

125

|

107

|

3 1 4 148 10 2 1 2 75 691

Table 1 shows that 82% of tuning rules have been defined for the ideal PI controller structure, with 42% of tuning rules based on a FOLPD process model. The range of PID controller variations has lead to a less homogenous situation than for the PI controller; Table 2 shows that 40% of tuning rules have been defined for the ideal PID controller structure, with 37% of tuning rules based on a FOLPD process model. Many of the controller structures are used in a variety of manufacturers' products (as outlined in Section 2.3); clearly a range of tuning rules are available for use with these products. 2.5 Organisation of the Tuning Rules The tuning rules are organised in tabular form in Chapters 3 and 4. Within each table, the tuning rules are classified further; the main subdivisions made are as follows: (i) Tuning rules based on a measured step response (also called process reaction curve methods), (ii) Tuning rules based on minimising an appropriate performance criterion, either for optimum regulator or optimum servo action, (iii) Tuning rules that give a specified closed loop response (direct synthesis tuning rules). Such rules may be defined by specifying the

Chapter 2: Controller Architecture

25

desired poles of the closed loop response, for instance, though more generally, the desired closed loop transfer function may be specified. The definition may be expanded to cover techniques that allow the achievement of a specified gain margin and/or phase margin, (iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built in to the design process, (v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimate cycling methods), (vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarter decay ratio or to achieve magnitude and frequency information at a particular phase lag. Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot be considered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. In the tables, all symbols used are defined in Appendix 1.

Chapter 3

Tuning Rules for PI Controllers

3.1 FOLPD Model G m (s) =

K„e" l + sT„

3.1.1 Ideal controller G c (s) = Kt

T s

v

>v

Table 3; PI controller tuning rules - FOLPD model G ra (s) = Rule

Kc

Ti

Callender et al. (1935/6). Model: Method 1

'0.568/K m x m

Process reaction 3.64xm

2

0.690/K m x m

2-45xm

Decay ratio = 0.015; Period of decaying oscillation = 10.5xm ' Decay ratio = 0.043; Period of decaying oscillation = 6.28xn 26

l + sT„

Comment

^=- = 0.3 T m

Chapter 3: Tuning Rules for PI Controllers Rule Ziegler and Nichols (1942). Model: Method 2

T;

Kmtm

3.33xm

x

Quarter decay ratio.

lTm

x

T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

l

x2

0.1 < ^ 2 - < 1.0 T Chiene/a/. (1952)servo. Model: Method 2; 0.1 < ^ 2 - < 1.0 T

3

Coefficient values x x2 m i

x

0.68 0.70 0.72 0.74 0.76 0.79 0.81 0.84 0.87

7.14 4.76 3.70 3.03 2.50 2.17 1.92 1.75 1.61

3

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.90 0.93 0.96 0.99 1.02 1.06 1.09 1.13 1.17

K„

1.49 1.41 1.32 1.25 1.19 1.14 1.10 1.06 1.03

*m

x2=-

•6t m -1.2T m

x2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

1.20 1.28 1.36 1.45 1.53 1.62 1.71 1.81

1.00 0.95 0.91 0.88 0.85 0.83 0.81 0.80

^2->3.5

1.66xm

xm » Tra

3.32Tra

^m«Tm attributed to Oppelt

K C (1)

0.8T ra /K m x m 0.91T m /K ra x m

3t m 3.3xm

attributed to Rosenberg

0-6Tm mTm

4x m

0% overshoot

0.7Tm Kmxm

2-33xm

20% overshoot

1.17Tm

0% overshoot

Tm

20% overshoot

K

0.35Tm 0-6Tm K

raxm

. 5 7 0 8 ^ + 1 I recommended K (0

K,(1>

i

x

T

x, = 0 . 5 ^ + 0.1

Chienefa/. (1952)regulator. Model: Method 2;

2xm

Kmtm x

Moros (1999). Model: Method 1

Comment

Kc 0.9Tm

T

Hazebroek and Van derWaerden(1950). Model: Method 2

Oppelt(1951). Model: Method 2

27

0.77K„

Handbook of PI and PID Controller Tuning Rules

28 Rule

Kc

Ti

Comment

Reswick(1956). Model: Method 2; K c , Tj deduced

0.20/K m

0.20x m

0.30/K m

0.22T m

0% overshoot regulator 20% overshoot regulator 0% overshoot - servo

from

graphs.

i s - = 6.25 T

0.15/K m

0.16x m

0.20/K m

0.14x m

0 . 9 ^ + 0.083

4T(2) i

20% overshoot servo Quarter decay ratio;

m

Cohen and Coon (1953). Model: Method 2

J _

Xl T m

Two constraints method - Wolfe (1951). Model: Method 4

"

X

0 < - ^ _ < 1.0 T

2Tm

T

Km m

*1

Coefficient values x2

T

*1

x2

0.78 0.30

1.28 0.53

T

Decay ratio is as small as possible; minimum error integral (regulator mode).

m

0.2 0.5 Two constraints criterion - Murrill (1967) -page 356. Model: Method 5

4.4 1.8

0.925

3.23 2.27

/

1.0 5.0

N 0.946

x

X

l mJ

(

0 583

A '

1.078 [T m J

Quarter decay ratio; minimum error integral (servo mode). 0.1<-^2L<1.0 T m

0.8T m Heck (1969). Model: Method 2

K

3im

mTm

0

x

Integral controller

2KmTm

Representative coefficient values - from graph - "Stoning" X x2 x2 x2 2 IHL 1B. T T1 T T m

0.11 0.13 0.14 0.17

2.94 2.86 2.78 2.63

2\

3.33-

-0.31

•(2)

1 + 2.22-

0.20 0.25 0.29 0.33

m

2.50 2.38 2.35 2.30

0.40 0.50 0.56 0.63

2.17 2.13 2.08 2.00

0.71

1.96

Chapter 3: Tuning Rules for PI Rule

29

Controllers

Kc

Comment

Tf

Representative coefficient values - from graph - "Fuhrung" Heck (1969)continued.

T

x2

m

x2

T 0.11 0.13 0.14 0.17

0.20 0.25 0.29 0.33

3.33 3.08 2.94 2.82

1.2408K m

Borresen and Grindal (1990). Model: Method 2 Klein et al. (1992). Model: Method 1

1.0Tm

deduced from

graphs

0.71

2.27

Model: Method 3

Also described by ABB (2001)

Kmtm

3^ r a

0.28T m

0.53T ra

Km(tm+0.1Tra)

Time delay dominant processes

^m

3 0.333T m K

l a . > 0.33 T

T

mTm X

2tm

Coefficient values - servo tuning Tm/Tm

0.1 Coefficients of K C ,T;

2.63 2.47 2.41 2.35

0.5x m

T T

x,/Km Hay ( 1 9 9 8 ) - p a g e s 197-198. Model: Method 1;

0.40 0.50 0.56 0.63

0.65T m

Km

Sakai et al. (1989). Model: Method 2

x2 T

m

3.92 3.85 3.64 3.51 0.56

Fertik and Sharpe (1979).

McMillan ( 1 9 9 4 ) page 25. Model: Method 5 St. Clair ( 1 9 9 7 ) page 22. Model: Method 5

x2 T

T

0.125 0.167 0.25

0.5

i

x2

"Cm Am

6.0 4.5 3.0 2.0 0.8

9.5 7.5 5.4 3.4 1.7

0.1

x

0.125 0.167 0.25

0.5

x

x2

l

4.0 3.2 2.2 1.3 0.5

10.0

8.0 6.0 4.0 2.0

Coefficient values - regulator tuning ^m/Tm

0.1 0.125 0.167 0.25

0.5 Shinskey (2000), (2001). Model: Method 2

I

x2

9.5 7.0 5.0 3.4 1.8

3.2 3.2 3.0 2.8 2.0

X

tm/Xi 0.1 0.125 0.167 0.25 0.5

i

x2

6.8 5.6 4.3 2.8 1.2

2.8 2.8 2.6 2.3 1.6

x

0.667T m KmTm

3.78x ra

T m

Handbook of PI and PID Controller Tuning Rules

30

Rule

Kc 0.95Tm K m x ra

Liptak(2001). Model: Method 2

—

Minimum IAE Frank and Lenz (1969). Model: Method 1

4^ m

|x,+x2-^

x

T

2 I

mJ

Representative coefficient values - deduced from a graph x x x2 x2 Tm/T m l *m/Tm i 0.37 0.42 0.44 0.45

0.2 0.4 0.6 0.8 Minimum IAE Pemberton (1972a), Smith and Corripio (1997) -page 345. Minimum IAE - Yu (1988). Model: Method 1. Load disturbance K e~STu model = —. l + sTL

K <3> =

Comment

i

Minimum performance index: regulator tuning N 0.986 f Model: Method 5; ( \ 0.707 T 0-984 (lm\ 0.1<^-<1.0 0.608 T T

Minimum IAE Murrill(1967)pages 358-363.

5

T

0.77 0.85 0.91 0.98

1.0 1.2 1.4

0.46 0.46 0.46

1.05 1.11 1.17

Model: Method 1;

T T

KmTm

0.1 <-^2-< 0.5 T m

5

T(3)

K C (3)

^ - < 0.35 Tm 6Kc(4)

T(4)

0.685 K„

T V m

T„

X(3)=-

. \ 1.977-!=—0.55 T I

•^-1 T

0.214 T,

mJ

0.874 K W _ K„

N -0.099+0.159—

V T1 m J

, ^ < 2.641^2- + 0.16. T T

N-1.041

T V m7 -4.515—+0.067,

-(4)

0.415 T

T

, 2.641- S L + 0 . 1 6 < - ^ - < l . T T

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Minimum IAE - Yu (1988)-continued.

7

31 Comment

T, T(5)

(5

KC >

i

T(6)

8 j / (6)

^ra/Tm>0.35

i

Minimum IAE Shinskey(1988)page 123. Model: Method 1 Minimum IAE Shinskey(1996)page 38. Model: Method 1 Minimum IAE Marlin(1995)pages 301-307. Model: Method 1. K c , Tj deduced from graphs: robust to ±25% process model parameter changes.

Minimum IAE Edgar e/a/. ( 1 9 9 7 ) pages 8-14,8-15. Model: Method 1 Minimum IAE Edgar e/a/. ( 1 9 9 7 ) page 8-15.

K

(5)

0.871 K„

1.00T m /K m x m

3.0t m

Tm/Tm=0.2

1.04T m /K r a x m

2.25i m

Tm/Tra=0.5

l-08Tm/Kmtm

l-45t r a

^m/Tm=l

1.39T m /K m x m

"Cm

^m/Tm=2

0.95Tm/KmTm

3.4x m

Tra/Tm=0.1

0.95Tm/KmTm

2-9x m

Tm/Tm=0.2

l-4/K m

0.24x m

*m/Tm=0.1I

l-8/K m

0.52t m

tm/Tra=0.25

l-4/K m

0.75t m

tm/Tm=0.43 tm/Tm=0.67

l-0/K m

0.68x m

0.8/K m

0-71i m

xm/Tm=1.0

0.55/K m

0.60t r a

xm/Tra=1.50

0.45/K m

0.54t m

xm/Tm=2.33

0.35/K m

0.49x m

Tm/Tm=4.0

0-4/K m

0.5x m

Time delay dominant

0-94T m /K m T m

4xB

Time constant dominant

0.67Tm/Kmxm

3-5i m

Model: Method 2

^-0.015+0.384^ f _

f

^-1.055

Ti

T

T V m J \ -0.217-!!— 0.213/ T

I

<(5)

"

0.444 T,

K

(6) _

% 0.867

)

T, T

-0.003—-0.084,/

0.513 K„

IX T

T

»

VTm

i

0.670 T V mJ

VTm y

Handbook of PI and PID Controller Tuning Rules

32

Rule Minimum IAE Huang et al. (1996). Model: Method 1 Minimum IAE Shinskey(1988)page 148. Model: Method 1 Minimum IAE Shinskey(1994)page 167. Minimum IAE Shinskey(1996)page 121. Minimum ISE Hazebroek and Van derWaerden(1950). Model: Method 2

'K™

Comment

Kc

Ti

KC(7>

T(7)

i

0.1<^-<1 Tm

0.58KU

0.81TU

im/Tm=0.2

0.54KU

0.66TU

x m /T m =0.5

0.48KU

0.47TU

v / L =i

0.46KU

0.37TU

fm/Tm=2

T 3.05-0.35-^

10 j (8)

Model: Method 1

0.55KU

0.78TU

Model: Method 1; xm/Tm=0.2

..Kc(9,

1.43Tm

^m/Tm<0.2

9

Ku

6.4884 + 4 . 6 1 9 8 ^ - + 0.8196

^

-5.2132

T

K„ V

m

-7.2712 K„

•y (')

T-1

-0.7241e' 'm J

\4

f

0.0064 + 3 . 9 5 7 4 ^ - 6 . 4 7 8 9 ^ 2 - 1 +9.4348

T

-10.7619

T V ra J

\6

f + T„

12\

10 y (8) _ ~

0 . 8 7 - 0 . 8 5 5 - ^ + 0.172

» K ' 9 ) = ^ M o . 7 4 + 0.3-T"

7.5146^

•l.lttb

T

Chapter 3: Tuning Rules for PI Controllers Rule

Kc x

Hazebroek and Van derWaerden(1950)continued.

K

Minimum ISE Zhuang and Atherton (1993). Model: Method I Minimum ISE - Yu (1988). Model: Method 1.

12

^ m

mTm

Coefficient values

0.2 0.3 0.5

Minimum ISE Frank and Lenz (1969). Model: Method 1

Comment

T;

lTm

X

Li. T

Minimum ISE Haalman(1965). Model: Method I Minimum ISE Murrill(1967)pages 358-363. Model: Method 5

33

x2

l

X

x

m

x2

i

T 0.7 1.0 1.5

0.80 7.14 0.83 5.00 0.89 3.23 0.67Tm

0.96 1.07 1.26

2.44 1.85 1.41

Tm

Kmxm f

N 0.959

i f T ^ — x1+x2-s-

Tm X

l

x2

2

2.0 3.0 5.0 M max

1.46 1.18 1.89 0.95 2.75 0.81 =1.9; A m =

2.36; *„, = 50°

T ( Tm" 0.492 [ T

1.305 f T O Km IT J

x

T

.0.739

0.1 < ^ s - < 1.0 T

x,+x2^l v

Representative coefficient values - deduced from a graph T x x x2 x2 TmAm m/T m l l 0.82 0.2 0.53 1.12 1.0 0.63 0.88 1.17 0.4 0.58 0.63 1.2 0.97 1.20 0.6 0.61 1.4 0.62 0.62 1.05 0.8 1.279

1.346 12

/

/

\ 0.945

0.586

—l

0-535

s 0.675

T ( m 0.552 [

—] K

\0.438

T(10)

(10)

K e~ Load disturbance model = —-— 1 + sT,

K

(10)

0.1<^_<1.0 T

\jmj

0.921 ( TL

\.\<^<2.0 T ^S-<0.35 T

0.181-0.205—,-

N -1.214

T

N 0.954—-0.49

TL 0.430 V

mJ

, - ^ < 2.3 1 0 ^ L +0.077. T T

Handbook of PI and PID Controller Tuning Rules

34 Rule

Kc

Minimum ISE - Yu (1988)-continued.

Comment

Ti (ii)

13

K

(11)

T

14

K

(12)

T(12) 1

15

K

(13)

T(13)

-^2- < 0.35 T

i

Minimum ITAE Murrill(1967)pages 358-363. Model: Method 5

/

N 0.977

,

.0.680

0.1<^L<1.0 T

Tm O 0.674 | j m J

0-859 (lm\

— (x, + x 2 i -

Minimum ITAE Frank and Lenz (1969). Model: Method 1

-^2- > 0.35 T

^fx1+x2^ x

2 V '»; Representative coefficient values - deduced from a graph T x x x2 x2 m/Tm i Tm/Tra l 0.2 0.4 0.6 0.8

0.32 0.35 0.37 0.38

0.77 0.80 0.86 0.93

1.0 1.2 1.4

0.38 0.39 0.39

1.00 1.05 1.10

1.157 T

K„

-2.532-=!-0.292 . T

( U )

=

.

0.359 T,

T

, 2 . 3 1 0 ^ + 0.077 < i < l T T

V 1m J H

K

(12)

1.07

,

N -0.065+0.234-5!- ,

N-1.047

T„,

T

T

V 1m J -1.112—!!!—0.094 , ,(12)

0.347 T 15

K

(13)

0.04+0.067Tm

1.289 V

T (13)

>

T

K„

T

m

v T

i

X

(T

\

0.596 V T m J

l<-^-<3. 0.372- 55 -- 0.44 ^

.0.46

,T V

my

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Minimum ITAE - Yu (1988). Model: Method 1. Load disturbance K e"STL model = — . l + sTL

Comment

Tj

16 ^

(14)

17 K

(15)

T(15)

18 K

(16)

T(16)

19

(17)

T(17)

K

35

j(14)

^-<0.35 T m

•^E->0.35 T m

Minimum ITSE Frank and Lenz (1969). Model: Method 1

—

\

(

X , + X

/

N

T

m J

K,

>T -1.055

x-0.011-1.945-

T

T V m J

-5.809^+0.241 , '(15)

x 0.196

^ < 2 . 3 8 5 - ^ + 0.112.

0.805 ,

s 0.304-5^-0.112^

T„

.(14)

0.735

+X 2 ^2T

T /•

(15)

X,

x2

0.272 -0.254-

V

17 j r

^

2

T ^

Representative coefficient values - deduced from a graph x x x2 x2 Tm/Tm ^m/T m i l 0.84 0.2 0.46 0.55 1.09 1.0 0.4 0.89 1.2 0.54 0.50 1.15 0.6 1.4 0.97 0.54 1.20 0.53 0.8 1.04 0.55

0.598 K„

(

T ^

T

N 0.901

T,

m

2.385^- + 0 . 1 1 2 < ^ < l . T T

0.425 T V 18

K

(16)

_

0.787 T T L Km

m J

0.084+0.154

T

{Tm

,

N-0.148-^-0.365/

N 0.901

-(16)

T

0.431 , T

l

, l
V m J 19

K

(17)

0.878 ( T L Km [T m

T V 1m / . (17)

\C228- 15 —0.257

Tm

0.794 T

T

Handbook of PI and PID Controller Tuning Rules

36 Rule

Kc

Minimum ISTSE Zhuang and Atherton (1993). Model: Method 1

1.015 K

/

T

\ 0.957

1.065

r

1

V m j /"

T

^-

V nJ

1.1 < ^ s - < 2.0 T A

m

Model: Method 1; 20 K

T(18)

(18)

0.1 < ^ s - < 2.0 T

i

m

Minimum ISTSE Zhuang and Atherton (1993).

Model: Method 31; 21

T(19)

(19)

K

1

0.1<^S-<2.0 T

i

A

Minimum ISTES Zhuang and Atherton (1993). Model: Method I

m

427

0-687 {Tm j

T

Minimum ISTSE Zhuang and Atherton (1993).

0.1<^2L<1.0 T

0.667

\0-673

T

x 0.552

f

T

V T nJ

m

Comment

i

1.021

,

N 0.953

T

Tjn.1

x

VTmJ 1.076 (T

^

m

T

\ 0.546

0.648

m

f

T

m

O.l^^-Sl.O T

m

0.629 \

0.442

LIS — < 2 . 0 T

0.650 T

Km Nearly minimum IAE, ISE, ITAE Hwang (1995).

(

Model: Method 35; 22

(20)

K

Kc(20)/u2Kucou

0.1<^-<2.0 T A

20 K ( , 8 ) _ f l - 8 9 2 K m K u + 0 . 2 4 4

,y

3.249KmKu +2.097 K,

(19)

4.126K m K u -2.610 5.848K m K u -1.06

22

^ = ( 1 - ^ , ,

*

0.706K m K u -0.227 \ 0.7229K m K u +1.2736 J

T(18)

Ku, V

A (19) Ku, ^

m

5.352K m K u -2.926

_

x 5.539K m K u

Tu.

+ 5.036

=1.14(l-0.482m.Tm^0.068a,^)

K u K m / l + KuKm 0.0694(-l + 2.1muTm-0.367co2uT^) H2

K

uKm/1

+

K

uKn

Decay ratio = 0.15.

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Nearly minimum IAE and ITAE - Hwang and Fang (1995). Model: Method 25 Gerry (2003) minimum error: step load change. Model: Method I

23 K

37 Comment

Ti

(21)

0.1 < -^2- < 2.0; T

T(21)

i

m

decay ratio = 0.12 x ±ZL>5 T

0.3 0.42xm

Kra X

x,/K m Minimum IAE or ISE - ECOSSE Team X fm/T m l (1996a). 0.11 1.1 Model: Method 1. 0.25 1.8 Coefficients of 0.43 1.1 K c , Tj deduced from 0.67 1.0 1.0 0.8 graphs 1.50 0.59 2.33 0.42 4.0 0.32 Minimum IE 24 K (22) Devanathan (1991). Model: Method 1.

2(tm+Tm)

Coefficient values x2 fm/Tm 0.23 0.23 0.72 0.72 0.70 0.67 0.60 0.53

0.11 0.25 0.43 0.67 1.0

x

T

tm/ m

0.2 0.4 0.6 0.8 1.0

Coefficients of K c , Tj deduced from graphs

i

x2

5.8 3.1 2.1 1.7 0.91

0.5 0.6 0.7 0.8 0.9

x

2Tu

Coefficient values x2 ^m/Tm 0.35 0.29 0.26 0.23 0.21

1.2 1.4 1.6 1.8

x2 0.19 0.18 0.18 0.17

2^

0 . 5 1 5 - 0 . 0 5 2 1 ^ - + 0.0254 T

Ku,

,(21)

(Kc(21)/Ku«u) 2^

0.0877 + 0.0918- 2 L -0.0141 T T

m

2; cos tan 24

K

1.57DD

0.16

(22)

K m cos tan

6.28T

_

Handbook of PI and PID Controller Tuning Rules

38 Rule

Minimum IAE Gallier and Otto (1968). Model: Method 47

Minimum IAE Rovira etal. (1969). Model: Method 5 Minimum IAE - Bain and Martin (1983). Model: Method 1 Minimum IAE Marlin(1995)pages 301-307. Model: Method 1 Minimum IAE Huang et al. (1996). Model: Method I Minimum IAE Smith and Corripio {1991)-page 345. Model: Method 1 Minimum IAE Huang and Jeng (2002). Model: Method 1

Comment Kc Ti Minimum performance index: servo tuning x x,/K m 2(tm +Tm) Representative coefficient values - deduced from graphs x x2 x2 fmAm Xl l •Cm/Tm 12 6.4 2.9

0.053 0.11 0.25 ,

0.95 0.91 0.84

0.67 1.50 4.0

% 0.861

T

0.758 f T m ^ K

1 Km

m lTmJ

0.1
m

1.020-0.323^Tra

0.3 + 0.3^2-

m

-^->0.2 T

T m +0.4x m

1.4

^ - = 0.11 0.72t m T Kra m K c , Tj deduced from graphs; robust to ±25% process model parameter changes 25 K

(23)

T

0-6Tm KmTm

0.1<^-<1 T

(23) i

0.1<^-<1.5 T

T m

m

Minimum IAE = 2.1038xm;

0.59Tm T

Kratm

m

T m /T m <0.2

N-1.0169

25 K

(23) _

_j_

0.75 0.66 0.55

1.15 0.65 0.45

•13.0454 - 9.0916— + 0.3053 T T 1 +

N3.5959

f

•1.1075 f

T ,

•, 3.6843 Tm

-2.2927

4.8259eTm

T

K„

V m

0.9771 - 0.2492 - ^ + 3.4651 T T

-7.4538 ^ M

+ 8.2567

^

-4.7536 U H L ] + 1 . 1 4 9 6 ^ -

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Minimum IAE Tavakoli and Fleming (2003). Model: Method 1 Small IAE - Hwang (1995). Model: Method 35

26

K

T(24)

0.1<^2_<10 Tm

Minimum A m = 6dB ; minimum * m = 6 0 ° Decay ratio = 0.1, 27

K

(25)

0-980

Kc(25)/n2Ku<»u

0.1<^L<2.0 T

Tm

0.1< — < 1 . 0 T

x 0.892

f

Minimum ISE Zhuang and Atherton (1993). Model: Method 1

Comment

Ti

(24)

39

(lm\

1.072 (T

^

m

0.690-0.155^T T

0.560

1.1<^2.<2.0 T

m

m

Km l T m J

0.648-0.114-^T m

Minimum ISE - Khan and Lehman (1996). Model: Method 1

26 j , (24)

28 ^

(26)

T

27 Kc(25)

=

(24) _

T

0.4849-2- + 0.3047

'

K„

m

0.4262^2. + 0.9581 T

T

i

0.01 < ^2-< 0.2 T

(26) i

n

(i _ ^ ) K u , n, = 1 -07(l-0.466 mu T m +0.0667co 2 u x^) ^ K

uKm/1

+

K

uKm

_ 0.0328(- 1 + 2.21couTm - 0.33&a>lz2m) r = 0.5; ^ ~ K u K m / l + K„K m l.ll(l-0.467
uKm/1

+

K

uKm

^

0.0477(-1 + 2.07c)uTm - O . ^ ^ x j )

=

K

uKm/1

+

K

^_ = 0.75;

uKm

1.14(l-0.466ffluTm+0.0647ffl2,T2J K

uKm/1

+

K

uKra

0.0609(-l + 1.97c i ) u T m -0.323( n ;T 2 n ) >r _ 1| K K u m / 1 + KuKm r = parameter related to the position of the dominant real pole. =

28

K

(26)

f 0.7388 ^ 0.3185 "I T m

tm

T

IK

T(2

' '

6)=T

0.7388Tm+0.3185xm -0.0003082Tm +0.5291x„

Handbook of PI and PID Controller Tuning Rules

40

Rule

Kc

Khan and Lehman (1996) - continued.

29 K

Minimum ITAE Rovirae/a/. (1969). Model: Method 5

(27)

0.586 (Tm^ K

m

V

T

Comment

Ti T

0.916

(27) i

0.2 < i s - < 20 T

T m

1.030-0.165 — T

m;

0.1
•* m

Approximately minimum ITAE Smith (2002).

0.586 Tm Model: Method 1

T 0.921

T

0.712 fT m ">

Minimum ISTSE Zhuang and Atherton (1993). Model: Method 1

K

m

V

T

0.968-0.247^T

m;

0.1
m ,

N 0.559

0-786

T

(lm\

0.883-0.158^2T

1.1< — < 2 . 0 T

m

Model: Method 1; Minimum ISTSE Zhuang and Atherton (1993).

0.361KU

(29)

,

T

s 0.951

0-569

(lm\

0.628

0.511

,

N

(29)

0.1 < i = - < 2.0 T T

1.023-0.179^T

Km I T J Km

^0.808

<2.0

Model: Method 31;

Minimum ISTES Zhuang and Atherton (1993). Model: Method 1

(27)

Q.\< — T m

31 K

29 j ,

30-r, (28) M

0.583

T 1

X

Km

m

1.007-0.1 6 7 ^ L T

J

0.1 < i 2 - < 1.0 T m

1.1< — < 2 . 0 T m

0.255 VT~T7 -(27).

0.808T m +0.51lT m -0.255VT~^ 0.095T m +0.846x m -0.38lVf^

30

T

31

K

28 (28)

T/ ' =0.083(l.935K m K u +l)T u f (29)

A

A

(29) = 0.0551 3.616Km K u +1 |TU 1.506K m K u -0.177 K u , T

3.341Km K„ + 0.606

Chapter 3: Tuning Rules for PI Controllers Rule Nearly minimum IAE and ITAE - Hwang and Fang (1995). Model: Method 25

Kc 32 K

41 Comment

T(

(30)

T

0.1 < -^2- < 2.0; T decay ratio = 0.03

(30)

Minimum performance index: other tuning Simultaneous servo/regulator tuning - Hwang and Fang (1995). Model: Method 25

33 K

Coefficients of Kc estimated from graphs

0.1<-i<2.0; T

T(31)

*i

m

decay ratio = 0.05 T

x,/K m

34

Minimum performance index Wilton (1999). Model: Method I.

(31)

Coefficient values b fm/T m

^m/Tm 0.1 0.1 0.1 0.1

i

b

750.0 17.487 4.8990 2.1213

0.0001 0.04 0.2 1

0.2 0.2 0.2 0.2 0.2

3.3675 2.2946 1.7146 1.1313 0.8764

0.008 0.04 0.2 1 5

0.4 0.4 0.4 0.4 0.4

1.6837 1.5297 1.2247 0.9192 0.7668

0.008 0.04 0.2 1 5

0.5 0.5 0.5 0.5

150.01 7.1386 2.6944 1.1314

0.0001 0.04 0.2 1

x

i

x

•}2\

32

K

(30)

0.438-0.110-2-+ 0.0376 T

Ku, (K c ( 3 0 ) /K u co u )

-(30)

2^

0.0388+ 0.108-2!--0.0154 T 2^

0.46-0.0835-21-+ 0.0305 T

K„

p (31)

=

_

(K C (31, /K U (D U ) 2V

0.0644 + 0.0759— - 0.0111 — T T ' Performance index = r[e 2 (t) + b 2 K m 2 (y(t)-y ocl ) 2 ]dt

Handbook of PI and PID Controller Tuning Rules

42

Rule Wilton (1999)continued

Minimum ITAE ABB (2001). Model: Method 7

Bryant et al. (1973). Model: Method 1

Kc

T

0.6 0.6 0.6 0.6 0.6

Coefficient values - continued T b x. m/T m 0.8980 1.1225 0.008 0.8 0.04 0.9178 1.1218 0.8 0.9798 0.2 0.7348 0.8 0.7778 1 0.7071 0.8 0.6572 0.6025 5 0.8

0.008 0.04 0.2 1 5

1.0 1.0

72.004 1.6657

0.04 1

tm/Tm

0.0001 0.2

1.0 1.0

N 0.977

3.6713 0.8485

0.68

1.4837Tm

^-<0.5 T

T

Direct synthesis: time domain criteria Critically damped T 0.37Tm dominant pole K t m m

0.40Tra

Damping ratio (dominant pole) = 0.6

T

mTm

0

x2KmTra

Coefficient values - deduced fr om graph x2 x2 ^m/T m

^m/X

12.5 11.1 11.1 9.1 7.1 6.3 5.6 4.3

0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67

1.0 2.0 10.0

6.7 4.2 3.0

Critically damped dominant pole

1.0 2.0 10.0

3.2 2.1 1.6

Damping ratio (dominant pole) = 0.6

X variable; suggested values: 0.2, K j l + XxJ 0.4,0.6, 1.0. X = 1.0 , OS<10%, Model: Method 10 (Poulin and Pomerleau, 2004). Appropriate for T m K T "small x m " m m(T L+0 ^Tm

Mollenkamp et al. (1973). Model: Method I

b

*1

0.8591 r

K

CKmetal, (1973). Model: Method 1

Comment

i

T

C

Chapter 3: Tuning Rules for PI Controllers Rule

Kc 0

43 Comment

Ti

Model: Method 1 *2*m Keviczky and Csaki x 2 coefficient values (estimated from graph) (1973); 0% 5% 10% 15% 20% OS values in first ^m/Tm OS row; T m /T m values 0.1 30 22 12 16 10 0.2 in first column; 18 11 9 7.5 6 Km=l. 0.5 8 6 5 4 3.5 1.0 5 4 3.5 3 2.5 2.0 4 3 2.2 2 2.5 5.0 3 2.4 2 1.8 1.6 10.0 2.5 2.1 1.7 1.8 1.5 5% overshoot - servo 0.52Tm 0.04 < i s - < 1.4 - Smith e< a/. (1975) T KmTm T - deduced from graph Also given by Bain and Martin (1983). Model: Method 1 1% overshoot - servo Model: Method 1; 0.44Tm - S m i t h s a / . (1975) T KmTm m 0.04 < i s - < 1.4 - deduced from graph T 1 % overshoot - servo

0.43Tra Bain and Martin (1983). Model: Method 1

K

T

mTm

- i s - > 0.2 T

T

"Very conservative tuning for small delay"

T

OS = 10%

1

Suyama (1992). Model: Method 1

0.5Tm Kmtm

5% overshoot - servo - Smith and Corripio (1997) page 346. Model: Method 1

0-5Tm T <j>m = 60° ; 0.25 < i s - < 1 (Voda and Landau (1995)) m

Viteckova (1999), Viteckova et al. (2000a) (Model: Method 9 Viteckova et al. (2000b)).

Closed loop response overshoot (OS) shown

T

*i m

T

Kmxm

i

OS (%)

0.368 0.514 0.581 0.641

0 5 10 15

x

Coefficient values OS (%) Xl 0.696 0.748 0.801 0.853

20 25 30 35

i

OS (%)

0.906 0.957 1.008

40 45 50

x

Handbook of PI and PID Controller Tuning Rules

44

Rule

Kc 0.5 lTm

Mann etal. (2001). Model: Method 31 or Method 33; Closed loop response overshoot (step input) <5%

K

Comment

Tm

0<-^2-<2

r

35 K

Viteckova and Vitecek (2002) Model: Method 1

T,

(32)

T(32)

x, = 0.4, x 2 =3.68

Ktm/Tm<2

x,=0.8, x 2 = 3.28

2
x,=1.2, x 2 = 2.88

4<^m/Tm

0

36 j <33> •M

37 K

(34)

T

(34)

For 0.05 < — < 0.8 , T; = Tm ; overshoot is under 2% (Viteckova and Vitecek (2003)) Pinnella e? a/. (1986). Model: Method 20

35

(32)

K

38

K

Critically damped servo response

(35)

0

1.0404 ^ - - x , + x ,

0.56 + 1.02K„

x„.

0.56 + 1.02-

Im_h.0404^2--x,+x2

,(32)

0 . 0 4 - 1 . 0 2 ^ + 1 ^ 2 - 1 . 0 4 0 4 ^ - xM , | + x,

K„

•(33)

1 "-

X

0.25 +T '

m 37 K

(34)

1 T

m

T

0.5

T

-J2m

+0.25 - - = - + 0.5

m

^-[TmTmx12+(2Tm+xm)x1+l]e^> 2

X, =

2

0.5

0.25

2 + T +VT T

T

T

^mTmx12+(2Tm+Tm)xl+l

„, (34)

2r , T :

X ^ m 1 ™ * ! +Tm 2

\2 38

K

(35)

4Kmxm'

T

Tm

TJ2J

T

T

0.5

+1 e

+ 1

J

Chapter 3: Tuning Rules for PI Controllers Rule

T,

Comment

Kmtm Tm

Tm

4=1

K ra x m

T

4 = 0.6

T

4=0.0

Kc Tm

0.368

Schneider (1988), Bekker etal. (1991), Khan and Lehman (1996). Model: Method 1

0.4 13

Tm

1.571 Regulator - Gorecki et al. (1989). Model: Method 1

McAnany (1993). Model: Method 24 Sklaroff(1992), Astrom and Hagglund (1995) -pages 250251.

39

K

(36>

2

=

T

41

K

1

(37)

40 K

Pole is real and has maximum attainable multiplicity

T(36)

(36)

39 K

Hang ef a/. (1991). Model: Method 1

T

i

(37)

0.16 < ^2-< 0.96 T

Servo response: 10% overshoot, 3% undershoot T(38) (38) TCL = 1.67Tm i Model: Method 28; Honeywell UDC 6000 controller

3 T Km

2+

m

fl + ^ 1

2T V

-1

2T m

2T m

m

2T

.(36)

3+

2+

2T

v 2T ,

2T V

12 + 2 40

K

(37)

45

15 + 14

m

1 1 - ^ + 13 T 37^--4 T

= :

2+

KU,T,

(37) _

0.2

1 1 ^ - + 13 T

11^2- + 13 T +i 37^-4 m

y

37^--4 T 41 K (38) _ (l-44Tm +0.72TmTro -0.43x m -2.14) K ^

72x m +0.36T m z +2

(3g)

Km(5 4T + 1 . 2 8 T - 2 . 4

2T

Handbook of PI and PID Controller Tuning Rules

46 Rule

i

Comment

5t m

ia-<0.33 T

Tm

-^2-> 0.33 T

Kc 0.5556Tm

Fruehauf et al. (1993). Model: Method 2

T

mKm

0.5Tm T

Kamimura et al. (1994). Model: Method 1

T

mKm

m

42 K

(39)

T(39)

43

(40)

T(40) 1

K

Servo response: 10% overshoot Servo response: 0% overshoot "Quick" servo response: "negligible" overshoot "Good" servo response; xm is "small" 4 = 0.9; 0.2<xm/Tm
i

44 K

(41)

T(41)

Zhang (1994). Model: Method 1 Davydov et al. (1995). Model: Method 6

T

45

K

(42)

T

0.5

0.5(T ra +T ra )

"Normale Einstellung"

1

0.7(T m +x m )

"Schnelle Einstellung"

Kuhn(1995). Model: Method 14

42

K

(39)

43 jr (40) _

0-5(Tm+Q T ' li K m [ 0 . 5 x m ( 2 T m + x J + 0.10] 0.375(T m+ T m ) K m [ o . 5 x m ( 2 T m + x J + 0.0781]

(39) _

1

T(40)_T

44

K

(42)

0.5x m (2T m +x m )+0.10

T

ir

T

+T

m +t

m

0.5x m (2T m +x m )+0.0781

0-425(T m+ T m )

(41)

K m [0.5x m (2T m +x m )+0.083] : T- (41) =T 45

K

(42)

,(42) K„

1.905--- + 0.826 T

+x

0.5x r a (2T m + x m ) +0.083

0.153-^ + 0.362 T . T

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Khan and Lehman (1996). Model: Method 1

Comment

Ti

46 K

(43)

47 K

(44)

47

0.01 <

T(43)

i

^L<

T

0.2

0.2 < -^2. < 20 T

T(44)

i

0 < V < 0.2 ; Abbas (1997). Model: Method 1 Valentine and Chidambaram (1997a). Biefa/.(1999). Model: Method 13

48 j r

1

Model: Method 1. \ = 0.707 49

m

0.5064Tm K

0.1<^-<5.0 T

0.413 + 0.8^2T

0.43-0.97^2T

Tm

mTm

Bi et al. (2000). Model: Method 13

0.5064Tm

Ettaleb and Roche (2000). Model: Method 29

1

46 T^ (43) _ _J_ni K„

T m +0.5x m

(45)

TmKm 0.5064

K ra x ra

T

MM1,0,723

m+Tm

TCL =

T

m + ^m

^ -0.404t m 2 +0.723T m x m -0.3852T m 2 -0.525T m 2 +0.4104T m t m -0.00024T m

47

K

0.404

(44) K„

0.256

0.1275

T_

V T m T .r 0.404T m +0.256t m -0.1275VTV^ 1

0.0808Tm +0.719x m - 0 . 3 2 4 7 ^

s -1.045

0.148 + 0.186 - ^ 48

K

(45)

Km (0.497- 0.464V 0590 ) 49

±1% Ts = 2.5Tra , 0.1 < -^s. < 0.4 ; ±1% Ts = 5Tm , 0.5 < - ^ - < 0.9 .

Handbook of PI and PID Controller Tuning Rules

48 Rule

Kc

Chen and Seborg (2002). Skogestad (2003). Model: Method 45

50 K

1

K

Model: Method 1

i

min(Tm,8Tm)

0.5Tm 51

Comment

Ti T(46)

(46)

T

(47)

Gorez (2003). Model: Method 1

o<^=—
(l-V>m+Tm

52 p, (48)

I

Tm

<

m

<1

^m + T m

50

K

(46)

1

Tm T m

K m

+2

TmTCL ~TcL

T

(46) _ J m ^ m + . 2 T m T C L - T C L

(T C L + xJ 2

T

0 < T r < T m + ^T

m + Tm

T/^se-v

+T x d

desired

Kc(TCLS + l)2

with XKmT„

1

x=— M s ( M s - l )- +

V

1.5M/-2 2

0.32MS (MS-1

x

5x„

+T

t m + T m X/M7j 2.5x„

0.5

MS(MS

v

(x m +T m yM sy

-0.4JM 52

K

(48)

withv = 1-0.5 XKmTn

Tm+Tm

1-0.4, M

0.1M„ M -1

-0.4, M 7.5-

1-0.4, M

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Sree et al. (2004), Chidambaram et al. (2005). Model: Method I Haen (2005). Model: Method 1

,

Comment

i

x-0.8915

0.9179(0

0.1<^-<1 T

53T(49)

m

54 K

0.96T m +0.46x m

(50)

0
T Direct synthesis: frequency domain criteria T x,/K m Maximise crossover Coefficient values frequency T

Hougen(1979)page 335. Model: Method 1; Coefficients of Kc

OT m

Regulator - Gorecki etal. (1989). Model: Method 1

55 K

0 m 0.6 0.7 0.8 0.9 1.0

(51)

x

l

1.1 1.0 0.8 0.75 0.7 Low frequency part of magnitude Bode diagram is flat

T(51)

i

10.59 -J2T V

l

7.0 3.5 2.2 1.8 1.4

X

0.1 0.2 0.3 0.4 0.5

estimated from graphs.

53 j (49) _ j

T

49

- 2 . 3 5 8 8 - ^ + 0.8985 T

mJ

0.7719

0.1 < - 2 - < 0.4; T

m

-3.6608

+ 6.5791 O T

J

-5.1652-^1- + 2.8059 T 0.4<-J2-<1.5 T

54

v

K„

(50) _

1

K.

0.407+ -

6.282

1 + 14.956

T

V in 55 K

(51)

1 l + 3(T m A m )+6(T m /x m ) 2 + 6 ( T m / Q : K„

l + 3(T m /T m )+3(T m /T m ) 2 T(5,)=x

2 3 1 + 3 T,„ A m ) + 6 ( T m / T m ) + 6 ( T m / x m )

2

1 + 2(Tm/xm)+2(Tm/xm) ]

Handbook of PI and PID Controller Tuning Rules

50 Rule

Kc *,/KmAm

Somaraeffl/. (1992). Model: Method 1; Suggested A m :

Comment

T, 56

x

2*m

Coefficient values X

x2

36.599 7.754 4.361 3.055 2.369 1.950 1.672

3.57 3.33 3.13 2.94 2.78 2.63 2.50

x

0.0375 0.1912 0.3693 0.5764 0.8183 1.105 1.449

1.75 < A m < 3 . 0

i

m/Tm

i

x2

1.476 1.333 1.226 1.146 1.086 1.042 1.011

2.38 2.27 2.17 2.08 2.00 1.92 1.85

m/Tm

x

1.869 2.392 3.067 3.968 5.263 7.246 10.870 T(52)

(52)

57 K

T

o.\<^-<\ T

Hang et al. (1993a), (1993b) -pages 6165.

•

F o r

i

a

L

< 1

,K

m Model: Method 31 or Method 33

58j(53)

opTm KmAm

e

Tm

.o^i KmAm

• + 1 or \( 1.886 + 4—si- - 1 . 3 7 3 T

0.9806

K,

1 + 1.886^1 ; f o r ^ > l , K VTr.

57

K

(52)

0.80 + 1.33-

KmAm

T(52)

__

* - ^ l KraAm

5T^ 1.04 0.80 + 1.33-

1 2co

p P

n

; - ^ < 0.3 , Yang and Clarke (1996). + — T

+ 8.669

Chapter 3: Tuning Rules for PI Controllers Rule O'Dwyer (2001a). Model: Method 1;

Kc

T,

T

Tm

51 Comment A m = 7t/2x, ;

K

m m

i s - > 0.3, T

4m = V 2 - x , 5 9

Representative results 1.048

Am 1.5

0.7854

2

m

x

Yang and Clarke (1996). O'Dwyer (2001a). Model: Method 1

x

i

K

l


Xi

30°

0.524

Am 3

60°

45°

0.393

4

67.5°

Am = V 2 x i ;

Tra

T

Tm u u

T

m VV+^X2

<> tm = V 2 - X l

xjm ^mKm

Chen et al. (1999b). Model: Method 1

Tm

0.50

Am 3.14

0.61

2.58

Coefficient values x Ms Am i 1.00 0.72 2.18 61.4° 1.10 0.76 2.07 55.0°

0.67

2.34

51.6°

1.20

0.70

2.24

50.0°

1.26

Xl

Gain margin and maximum closed loop magnitude - Chen et al. (1999a). Symmetrical optimum principle Voda and Landau (1995). Model: Method 1

r



Ms 1.30

46.5°

1.40

44.1°

1.50

•I'm

0.80

l»r m

4

2

1

60

1

4.6

3.5C} ( j ° w ) P 1 2.828 G p ( J ° W )

Model: Method 1

1

T

Km

1.96

4 ffl

i 35°

^ - < 0.1 T 0

-

«

/ n u

T

'Given Am , ISE is minimised when <|>ra = 68.8884-34.3534A m + 9.1606for servo tuning (Ho et al. (1999)); 2 < A m < 5 ; 0.1 < T m /T m < 1 : Given A m , ISE is minimised when <|>m = 45.9848A m a2677 (T m /T m ) a2755 for regulator tuning (Ho et al. (1999)); 2 < A m < 5 ; 0.1 < T m /T m < 1 7t 4x„

2r,A n ,A

m

)2-l

52

Handbook of PI and PID Controller Tuning Rules Rule

T,

Comment

(54) i

0.15<^-<1

Kc

Voda and Landau (1995)-continued.

61

Friman and Waller (1997). Model: Method I

K

(54)

T

1

0.1751 Gp(j» 135 <.)

^ > 2 .

"Normal" design Butterworth filter. Model: Method 21 "Sharp" design Tschebyscheff filter (0.5dB). Model: Method 21 Minimum ITAE (Henry and Schaedel, 2005). Model: Method 22

i

Km(T,-T<55>)

0.375T/ 5 6 '

63

T

(56) i

K^T.-T,^) 63

0.375Ti (57)

(57)

T

i

Km(Tl-T<">j Henry and Schaedel (2005). Model: Method 22; Optimal regulator

64 K

"Normal" design Butterworth filter

(58) *i

(58)

Am>3

m

135°

T(55)

0.5T,<55»

62

Schaedel (1997), Henry and Schaedel (2005) Optimal servo response.

03

T

response; —— < 0.2

61 ^

1

(54) ___

1.15JGp(jm135o)| + 0.75Kn,

,(54)

4.6 G p (ja> 1 1 ,„)-0.6K n 62

(Bl35„[2.3|GI)(j(D135„)|-0.3Kra]'

Ti(55)=A/T,2-2T22, — + T m ,T 2 2 =1.25T m i a m +

InL_

1 + 3.45 0.7Tm

-Tm>T2

i - ^ f - > Tj ,(58)

64 j ,

(58)

1

.5

2 T m

;

0 < ^ < 0.104;

-3.45T

T m -0.7x a m =T

T m + 0

0.5

0.7T,

2

=TmTm +

-0.5t m 2 , i s . > 0.104.

Tm-0.7x^

=-0.64T1+1.64T,Jl-1.2|^2

-\

T

Z/'

-1

T, I

T

2 ^

T,2 j

Chapter 3: Tuning Rules for PI Controllers

Kc

Rule Henry and Schaedel (2005) - continued.

Leva (2001). Model: Method 1 Minimum §m = 50°

53

T;

Comment "Sharp" design Tschebyscheff filter (0.1 dB) Minimum ITAE

65 £

(59)

T<59)

66 £

(60)

rp(60)

67 £

(61)

20

T

Mc

Tm+5Tm

68 j ^ (62) c

69 K

70 £

(59)

2

(60)

, T: "2

0.69 K„

.

m

m

1-1.46^

m

' 0.698 lT m

50T„, I<'

(r m

]

+ 5T m )J

0.5 Gp(jw,35o)

1 + 2.48 J

(64)

3.86^

Km(Tm+5Tm)_

• (63)

K

J

20 T

| G p(j W 135»] 1.14 + 13.24

70

V

•i

*• in \

(63)

T = 3 . 1 1 - ^ 1-1.43 T,

2

T

KmTm

T^

(64)

J

K

69

(59)

j (60) _

-1 T

"0.69£

(62)

i s - < 0.1 T

"J

V

68 „

T

"\

0.7

"rp (59) 66 £

(64)

20 +5T

0.1 < i s . < 0.15 T

•(59) 65 ^

T(63)

(63)

i

Potocnik et al. (2001). Model: Method 39

x

*M#).

in AnpMA

^ + 17.40

135°

K -2|G p (ico 135 „] 2Km[Km-2|Gp(JaW)|]' -(64) .

[ K m + V2|G„ (jco 1 3 5 .)[][2|G P (jiQ 1 3 5 . )| ~ K m ] Q)

135»| G p(j £ 0 135»)|[| G P ( j C D 135»>|- K n 1 l

Handbook of PI and PID Controller Tuning Rules

54 Rule

Kc

Potocnik e< a/. (2001) - continued.

71

K

(65)

T (65)

i

Cluett and Wang (1997), Wang and Cluett (2000) -page 177. Model: Method 1;

Comment

T,

T (66)

72 jr (66)

0.1<^-<1 Tm TCL = 4xm . A m e [7.50,8.28],

(t)me[78.50,79.70] T

73 v

(67)

T (67)

i

0.01<-^2-<10 T

= 2x m . A m e [4.35,5.02], CL

<S)m e\l0.l\74.0°]

m

TCL = 1.33xm. T (68)

74 jr (68)

i

A m e [3.29,3.89], (|)me[64.40,70.60] TCL

T (69)

75 jr (69)

=

T

m•

A m e [2.74,3.30], (t)mg[57.80,68.50] TCL = 0.8tm .

76 K

(70)

T (70)

A m e [2.39,2.93], * m e [49.6°,67.1°]

71

K

(65)

2K„

0.75 + 1.15

|GpOroi35°)

(65)

T:

0.75 + 1.15^

= G°w) K„

K„

72

K

(66, _0.019952T m +0.20042T m

^

m

0.099508Tm+0.99956Tm 0.99747xm -8.7425.10" 5 T m

73 K (67) _0.055548T m +0.33639T m

T_(67)

0.16440Tm +0.99558Tm 0.98607T m -1.5032.10" 4 T n ,

74 75

K

K

(68) _0.092654T m +0.43620T m j Kmxm ' ' (69) _0.12786T m +0.51235T m Kmxm

T m

m

0.96515Tm+4.2550.10"3Tm =

76 Kc(70) _0.1605lT m +0.57109Tm m >TT( T O ) _ K

mTm

0.20926Tm+0.98518Tm 0.24145Tm+0.96751Tm 0.93566xm +2.2988.10" 2 T m 0.26502Tm+0.9429 lTm

0.89868xm+6.9355.10"'Tm

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Cluett and Wang (1997), Wang and Cluett (2000) continued.

(71)

T(71) 1

i

K

(72)

T

(72) i

0.5Tm

Smith (1998). Model: Method 1

0.35

79 K

(73)

0.19067tm +0.61593Tm

0.5

0.42xm

Km

Wang et al. (2000a). Model: Method 31 or 32

(71)

A m G [2.11,2.68], 4>me[38.1°,66.3°]

78

Tm

K

Comment

T|

T C L =0.67T m . 77 K

Modulus optimum principle - Cox et al. (1997). Model: Method 16

77

55

t —2L<1 T m I»L>1 T 6 dB gain margin dominant delay process

Tm

.(71)

_

1.3 <M S < 2

0.28242Tm + 0.9123 lTm 0.85491-r +0.15937T„

TmJ+VTm+0.5TmT^+0.167T,

3^

T m 2 T m +T m T m 2 +0.667T m 3 / • (72)

_

Tm3+Tm2Tm+0.5Tmxm2+0.167Tm3^ T m 2 +T m T m +0.5x m 2

79

K

(73)

1.451-

1.508 M.

Handbook of PI and PID Controller Tuning Rules

56 Rule

Kc

Wang and Shao (2000a). Model: Method I Chen and Yang (2000). Model: Method 8

80 K

Comment

Ti

A.e [1.5,2.5]

T.(74)

(74)

M s =1.26;

0 7T v. / i

Tm

m

Am=2.24;

KmTm

*m=50°

0.25Tm Hagglund and Astrom (2002). Model: Method 5; M.=1.4

0.1Tm 0.15 —+ Kmtra Km 0.15

0.3x m +0.5T m

Km

0-3Tm

Hagglund and Astrom (2002). Model: Method I

80

K

81 K

0.2<^L<1 T

0.8Tm

^mTm

Tm T m

M s =1.4;-is->0.5

T(75)

(75)

1

(74)

f2(WQno)-*-fl (WQn« ) K„

fl ( ffl 90» )

-[(Tm + { 1 +

:

fr + W } '

(O

90 l > 2T m 2 }' t m) Sm (- CO 90»' C m ~ t a n

'"V7™)]

,"1 ,V.5 ko° T m 2 cosC-^so"1", " tan"1 co90„Tm)J,

(l + co90„2T,

2

[(Tra +{l + co9o02Tm2}Tm)cot(-co90„xm -tan" 1 co90„Tm)J

f2(«90«):

1+ ^ X

2

m

i ,

90oTni 2T

1+C , (74) _

T 2 2 «9o»l n +(l + ©90 )xm]cos0 + (l + 2co9Oo2Tm2)sin( Qnu» T 90° L m 1 mm / m J

V

T m

2

w

-co „ 3 T m 2 cos9 + co9no2[Tm +(l + co909no2Tm m2y )Tm mJ ]sin( -C090oTn,-tan"lc090»Tm81 j r (75) ,

0.14 + 0 . 2 8 ^ - 1 , T ; (75) = 0.33xm + K„

6 8T - ™T" 10x m +T m

Chapter 3: Tuning Rules for PI Controllers Rule

Kc 82

K

(76)

83 K

(77)

Leva et al. (2003). Model: Method 1 x

83

K

84

(77) _

mm lmum

'Acceptable'

Comment

Tm

minimum <j>m ;

T

minimum ()>m = 50

T1

x^O.Srt-^84

1.25 T

1.6 <MS <1.9

lTm

m

K ra x ra

Leva et al. (1994). Model: Method 43

K ( 7 6 ) = minimum

Ti

maximum coc

Matausek and Kvascev (2003). Model: Method 1 Kristiansson (2003). Model: Method 11

82

57

85 ^

(78)

86

(79)

K

T

~Tm(0.5n-$m)

m (79)

5T„ K m T CL

SaT,,, 0.698T„, ,TCL = K m x ra ' K m ( T m + 5 T j

values of

x, :

0.5<x,<3.

Tm+5Tm

;ae[4,10].

'Recommended'

values of

x, :

0.32 < x , < 0.54 (59° < <|>m <72°). Choose x, = 0.32 when large variations in process parameters are expected; choose x, =0.54 to give 6 % < O S < 1 3 % and 1.4 <M < 2 .

85

K

(78) _ ^ 1 5 0 °

0.075

0.2 + -

K,,„o

K„

+ 0.05

K„

86 j ^

(79) _

M

cn

x

i

Km

l + «c„ 2 T m 2 ii+ajnr]]2

tan

-^+

T

m
,(79)

2.82with a>„

• tan

M

cn

T

m)

Handbook of PI and PID Controller Tuning Rules

58 Rule

Kc

Ti

Tan et al. (1996). Model: Method 30 Xu et al. (2004). Model: Method 31

87 jr (80)

T(80) 1

^Kn^m

Tm

Huang et al. (2005). Model: Method 42

0.50T m +0.22x ra

0.9T m +0.4x ra

Comment

i

T X e [1.5,2.5]

KmTm Robust A>1.7x m , ^K m

T

Tm K m (x m +A)

Tm

T m +0.5x ra Km(A + x ra )

T m +0.5x m

A>0.1T m .

m Rivera etal. (1986). Model: Method 1 A, > 1.7xm , X > 0.2Tm (Luyben (2001)); X = 2xm (minimum ISE deOliveirae/a/., (1995)) 2T m +x m A>1.7x m , T m +0.5x m 2XKm A>0.1T m .

Chien (1988). Model: Method 1 Brambilla et al. (1990). Model: Method 1

A = Tm (Chien (1988)) 88

No model uncertainty: A, = xm , 0.1 < x m /T m < 1 ; A = [l-0.51og10(xm/Tm)]xra)l<xm/Tm<10.

K

(80) .

A m ) /l + (pTi(80)coJ

T ' '

K

tanl-tan- 1 pTmco<, -pxmco<, -j'

co,,, < cou. (3 = 0.8,is- < 0.5 ; p = 0.5,^2- > 0.5 . ra

^m

X = 2xm (Bialkowski (1996)). X > Tm + xm (Thomasson (1997)). X e [0.45xm,0.8xm] (Huang et al. (1998)). A = 2xm (aggressive, less robust tuning) (Gerry, (1999)); X = 2(Tm + x m ) (more robust tuning) (Gerry, (1999)); A = a.max(T m ,x m ): a > 3 ... slow design, 2 < a < 3 ... normal design, 1 < a < 2 .. .fast design, a < 1 ... faster design (Andersson (2000) -page 11). Model: Method 10. A 6 [0.1T m ,0.5T r a ],^ < 0.25 ; A = 1.5(xm+Tm),0.25 < ^n_ < 0.75 ; * = 3(xm + T m ) , ^ - > 0.75 (Leva (2001)). A e [ T m , x J (Smith (2002)).

Chapter 3: Tuning Rules for PI Controllers Rule

Comment

M^ m + T m )

Model: Method 1

0.7T ra K

graphs

Chen etal. (1997). Model: Method 26

X

2Tm

Coefficient values i

x2

T

0.9 0.6 0.7 0.47 0.47 0.36 0.4 0.33

1.3 1.6 1.3 1.7 1.3 1.8 1.3 1.8

2.0 10.0 2.0 10.0 2.0 10.0 2.0 10.0

x

T 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0

Thomasson (1997). Model: Method 1

mTm

x,/Km

Ogawa(1995). Model: Method 1; Coefficients of K c ,Tj deduced from

Tf

Kc

Pulkkinen et al. (1993).

*1

x2

0.45 0.4 0.4 0.35 0.32 0.3 0.3 0.29

2.0 7.0 2.2 7.5 2.4 8.5 2.4 9.0

mi

0-5Tm

minim u m ( T m , 6 T m )

xra>0.5

K r a max(i m ,0.5)

3

Tm<0.5

Tm

1

^

T

=

T

CL

Desired closed loop

Tm

2(A. + 0

2

T

1 m

+ 0.25x m

T

Tm

2(* + 0 + 0.25i m specified

T m (x m + 2 X ) - ^ 2 2

T m (x m+2X)-X2 xm+Tm

Representative A.: X = 0.4T m a e [Atm,1.4Atm];

Tm

Smith (2002). Model: Method 1

T

2K m ( t m + ^ )

Kr „(xm+X) Zhong and Li (2002). Model: Method 1

3 3 % uncertainty in process parameters 50% uncertainty in process parameters 60% uncertainty in process parameters

0-5x m

K Isaksson and Graebe (1999). Model: Method 1 Chun et al. (1999). Model: Method 34

20% uncertainty in process parameters

*m »

*m

2

Lee etal. (1998). Model: Method 1

59

(a + 0

K ra

Tm

Axm = time delay uncertainty.

1

T Tm

60

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Comment

T, Ultimate cycle

McMillan (1984). Model: Method 2 or Method 44

89 K

(81)

Tuning rules developed from

T(81) 1

i K

Boe and Chang (1988). Model: Method I Geng and Geary (1993). Model: Method 1

X 1.935Tm [ m T

0.54K U

Quarter decay ratio

0.7 lT m

m

K u tan 2 4>m

0.1592 T u tanm

2

Vl + tan c() m

Matsubaefa/. (1998).

0.097

0.9

0-99 JX^

Model: Method 1

i a - < 0.167 T

3.33x m

Km^m

Feric etal. (1997). Model: Method 40

0.1 < ^ 2 - < 1.0 T

2.63TJ^ \ lm J

Huang et al. (2005). Model: Method 42

89 ^

(81)

90 K

LT

K

(82) _

( 8 1 , ;

=1.67TJI+|

T

"

Tm

0.9, 90

m

T(82)

(82)

1-881 T m Km

u'Tu

0.654

K„

-1

0-55

+ 0.4

Km TC

- tan

K„

-1

X ( 8 2 ) =0.159T„ 0.9,

K„

- 1 + 0.4-U - tarT

K„

-1

Chapter 3: Tuning Rules for PI Controllers

61

3.1.2 Ideal controller in series with a first order lag G c (s) = K, 1 + TjS,

R(s)

>

E(s)

y

K,

V

U(s)

1 1 + Tfs

1 Tfs + 1

Kme-"» •

^

l + sTm

i

Table 4: PI controller tuning rules - FOLPD model Gm (s) = Rule Umetal. (1985). Model: Method I

K„,e~

l + sT„

Comment T; Direct synthesis: time domain criteria T T 2 Tm X j CL2 K m (24T C L 2 + x m ) 2 T +-C Kc

^ CL2

Rivera and Jun (2000). Model: Method 1

Tm K m (2T m +*.)

Robust T

Desired closed loop transfer function = T C L 2 V+2^T C L 2 s + l

1 m

TmX ; 2Tm+X X not specified

Tf=

Handbook of PI and PID Controller Tuning Rules

62

3.1.3 Ideal controller in series with a second order filter G c (s) = Kc 1 + J_

2 A

l + bfls + bf2s

TiSy U(s)

E(s) R(s)

2

®+K

1+TiSy v

+ X

1 + 3. j-jS H~ 3. j-2^

l + sTm

/

Table 5: PI controller tuning rules - FOLPD model G m (s) = Rule

Kc

Y(s)

K e"

l + bfls + bf2s ^

Kme~ l + sT„

Comment

Ti

Direct synthesis: time domain criteria x

Tsange/a/. (1993). Model: Method 15

K

lTm

2

T

raxm

Coefficient values x

\ 0.0 0.1 0.2 0.3

i

1.851 1.552 1.329 1.160

x

\ 0.4 0.5 0.6 0.7

l

1.028 0.925 0.841 0.768

x

l

0.695 0.622 0.553

\ 0.8 0.9 1.0

Robust Lee and Shi (2002). Model: Method I

2

3

3

K

T

(83)

l < y < ^2T 2_

Y

b f l = 0.5xm , b f 2 = 0.0833x m 2 , afI = 0.2x m , a f2 = 0.01xm2 K

«>bwTm

(83)

;bfl=Tm+0.5Tm;bf2=0.5TmTra;

K m (co b w yT m +lJl + ^ ^ t a n t m +2co b w T m T m +2T m 2

KwY^m+l)

;a f 2

0.6 Tmxro , suggested co = 2(cobwYTm +1)

Chapter 3: Tuning Rules for PI Controllers

63

3.1.4 Controller with set-point weighting U(s) = K v

T s

E(s)-aK c R(s)

>y

<xK„

R(s) + t ~

K,

E s

()

T

v

is;

+® +

U(s)

1 + sT

Table 6: PI controller tuning rules - FOLPD model Gra(s) = Rule

Y(s)

Kme~

K m el + sT„

Comment T Kc i IVlinimum performance index: other tuning Tm/Tm<1.0. 4K (84) T (84) Overshoot (servo step) <20%

Servo/regulator optimisation - Taguchi and Araki (2000). Model: Method 1

2

a = 0.6830 - 0.4242-21- + 0.06568 -^Direct synthesis: time domain criteria

Chidambaram (2000a). Model: Method 1

K

(84)

T-i 'T' (85)

T1

CL

0.1098 + K„

2

++ T ( 8 5 ) xx M

5j

(85)

Choose T CL ,^

m

0.7382

2™ • 0.002434 T 2

.(84)

5 j (85)

=

TmTm + 2 T C L ^ - TCL

= T„

3\

+ 1.205

0.06216 + 3.171—s-- 3.058 T T T _ m_ _m_ J Tj(85) - £Jt KC. KJ!1J T ^ ' + T J - T / 8 5 ' ,(85)

^ora= " "

-(85)

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Srinivas and Chidambaram (2001).

6 0.9Tra K

Comment

Ti 3.33xm

Model: Method 1

mTm

Direct synthesis: frequency domain criteria Astrom and Hagglund (1995) - dominant pole design-page 204-208. Model: Method 5 or 14

0.29e- 2 ' 7t+3jT! T m K

mxm

g _9Tine-6.6,+3.0T»

0.79Tme-

Qr

L4l+2 4t2

-

0.14<^=-<5.5; Tra M.=1.4

a = l-0.81e°' 73,+L9T2 0.78e-4lT+5'7t2Tm Kmtm

8.9x m e^ + 3 - 0 T 2 or 0.79Tme-

L4x+2 4T2

-

0.14<^L<5.5; Tm Ms=2.0

a = l-0.44e a78t -°- 45t2 Astrom and Hagglund (1995)modified ZieglerNichols - page 208.

0-4Tm K

0
m

a =0.5; Model: Method 5 or 14 0.35Tm

Hagglund and Astrom (2002). Model: Method 5; M s =1.4;

0.1 < — < 2 T

0.7Tm

mTm

0.6

0.35Tm K

7xm

-^=-<0.11 T

0.8Tm

0.11<^=-< 0.17 T

0.8Tm

0.17<^-<0.2 T

m

0.6

mTm

K

m

0.25Tm 7

R

(86)

T(86)

^2-<0.5 T m

Ms = 1 . 4 ; 0 < a < 0 . 5

Sample K c , Tj taken by the authors (corresponding to the process reaction curve tuning rules of Zeigler and Nichols (1942)); oc = -

— - — = 0.30-1.11-^2 for K

these tuning rules. 7

K

(86)

— 0.14 + 0 . 2 8 ^ - 1 , T- (86) Km 1 T

:0.33t„

10x m +T m

c

K

m

T

m

Chapter 3: Tuning Rules for PI Controllers 3.1.5 Controller

with proportional

65

term acting on the output 1 K„

U(s) = - ^ E ( s ) - K c Y ( s ) T:S U(s) R(s)

?)

+

F-

• E(s)

+

Kc

P

T;S

-

VsJ *

Y(s)

K m e-™

^-

l + sTm

Kc

t Table 7: PI controller tuning rules - FOLPD model Gra(s) = Rule

Comment Kc Ti Minimum performance index: other tuning x

Minimum ITAE Setiawan et al. (2000). Model: Method 1

K

(87)

lTm

Kmtm la T 0.5 0.75 1.0 1.5 2.0

Chien etal. (1999). Model: Method 1

K el + sT„

x

l

^ m

x2

Coefficient values X x2 Ira. l T

x

la T

i

m

x2

0.554 1.805 3 0.900 0.547 7 1.740 0.413 0.570 1.373 4 1.090 0.480 8 1.930 0.398 0.594 1.117 5 1.290 0.445 9 2.120 0.385 0.645 0.822 6 1.520 0.428 10 2.410 0.389 0.720 0.679 Direct si,'nthesis: time domain criteria Underdamped system 8 K (87) T(87) response - ^ = 0.707 .

-T C L /+1.414T C L 2 T m + t m T m KmTf,,/+1.414Tr,,Tm+T„

*•» > 0.2Tm

r(87)

-1.414T CL2 T m+ T m x n T„+T„

Handbook of PI and PID Controller Tuning Rules

66

3.1.6 Controller with proportional term acting on the output 2 U(s) = Kc

E(s)-K,Y(s) V

R(s)

+

»

1

i

,s7

U(s)

E(s)

>6*

T

i "1

(

+

Y(s)

Kme-""

Kc 1 +

' P"

l + sTm

T Table 8: PI controller tuning rules - FOLPD model G m (s) = Rule Lee and Edgar (2002). Model: Method I

K,

K m ( ^ + Tm)

Kc

9

K

(88)

Kme" 1 + sT,.

T, Robust

Comment

T (88)

Kj =0.25KU

T m -0.25K u K m T m ^ T 2(X + t m ) .(88)

m-0.25KuKmxm

l + 0.25K u K m

2{X + zJ'

Chapter 3: Tuning Rules for PI Controllers

3.2 F O L P D Model with a Positive Zero G m (s)

67

Km(l-sTm3)e1 + sT,ml

3.2.1 Ideal controller

G c (s) = Kc T s

v E(s) R(s)

i y

U(s)

< 2 > — Kc

.

Km(l-sTm3>-"l + sT„

T s

v

Y(s)

i y

Table 9: PI controller tuning rules - FOLPD model with a positive zero Km(l-sTm3>-^ G m (s) = l + sT„

Rule

Kc

Comment

Ti

Direct synthesis: time domain criteria Sree and Chidambaram (2003a). Model: Method 1

1

i

YT ml T(89)

KmTm3

K „ T mm3

y = value of the inverse jump of the closed loop system; y < — m

Tmi

^[Tm3(l-xl)-0.5xm(l • (89) _

+

x,)]

Tm3

yTm

-(l-X,)-X,

L

m3

— x, e 0 . 9 5 — T T Y

0.1-

ml+

^.98—ITJHI—

YT ml

yT m i+T m3

0.1 < - = ^ < 0.3;

YTml+Tm

m3 0 3_

YT m ,

yT m i+T m

->1

68

Handbook of PI and PID Controller Tuning Rules

3.2.2 Ideal controller in series with a first order lag 1 G c (s) = Kc 1 + — TjsJl + TfS U(s)

E(s) R(s)

^ K r o (l-sT m 3 )e-""

•®->K v

i v 1 + Tfs

T s

Y(s)

l + sT„

Table 10: PI controller tuning rules - FOLPD model with a positive zero Km(l-sTm3)e-^ Gm(s) = 1+sT^ Comment T; Direct synthesis: time domain criteria

Rule

Kc

Sree and Chidambaram (2003a). Model: Method I

K m (2T m3 +X + Tm)

Tmi Tmi

Tf = m3^-0 2T m 3 +^ + t m T

Chapter 3: Tuning Rules for PI Controllers

3.3 FOLPD Model with a Negative Zero G m (s)

69

_ K m ( l + sT m3 )e1 + sT,ml

3.3.1 Ideal controller in series with a first order lag f

G c (s) = Kc

^6?S

1 A 1 1+— v T i S Jl + Tfs

U(s) K v

T

i

s

K m ( l + sT ra3 )e-

l 1 + Tfs ;

Y(s)

l + sT„

Table 11: PI controller tuning rules - FOLPD model with a negative zero K m (l + s T m 3 > - ^ G m (s) = l + sTm Rule

Kc

Change/al. (1997). Model: Method 2

Tm, K m (T CL + Tra)

Ti

Comment

Robust Tml

Tf = Tm3

Handbook of PI and PID Controller Tuning Rules

70

3.4 Non-Model Specific

3.4.1 Ideal controller Gc (s) = Kc v

R(s)

> *

L

i y

U(s)

m

l> y

T s

Kc

Non-model specific

fi + -Ll I TiSJ

Y(s)

Table 12: PI controller tuning rules - non-model specific Kc

Ti

Comment

Ziegler and Nichols (1942).

0.45KU

Ultimate cycle 0.83TU

Quarter decay ratio

Hwang and Chang (1987).

0.45KU

Rule

1 ("5.22 T

5.22' T

P. 1 u

T

i ,

p, ,T, = decay rate, period measured under proportional control when Kc = 0.5KU Parr (1989) -page 191. Parr (1989)-page 192. Hanged a/. (1993b)page 59. Pessen(1994). McMillan (1994)page 90.

Astrom and Hagglund (1995) -pages 141142.

0.5KU

0.43TU

0.33KU

2T

0.25KU

0.25TU

0.25KU

0.167TU

0.3571KU

T

0.4698K„

0.4373TU

Am=2,(j)m=200

0.1988KU

0.0882TU

A m = 2.44,

0.2015KU

0.1537TU

A m = 3.45,

dominant time delay process dominant time delay process

4>m = 46°

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Ti

Calcev and Gorez (1995).

0.3536KU

0.1592TU

0.59KU

0.81TU

Edgar e/a/. (1997)page 8-15. ABB (2001).

71 Comment

0.5KU

0.8TU

0.3KU

1 T (90)

0.3KU

1 T (91)

4>m= 45°, smallxra (j)ra= 15°, largex m Minimum IAE regulator Minimum IAE servo Good response regulator

Robbins (2002).

Direct synthesis Wang et al. (1995b).

2^(92)

T (92)

VranCicefa/. (1996), Vrancic(1996)page 121.

0.5A3 (A,A2-KraA3)

A3 A2

Vrancic(1996)page 140.

0.45KU

A3 A2

A m >2, m >60° (Vrancic et al. (1996)) Modified ZieglerNichols method

Friman and Waller (1997).

0.4830

3.7321

Am>2,

G

p(j f f l 150»)

K > 15°

1

T/ 90 ' = (0.24 + 0.11 lK u K m )TU ; T;(91) = (0.27 + 0.054K u K m )r u .

2

For a stable process, K m is assumed known; for an integrating process, K m and i are assumed known. K c

(92)

=

^_

I m

G p (jffl CL ){l-G CL (jco CL )}J' Im G P (JCO CL ){1-GCL(JCOCL)}_

. (92)

JWCLGCL

Rl G

G«>CL) ffl

I

p(jffi>c L ){l-GcL(j CL)}_

72

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Ti

Comment

1.18K u K m -1.72

3 T (93) i

0.1
Kristiansson and Lennartson (2000).

K

uKm

0.50K u K m -0.36 K

U

3

T

KuKm>0.5

(94)

K m

KuKm<0.1;

4T(95) i

20K135„Km-160

K 135 oK m <0.1

K, 3 5 oK m 5.4K135„Km-13.6 K

Kristiansson and Lennartson (2002).

4

KuKm<0.1;

(96)

K 135 oK m >0.1

135oKm

0.33KuKm -0.15 K m (0.18K u K m +l) 5

Kristiansson (2003).

T

KC<97»

6

K

(98)

KuKm co u (0.18K u K m +l)

KuKm<10

T(97)

KuKm>10

T(98)

1.6 <M S <1.9

Other tuning rules

Aikman(1950).

1.29T37%

[0.63K33„/o,0.97K33„/o]

[1.1T33%>1.3T33%

33% decay ratio

0.90K 37%

T 37%

37% decay ratio

1.19K 37%

0.92T 37%

37% decay ratio

1.02K 37%

0.92T 37%

37% decay ratio

^37%

<1.1T U

37% decay ratio

Rutherford (1950). MacLellan (1950). Young (1955).

0.50K u K m -0.36 (0.33K u K m -0.17)o) u

3 j (93) _ 1.18KuKm -1.72 (94) • (0.33K u K m -0.17)co u ' ' 4

T

20K135oKm-160

(95)

-(96)

(0.315K 1 3 5 „K r a -0.175) f f l u 5 v

K

6

K

0.09K,„„Km+0.25

(97)

,(97)

° 0 . 0 9 K , „ 0 K m +1.2

0.075

0.2+ K„

150

K„

- +0.05

5.4K135„Km-13.6 (1.32K135.Km-3.2)mu

135

(98)

J

37% decay ratio

0-81K 37%

K 135 oK m , 5 ol0.09K 1 3 5 „K m + 1.2)

>Ti

(98)

K,

=< D 150° 0.06 + 1.6-

Kn

^150°

-0.06 V

K

m J

Chapter 3: Tuning Rules for PI Controllers Rule K

Kc

Ti

73 Comment

x%

Tx%

0.5 + 0.3611nx

1.2Vl+0.0251n 2 x

0.999K 2 5 %

0.814T 25 „ /o

Parr (1989) -page 191. McMillan ( 1 9 9 4 ) pages 42-43.

0.667K 2 5 %

T 25 %

0.42K 2 5 %

^25%

'Fast' tuning

0.33K 25%

^25%

'Slow' tuning

Wade (1994) -page 114. ECOSSE team (1996b). Hay (1998) -page 189. Bateson (2002) page 616.

0.7515K 25%

0.7470T 2 5 „ / o

0.65K 50%

0.7917T50%

Chesmond(1982)page 400.

K

25%

^25%

K25%

T

sin(|)m

tan(|) m

Gp(jav)

co90»

Astrom (1982). Leva (1993).

7

'Modified' ultimate cycle method

1

tan((j)m - (j)ra - 0.5TI)

i

4

G p (jra 135 o) 0.25

1.6

G p (jco 135 „)

co 1 3 5 „

0.25KU

0.2546 Tu

Hagglund and Astrom (1991).

Example: x = 25 i.e. quarter decay ratio Quarter decay ratio

T(99)

KC<99)

0.5 Astrom and Hagglund (1995) -page 248.

x = 100(decay ratio)

>0 Alfa-Laval Automation ECA400 controller Alfa-Laval Automation ECA400 controller - large delay Model: Method 2

0.2hTusin(t>m

Cox et al. (1994). Jones ef a/. (1997). Model: Method 2

A

0.16T u tan* m

P U

0.5787 P

tan(.•,-0-5rc) 2

a e [0.7958,1.5915]

aT u A

|Gp ( j a ) ) ^ tan ( ^ - < ! . , „ - 0 . 5 T I ) '

(•»)

T

'

=

tan((|) m - <|)m - 0.5TC) m

Handbook ofPIandPID

74 Rule

NI Labview (2001). Model: Method 2

Controller Tuning Rules

Kc

^

0.4KU

0.8TU

0.18KU

0.8TU

0.13KU

0.8TU

2.5465hsinm

tan<|)m

Comment Quarter decay ratio 'Some' overshoot 'Little' overshoot 0.81 <x, < 1 ;

Tang et al. (2002). Model: Method 4

A p x, co90o A m

C0

90°

x, = 1, 'small' x m ; x, =0.91, 'large'T m

Chapter 3: Tuning Rules for PI Controllers

75

3.4.2 Controller with set-point weighting U(s) = Kc

E(s)-aK c R(s)

T s

v

iv

aK„

+



Non-model specific

*K,

-

E(s)

T

v

is/

+

R(s)

Y(s)

U(s)

Table 13: PI controller tuning rules - non-model specific Rule

Kc

Comment

Ti

Direct synthesis 2 9K 2 6K2

0.053Kue - - Astrom and Hagglund (1995) -page 215.

M s =1.4

0.90T„e-4'4K+2'7K2

a = l-l.le-0006lK+L8K2;0
M s = 2.0

O^OT^- 4 ' 4 " 2 - 7 ^

a = 1 - o.48e 0 ' 40 ^ 017K2 ; 0 < K m K u < oo . Vrancic (1996)page 136-137.

8

c

K X

2

"(l-b K2A3 (100)

K

rp (100)

(100)

A,A2-KmA3-x

(100)

K

8

+

.KmA3-A,A2<0;

A, - 2 K m A , A 2 J

A,A2-KraA3+x 2 3 (l-b2J(K ^mm "A 3 1 +T A ^ ,l - 2 K m A 1 A ,

y

KmA3-A,A2 >0; b = l - a ;

= V(KmA3-A1A2)2-(l-b2)A3(Km2A3+A13-2KmA1A2); A,

. (100)

Km+-

K 2K

(100)

a = [0.2,0.5] - good servo & regulator action (Vrancic (1996), page 139)

(I00)

K

2

(,-„>)

76

Handbook of PI and PID Controller Tuning Rules

3.5 IPD Model G m (s)

3.5.1 Ideal controller G c (s) = Kc T,Sy

R(s)

U(s)

E(s)

+

K

1+-

Xs

Table 14: PI controller tuning rules - IPD model G m (s) = Rule

Kc

Ziegler and Nichols (1942). Model: Method 2

0.9

Two constraints method - Wolfe (1951). Model: Method 2 Coon (1956), (1964). Model: Method I Astrom and Hagglund (1995) -page 13. Model: Method 1 Hay (1998)-page 188. Hay (1998)-page 199. Model: Method 1 Kc and T> deduced from graphs

Y(s)

K m e"

K m e"

T,Process reaction

Comment

Kmxm

3.33xm

Quarter decay ratio

0.6 Kmtm

2.78xm

Decay ratio = 0.4

4.35xm

Decay ratio is as small as possible

0.87

Kmtm Minimum error integral (regulator mode). Quarter decay ratio; 1.0 0 Kc,Tj deduced from graphs Ultimate cycle 0.63 Ziegler-Nichols 3-2xm K mTm equivalent 0.42 Model: Method 3 5.8xm K X m m

7.5

Kmtm=0.1

3.5

Kmxm=0.2

2.5 2.2

4.5Kmxm

K m T m =0.3 Kmxm=0.4

Chapter 3: Tuning Rules for PI Controllers

77

Rule

Kc

Hay ( 1 9 9 8 ) continued.

2.0

Kmtm=0.5

1.7

Kmxm=0.6

Comment

Ti

*i

Bunzemeier (1998). Model: Method 4

1^

0% overshoot (servo)

m X2Tm

x3

c

1( J n ,

Kp "

s(l + s T p )

10% overshoot (servo)

n

x2

*i

0.18 0.24 0.22 0.21 0.20 0.20 Minimum IAE Shinskey ( 1 9 8 8 ) page 123. Minimum IAE Shinskey ( 1 9 9 4 ) page 74. Minimum IAE Shinskey ( 1 9 8 8 ) page 148. Model: Method 1 Minimum ISE Hazebroek and Van derWaerden(1950). Minimum ISE Haalman(1965). Model: Method 1

x3

Coefficient values n x i

x2

X

3

n

1.350 0.23 0 0.20 1.477 0.25 6 3.050 0.40 1 0.24 0.19 1.463 7 1.910 0.30 2 0.19 1.453 0.24 8 1.690 0.28 3 0.18 1.385 0.24 9 1.584 0.26 4 0.24 0.18 1.378 10 1.557 0.26 5 Minimu m performance index: regulator tuning ().9524 Model: Method 1 4T 1 ().9259 4T

1

C.61KU

Model: Method 1

Tu

Also specified by Shinskey (1996), page 121.

5.56x m

Model: Method 2

1.5

M,=1.9,

().6667 0

Am=2.36, *m=50°

Minimum ITAE Poulin and Pomerleau (1996). Model: Method 1

4.5804x m

Process output step load disturbance

3.8853x m

Process input step load disturbance

(X5264

().5327 K

T

m ra

Minir num performance index: other tuning Skogestad (2001). Model: Method 1

0.28 K

mTm

7tm

M m a x =1.4

Handbook of PI and PID Controller Tuning Rules

78 Rule

T,

Comment

7xm

M m a x =1.7

3.77xm

M max = 2.0

Kc 0.404

Skogestad (2003). Model: Method 1

K

Skogestad (2001). Model: Method 1

mTm

0.49 Direct synthesis: time domain criteria Maximum closed 0.487 loop log modulus 8.75xm K ra x m = 2dB; TCL = 2.2TU 0.31KU

Tyreus and Luyben (1992). Model: Method 1 or 9 Fruehauf et al. (1993).

0.5 5xm 0.75

Rotach(1995). Model: Method 5

Kmxm

Wang and Cluett (1997). Model: Method 1

Cluett and Wang (1997). Model: Method 1

2-4lTra

K

(ioi)

T

2

K

(102)

•j- (102)

K

Damping factor for oscillations to a disturbance input = 0.75. 4 = 0.707; TcL^m^xJ 4 = 1; T

CL 6 k.>16xJ

0.9588/Kmxm

3.0425xm

0.6232/Kmxm

5.2586xm

T

CL

= 2xm

0.4668/Kmxm

7.2291xm

T

CL

= 3xm

0.3752/Kmxm

9.1925xm

TCL

= 4t m

0.3144/Kmxm

11.1637xm

T

CL = 5xra

0.2709/K m t m

13.1416xm

T

CL

Poulin and Pomerleau 2.13 0.34KU or (1999). KmTu Model: Method 1

i

(ioi) i

i

Model: Method 2

(ioi)

K m i m (0.7138T CL +0.3904) 1

K m x m (0.5080T CL +0.6208)

1.04TU

, T,m)

TcL

=

T

m

= 6xm

M max = 5 dB

= (l.4020TCL +1.2076)xm .

Tj 0 0 2 ' = (l.9885TCL +1.2235)xm .

Chapter 3: Tuning Rules for PI Controllers Rule

Kc X

Viteckova(1999), Viteckova et al. (2000a). Model: Method 1

x

OS

0.368 0% 0.514 5% 0.581 10% 0.641 15% 1.1111

Chidambaram and Sree (2003). Model: Method I

Comment

T; 0

l/KmTm

l

79

Coefficient values OS x i 0.696 0.748 0.801 0.853

20% 25% 30% 35%

i

OS

0.906 0.957 1.008

40% 45% 50%

x

4-5xm

Kmtm 0.23

Huba and Zakova (2003). Model: Method I

K

2.914xm

mTm

0.281 3.555xm

KmTm

Km(TCL+0 0.5

3

Direct synthesis: frequency domain criteria (103) Maximise crossover 0 frequency

K

Model: Method 6; Am=2

0.67075 K

3.6547tm

T

ra m

m

Gain and phase margin - Kookos et al. (1999). Model: Method 1

'good' robustness TCL=xm,5 = l

8t m

mm

Hougen(1979)page 333. Model: Method 1 Chidambaram (1994), Srividya and Chidambaram (1997).

Suggested % = 0.7 or 1

4? 2 (T CL +0

1 Skogestad (2003), (2004b). Model: Method 1

1

p

C0p (0.571-CO p T m )

AmKm 0-942/Km xm

Representative results 4.510xm

A m =1.5;
3

0.698/Kra xm

4.098tm

Am=2;4,m=30°

0.491/Km r m

6.942xm

A m = 3; *„,= 45°

0.384/K m x m

18.710xm

Am = 4 ; ^ m = 6 0 0

Values recorded deduced from a graph: 0.1 0.2 0.3 0.4 ^m^m K

(103)

7.0

3.5

2.2

1.8

0.5

0.6

0.7

0.8

0.9

1.0

1.4

1.1

1.0

0.8

0.75

0.7

80

Handbook of PI and PID Controller Tuning Rules

Rule

Chen et al. (1999a). Model: Method 10

Comment

Kc

T;

•L+fta-i)

4rp(104)

f

1.0607~] Am

KmTra(Am2-l) 5

(105)

K

j(105)

Cheng and Yu (2000). Model: Method 1

0.5236/Km xm

O'Dwyer (2001a). Model: Method 1

1< m T m

Am=2.83; 8^m

* m = 46.1°

*1

4

T

(104)

*2*

Representative coefficient values x x2 Am 4>m i 0.357 1.5 4.3 46.2°

*1

x2

0.558

1.4

0.484

1.55

2.0

45.5°

0.458

3.35

3.0

59.9°

0.305

12.15

4>m

Am 4.0

60.0°

5.0

75.0°

Amz-1

__n_

•0.5ir(Am-l)

+ 0.57t(Am-l)

0.57t-

Am 5 r,

[6]

m

Tl71!2^,,, - l )

(105)

2

-(105)

2

4Kmxm(r1 Am -l) x2

IT-

4x,

^(r.'A^-l)

__

2

2

;Cr1 Am(r1Am-2X2r1Am-l)

-\2

i"., Tt

'

-1

7t

x2

A„ 71

1+-

m = t a n '

1

7t

7t

"2 + ^T _ ^7 ^0.5x!2x22 +0.5x1x2^/x,2x22 + 4

2 0.5x, +0.5^J-A/x12x22 + 4 M

Chapter 3: Tuning Rules for PI Controllers

81

Rule

Kc

T,

Comment

O'Dwyer (2001a). Model: Method 1

aK„

bT u

[']

Robust Chien (1988). Model: Method I

1 ( 2X + zm ^

X

Ogawa(1995). Model: Method 1; Coefficients of K c and Tj deduced

from

graphs Alvarez-Ramirez et al. (1998). Model: Method 1 Zhong and Li (2002). Model: Method 2

7

l / K m Tra

l

x2

0.45 0.39 0.34 0.30 0.27

11 12 13 14 15

x

K m lT C L T m J

8

2X + rm X

20% 30% 40%) 50% 60%

2t

m

uncertainty uncertainty uncertainty uncertainty uncertainty

in process in process in process in process in process

parameters parameters parameters parameters parameters

Tm+^m

T CL > t m

2 a + xm

a not specified

2a + i m Km(a + tm)2

2b

71

Tea

4b

Am = 71

1+-


tan

—+ 2

7t

4b

27: 2 a 2 b 2 +27tabV47t 2 a 2 b 2 +4

- . . 0 . 1 2 5 T C V + — V 4 7 i 2 a 2 b 2 +4 11

16b X = [l/K m ,x m ] (Chien and Fruehauf( 1990)); X = 3xm (Bialkowski (1996)); I > t m +T m (Thomasson (1997)); I = [l.5xra,4.5Tm] (Zhang era/. (1999)). From a graph, X = 1.5tm ....Overshoot = 58%, Settling time = 6xm X = 2.5Tm,....Overshoot = 35%, Settling time = 1 lxm X = 3.5Trn, ....Overshoot = 26%, Settling time = 16xm X = 4.5xm .Overshoot = 22%, Settling time = 20xm . X = e„„ /100K ; emax = maximum output error after a step load disturbance max V/100K„m (Andersson (2000) -page 12); Xse K c (^s + l)

• (Chen and Seborg (2002)); X = xra (Smith (2002)).

Handbook of PI and PID Controller Tuning Rules

82 Rule

Smith (2002). Model: Method 1 Skogestad (2004a). Model; Method 1

Kc 1

Ti

Km^m

%

Comment

4 9

K

(106)

(1 6)

K c ° K ra Other methods

Penner(1988). Model: Method 1

0.58 KmTm

10tm

Maximum closed loop gain = 1.26

5.9xra

Maximum closed loop gain = 2.0

0.8 KmTm

9

K

(106) >

lAd0l A

I Y max I |Ad0| = maximum magnitude of a sinusoidal disturbance over all frequencies, I Ay maX | = maximum controlled variable change, corresponding to the sinusoidal disturbance.

Chapter 3: Tuning Rules for PI Controllers

83

3.5.2 Ideal controller in series with a first order lag G c (s) = K( 1+-

TiSy

U(s)

E(s)

R(s)

1

K, 1 + — T s v i /

Y(s)

K m e"

1 + Tfs

Table 15: PI controller tuning rules - IPD model G m (s) = Rule

1 1 + Tfs

Kc

K m e" Comment

Ti

Robust H„ optimal Tan et al. (1998b). Model: Method 1 Rivera and Jun (2000). Model: Method 1

1 £

(107)

0.463X. +0.277 Kmxm

Km(2xm +4zmX

0.238X +0.123 '

1K

2(x1B+X)

(107)

2{Tm+X) 2

A. = 0.5;

tm

2

+ X )'

,Tf=f

x ^ 2

2xm +4TmX + X2

5.750A. + 0.590 X not specified

84

Handbook of PI and PID Controller Tuning Rules

3.5.3 Controller with set-point weighting U(s) = Kc

E(s)-aK c R(s) V

T

>s/

KaK

K8> -

R(s)

xg>

K, E(s)

T s

v

. y

Y(s)

K m e" U(s)

K e" Table 16: PI controller tuning rules - IPD model G m (s) = - m T Comment Kc i Minimum performance index: servo/regulator tuning a = 0.6810, 0.7662 Tm/Tm<1.0. 4.091xm KmTm Overshoot (servo step) < 20% X T x,K u Km=l 2 U

Rule

Taguchi and Araki (2000). Model: Method I Minimum ITAE Pecharroman (2000). Model: Method 10

Chidambaram (2000b). Model: Method 1

Coefficient values *i

x2

0.049

2.826

0.066

2.402

0.099

1.962

0.129

1.716

0.159

1.506

0.189

1.392

a

a

*1

x2

0.506 -164° 0.512 -160°

0.218

1.279

0.250

1.216

0.522 -155° 0.532 -150°

0.286

1.127

0.330

1.114

0.578 -125° 0.579 -120°

0.544 -145° 0.555 -140°

0.351

1.093

0.577 -118°

4>c

4>c

0.564 -135° 0.573 -130°

0-64/Kra xm

Direct synthesis 3.333Tta

a = 0.65

0.487/Km xm

8.75xm

a = 0.557

In general, a ~ 0. 6 (on average) or a = 0.' T

Hagglund and Astrom (2002). Model: Method 2

M s = 1.4;

0.35 K

T

i

7?m

0.3
Chapter 3: Tuning Rules for PI Controllers

85

3.5.4 Controller with proportional term acting on the output 1 K„

U(s) = -£-E(s)-K c Y(s) R(s)

Kc

•6^ *\ F-

E(s)

~L6?V—*

T,s

-+

U(s)

Kme-"» s

Y(s) •

Kc

4 Table 17: PI controller tuning rules - IPD model G m (s): Rule

Comment Kc T; Minimum performance index: regulator tuning 0.6330 Model: Method I 2.3800im K ra ^ m

Minimum ISE Arvanitis et al. (2003a). 2

Arvanitis et al. (2003a). Model: Method 1

3

0.6114 KmTm

2.7442xm

0.6146 Kmxm

2.6816xra

Minimum |jerformance index: servo tuning 0.6270 Model: Method I 2.4688xm K x

Minimum ISE Arvanitis et al. (2003a).

m m

2 0.6064 Kmtm

Arvanitis et al. (2003a). Model: Method 1

mTra

Minimum f[e2(t) + K m V ( t ) ] d t .

Minimum

1

2.8489tm

3 0.6130 K

1

K m e"

e2(t)+K 2

rdu^ 2

- Ut

dt.

2.7126tm

Handbook of PI and PID Controller Tuning Rules

86 Rule

Kc

Chiene/a/. (1999). Model: Method 2 Arvanitis et al. (2003a). Model: Method 1

4jr

Comment

Ti

(108)

Direct synthesis 1.414T CL2 +t m

4 (8-P)K m x m

4tm

P

Underdamped system response - £, = 0.707 .

P

%Al

Robust Arvanitis et al. (2003a). Model: Method 1

4

v

(108)

^r'=—r

5

K

(109)

0.(712 - a j

1.414T CU +T m

J .CL2 , , 2 + 1 . 4 1 4 T CL2 T mT+ xl m„ 2 ) ' r , I , '•m

5

K

(109)

4K1 2 2

[(8-a>i +a ]Kr

with 0 < a < %2 :

recommended a = — 1 - 1 2

'TI2(452+I)

, £> 0.3941

Chapter 3: Tuning Rules for PI Controllers

87

3.5.5 Controller with proportional term acting on the output 2 U(s) = Kc 1 +

E(s)-K,Y(s) Ti^y

U(s)

E(s)

R(s)

"KgH-

K, 1 + T;s

+ *

Y(s)

K„e-ST™

K,

I Table 18: PI controller tuning rules - IPD model G m (s) = Rule

Kc

K„,e"

Comment

Ti

Robust Lee and Edgar (2002). Model: Method 1

6

K

(HO)

6

K

T (110)

(HO)

0.25K,,

T(110)

fc + O u K

K

m

K, =0.25KU

i

2(X + T m )

_ _ i _ _

KuKm

T

m

,

T

m

2(\ + T J "

88

Handbook of PI and PID Controller Tuning Rules

3.5.6 Controller with a double integral term G c (s) = K E(s) R(s)


K,

\

1

1 ^ U(s)

'

l l ^ 1+—+—T;,s X 9 s 2 Y(s)

K„e"

1+ + V T ii s T i 2 s 2 y

Table 19: PI controller tuning rules - IPD model G m (s) = Rule

Kc

Belanger and Luyben (1997). Model: Method 2

0.33KU

Ti Ultimate cycle T„ = 2.26TU ; Ti2 = 20.5TU2

K„e-

Comment Maximum closed loop log modulus = +2dB.

Chapter 3: Tuning Rules for PI Controllers

3.6 FOLIPD Model G m (s) -

89

Ke" >(l + sTm)

3.6.1 Ideal controller Gc (s) = K( v R(s)

?)

y

E

(s),

*

T s

i y U(s)

Kc

k

s(l + sT m )

T s

I

Y(s)

Kme"^

iJ

Table 20: PI controller tuning rules - FOLIPD model G m (s) = Rule

s(l + sT m ) Comment

^

Kc

K„e"

Process reaction x

Coon (1956), (1964). Model: Method 1; Coefficients of Kc deduced from graphs

Minimum IAE Shinskey(1994)page 75. Minimum IAE Shinskey(1994)page 158

i

Quarter decay ratio criterion

0

Km(^m+Tm)

Coefficient values ^m/T m

x

i

tm/Tm

*1

tm/ T m

x

i

0.020 0.053 0.11

1.1 5.0 0.25 2.2 4.0 4.0 0.43 1.7 3.0 1.0 1.3 Minimum pe rformance index: regulator tuning 0.556 Model: Method I 3.7(x ra +T m ) Km(xm+Tm)

0.952 Km(Tm+Tj

4(Tm

+ 0

Model: IMethod 1

Handbook of PI and PID Controller Tuning Rules

90 Rule

Minimum ITAE Poulin and Pomerleau (1996). Model: Method 1

i

T

Process output step load disturbance

Process input step load disturbance

K

(I")

m/T m 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

K

° 2

X

(111) _

5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397

T

2

x,(xm+Tm)

Kc and Ti coefficients deduced from graph x2 0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314

*1

4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099

0

Maximise crossover frequency

1.04TU

M max = 5dB

m

Km(xm+Tm)^Xl(Tm+Tm)2

x, values recorded deduced from a graph: 1 2 3 4 5 *m/Tm

x,

Comment

Coefficient values x2 tm Am 1.2 0.5231 0.5237 1.4 0.5241 1.6 0.5245 1.8 2.0 0.5249 1.2 0.5320 1.4 0.5315 0.5311 1.6 0.5311 1.8 0.5311 2.0 Direct synthesis

*1

Hougen(1979)2 1.26X! page 338. K mTm Model: Method 1 Poulin and Pomerleau 2 13 (1999). 0.34K,, or Model: Method 1 KmTu

1

Ti

Kc

0.31

0.19

0.14

0.11

0.09

6

7

8

9

10

0.078

0.068

0.06

0.053

0.05

Chapter 3: Tuning Rules for PI Controllers Rule K

Comment

^

Kc 3

91

(112)

0

Representative tuning rules; tuning rules for other x m /T m values may be obtained from a plot.

Huba and Zakova (2003). Model: Method 1

0.193/Kraxm

3.717xm

0.207/K m t m

3.386xm

Tm Am =1

0.219/K m x m

3.127xm

^m/Tm=2

tm/Tm=0.5

Ultimate cycle McMillan (1984). Model: Method 1

4

v

JS..

(113)

Ku tan2
Peric etal. (1997). Model: Method 8

3

K

2

yj\ + tan
Tuning rules developed from KU,TU

-p (113)

0.1592 Tu tan


-2Tm+^/xm2+4Tm2

(112) —

1— 2T m +x m - > /T ni 2 +4T m

4K.(H3)_l-477Tm

1

Km xm2 1+| — m I

r

• , Ti1"-" = 3.33xm 1 +

i Y'65

Handbook of PI and PID Controller Tuning Rules

92

3.6.2 Controller with set-point weighting U(s) = Kc 1 + -

E(s)-aK c R(s)

TiS, * aKc

R(s)

+ T

K, 1 + -

Xs

E(s)

U(s)

+

s(l + sT m )

Table 21: PI controller tuning rules - FOLIPD model G m (s) =

Taguchi and Araki (2000). Model: Method I

X

x2

0.049

2.826

Km = l; Tm = l

0.066

2.402

0.05
0.099

1.962

0.129

1.716

0.159

1.506

0.189

1.392

(114)

0.1787 + K„

x2Tu Coefficient values

1KU

Xl

K

K m e" s(l + sT m )

Comment Tj Kc Minimum performance index: servo/regulator tuning ^n/Tra<1.0. 5 Kc0M, T (114) Overshoot (servo step) < 20%

Rule

Minimum ITAE Pecharroman (2000). Model: Method 4

Y(s)

K„e"

a

a

i

i

x2

0.506 -164° 0.512 -160°

0.218

1.279

0.250

1.216

0.522 -155° 0.532 -150°

0.286

1.127

0.330

1.114

0.578 -125° 0.579 -120°

0.544 -145° 0.555 -140°

0.351

1.093

0.577 -118°

4>c

x

0.564 -135° 0.573 -130°

0.2839 -0.001723

.(114)

V " " =4.296 + 3.794-^- + 0.2591 - ^ T T

a = 0.6551+ 0.01877-

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

93 Comment

Ti

Direct synthesis Q41e-0.23T+0.019t2

Astrom and Hagglund (1995) -pages 210212. Model: Method 3 0.14 < ^ 2 - < 5.5 T

Km(Tm+iJ

5.7xrae,J-°-69T2

M s =1.4

a = l-0.33e25'-'^2 agle-l.lt+0.76T'

Km(Tm+Tm)

3 4T

e°-

28

i-00089^

a=l-0.78e- L 9 T + 1 ' 2 x !

Ms=2.0

Handbook of PI and PID Controller Tuning Rules

94

3.6.3 Controller with proportional term acting on the output 1 K„ U(s)=-*-E(s)-KcY(s) T:S R(s)

d

•

y + T F-

'

K« T,s

E(s)

Y(s)

k- K m e ^

^6?i . f

U(s)

s(l + sT m )

Kc

T Table 22: PI controller tuning rules - FOLIPD model G m (s): Rule

K m e" s(l + sTra)

Comment Kc Ti Minimum performance index: regulator tuning 4 4xm

Arvanitis et al. (2003b). Model: Method 1

(8-p)Kmxm

P Minimum ISE 6

3

P = 2.6302-2.1248

i

+ 2.5776

, T

T

•1.8511

-0.23641 • I'm/

-0.044128 -siVTm J

+ 0.82557

T V

-0.0053335^2- + 0.00040201^1V mJ V m7 10

-1.7164.10"

, T

V mJ P = 1.7110073-0.003554

+ 3.1685.10"7 ±2L T

vT

T

m J

, 0
-4.1 |, -21-> 4.1.

•

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

7 8

Minimum performance index ,

Minimum performance index 9 , l 0

7

Performance index = He2 (t) + Km V (t)Jdt .

!

B = 2.1054-0.76462^=- +0.79445f^=-

K + 0.010927

-0.51826

'

IT, *

'T

T

-0.0013006

T

+ 7.6235.10"

T

+ 0.21771

T

-9.717.10"

T

T

A

('_

-4.132.10"

Comment

Ti

Arvanitis et al. (2003b). Model: Method 1 (continued)

-0.06

95

, 0<-S-<4.6; T

T

* m

B = 1.69667833406 - 0.00148768

-4.6

T

•>4.6.

V m

e 2 (t) + K

Performance index:

2 m

f

6 = 1.6505623 - 0.0034983 - ^ - - 8

8 = 2.1806-0.6027:

-0.0042243 V,

+ 4.2603.10"7

-0.14955

T

+ 0.034535

+ 5.0912.10" 5 |^-

-7.283.10"

-9.5824.10 I — m J

T V

m

\ T

m

>T

9

(,

T

V

V

m J

V

X

T

m J

+ 8.2395.10"

T

>2;

+ 0.3875 V

dt

, - 21 - range not defined.

m J

96

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Arvanitis et al. (2003b). Model: Method 1

nK(ii5)_

II

Comment

i

Minimum ISE 12 Minimum performance index '3, 14

. 12.5664(Tm+Tj-4 Km[2(xm +T m )(l2.5664(x m +T m j - 4 ) - (l.5Tra2 + 3xmTm + 2T ra 2 ^J 3.4674T m +3.1416T m -l

with a = [0.5708 + 12

K

T

T(H5)

(H5)

% = 0.99315 + 1.9536

, T

-2.5157

k;T|(115)_12.5664(xm+Tm)-4

( 1.9163

T

\3 0.8893

T

, T V

-0.049809-^2-

+ 0.26167 - E -

+ 2.0083.10"

T

- 3.7368.10 -7

T

Xm

V

m J

0<^_<6.0; T

, T

X = 1.96705757 + 0.01308189

- 0.00046593^-

+ 0.0061098-2-

V mJ

, m /

6 1, -^2- > 6.0 .

T

' Performance index = l[e2 (t) + K m V(t)]di A-1

X = 0.73114 + 2.346ip2- - 2 . 8 5 0 9 P S

Un, J

+ 2.0965

(jra

m / f

+ 0.27639 ^ L

lTm J

+ 2.0611.10~5

f.

_ 0.052089 V

T

,

-0.95252

m J

>J V

x^

, 0<-!2-<4.5; T

T m J

X = 1.916423116 + 0.018175439^- -4.5 V 'in

V

^7

f

V

\

T

+ 0.0063413 - ^ L | -0.00048064 T,

—

{Tm j

-3.8178.10~ 7

(~

• > 4.5 .

m J

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

97 Comment

T,

Arvanitis et al. (2003b). Model: Method 1 (continued)

Minimum performance index 15, 16

Minimum iperformance index: servo tuning 4 4T (8-p)K m T m P

Arvanitis et al. (2003b). Model: Method 1

Minimum ISE "

' Performance index = f e 2 (t) + K m 2 — ra

J

I dt

UJ|

oL

J

X = 0.60896+ 3.0593

-3.9339

, T

,T V

r + 0.38696 V

-0.07277:

T

\

6

m /

+ 0.0088395

T

-1.3353 m

T

-5.2941.10'

T V

V

it

-0.0006686

0<-m_<4.5; T

>T

-4.5

T ^

* T

V

-0.092452^vT

+0.017215

-0.0020701 T

--5.5

T

+0.32269 [ ^ , T

\T m J

-0.00015505

, 0<^-<5.5; T

T V

m J

(3 = 1.6281745-0.001171

0.72106M M T

m J

+ 1.2025.10"

T

-^->4.5. T

m

p = 1.997- 0.82781| - ^ - |+1.0002

V

m

m

X = 1.9283751 + 0.016846129

-6.5691.10"

T V

m J

-2.8623.10"

17

x„

* + 2.933 , V

'T

m J

->5.5.

T

98

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Comment

Ti

Arvanitis et al. (2003b). Model: Method 1 (continued)

Minimum performance index l8,19 Minimum performance index 20, 21

00

1

Performance index = f[e2 (t) + K m V (t)]dt \2

f

'(3 = 1.5693 + 0.1066^2-1-0.10101

T

-0.016147

V 1m J

•1.2274.10"7

VTm

m

\

r

T

Performance index =

+ 3.7365.10"

T

*"**.'i£

P = 0.780651417 + 4.7464131-^--0.1 T

J

, T

10

, 0<-^-<4.6. T dt

0<^_<0.3; T

P = 1.729934 -0.051535 -^--0.3 , 0.3<-SL<2. T T V1m

T V

+ 0.0030412 — I -0.00032187 -^2-1 +1.2181.10"5 V

\4

e + 0.05217

m J

+ 9.9698.10"

T

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Arvanitis et al. (2003b). Model: Method 1

99 Comment

Ti

(116)

22 K

i

Minimum ISE 23 Minimum performance index 24 , 25

22

v

12.5664(x m +T m )-4

(116)

Km[2(xm +TmXl2.5664(Tm + T m ) - 4 ) - ( l . 5 x m 2 +3x m T m + 2T m 2 >]

wkha = [0.5708 + i p Z l ^ H l l K E i ] x , T(..« =

23

r A3 x = 0.79001 + 2.114|^2- - 2 . 5 6 0 1 P 2 - ! +1.8971 T Tmy 1 IT,

12.5664(x m+ T m )-4 f

0.86871

-0.048068

.9231.10"

+ 0.0058771 • ^ M -0.00044707

T

Y m 7

-3.5718.10"

T

T V m y

V m J

+ 0.25371

\4

x„

0< J L < 4 . 5 ;

T

T

1

V m J

V m

^ M V m 7

X = 1.903177576 + 0.0181584p2--4.5 , ^ - > 4 . 5 . V m

/

m

00

1

25

Performance index = f[e2(t) + K r a V(t)jdt .

x = 0.48124 + 2 . 5 5 4 9 ^ - 1 - 2 . 8 9 6 \5 'T + 0 . 2 6 0 9 4 ^ - - 0.048775 Y m )

+ 1.9095.10"

V Tm j

+ 2.0495

T ^

+ 0.0059064

i T , V mJ

-3.5335.10"

X = 1.87054524 + 0.021802| — - 4

-0.911151 —E-

vTm T

-0.00044618

0<-^-<4.0; T

T 1

V m J

->4.0.

Handbook of PI and PID Controller Tuning Rules

100 Rule

Kc

T

Arvanitis et al. (2003b). Model: Method 1 (continued)

Comment

i

Minimum performance index

26 27

,

Direct synthesis Arvanitis et al. (2003b). Model: Method 1

4 (8-P)K m x m

4tm

28K(U7)

(l + 4 ? 2 ) k + T j

00

'x

^

T V

r + 0.52906

A

dt. f

- 5.0294

m J

0.10079 -S2-

+ 3.8589 I™. T

T V

5

T

^

f

' Performance index = I e 2 (t) + K m 2 —

X = 0.31373 + 3.7796

p =

p

m

\

^

( 1.7956

\4 T f

+ 0.012369 ^ 2 - | -0.00094357

\ T

T

V 1m J

+ 4.0678.10- 5 |-E-

-7.5694.10

X = 1.86144723+ 0 . 0 2 1 9 8 9 ^ - 3 . 5 28 j r

(117)

, — > 3.5 . j

V*m

0<-E-<3.5; T

T

*m

2

(l + 4 4 k + T m ) K r a k 2 ( 0 . 5 + 84 2 )+x ra T m (l + 1 6 ^ ) + 8 ^ 2 T m 2 ] '

Chapter 3: Tuning Rules for PI Controllers

101

3.6.4 Controller with proportional term acting on the output 2

(

11

U(s) = Kc 1 + — E(s)-K,Y(s) V T.sy b(s)

R(s) k

+

I

T s

iJ

vy »;

f

Y(s)

Kme-" •"W

w

s(l + sT m )

1

t Table 23: PI controller tuning rules - FOLIPD model G m (s) = Rule

Kc

Ti

K e~ s(l + sTm) Comment

Robust Lee and Edgar (2002). Model: Method 1

29 ^

(118)

29 K

(118)

T(118)

0.25K,,

(A- + 0

KuKm

2(X +

>T ; T„

K, =0.25KU

(118)

2(^ + 0 '

Handbook of PI and PID Controller Tuning Rules

102

3.7 SOSPD model

K m e" K e" T m l V + 2 ^ T m l s + l or-(l + TmlsXl + Tm2s)

3.7.1 Ideal controller G (s) = K

1+ — T s

V

RW^E(S) . +

Kc

1

i

i 7

U(s)

f. + -L)

Y(s)

SOSPD model

Table 24: PI controller tuning rules - SOSPD model K-e-"Kme-'of TmlV+2i;mTmls + l (l + TmlsXl + Tm2s) Comment Kc Ti Minimum performance index: regulator tuning

Rule Minimum IAE Shinskey(1994)page 158.

1

K

T(119)

(119)

2-83(T m +T m2 )

T - H i = 0.1 Tml

2.65(T m +T m2 )

^Jsl = 0.2 Tml

0.77Tm, Minimum IAE Shinskey (1996) page 48. Model: Method 1 -^2-= 0.2 Tml

0.70Tml K

mXm

0.80Tml K

2.29(t m +T m 2 )

mTm

0.80Tral Kmxm

1

K

(119)

Model: Method 1

i

l-67(T m +T m2 )

^ 2 1 = 0.5 Tml T i==- = 1.0 Tml

100T„. K m ( x m + T m 2 ) 50 + 55 ! _ e ^

+T

3T m ,

-019) _ .

0.5 + 3.5 1-e

Chapter 3: Tuning Rules for PI

Rule 2

(HO)

K

103

Comment

T,

Kc

Minimum IAE Huang et al. (1996). Model: Method 1

Controllers

T

0
(120)

ml

0.1<-±2-
Note: equations continued into the footnote on page 104 K

(120) _

1

6.4884 + 4 . 6 1 9 8 - ^ 2 - - 3 .49i_i!i2__ 2 5.3143

Kn

T

Km

Km 1

f

•sO.7204

-18.0448 ^L\ vT m i

\ 0.5961 X,

-7.2712 L

VTmiy

ml J x 1.0049

,

+ 5.3263^-

,

0.4937^- +19.1783^

+ 12.2494V T ml V

^n,|

J_

1

8.4355-

\ 1.005

+ 13.9108—212-

lTral

Km m2

-17.6781T

'ml

1

-0.063

VTmiy

J_

™2

Tml

•5.2132

0.8196

TmT

ml

A -0.9077

( _

1

T

ml

V^ml

-0.7241eT"» -2.2525e T m l +5.4959e T "" 2

K„

, (120)

=T

tm , „ im2 /1-709 J E " 0.0064 + 3.9574—2+ 4.4087-2^-
+ T„

-12.8702

TmTm2

-1.5083 ' ^ 2 i l |

T1 m l

+9.4348

V^ml

V^ml

+ 15.9816-

+ T m l 17.0736-

V T ml 7

-3.909 V T ml J

T 1

"l

m2

V T ml y + T„

•10.864Tm,

•10.7619 V •'ml

+ Tml

-6.6602

x„ ^ml V

X t" 1 J

-22.3194 Tm,

^

V^ml

+ 6.8122|-i2l + 7.5146-21 T,ml J ml /

V^ml

Handbook of PI and PID Controller Tuning Rules

104 Rule

Kc

Minimum IAE Huang etal. (1996). Model: Method 1

Comment

T,

0.4 < 5 m < 1; 3

T(121)

(121)

K

0.05 < —

2,

+ T„

2.8724-

+ Tm

T ml V

+ 11.1207

-4.3675 l

V^ml

V *-m\

.2174, ml J

\2 (

N3

11.4666

(T m 2 x

<1

^m

V tmT

V^ml

-2.2236

V *ml

ml J 5

+ T„

-0.112^-^2-

+1.0308-^

•1.9136 V ^ml

+ T„.

-3.4994U^ 3 - I 1-=^T , , T x

l ,ml y

\

ml

-1.5777

T m2

-

T

V ''ml

I + 1.1408^4 i 2 ^m] V

y

' Note: equations continued into the footnote on page 105. K

(121)

K„

10.4183 - 20.9497^2- - 5.5175^m - 26.5149^m - ^ lml s

42.7745 K„

T

ITT

f

N 1.4439

+ 10.5069

V lml

\0.1456 Tm

+15.4103[ T "

!

V*ml

+ — [34.32364m3'7057 - 1 7 . 8 8 6 0 C 5 3 5 9 -54.0584^ m L9593 ] Km -0.0541

+ — I 22.4263^mf ^=_) K„

T

f

-\ 4.7426

+2.7497^, ' T'

50.2197^m'-8i88 - 2 - -17.1968—=s-4m2-7227 + 1.0293£,m -JBL K„

Kn ,(121)

•16.7667e'ml +14.5737e?m -7.3025e

5m^

11.447 + 4.5128-S- - 75.2486^m - 110.807

12.282^n V^ml

2 + T„. 345.3228^m +191.9539 ^ |

+359.3345^n

ml .

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Ti

Minimum IAE Shinskey(1996)page 121. Model: Method 1

0.48KU

0.83TU

105 Comment ^ - = 0.2, Tm, T - ^ - = 0.2 Tmi

Minimum ISE McAvoy and Johnson (1967). Model: Method 1; ^ Coefficients of Kc and Tj deduced 1 from graphs 1 1

+ T„.

X

*i/Km

T1

x

ml

X

i

2

2^m

Coefficient values ^ Tm, x i x 2

^m

^m

0.5 0.8 1.82 4.0 5.7 12.5 10.0 13.6 25.0

4 4

Iml

7 7

-414.7786^m2 U * l

+ 485.0976^m3

ml J

+ T„

864.51954m +55.4366 - s - | + 222.2685i;n

+ T„

275.116^

21 Tm

' '-205.2493^-

Um, '

'ml

c e n -473.1346^m 5 -6.547

2

_Tn

T

+ Tm 1 6 0 . 7 7 1 4 ^ m 'ml

5

\J

-479.5627 V^ml A

-43.28224m|-^

+ Tm - 7 3 . 5 6 6 6 1 ^ 1 4 m "56.4418 ^ - j

x2

0.5 7.8 3.85 4.0 51.2 5.88

-158.7611—4 m 2 -770.2897^ m 3 -153.633

-412.5409£,n

i

•Cm

0.5 4.3 3.45 4.0 27.1 6.67

Tmi

+ T„

x

+ 99.8717£,n

'ml

^J -37.4971 ^

| ^

Handbook of PI and PID Controller Tuning Rules

106 Rule

Kc

Minimum ITAE x,/K m L o p e z e s / . (1969). Model: Method 7; x2 lm *1 Coefficients of T ml Kc and T; deduced 0.5 0.1 3.0 2.86 from graphs 0.5 1.0 0.2 0.83 0.5 10.0 0.3 4.0 4 K (122) Minimum ITAE Chao etal. (1989). Model: Method 1 0.7 < £ Minimum ITAE Hassan (1993). Model: Method 7

4 R

(m)

Comment

^ x2Tm Coefficient values x x2 l Sm 1 1 1

Tml 0.1 7.0 2.00 1.0 0.95 2.22 10.0 0.35 5.00

44 4 4

Xl

*m

x2

Tm, 0.1 40.0 0.83 1.0 6.0 3.33 10.0 0.75 10.0

T (122)

i

< 3 ; 0.05 <

Tm

2^mTml

< 0.5 0.5<£.m<2,

5

(123)

K

T

(123)

0.1<^S-<4 Tra,

-1.108+0.2558l;m-0.0579l;m

_ 0.09578+ 0.6206£, m -0.1069i; n K„

V^Sm

ml J -0.6441+0.76265 m -0.11935„'

T,<122) = 2i;mTml(l.31-0.0914£.m + 0.07464J;m2 J

-Is_

V^Sra 'ml , 5

Formulae correspond to graphs of Lopez et al. (1969). Kc(123) is obtained as follows:

log[KmKc<123)]= -0.0099458 + 1.9743700E,m - 3.8984310^2- - 1.1225270E,m2 + 2.3493280 — VTml

V

\2

f -0.5594610^,

+ 1.9126490§m -is_ + 0.0754568£,m2

VTml J

-0.3567954

ml .

- 0.7930846£,iT12 - i * - + 0.2412770£,m3 Tmi + 0.04785934n

-0.0024730£,n

V 'ml J

V 'ml .

-0.1354322E. m 3 -^--1.1756470£. m 3 'ml

- 0.0096974^,

is obtained as follows (equation continued into the footnote on page 107):

• (123) '

log

3

vTmi

,(123) ,

T;

/

= 0.9321658-0.7200651E,ra -3.4227010^2- + 0.0432084£,m2 Tmi

X„

ml /

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Minimum ITSE Chao etal. (1989). Model: Method 1

6

K

107 Comment

Tf

(124)

T (124)

0.7 < £m < 3 ; 0.05 < — ^ — < 0.5

Nearly minimum IAE, ISE, ITAE Hwang (1995). Model: Method 10

7

K

(125)

Decay ratio = 0.2

(125)

T

i

+ 1.4965440-^2T

+5.3636520^ m -^--0.0848431£ m 2

f

^

V m

- 2.1777800^m2 -2- + 0.0404586^m3

-1.6011510£,n V^ml

-0.1769621-^2-

lTn,l,

-0.09120084m -21-

+ 0.15014744n 2

r + 0.3033341^ra3 -^2- + 0.1753407^m3 ^-)

K

(124)

\lmlJ

V ml 7

-1.0167+0.1743^1T,-0.03946^m

0.3366 + 0.6355E,m -0.1066^ n K„

-0.0622614^ m 3 f^2

I

Tml

t„

2^mTm -0.5848+0.72945m -0.1136l;m

T/ 124 ' =2^ m T ml (l.9255-0.3607^ m +0.1299^n 7

2^Tra

0.2 < —— < 2.0 , 0.6 < £ra < 4.2 . Equations continued into the footnote on page 108. K

(125) _

0.622[l-0.435coHTm +0.052(coHTm)2 [ Kt K H K m /(l + K H K m ) . (125)

K c (125) (l + K H K m ) 0.0697fflHKra(ll + 0.752co H T m -0.145co H 2 x m 2

K

(125)

0.724[l-0.469coHTm +0.0609(co H T m ) 2 | K, K H K m /(l + K H K m ) ,(125)

K c (125) (l + K H K m ) 0.0405(DHKm(l + 1.93(o H T m -0.363co H 2 T m 2 j

K

(125)

•, s < 2.4 ;

1.26(0.506)mHl'"[l-1.07/s + 0.616/s 2 ]' K^ KHKm/(l + KHKm)

2.4<e<3;

Handbook of PI and PID Controller Tuning Rules

108

Comment Kc ^ Minimum performance index: servo tuning x T - Tx x,/K m 2(Tmi +Tm2 + x m ) 'ml m2 Representative coefficient values - deduced from graphs x T x2 x2 Xl l ^m m

Rule

Minimum IAE Galher and Otto (1968). Model: Method 19.

2Tml 0.053 0.11 0.25

Minimum IAE Huang et al. (1996). Model: Method 1

.(125)

K

x

(125)

8jr

3.6 2.3 1.35

2Tml 0.67 1.50 4.0

1.35 1.17 0.93

(126)

T

0.72 0.60 0.51

0.76 0.53 0.43

0
(126)

;

Tm, 0.1<^-<1 T

KC(I25)(1 + K H K J -. = =rfr—-5, 3 < s < 20 : 0.066 ko H K m (l + 0.824 ln[coHtm])(l + 1.71/s-1.17/E 2 j

1.09[l-0.497a) H x m+ 0.0724((o H T m ) 2 l N KH, K„Kj(l + K H K j

Kc<125)(l + K H K m ) , 8 > 20 . 0.054fflHKm(-1 + 2.54a,HTm -0.457co H 2 x m 2 ) ' Note: equations continued into the footnote on page 109.

X

K

(125)

,T,m2

(126)

K, 1

-13.0454 - 9.0916^2- + 2 . 6 6 4 7 - ^ + 9.162

T T

'ml 3.5959

0.3053

+ 1.1075

•2.2927 l

1 K

m

1 Km 1

-31.0306 U ^ -

lTml

13.0155|^sl

X ml

+ 9.6899 i ^ -

lTml

/

- 0.6418^2-+ 18.9643^ 2 I xm T

-39.7340-

T

'ml V

'ml V 'ml

28.155^-1 -V^-

-2.0067-

T

T V 'ml J

T„,

1

T„,;

ml J

1mTm.

4.8259eT"" + 2.1137eT"" +8.45lie T""2

ml .

Chapter 3: Tuning Rules for PI

Rule

Kc

Minimum IAE Huang et al. (1996). Model: Method 1

.(126)

109

Controllers

Comment

Ti

0-4<^m
R

(127)

T(I27) 1

i

0.05 < - ^ 2 - < 1 Tml

0.9771 - 0 . 2 4 9 2 - ^ - + 0 . 8 7 5 3 - ^ + 3.4651 - ^ 'ml

'ml

V

- 3.8516

TmT

T2

rr

3

7.5106

-7.4538

T

Tmi / ^ - 1 v"Tmi

-10.9909-

+ T„

+ 11.6768-^ Tr V ^ m l

-16.1461 - ^

+ 8.2567^

V^ml )

V^n-

+ 6.2208

-18.1011V^ml

-21.9893-

-4.7536

15.8538

VTml .

V^ml

+ 14.5405l

V^ml

ml J 2

m

-2.2691

+ T„

VTml J

-8.387

ml J

t

m

'ml

T„

/

i

16.651 Is!_ If I s L I -7.1990 I n n ] T

V

'm

5

4

+ Tml

^3

f

1

+

u496p2_

T 1

ml ) \ m l

+ Tm - 4 . 7 2 8 -

+ 1.1395

T V

ml

-0.6385 - ^ s -

lTml

V 'ml 2 /

.08851 ^ _ I Y^A + 3.16151 ^ Tm, J I Tml { Tml„ ral / J V

V^ml

r

1

m2

+ 4.5398T

V 'ml

V

ml

' Note: equations continued into the footnote on page 110

K

(127)

—

- 10.95 - 1 . 8 8 4 5 ^ 2 - - 3.41234 m + 4.59544 m - ^ L . - 1.7002 \

-2.13241JJ-*K„

^ml y

-14.4149^ m 2 | ^

1-0.7683^,

3

X

ml J

110

Handbook

/•

K„

of PI and PID Controller

Tuning

Rules

N0.421

+ 3.7291 lis-1 V^ml

7.5142^2VTml j

+5.3444| J V ml j

+ J_[_0.0819^m19'5419 -3.603^m10749 +7.1163^m11006]

K„

K„

T2

i

(127) __

3.206^n

-7.8480£ n - ^ 5 - | V *ml

+ 11.3222!;

L9948

V ml .

2.4239e T »' + 3.4137e^ +1.0251e T°" - 0 . 5 5 9 3 ^ ^ 2 1

r

1T

\2

ml 2.4866 - 2 3 . 3 2 3 4 - ^ - + 5.3662£ m + 65.6053 Tmi Tml

+ T„

+ Tm

29.0062^m^--24.1648);m2 - 8 3 . 6 7 9 6 ^ T ml I Tm + 43.1477-^a-^m2 + 51.9749^

-135.9699£ n L

ml J A3

r + T„

86.0228

+ 153.4877^ 2

+ 70.4553£„ T ral

V T ml^

+ T„

- 1 2 5 . 0 1 1 2 ^ ^ m 3 -68.5893^m4 -62.7517 T Tmi V ml

+ T„

27.6178^m - J 2 -

\3

f l

-152.7422^n

VTml J

ml J

*n

T rol

- 2 0 . 8 7 0 5 ^ = - ] £„

+ T„

54.0012 U ^ i - C + 5 8 . 7 3 7 6 C + 1 3 . 1 1 9 3 | ^ 'ml .

+ Tm

20.2645^|^|

-23.20644m6-61.6742 V Xml

f

\3

+ Tm 136.2439

f Z.J - 9 5 . 4 0 9 2

V^mlV

Tm,

\

C+20.4168^C lml

Chapter 3: Tuning Rules for PI Controllers Rule

Kc x

Minimum ISE Keviczky and Csaki (1973). Model: Method 1 Representative K,.,^ coefficients estimated from graph; Kra = 1

0.1 0.2 0.5 1.0 2.0 5.0 10.0

10

Minimum ITSE Chao etal. (1989). Model: Method 1

li

K

(128)

x

i

0.26 0.21 0.15 0.12 0.1 0.1 0.15

K

2*m

Coefficient values 0.2 0.5

0.1

^m/X I

Minimum ITAE Chao etal. (1989). Model: Method 1

10

i

x2

X

x2

l

x

i

x

1.0 x

2

i

1.5 0.55 2 2.2 3.5 6 1.3 0.4 1.5 1.5 2.8 3 2 1.2 0.25 1.3 0.8 2 1.1 0.2 1.2 0.65 1.8 1.2 1.0 0.16 1.2 0.4 1.9 0.9 1.2 0.16 1.4 0.4 3.0 0.65 1.6 0.2 2.0 0.45 5.5 0.6 Other coefficient values

i

x2

\

30 13 30 7 14

15 14 30 14 30

5.0 5.0 10.0 5.0 10.0

x

Comment

Ti x

%

Tm, 0.2 0.5 0.5 1.0 1.0

111

(128)

Tml 2.0 2.0 5.0 5.0 10.0 10.0

2.0 x2

x

i

14 6 4.5 9 3.5 5 3.3 3 3.5 1.8 5.0 1.0 7.5 0.7

x2 9 8 7 6.5 7 8 10

i

x2

4

4 8 1.9 3.4 1.1 2.0

15 30 17 32 19 34

5.0 10.0 5.0 10.0 5.0 10.0

x

(128)

T

0.7<^ m < 3 ; 0 . 0 5 < — ^ — <0. 5 2i;mTml Kc(.29,

(129)

x

0.7<£ m < 3 ; 0 . 0 5 < — ^ — <0. 5 2^ m T ml

0.3772-0.24141; m +0.033421; m

0.2763 + 0.3532^m - 0.069554n It

Tml

J -0.3276+0.3226^,-0.05929^„

Tj" 28 ' = 2^ m T ml (o.9953 + 0.07857^m + 0.005317£,m2| — 2^mTral 11 v (.29)

0.5629-0.11874m+0.01328^„,2

0.3970+ 0.43764 m -0.08168^ K„

^

m

T

m

-0.42B7+O.I981^m-0.01978E.m -(129)

^ r a T r a l ( l1.3414-0.1487^m +0.07685E,n

^mr„

Handbook of PI and PID Controller Tuning Rules

112 Rule

Kc 12 ^

T(130)

(130)

i

Nearly minimum IAE, ISE, ITAE Hwang (1995). Model: Method 10

For^ m <0.776 + 0.0568-^- + 0.18 'ml

x

x2

i

X

Coefficient values X x4 5

0.122

8

-0.441

0.0569

0.0172

X

X

X

15

-0.986 x

0.558 x

22

-0.466

16

23

0.0647

X

X

0.0142

X

6.96

x

n 4.62

10

X12

X

-0.823 X|9

0.996

2.13

x

24

0.0609

X

25

1.97

X

6

X

13

14

0.542

1.28 X

7

0.786

-1.77

I8

x

17

0.0476 x

V

3

-0.549 x9

0.822

12

Comment

T,

X

20

21

1.14

-1.13

26

-0.323

Decay ratio = 0.1; 0.2 < -^2- < 2.0 ; 0.6 < £,m < 4.2 . Tmi

K

(130)

(

2 Xl [l + x 2 a) H T m +x 3 (co H T m ) J Kt KHKm/(l+KHKj •(130)

K c (l30) (l + K H K m ) X4a H K m (l + x5coH t m

K

(130)

+X6fflH

K c (l30) (l + K H K m ) f ^ ^

7—r\, 2.4 < E < 3 ;

X l 0 « H K m l l + X„C0 H T r a + X | 2 C 0 H T m (130)

K c (130) (l + K H K m ) X

(130)

j

x13x14m"T"'[l + x l 5 /s + x 1 6 / e 2 f KH, K H K m /(l + K H K m ) •(130)

K

E<2.4 ;

X7[l + X8(0HTm+X9(cOHTm)2J KH. K H K m /(l + K H K m ) . (130)

K

r,

T„

1 7 ( 0 H K m ( 1 + X I8 l n [ C O H- C m]X 1

+ X

19/S +

3 < s < 20 ; X

2o/s2)

X21[l + X22(0HTm +X23(cOHTm)2J Kv K H K m /(l + K H K m ) ,(130)

KC(130)(1 + K H K J ^ H K r a ( - 1 + X25COHTm + X 26 CO H 2 T m 2 j

e>20.

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

T

113 Comment

i

2

Nearly minimum IAE, ISE, ITAE Hwang (1995)continued

x

i

0.794 X

8

-0.415 X

15

-0.959 x

22

-0.466

For lm > 0.889+ 0 . 4 9 6 ^ - + 0.26 Tmi Coefficient values X x x2 x4 3 5 -0.541 X

9

0.0575 X

16

0.773 X

23

0.0667

0.126 X

10

0.0124 X

17

0.0335 x

24

0.0328

0.0078 x

u 4.05 X

18

0.947 X

8.38 x

13

X14

0.564 x 2) 1.07

x

20

-1.07

26

-0.338

i-p

and

ryi

V 1ml J

^ml

Em < 0.889 + 0 . 4 9 6 ^ - + 0 . 2 6 ^ -

x

i

0.789

X

2

8

-0.527 x9

-0.426

0.0551

x

X

15

-0.978 X

22

-0.467

X

16

0.659 X

23

0.0657

Coefficient values X x4 3 5 0.11 9.7 0.009 x

x

10

0.0153 X

17

0.0421 x

24

0.0477

x

u 4.37 X

18

0.969 X

25

2.07

7

0.738

1.15

X

12

For £m >0.776 + 0.0568^2- + 0.18 — M -'ill

x

6

-1.97

-0.63 x 19 1.9 x

25

2.21

X

X

12

-0.743 x19 2.02 X

26

-0.333

:

X

6

-2.4

X

7

0.76

13

X14

1.22

0.55

X

X

20

-1.11

X

21

1.11

Handbook of PI and PID Controller Tuning Rules

114 Rule

Kc

Comment

T| Direct synthesis

0 Bryant et al. (1973). Model: Method 1

x

K

x

2KmTml

Tmi>Tm2

Tm,

i

Tmi

>

Tm2

mXm

Coefficient values - deduced from graph tm/Tml 0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67

13

x

i

0.08 0.09 0.09 0.11 0.14 0.16 0.18 0.23

tm/Tml 1.0 2.0 10.0

0.15 0.24 0.33

Critically damped dominant pole

1.0 2.0 10.0

0.31 0.47 0.64

Damping ratio (dominant pole) = 0.6

x

i

Representative x 2 values, summarized as follows, are deduced from graphs: x 2 value ^ 0.4 0.6 0.8 0.9 1.0 1.1 1.2 1.5 1.9

0.5

1.0

2.0

•tmAml 3.0

4.0

5.0

6.0

7.94 8.93 9.80 12.2 15.4

5.13 5.99 5.81 9.01 9.90 10.8 13.3 16.7

5.81 6.94 7.30 7.30 11.5 12.3 13.2 15.4 18.9

7.46 8.40 9.17 10.0 13.9 14.5 15.4 18.2 21.3

9.17 10.0 11.9 14.1 16.4 16.9 17.9 20.0 23.8

10.9 11.8 15.4 19.6 18.9 19.6 20.4 22.7 25.6

12.7 14.1 20.0 28.6 -

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Bryant e/a/. (1973) continued Hougen(1979)pages 345-346. Model: Method 1

1

1 4 X

15

K

l/

K

x

m

(131)

115 Comment

T| 2K m T m l

Tmi

>

Tm2

Maximise crossover frequency

T(131)

Representative x, values, summarised as follows, are deduced from graphs: x, value

T

1.0

km

2.0

m/*ml

3.0

4.0

5.0

6.0

0.4 0.006 0.076 0.103 0.122 0.115 0.6 0.039 0.114 0.093 0.122 0.120 0.8 0.114 0.055 0.110 0.127 0.093 0.075 0.9 0.139 0.147 0.122 0.093 0.070 0.052 Representative x 2 values, summarised as follows, are deduced from graphs: x 2 value

15

K

*m/Tml

5m

1.0

2.0

3.0

4.0

5.0

6.0

0.4 0.6 0.8 0.9

5.13 5.99 5.81

5.81 6.94 7.30 7.30

7.46 8.40 9.17 10.0

9.17 10.0 11.9 14.1

10.9 11.8 15.4 19.6

12.7 14.1 20.0 28.6

(131)

0.7T m 2 (T m l +T r a 2 )+T m ,

1+K„

2~\

Tm, +0.7T„

1ml *m2

Ti ( 1 3 1 ) =(T m l +0.7T m 2 | 1 + 0.1 x, values are obtained as follows: x, value Tm2/Tml 0.1 0.3 0.5 1.0

c

0.2

0.3

0.4

0.5

0.31 0.6 0.65 0.7

0.19 0.42 0.5 0.6

0.15 0.35 0.45 0.5

0.11 0.28 0.46 0.45

0.09 0.23 0.32 0.38

0.6

0.7

0.8

0.9

1.0

0.085 0.2 0.29 0.34

0.075 0.18 0.26 0.32

0.07 0.16 0.24 0.31

0.065 0.155 0.23 0.29

0.06 0.15 0.22 0.28

C

' m2 /T m l 0.1 0.3 0.5 1.0

ra/^ml

0.1

m/'ml

Handbook of PI and PID Controller Tuning Rules

116 Rule

Kc

Hougen(1988). Model: Method I Somanie?a/. (1992).

16 K

T

(132)

Comment

i

(132)

T

i X|/

17

K

m

A

x

m

Five criteria are fulfilled

2Tm

Coefficient values - see page 117

Equations for Kc

deduced from graph;

731og,0 -=H +0.65

K (>32, = _1_ ] 0 !

:0.1 0.611ogl0 -1H1 +0.29

K

(132)

= 0.2; K„ 0.48 l o g j - ^

Kc<

132

+0.05

»=^10

= 0.5; 0.41 l o g j - m i -0.06

K

(132)

For

^ . = l.T i ( 1 3 2 ) ==T m l +0.7T m 2 + 0.12T n

-10

K, 25T ml T.

'ml

«1.

Ti

K„

<X

0.9806 KmAj

^

2.944 - tan"

1ml

Am

2.944 - tan" J_m_L + J_m2_

1+

K

2 * ^ - U i + ' T . " 2.944 -

T

T m

For

25T ,T

X

0.9806

• » 1, Kc

K

ml

m

0.5

T V

ml

Tm2

+,

w

m

T 1

m2 J

2,944 -

Tml

-

Tm2

''m

(T

1+

A„

0.5 V

1+

j

+ 0.945+ ,0.25 V^ml

Tm2

+ 0.945-,|0.25| ^is_ + JEJS_ Tmi Tm2 j

-0.945^ +

lTml

T— m2J

-0.9451 -^=- + - ^ 2 Trai Tm2

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Sorrnmi etal. (1992) - continued. Model: Method 1; Suggested A m : 1.75 < A m < 3 . 0

T

i

x2

11.707 10.091 10.315 11.070 12.207 13.337 11.318 8.613 8.510 8.997 9.840 10.707 14.494 6.504 5.136 4.961 5.174 5.502 14.918 4.353 3.274 3.025 3.038

12.5 8.33 6.25 5.00 4.17 3.70 12.5 8.33 6.25 5.00 4.17 3.70 11.11 8.33 6.25 5.00 4.17 3.70 8.33 6.25 5.00 4.17 3.70

x

T m2

0.1 0.1 0.1 0.1 0.1 0.1

0.147 0.440 0.954 1.953 4.587 13.89 0.121 0.404 0.897 1.842 4.219 11.628 0.064 0.256 0.667 1.406 3.021 6.452 0.062 0.376 0.909 1.880 3.401

0.5 0.5 0.5 0.5 0.5

18 K

Comment

Ti Coefficient values

^m

0.125 0.125 0.125 0.125 0.125 0.125 0.25 0.25 0.25 0.25 0.25 0.25

Tan et al. (1996). Model: Method 2 ?ou\in etal. (1996). Model: Method 1

117

18 jr (133)

m x

T ra2 1.00 1.00 1.00 1.00 1.67 1.67 1.67 1.67 1.67

2.5 2.5 2.5 2.5 5.0 5.0 5.0 5.0 5.0 10.0 10.0 10.0 10.0 10.0

-p (133)

0.082 0.437 1.016 2.075 0.169 0.581 1.242 2.445 4.651 0.156 0.338 0.559 1.182 0.075 0.238 0.433 0.667 0.951 0.073 0.235 0.424 0.649 0.917

12.389 3.462 2.370 2.087 6.871 2.773 1.930 1.624 1.957 7.607 4.015 2.797 1.848 16.187 5.623 3.448 2.52 2.014 17.650 5.996 4.524 2.641 2.093 T - T i

T

Tml

T

m( ml + m )

(133) P ^ S V 1 ^ 7 - " * ) ' 33

Am)/l + (3T<' >coJ

ml

Tmi

Tm, K

tp_as>tm T

'

" ->

x2

i

6.25 5.00 4.17 3.57 5.00 4.17 3.57 3.13 3.13 4.55 4.17 3.85 3.33 4.17 3.85 3.57 3.33 3.13 3.85 3.57 3.33 3.13 2.94

'm2 >

5T m2 ;

Min lmum §„ = 58°

118

Handbook of PI and PID Controller Tuning Rules Rule

Kc 19 K

(134)

T

(134)

"Normal" design

(135)

"Sharp" design

i

Schaedel (1997). Model: Method 1

20 K

(135)

T

i

Pomerleau and Poulin (2004). Model: Method 5 Leva et ah (2003). Model: Method 1

Comment

Tj

Tml Km(Tml+tnJ

l-STml

Tml 2Km(Tm2+T,J

4(T m 2 +x m )

T m l =T m 2 ;OS<10% >>

'ml

Tm2 + Tm '

Symmetrical optimum principle

Robust

Brambilla et ah (1990). Model: Method 1

T m l +T m 2 +0.5T m Kmxm(2A + l)

19

K

'ml

m +

X

Tm2

0.1 0.2 0.5 1.0

0.5T;

(134)

Tm

0.1 <

<10

Tmi + Tm2

Coefficient values T

X values obtained from graph

T m l +T m 2 +0.5T m

*m T m ,+T, n2

X

2.0 5.0 10.0

0.4 0.2 0.2

3.0 1.8 1.0 0.6

(134)

Ti° 34) = V & m T m l + x , J -2(T r a l 2 +2^ m T m l t m +0.5T,,, 2 ). 20

K

(.35) _ 0.375 4Z,m'Tml* +2^ m T m l T m +0.5T,/ - T m l 2 K

m

T m l 2 +24 m T m l T m +0.5T m 2 .(135)

. 4s

m

'ml

+

^Sm'ml

T

2s m T m i

m

+

^-5

+T m

T m

'ml

Chapter 3: Tuning Rules for PI Rule

Kc

Kristiansson (2003). Model: Method 1 Panda et al. (2004). Model: Method 1

21

K

2^ m T„

(136)

Controllers

T

Comment

i

21 K

(136)

22

K

(137)

23 K

(138)

j(138)

25 K

(139)

T (139)

2^mTml

Suggested M m a x = 1 . 7 , p = 10 24

a-

la 2 - 1 + -

, a = l+-

1-1 M„

M„

22

v

K

2^ ra T ral

(137)

K m x ra

1 + 0 . 1 9 1 3 ^ - - J o . 3 4 6 + 0.38261s!2-+ 0.0366-is 2 -

23 v (138) _ 8 ^ m T r o l + 2 ^ m T m + T r o l K„

T(n8)

_ 4^ro Tml + 4 m T m +0-5Tml

2MC

2^m

2 X, > max imum 0 . 8 2 8 + 0.812-2! - + 0.172T m l e -(69T„, 2 /T ml )

0.828 + 0 . 8 1 2 ^ ^ + 0.172T m l e 25

v

K

119

-(6.9Tm;/Tml) ,

(139) _

,

7

0-558T m 2 T m Tml+1.208Tm2

|

0.Sxn

A.K„ T i (139) =0.828 + 0 . 8 1 2 ^ + 0.172T m ,e- (6 - 9T ^ /T -'+ ° - 5 5 8 T ^ T " " Tml T m ,+1.208T m2

+0.5tn

120

Handbook of PI and PID Controller Tuning Rules

3.7.2 Controller with set-point

weighting E(s)-aKcR(s)

U(s) = Kc v

T s

i y

*aK„

Y(s) K e"

+ T

*K,

E(s)

v

T s

.y

R(s)

+

U(s)

Table 25: PI controller tuning rules - SOSPD model Rule

(140)

0.3717 + K„

•(140)

K m e" 2 2

T m l s +2i; m T m l s + l Comment

Kc i Minimum performance index: servo/regulator tuning T

Taguchi and Araki (2000). Model: Method I

K

T^V+l^T^s +l

1K

(140)

T

5m = i ; Tm/X<1.0; Overshoot (servo step) < 20%

(140)

0.5316 + 0.0003414

2.069-0.3692-S1- + 1.081 T

-0.5524 f ~

a = 0.6438 - 0.5056^2- + 0.3087 T T

\

U ^

-0.1201

T

V Xm J

Chapter 3: Tuning Rules for PI Controllers Rule 2 ^

(141)

Minimum ITAE x i Pecharroman (2000), 0.147 Pecharroman and Pagola (2000). 0.170 Model: Method 11

2

K

m = 1 i T m i = 1 •>

<141>

--

T(141)

^m=0.5

i

x2 T Coefficient values

x2

a

i

1.150

0.411

-146°

*i

x2

a

4>c

0.280

0.512

0.281

-73°

1.013 0.401 -140° 0.291 0.1713 1.0059 0.4002 -139.7° 0.297 0.195 0.880 0.386 -133° 0.303

0.503

0.270

-63°

0.483

0.260

-52°

0.462

0.246

-41°

0.210

0.720

0.234

0.672

0.249

0.610

0.262 0.274

0.1000 + K„

Comment

1

x,K u

K

1

Kc

Taguchi and Araki (2000) - continued

121

0.342 -125° 0.345 -115°

0.307

0.431

0.229

-30°

0.317

0.386

0.171

-19°

0.324

0.302

0.004

-10°

0.568

0.323 -105° 0.308 -94°

0.320

0.223 -0.204

-6°

0.545

0.291

-84°

0.05627 - +0.06041]2 3

4.340-16.39^2-+ 30.04 T

f

a = 0.6178 - 0.4439-^ - 7.575 T T

r

-i4^

+ 8.567 - ^ T

-25.85 \l

3

•3.182

+ 9.317 vTm

T

V im.

Handbook of PI and PID Controller Tuning Rules

122

3.8 SOSIPD Model - Repeated Pole

s(l + Tmls)2

3.8.1 Controller with set-point weighting U(s) = Kc

'. + -L' E(s)-aK c R(s) v

T s

i y

aK Y(s)

&

E(s)

R(s)

x8>

*K, T S

V

i V

K„e"

U(s)

s(l + Tmls)2

Table 26: PI controller tuning rules - SOSIPD model G m (s) =

Taguchi and Araki (2000). Model: Method 1

Coefficient values Xl

x2

0.049

2.826

Km = 1; Tml = 1

0.066

2.402

0.1 <x m <10

0.099

1.962

0.129

1.716

0.159

1.506

0.189

1.392

K

(142)

0.07368 + K„

s(l + TraIs)

Comment Kc T; Minimum performance index: servo/regulator tuning ^/Tml<1.0; 8.549 + 4 . 0 2 9 ^ 4 Kc(.42, Overshoot (servo Tm, step) <20% X T x,K u 2 u

Rule

Minimum ITAE Pecharroman (2000). Model: Method 2

K e"

a

a

i

x2

0.506 -164° 0.512 -160°

0.218

1.279

0.250

1.216

0.522 -155° 0.532 -150°

0.286

1.127

0.330

1.114

0.578 -125° 0.579 -120°

0.544 -145° 0.555 -140°

0.351

1.093

0.577 -118°

0.3840

- + 0.7640

*c

x

, a = 0.6691+ 0.006606-

4>c 0.564 -135° 0.573 -130°

•

Chapter 3: Tuning Rules for PI Controllers

3.9 SOSPD Model with a Positive Zero G m (s) =

123

Km(l-Tm3s)e-^ (l + sTmlXl + sTm2)

3.9.1 Ideal controller Gc (s) = Kc 'i + -L^ v R(s)

+

<^

E(s)

i

c

K,

1+-

N

\

TjSy

T s

i y

U(s)

K m (l-T m 3 s)e(l + TralsXl + sTm2)

Y(s) — •

Table 27: PI controller tuning rules - SOSPD model with a positive zero Km(l-Tm3s)r^ G m (s) = (l + sTmlXl + sT m2 ) Rule

Kc

Tj

Comment

Direct synthesis: time domain criteria Pomerleau and Poulin 1 Tml (2004). OS<10%;T m l =T m 2 l-5Tml K m T m l +T r a 3 +T m Model: Method 3

Handbook of PI and PID Controller Tuning Rules

124

Comment ^ Kc Direct synthesis: frequency domain criteria 5 K (143) Recommended T (143) Mraax=0dB;

Rule Khodabakhshian and Golbon (2004). Model: Method 1

Tmi

Luyben (2000). Model: Method 1 T

m3/Tmi

Ku x

values

x, values

5

(143)

K

Ultimate cycle 6 Tu T1

i

K m

0.2

0.4

0.8

m

•10

\

Tm2

Tmi - Tm2

1.6

3.9

3.3

2.8

3.0

T r a /T r a 3 =0.2

3.8

3.4

3.0

2.9

^ m /T r a 3 =0.4

4.1

3.6

3.2

3.1

x m /T r a 3 =0.6 x m /T r a 3 =0.8

4.3

3.8

3.4

3.2

4.8

4.3

3.7

3.3

x m /T m 3=1.0

5.2

4.6

3.9

3.4

T m /T m 3 =1.2

5.5

5.0

4.4

3.7

x r a /T m 3 =1.4

6.0

5.4

4.7

3.8

X T

T,"43) J(T r o l T m 2 )V+(T m l 2 +T m 2 2 )to K

>

(TmJm3)

2

4 ra

2

4

2

'ffl2 2

J m 3 =1-6

with o> given by solving

+(Tm, +Tm3 > +l

-0.1M„„ A

= 0.5TI + tan"1 (Ti<143)co)- tan"1 (Tm3co)- tan"1 (Tm2co) -tan-'(T m l co)-T m co:

.(143) __J_ i

T

+ 0.2-

1 + 0.175—S2- + 0.3|

-<2:

A ml

,(143)

(T

\

0.65 + 0.35—— + 0.3

+ 0.2V T ml J

0.5 + 1 . 5 6 ^ -

+ 3.44-1.56- Tm3 '

Tm

->2.

Chapter 3: Tuning Rules for PI Controllers

125

3.10 SOSPD Model (repeated pole) with a Negative Zero K m (l + Tm3s)eG m (s) : (l + sT ml ) 2 3.10.1 Ideal controller in series with a first order lag G c (s) = Kc 1 + -

T|S y 1 + T f s

R(s) ^ E ( s ) -•

Table 28: PI controller tuning rules - SOSPD model with a negative zero K m (l + T r a 3 s > - ^

Gm00 =

Rule Pomerleau and Poulin (2004). Model: Method 3

Kc 1 K

m

T

ml

(l + sT ml ) 2 T

i

Comment

Direct synthesis: time domain criteria Tml Tf =T m 3 ;OS<10% l-5Tml + T m

Handbook of PI and PID Controller Tuning Rules

126

3.11 Third Order System plus Time Delay Model l + b,s + b 2 s 2 +b 3 s 3 Gm(s) = Kr 'or l + a,s + a2s3 +a 3 s K„e"

G m (s) =

(l + sTmlXl + sTm2Xl + sTm3)

3.11.1 Ideal controller G c (s) = Kc 1 + — E(s)

\ 0

R(s)

U(s)

Kc 1 + —

%-*

+ ?_

T s

I

iJ

Y(s)

Third order system plus time delay model

Table 29: PI controller tuning rules - Third order system plus time delay model „ , , „ l + b,s + b,s 2 +b-,s3 _,, _ ,N K m e-"" 2 G m (s) = Km ! ~ ^ e st " or G m (s) = (l + sT ml Xl + sT m2 Xl + sT m 3 )l + a,s + a,s +a,s Rule

Kc

T,

Comment

Direct synthesis Hougen (1979) pages 350-351.

0.333

0.7

^ - > 0.04 ;

Km

\Xm

)

T

(H4) Tml -

Model: Method 1 I

1K

(144)

2Kn

0.7|i=i-

K

-

Tm3

•^2- < 0.04 ; Tml Tml>Tm2>Tm3

(H4)

+ 0.8

'•ml

Tmi + Tm2 + Tm3 vTmiTm2Tm3 r

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Vrancic e/a/. (1996), Vrancic (1996)page 121. Model: Method 1

0.5A3 (A,A2-KmA3)

Vrancic et al. (2004a). Model: Method 1

0.5A3 (A,A2-KmA3)

3

4

R

T;

Comment

A3 A2

Am>2,4L>600 (Vrancic et al. (1996))

(145)

x

i

A,A 3 A[A 2 - K m A 3

0

Vrancic et al. (2004b).

2

2

127

(146)

T

Model: Method 1

(146)

A, = K r a ( a 1 - b 1 + x m ) , A 2 = K m ( b 2 - a 2 + A , a , - b , t m + 0 . 5 x m 2 ) , A 3 = K m (a 3 - b 3 + A 2 a, - A,a 2 +b 2 x m -0.5b,x r a 2 +0.167x m 3 ).

3

T

A,A 3 (A,A2 - K m A 3 ))

(145) 2

(A,A 2 - K m A 3 ) + K m A 3 ( A I A 2 - K m A 3 ) + 0.25K m 2 A 3 2 4

K

(146)

A,A 2 - A 3 K m -[sign(A,A 2 - A 3 K m )]A,VA2 2 - A,A 3 A,Km2-2A,A,Km+A,3 -(146) _

2A,

A, A2 - A3Km - [sign(A1 A2 - A3Km )]A, JA22 2

3

(A3Km - 2A,A2Km + A, )(l + K C

(146)

- A, A3 KJ 2

12 8

Handbook of PI and PID Controller Tuning Rules

3.11.2 Controller with set-point

weighting 1 "1 U(s) = Kc ( 1 + — E(s)-aK c R(s) T s V i J

aK„

Y(s) Third order system plus time delay model

K, v R(s)

T s

i y

E(s)

U(s)

Table 30: PI controller tuning rules - Third order system plus time delay model

O.WK

•*b-+--:+b-:.--.,o.w.

l + a,s + a 2 s 3 +a 3 s 3

Rule

Comment Kc Ti Minimum performance index: servo/regulator tuning

Taguchi and Araki (2000). Model: Method 1

5

K

(147)

0.2713 + K„

l + sTmlXl + sTm2Xl + sT m 3 )'

5 K (147)

(147)

T

i

^2-<1.0 T

Overshoot (servo step) < 20% ; Tra] = Tm2 = Tm3

f

0.7399 - + 0.5009

T

\2

, a = 0.4908-0.2648-^ + 0.05159 T T

(147)

V m J

=

T

2.759-0.003899-^2-+ 0.1354 T

Chapter 3: Tuning Rules for PI Controllers Rule

129 Comment

T;

Kc

Direct synthesis Vrancic(1996)page 136-137. Model: Method 1

6 jr

(148)

6

(148)

(148)

b = [0.5,0.8]-good servo and regulator response (page 139); b = l-oc

(148)

T

i

A,A2-KroA3-X 2 3 n A3+A1 -2KmA1A2j

V^¥

X

K

K

K m An ,

•~m i

- An, An , < 0 ; \

l

= V(KmA3 - A,A 2 ) 2 -(l-b 2 ]A 3 (K r a 2 A3 + A, 3 - 2 K m A , A 2 ) ;

A,A2-KmA3+x

¥^¥

2 m

KmA3-A,A, >0. A ,rt+-r^i.| A , 3 - 2 zK. x„vA^, ,A ,)' ~ ™ - 3 — ' " 2 rv 3

Km+-

K <148>K J

1 2K

m

2

A,

.(148)

(148)

with

&-„')

A, = K m ( a i - b , + T m j , A2 =• K .m( ^bb 2 - a 2 + A 1 a 1 - b 1 T r a + 0 . 5 T m '). A 3 =K m (a 3 - b 3 + A 2 a, - A,a 2 +b 2 x m - 0 . 5 b , x j +0.167x m 3 ).

Handbook of PI and PID Controller Tuning Rules

130

3.12 Unstable FOLPD Model G m (s)

K m e- sx ™ T„s-1

3.12.1 Ideal controller G c (s) = Kc 1 + T iSy R(s)

?)

E(S)

r

Kc

i

U(s)

M

Y(s)

Kme-Tras-1

T s

I

iJ

K m eTable 31: PI controller tuning rules - unstable FOLPD model G m (s) = T m s-1 Rule

Comment Kc T, Minimum iDerformaiice index: servo tuning

Minimum ITSE Majhi and Atherton (2000). Model: Method 1 Minimum ITSE Majhi and Atherton (2000). Model: Method 3

1 v (149) _ _ J _ K„

0.889 +

v 2

K

(150)

J_

0.889 +

1K

(149)

0
(149)

T

i

2

K

(150)

0 < I s . < 0.693 ; Tm K = -Ap/Kmh

x(ISO)

i

- ' A - 0.064 - 0.990

T T e -/ ™

1 K(I + K)

T

(150)

T(i49)

'

_2.6316T m (e T -/ T - - 0 96e) (e-'-^"-0.377) T m K(l + K)

0.5(l-0.6382K)tanh _1 K '

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

/

(151) K„

cos

TmjTm

\t„

-(151)

(152)

=

_

Km|

A ra = 2 ; ^ < l

T(151)

m

i

^S-<0.67 T

25(T m -xJ

m

-^-<0.6 T

25T m -27x m

m

Model: Method 1

0.4015Tme5'8t'"/T"'

5

0.4015Tme

1--=- | sin T,

"' "'

^2-<0.7 T

T

T

T

V

1+ 0.98J1 +

0.04T^

(Tm-xm)

25 2

^ = 0.333;

58T /T

, <|> = t a n

/ 4 ^

Comment

T; Direct synthesis

De Paor and 3 K (151) O'Malley (1989). Model: Method 1 Venkatashankar and 4 K (152) Chidambaram (1994). Model: Method 1 Chidambaram if T^ (1995b). — 1 + 0.26-^Model: Method 1 K„A ^J 1.678, In Chidambaram (1997). Km Valentine and Chidambaram (1998). 2 . 6 9 5 -1.277Tra/Tm Model: Method 1 Km

K

131

|p(T m -x m )

T

my

P2T2

l+ P 2 ^ | ( T m - x J 2

B = 1.373, — < 0.25 ; B = 0.953, 0.25 < — < 0.67 . 5

+1% Ts = 7.5Tm , x m /T m = 0.1; ±1% Ts = 8.75Tm, x m /T m = 0.2 ; ±1% Ts = H.25T m , x m /T m = 0 . 3 ; ±1% Ts = 15.0Tm, x m /T m =0.4; ±1% T s = 17.5Tm , x m /T m =0.5 ; ±1% Ts = 20.0Tm , x m /T m =0.6.

Handbook of PI and PID Controller Tuning Rules

132 Rule

Ho and Xu (1998). Model: Method 1

Comment

Kc

Ti

»pTm

6T(153) i

AmK.m

^ T

< 0.62 m

Representative results 0.5775Tm K

T T

0.3212T m -x m

T

m m

0.6918Tm K

T T

0 3359T - T

raTm

Good servo response. A m = 2.3 , *m=25° Good regulator response. A m = 1.9 ,
x

-T

] T m / K m xm

Other representative results: coefficient values X x Am Am l i K 3 0.5236 1.0472 1.5 30° 0.7854 Luyben(1998). Model: Method I

2

0.3927

45°

2.2TU

0.31KU

*m 60°

4

67.5° TCL = 3.16xm

Maximum closed loop log modulus = 2 dB

Gaikward and Chidambaram (2000). Model: Method I Chidambaram (2000c). Model: Method 1

K

(154)

x

(154)

8 ^

(155)

T

(155)

7

0.1<^2-<0.6 T 'small' ^ Tm

6 j (153)

1.57cop-cop xm - — 7 v (154) _ _J_ K„ 0.2619 + 0.7266-^Km -(155) 8

K

(155)

0.04TS242+Ti(155)tr

- T-

(155)

,(154)

=T

r

0.193 + 4.19—2-4-8.214 T T

_ 0.04TS2^2+Tmtm+0.4TS^

\1

Chapter 3: Tuning Rules for PI Controllers Rule

Comment

Ti

Kc

'Chandrashekar et al. (2002). Model: Method 1

133

4 9 . 5 3 5 e-21.644xm/Tm

Km /•

\ -0.8288

i s - < 0.1 T

0.1523Tme7'9425T">^

tn

0.1 < i s - < 0.7 T

0.8668 f x j T

Km t mJ

Coefficients of Kc and Tj recorded in tables; TCL2 as in footnote

Sree et al. (2004). Model: Method 1

6.0000/Km

Alternative tuning rules 0.6000Tm

xm/Tm=0.1

3.2222/Km

1.4500Tm

T m /T m =0.2

2.2963/Km

2.6571Tm

xm/Tm=0.3

1.8333/Km

4.4000Tm

xm/Tm=0.4

1.5556/Km

7.0000Tm

*m/Tm=0.5

1.3265/Kra

14.6250Tm

tm/Tm=0.6

1.1875/Kn

31.0330Tm

X m /T m =0.7

0-8624 f x j

-0.9744

10 T (156)

0.01 < i a - < 0.6 T

Km | j m J

m

Robust Rotstein and Lewin (1991). Model: Method 1

Km

1

^2Km uncertainty = 50%

Desired closed loop transfer function =

^ m /T r a =0.2

Ae[0.6T m ,1.9T m ]

fas+!>-"- , (TCL2S + 0 2

0.025 + 1.75-

A. obtained graphically - sample values below

X

T ra ^

'

Tm,i2.<0.1;TCL2=2xm,0.1
T

10 J (156) _ rj,

143.34

-73.912

+ 19.039 - 2 - -0.2276 T

T

Handbook of PI and PID Controller Tuning Rules

134

Rule

Kc

Rotstein and Lewin (1991) - continued Tan et al. (1999c). Model: Method 1

K

Km uncertainty = 30% 11 K

,(157)

K„

0.1486-^ + 0.6564 T

Comment

* m /T m =0.2

^G[0.5T m ,4.5T m ]

* m /T m =0.4

^e[1.5T r a ,4.5TJ

^ m /T m =0.6

^6[3.9T m ,4.1T ra ]

T(157)

(157)

(157)

Ti

^*-<0.5 T

12.7708—2- + 3.8019 T 0.1486-2!- +0.6564 T

Chapter 3: Tuning Rules for PI Controllers

135

3.12.2 Controller with set-point weighting E(s)-aK c R(s)

Table 32: PI controller tuning rules - unstable FOLPD model Gm (s) = Rule

Kc 12

Chidambaram (2000c). Model: Method 1

Comment

T, Direct synthesis

(158)

'small' x m /T m

(158)

T

i 13 2 . 6 9 5

-1.277t m /T m

Kra 14

Jhunjhunwala and Chidambaram (2001). Model: Method 1

K

K

(159)

0.4015Tme5-8x™/T'"

'Dominant pole placement' method; E, = 0.333

(159) _

T

^2_<0.2 T

i f

\ -0.898

0.876^

0.324Tme

K m eTms-1

6.461tm T

m

"

0.2 < -is- < 0.6 T m

-(158) 12 j r

(.58)

(158) •.

O.O^s^HTi" 58 '!,,

T

0.04Ts^2+TmTm+0.4Ts42

i

<x = l-

r

a = 0.733+ 0.6^2--0.36 T T

\2

V ra J 14

v

K„

(159) _

1

10.7572-35.

0.2T s 4 2 (T m -T m ) 0.04T S 2 ^ 2 +T m x m +0.4T S ^

0.1 < - 2 - < 0.6. T Kc""'Km(2-Tj+(Kc'l"'Km-l)ri"^ 2K<159)K

Handbook of PI and PID Controller Tuning Rules

136 Rule

49.535

15

Chandrashekar et al. (2002). Model: Method I

-21.644 Tm /T m

e

ra

m

'

0.1523Tme7'9425t"/T™

Kra -0.8288 0.8668 f O

Km Coefficients of Kc and Tf recorded

Ti

Kc

Comment is-<0.1 T m

0.1 < -^s- < 0.7 T

[TJ Alternative tuning rules a

Kc 6.0000/Km

Ti

0.6000Tm

0.6667

^m/Tm=0.1

in tables; TCL2 as in 3.2222/Km footnote 2.2963/Km

1.4500Tra

0.7246

^m/Tm=0.2

2.6571Tm

0.7742

x m /T m =0.3

1.8333/Kra

4.4000Tm

0.8182

im/Tm=0.4

1.5556/Km

7.0000Tra

0.8571

tm/Tm=0.5

1.3265/Km

14.6250Tra

0.8974

^m/Tm=0.6

1.1875/Km

31.0330Tm

0.9323

tm/Tm=0.7

Hagglund and Astrom (2002). Model: Method 1; M, =1.4; 0.3
15

8T m

0.28/K m l

m

7.8xm

0.2/K m Tm

ll^m

•^2- - 0.02 T m

10t m

i^-0.05 T

m

7tm 19t m

is--o.lO Tm

m

135tm

^m/Tm=0.15

0.28/KJ

m

0.13/KJ

m

0.28/KJ 0.28/KJ

Desired closed loop transfer function = —

—

(TCLZS + I ) 2

TCL2 = 0.025 + 1.75-

Tm,^<0.1;TCL2=2xm,0.1<^<0.5; TCL2=2tm+5xm|^-0.5

a = 0.505 + 3 . 4 2 6 ^ -10.57 ^ T T

, 0.5<-^<0.7. T

•<0.2:

a = 0.713+ 0.232-^-, 0 . 2 < i 3 - < 0 . 7 T T

Chapter 3: Tuning Rules for PI Controllers 3.12.3 Controller with proportional

137

term acting on the output 1 U(s) = ^ E ( s ) - K c Y ( s ) T:S

d

Kc

+ r - E( ) S

T

i

- X u(s)

s

Y(s)

K m e- St ™

^6?V——fc.

•

Tms-1

t Table 33: PI controller tuning rules - unstable FOLPD model G m (s) = Rule

Comment T; Minimum performance index

Kc

Minimum ISE Paraskevopoulos et al. (2004). Model: Method 1

1K

K m e" T m s-1

0.05 < ^ - < 0.85; T regulator tuning

i.9i6,m+4-;^2

(160)

•'rn

0.88--^T 2.725xra +

L ?

^

2

0.05 < ^ s - < 0.85 ; T servo tuning

0.88-^ T m

£

(160) _

with K

/ ] £ (min)j^

(min)

m

minTi

)l +

K„

l+

m

m

min2Tm2 mi„

T

R

fi>maxTi

(max) _

K„

i

1 +

K~

T

2xm T

T +T

^ " m a x ^ m

T

i

w

max

2T

^m

i + ^

T m

m j

. ™n T

T

rt

i

^

2

i

^ 2

Handbook of PI and PID Controller Tuning Rules

138

Comment Kc Ts Minimum |performance index: other tuning

Rule

2

Paraskevopoulos et al. (2004). Model: Method 1

1

K

3T(160) i

(160)

m

(page 137)

3 r-p (161) i

Minimise dt

0.3 < i s - < 0.85 T

2.58, m + 3 - 6 V

1 j , (160)

f

0.01 < — < 0.3 T

T

(page 137)

0.03 < ^ - < 0.85; T m

0.88-^-

iraskevopoulos et al. (2004).

Model: Method 1

m

Direct synthesis -j. (min) 4 i

Paraskevopoulos et al. (2004). Model: Method I

1

K

0.02 <^S-< 0.7 T

(160)

m

.

(page 137)

l-35xm

0.1<^-<0.9 T

2

V X

l^m

+

mJ

X

2

T

V mTm

i s - < 0.3 T

Coefficient values l

x2

4

0.7269 1.7016 3.0696

2.5362 3.0585 3.7509

0.5 0.75 1

X

:

i

x2

S

6.975 12.452

5.7053 8.4358

1.5 2

x

Minimise performance index = l[e2(t) + ( u ( t ) - u „ ) 2 jit 1

3 j

(160)

_ ~

f

\3

0.747 + 3 . 6 6 - ^ + 18.64 T T V

Ti(min) =T m [1.3146-^- - 2 . 3 8 0 5 ^

.T(161)

_ _

(%

4.52-!2- +0.476 - ^ T T

\

0.88-

m J

(+ 57.622

\ -158.99

T V

(. T

m J

' ^

+ 173.41

T

Chapter 3: Tuning Rules for PI Controllers Rule 1

K

Comment

Ti

Kc

Paraskevopoulos et al. (2004) continued.

139

(160)

Xitm+x2Tm +

(page 137)

X

6T

T

1

3 m

0.2 < ^ 2 - < 0.7 T

-5 m

Coefficient values l

x2

4.59 6.16 8.21

-0.19 0.028 0.37

X

1

Paraskevopoulos et al. (2004). Model: Method 1

K

3

I

8.28 9.90 16.88

0.5 0.75 1

X

(160)

5

i

x2

12.41 19.91

1.89 3.44

x

T

3

4

51.95 95

1.5 2

X

(162)

^2L<0.2 T

(page 137) Coefficient values i

x2

0.409 0.717 1.142

0.2817 0.6542 1.177

x

1

K

TcL2 0.5 0.75 1

x

(160)

T

(page 137)

i

2.353 4.07 8.88 r \ 3~

x, + x 3

T

x2

TcL2 1.5 2 3

2.67 4.76 10.73

0.2 < ^ 2 . < 0.7 T m

Coefficient values i

x2

1.518 3.34 5.81

10.54 16.89 29.17

x

x i TcL2 13 0.5 23.08 0.75 51.94 1 Robust

2

* CL2

49.66 82.87 176.8

1.5 2 3

x

^ < Tmi i for Zhang and Xu (2002). Model: Method 1

6

K

(163)

T

(163)

^2-<0.5, T X. = 2 - 5 x m

• (162)

= T„

T 6

v

iV

(163)

^ + 2XTm + T m T m (163) ; T-, > ^i

K ^ + TJ

^+2XT m +T m T m

Handbook of PI and PID Controller Tuning Rules

140

3.12.4 Controller with proportional term acting on the output 2 E(s)-K,Y(s)

Table 34: PI controller tuning rules - unstable FOLPD model Gm(s) = Rule

Kc

Chidambaram (1997). Model: Method 1

T

Comment

T, Robust 7j

m - t m

KmeT m s-1

(164)

X>1.7;is-<1 T

i

m

Lee and Edgar (2002). Model: Method 1

T 1

7 j

8

K

-

(T

-55-

8

111

1.

v 1 v . - , . m , xM

n ^T

/ „

L Kr>K m -i

Tm-K,(I65)K K, K

(165)

(165)

T

T

'

2

1

Km

1

2(^ + TJ_

2(X + ' J '

Km-l

(165)

\0.5

xm2

K/l65,Km-l K m (X + T m )

-p (165)

T

0.5

'

(164) _

(165)

(165)

K

C

°

S

\

T V

T m J *m

1

Tm

H

,

i

Chapter 3: Tuning Rules for PI Controllers

141

3.13 Unstable FOLPD Model with a Positive Zero G m (s) =

3.13.1 Ideal controller Gc (s) = K

K m (l-sT m 3 )eTmlS-1

1+Ti^y

E(s) R(s)


f

1i

>

\

U(s)

1+-

Kra(l-sTm3)eTmls-1

T S

V

Y(s)

i V

Table 35: PI controller tuning rules - unstable FOLPD model with a positive zero • Km(l-sTm3>-"G m (s) = ~ : Rule

Kc

Ti

Comment

Direct synthesis: time domain criteria Sree and Chidambaram (2003c). Model: Method 1

YTml

1 K

T

raTra3

(166)

y = the negative of the value of the inverse jump of the closed loop system; y > —I!^Tmi T

Y m -iT 3(l-x,)-0.5T (l + x,)] m m -(166) _

im3

, X, >

^-(l-xj+x,

YTrol YT m l -T m 3

142

Handbook of PI and PID Controller Tuning Rules

3.13.2 Ideal controller

in series with a first order

lag

1+ —

G c ( s ) = Kc

V U(s)

E(s) R(s)

l K, 1 + T s 1 + Tfs v i y

+ T

> V 1 + Tfs

T S

Y(s)

Km(l-sTm3)eTm,s-1

Table 36: PI controller tuning rules - unstable FOLPD model with a positive zero •

Rule

Kc

Sree and Chidambaram (2003b). Model: Method 1

2

2

K

Comment T; Direct synthesis: time domain criteria

(167)

T

(167)

ISL. - JEHL. < o.62

Tmi

( l - s T )(l + T. (167) s)e~ Desired closed loop transfer function = -—*—^-^——-1 EL,

(I + T& S J(I + T & S ) .(167)

K

(167)

K r a lT(' L > 2 + T< 2 L ) 2 + T m

. (167) _

T

T

T

T

T1

T

)•

ml j CL2 CL2 ~ T m 3 m + [ CL2 + CL2 + T m3 + Xm F m l j T

Tf =

, (167) T

ml 2 +T r a 3X m -(-C r a +T m 3 )T m l

_ T (167) m3 T m x i

T (T (l) + T ( 2 ) + T + T iml^CLJ + X CL2+ 1 m 3 + T m

_T(167)V l i )

'

Tm3

Chapter 3: Tuning Rules for PI Controllers

143

3.13.3 Controller with set-point weighting U(s) = Kc

E(s)-aK c R(s) T s

v

i y

aK„

U(s)

R(s)

+ T-

E(s)

T s

v

Km(l-sTm3)e-

v8»

Kr i y

Y(s)

Tmls-1

Table 37: PI controller tuning rules - unstable FOLPD model with a positive zero • Km(l-sTm3)r"» G m (s) = T m l s-1 Rule

Kc

Sree and Chidambaram (2003b). Model: Method 1

3

3

K

Comment Ti Direct synthesis: time domain criteria ry (168)

(168)

-^s—-^2-< 0.62 Tmi Tm3

(1_ S T )(l + T. (168) sV st Desired closed loop transfer function = -—*— m3 ?\ y '—,„; \ (l + T & s J l + T & s ) .(168)

K

(168) (i) K ^ l' T CL2

. (168)

T1

, <2) CL2

+ TT

(T(D T(2) _ mU1CL21CL2 T

mi

f=:

T, =

T

a

IT") + T

(2)

T1

+ T

m3 +

T

. (168) m ~ *i

,|T(l) +T(2) + T T m 3 1 : m + I ^ C I ^ + X CL2 + i m 3 + Tm3tm-(xm+Tm3JTm|

T(168)

+T

+T

i 1 C L 2 + ^CL2 + im7,+Xm

_T l

\

a = 1-0.5

(168)

)

(r l ) m\ m\]

++ t

x

Ti 4- T^ 1 ) 4- T ( 2 ) m 3 + X CL2 + XCL2 ,(168)

V

Handbook of PI and PID Controller Tuning Rules

144 Rule

Kc

Sree and Chidambaram (2003c). Model: Method 1

4

4

Comment

T|

yT m , T

K ra T ra3

(169)

y = the negative of the value of the inverse jump of the closed loop system; ^ - [ T m 3 ( l - x 1 ) - 0 . 5 T m ( l + x1)] > Tm3 _Tm1

^, (169) _

'

'

"

Tnl3

^-(l-xj+x, 1

yTm yTml-Tm

m3 1

ct = l - 0 . 5

Desired closed loop transfer function =

m 3 "*" X CL2 "*" 1 C L 2 T < 169 )

(l-sT n 3 )(l4.T,»»),)b-". (Sree and Chidambaram (2003b)).

Chapter 3: Tuning Rules for PI Controllers

145

3.14 Unstable SOSPD Model (one unstable pole) G m (s) =

K e" (T ml s-l)(l + sTm2)

3.14.1 Ideal controller G c (s) = K( 1 + ^J

R(s) . o .

E(s)

Ke~ (T ml s-lXl + Tm2s)

Y(s)

Table 38: PI controller tuning rules - unstable SOSPD model (one unstable pole) K m e" s t " G m (s) =

(Tmls-lXl + sTm2)

Rule Minimum ITAE Poulin and Pomerleau (1996). Model: Method 1

Process output step load disturbance

Process input step load disturbance

Comment Kc Ti Minimum performance index: regulator tuning 1 K (170) 4T m l (T m +T m 2 ) Coefficients of KC,T; X T l m l - 4 ( T m + T m 2 ) deduced from graph T m /T m 0.05 0.10 0.15 0.20 0.25

i

x2

0.9479 1.0799 1.2013 1.3485 1.4905

0.05 0.10 0.15 0.20 0.25

1.1075 1.2013 1.3132 1.4384 1.5698

X 1 !„.

K

x

T

m( l rr

4(T m +T m 2 -4[xm+T^

x

i

x2

2.3546 2.4111 2.4646 2.5318 2.5992

tm/Tm 0.30 0.35 0.40 0.45 0.50

1.6163 1.7650 1.9139 2.0658 2.2080

2.6612 2.7368 2.8161 2.9004 2.9826

2.4230 2.4646 2.5154 2.5742 2.6381

0.25 0.35 0.40 0.45 0.50

1.6943 1.8161 1.9658 2.1022 2.2379

2.7007 2.7637 2.8445 2.9210 3.0003

x

Handbook of PI and PID Controller Tuning Rules

146 Rule

Kc

Comment

T; Direct synthesis 2

<°T ml

T(171)

minimum i?m = 25

i

Pouline/a/. (1996). Model: Method 1

T m 2 + t m <0.16T mI ;

3.53

Tml

Kra

me[25°,90°]

Ultimate cycle

McMillan (1984). Model: Method 1

2 x('71)

3

K

4T m l (T m +T r o 2 ) 1.3Tml-4(Tm+Tm2)

Tuning rules developed from KU>TU

T (172)

(172)

. 0.16Tm, < Tm2 + xm < 0.3Tml ; co = frequency

where phase of Gm(jco)Gc(j(n) is maximum

72) 3 £ (172)

1.477 T^T K

m

0.65

Xm2

1+

l^rol (Tml

_

T

+

^m2/^ml ^m2

m 2 X T ral

_

T:(172) = 3 . 3 3 x j l +

T

m i 1 !! _

Umi l T ml

+

Tm2jTmlTm2

Tm2j(Tml - T

m

JT r

Chapter 3: Tuning Rules for PI Controllers

147

3.15 Unstable SOSPD Model with a Positive Zero G m (s)

_

Km(l-Tm3s>2„2

TmlV-22;mTmls + l

3.15.1 Ideal controller R(s)

E(s)

3

f-

+

>

Gc(s) = K

1+ —

U(s) Kc

Y(s)

Km(l-Tm3s)e-^

—•

T m l 2 s 2 -2^ m T m l s + l

Table 39: PI controller tuning rules - unstable SOSPD model with a positive zero Km(l-Tm3s)e-^ Gn(s) = Tml2s2-2^mTmls + l Rule

Kc

Comment

Ti

Direct synthesis: time domain criteria Sree and Chidambaram (2004). Model: Method 1

4

4

R

(173)

T

(173)

( l - s T yi + T('73)s)e"STm Desired closed loop transfer function = -f —^ 7-7-17 r. (l + s T ^ l + s T ' ^ l + sx,) (173)

J (173) _

T

m l \ T CL + T CL

+ X

l + T m3 + T m j ~

T

CL T CL X l

^ ^

I i r ^ m ~~ ^ml

Tm,(T^+Ta)+Tm3+Tm)(-2^mTm3Tm+Tml(T

x X

'

2

3+Tm))-x2

(- 2^ ra T ml + Tra3 + xm Jr^Tg,' - T ml + (Tra3xm - T m l )( T g + T<2> + 2^ m T m ] )'

and with x 2 = (Tm3xm - T ^ ) ( T # T & > -T m T m 3 );

2

2^ m V Tmi

< 0.62 . 1

m3 /

Handbook of PI and PID Controller Tuning Rules

148

3.15.2 Controller with set-point weighting E(s)-aK c R(s)

U(s) = Kt aK

+

®—

R(s)

T

v

;sy Y(s)

v

Km(l-sTm3)e-"-

•
K, T s

i y

Tml2s2-2^mTmls + l

U(s)

E(s)

Table 40: PI controller tuning rules — unstable SOSPD model with a positive zero Km(l-Tm3s)r^ G m (s) = Tm,V-25raTmIs + l Rule

Kc

Sree and Chidambaram (2004). Model: Method 1

1

5

K

Comment T; Direct synthesis: time domain criteria

(174)

i

Desired closed loop transfer function =

K

(174) _

T 2 ( TX( 1 ) ++ TJ < 2 ) ++XX ' m l V CL CL 1

+4-T

*m3Xm

.,

(l + s T ^ l + s T ^ l + s x , ) '

-(174) ) K ^| T S + T g . + x 1 + T i n 3 + T i n - T i

•p (174) _

(174)

T

^ 3 —

+ +X

T m)

(174)

< 0.62 ; 'ml

1 -XT ( I X) T ( 2X) I CL CL l

Tm3y

^

'ml

Tm,(T^+Tg+Tm3+Tm)(-2^Tm3Tm+Tml(T

+TJ)-X2

(- 2^ m T ml + Tm3 + xm )r^>Tg.> - T ml 2 + (Tm3xm - T ml 2 )(T<'L> + T<2L> + 2^ m T m l )' andwithx24m3Tm-Tm,2fe^^^

Chapter 3: Tuning Rules for PI Controllers

149

3.16 Delay Model G m (s) = K m e

3.16.1 Ideal controller G c (s) = K

'i + -L^ v

T s

i y

R(s)

Table 41: PI controller tuning rules - G m (s) = K m e Rule

Callender et al. (1935/6). Model: Method 1 Two constraints method - Wolfe (1951). Model: Method 1 Haalman(1965)minimum ISE. Model: Method 1 Gerry (1998)minimum error: step load change. Gerry and Hansen (1987) - minimum IAE.

Kc

Ti

'0.568/K m T m

Process reaction 3.64xm

2

0.65/K ra x m

2.6xm

3

0.79/K m T m

3.95im

4

0.95/K m t m

3.3xm

0.2 K

T

m m

0.3im

Comment

Representative results deduced from graphs

Decay ratio is as small as possible

Minimum error integral (regulator mode) Minimum performance index: regulator tuning 0 l-5xm M , = 1 . 9 ; A m = 2 . 3 6 ; i)m = 50° 0.3 Km

0.42t m

Model: Method I

0.455im

Model: Method 1

0.345 Km

Decay ratio = 0.015; Period of decaying oscillation = 10.5xn 2

Decay ratio = 0.1; Period of decaying oscillation = 8xm

3

Decay ratio = 0.12; Period of decaying oscillation = 6.28xm

4

Decay ratio = 0.33; Period of decaying oscillation = 5.56im

Handbook of PI and PID Controller Tuning Rules

150

Kc 0.43

^

Comment

Km

0.5xra

Model: Method 1

0-4/Km

0.5xm

page 67

0

1.6Kmxm

page 63

Km

0.25TU

Model: Method 1

0.4KU

0.25TU

Model: Method 1

Rule Shinskey (1988)minimum IAE - page 123. Shinskey (1994)minimum IAE. Model: Method 1 Shinskey (1988)minimum IAE -page 148. Shinskey (1996)minimum IAE -page 167.

0.4255

Minimum performance index: servo tuning Minimum IAE = 0.47xm 1.377xm. A r a = 2 ; 0.36

Minimum IAE Huang and Jeng (2002). Model: Method 1 Minimum ISE Keviczky and Csaki (1973) Model: Method 1

Km

Astrom and Hagglund (2000). Model: Method 1

Astrom and Hagglund (2004). Model: Method 1

0

1.33xm

Minimum ISE = 1.53xm; K r a = l

0.5

0.635xm

Km=l

Minimum ijeri'ormance index: other tuning 0.35/Km Kc,Tj values 0-4xm deduced from graph Maximise 0.25 0.35x K /T m c i subject to K

Fertik(1975). Model: Method I

Skogestad (2001). Model: Method 1 Skogestad (2003). Model: Method 1 Skogestad (2001). Model: Method 1


m

Mmax=2 0.16/Km

0.340xra

M m a x =1.4

0.200/Km

0.318xm

M m a x =1.6

0.26/Km

0.306xm

M m a x =2.0

x,/K m

x2tm Coefficient values

*i

x2

0.057 0.103 0.139 0.168 0.191

0.400 0.389 0.376 0.363 0.352

M

max

*1

x2

1.1 1.2 1.3 1.4 1.5

0.211 0.227 0.241 0.254 0.264

0.342 0.334 0.326 0.320 0.314

M

max

1.6 1.7 1.8 1.9 2.0

Chapter 3: Tuning Rules for PI Controllers Rule

Kc

Van der Grinten (1963). Model: Method 1

(175)

K

T

0

l-45Kmxm

0.075/Km

0-Hm

0.53/Km

Stochastic disturbance M m a x =2dB

T

Representative results; M m a x =2dB

m

0.55/Km

1.59im

0

2-70Kmxm

0

2-50K m i m

Bryant et al. (1973). Model: Method 1

0

X

Critically damped dominant pole Damping factor (dominant pole) = 0.6

2^m

Coefficient values OS OS x2

x2

OS

2

4%

x2

OS

x2

0.64

105%

1.00

x

2xm Coefficient values x x OS OS x OS i i ! 0.72 10% 0.78 20% 0.75 15% OS = 10% 2K m t m OS = 0% 2.6667K m t m

i

OS

x

> 0.68

5% 0

Kamimura et al. (1994). Model: Method 1 Hansen (2000). Model: Method 1

Step disturbance

(175) i

Buckley (1964). Model: Method 1

Keviczky and Csaki (1973). Model: Method 1; Representative Kc,Ti values estimated from graphs; Km = 1

Comment

Ti Direct synthesis 0.5xm

0-5/Km 5

151

0

50%

1.5

0

2.3529Kmxm

0-2/Km

0.3tm

17%

"Quick" servo response; "negligible" overshoot

Robust Model: Method 1

Bequette (2003) page 300.

V K m (2^ + x m )

0.5xm

Skogestad (2003). Model: Method 1

0.294/Km

0.5xm

IMC based rule

Skogestad (2003).

0.25/Km

Other tuning 0.333xm

Model: Method 1

(•75) = e - m ^ A p s e - ^ J

T

(175) = _ L s _ (t _ 0 .5e-»^»).

152

Handbook of PI and PID Controller Tuning Rules

3.17 General Model with a Repeated Pole

/

3.17.1 Ideal controller Gc (s) = Kc1 +

R(s)

E(s)

<S>

+ T

f

1 ^

1

U(s)

K, 1 + -

T;s

K m e"

Y(s)

d + sTm)"

Table 42: PI controller tuning rules - general model with a repeated pole G m (s) = Rule

Schaedel (1997). Model: Method 2

Kme-nd + sT m ) n

Comment T Kc i Direct synthesis: frequency domain criteria x +T 0.5 m ar 05 'Normal' design

K m [r . - l J

0.375 "n + l" n Km _ - l .

(n + l)(T m +T a r ) 2n

'Sharp' design

Chapter 3: Tuning Rules for PI Controllers

153

3.18 General Model with Integrator

G m (s) =

3.18.1 Ideal controller

K, s

n(Tm,is+i)n(Tm,2s2+2^iTm2iS+i) n( T m 3i s + i )n( T m / s 2 + 2 ^ d i T m 4 i s + i) e

G c (s) = K

1+T s

i y

E(s)

T s+i T s2+2 iT s+i

Table 43: PI controller tuning rules - general model with integrator K„

Gm(s) =

s

n( ^ n( -/ ^ ^ ) n

(T->'S+^n t^

Rule

Kc

v+2

^«..iTm,is+!) Comment

Ti

Direct synthesis 6

6

l + sin
Q

K

=

Symmetrical optimum method Non-symmetrical optimum method

(176)

Kc(177,

(176)

K

T

K

aJQ

Loron(1997). Model: Method 1

cos<|>n

T

mVo7 Q

^Tmji+2^^mdiTm4i - ^Tm|i+2^^mniT,

(177)

X= K

mV a 2 T Q

1 + SlnA

M„

M/-1

^l . A ^ c o s ^ A ) .

+ T„

, M c = desired closed loop resonant peak,

Chapter 4

Tuning Rules for PID Controllers

4.1 FOLPD Model Gm(s) =

K e~ l + sT„

4.1.1 Ideal controller G c (s) = Ki

/ T

V

1

\

is

J

R(s) , o E ( s )

Table 44: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

Callender et al. (1935/6). Model: Method 1 Ziegler and Nichols (1942). Model: Method 2

i 1.066 K

mTm

aTm KmXm a s [1.2,2]

Td Process reaction 0.353xra or 1.418xm 0.47x Ti

ra

2T

m

0.5xm

Decay ratio = 0.043; Period of decaying oscillation = 6.28xn 154

K m e~ l + sT„.

Comment

^ - = 0.3 T

Quarter decay ratio

Chapter 4: Tuning Rules for PID Controllers Rule

Kc 0.95Tm Kmtm

Chien et al. (1952)regulator. Model: Method 2

155 Comment

Ti

Td

2.38xm

0.42xm

0% overshoot; 0.1<^2-<1 T m

20% overshoot;

l-2Tm 2xra

Kmxm

0.42xm

0.1<^-<1 T m

0% overshoot;

0.6Tm Chien et al. (1952)-servo. Model: Method 2

Kmxm

Tm

0.5xra

1.36Tm

0.47xm

0.1<-^ L <1 T

0.37tm

Quarter decay ratio.

0.1<^-<1 T 20% overshoot;

0.95Tm

Cohen and Coon (1953). Model: Method 2

1K

T(178)

(178)

i

1 + 0.19^T

0<^-
m

Three constraints r x 0.738 method - Murrill T T 1.370 2 T (179) (1967) -page — 1.351 Km x m J 356. 0.1<^-<1 lTnJ Model: Method 5 T m Quarter decay ratio; minimum integral error (servo mode). Model: Astrom and 0.5Tm 3 ^90% Hagglund(1988) Method 23 Km -pages 120-126. T90% = 90% closed loop step response time (Leeds and Northrup Electromax V controller). Model: Parr (1989)1.25Tm Method 2 page 194. 0.4t 2.5x m n K t m m

Borresen and Grindal (1990).

Tm 3^ m

Model: Method 2

0.5tm

rx

K

(178)

1.35—!2- + 0.25 K„

2 ^(179)

=

0 3 6 5 T I I

,(178)

^

2 . 5 - ^ + 0.46 T T \

1 + 0.61-

m/

Handbook of PI and PID Controller Tuning Rules

156 Rule

Kc

Sain and Ozgen (1992). Connell(1996)page 214. Model: Method 2

3 K (180)

Td

Comment

(180) *d

Model: Method 24 Approximate quarter decay ratio

Ti (180)

T

T

i

l-6T m K

1.6667xm

mTni

x,/Km

x

Hay ( 1 9 9 8 ) pages 197.198. Model: Method 1

0-4Tm X

2Tm

3t

m

Coefficient values - servo tuning x

T Coefficients of Kc,Tj,Td

0.125 0.167 0.25 0.5

deduced from graphs

i

6.5 5.0 3.5 1.8

IlIL T

x

9.0 6.5 4.2 2.2 Coefficient

i

x2

9.8 7.2 4.5 2.2

2.0 1.8 1.6 1.4

x

0.125 0.167 0.25 0.5

x2

x

T

3

0.125 0.47 0.167 0.45 0.25 0.40 0.5 0.35 values - regulator x

5.0 3.7 2.5 1.1 tuning x

T

3

0.52 0.50 0.45 0.35

0.1 0.125 0.167 0.25 0.5

x2

l

x

3

2tm

0.42T m

2.2 0.35 2.2 0.35 2 0.34 1.8 0.32 1.4 0.28 attributed to Oppelt

Kmxra

2^ m

0.44x m

attributed to Rosenberg

0.85T ra

l-6x m

0.6x m

Kmxm

10.0 8.0 6.0 4.0 2.0

1.2Tm

Liptak(2001).

3

0.34 0.32 0.30 0.27

x2

i

l-2T m Moros (1999). Model: Method 1

9.8 7.0 4.8 2.5

x

Model: Method 2

Kmxm

ControlSoft Inc. (2005).

2

4

^m+Tm

T :

Model: Method 10

(181) d

Km

0.6939-=!-+ 0.1814 | , T,

(180)

0.8647Tm + 0 . 2 2 6 T „

K„

^

+ 0.8647 . (180)

0.0565T^ 0 . 8 6 4 7 ^ - + 0.226

4

-T,

Ty

(181)

' =max

T

m Tm

(slow loop); Td

=min

tm

Tm

(fast loop).

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IAE 1.435 Murrill(1967)pages 358-363. Km

Minimum IAE Marlin(1995)pages 301-307. Model: Method 8

157

Comment Td Minimum performance index: regulator tuning Model: T i ^0•749 5 T (182) Method 5; x Ti

—]

0.878 [x m J

d

l-3/Km

0.7xm

0.01xm

0.1<^-<1 Tm x m /X m =0.43 x m /X m =0.67

l-0/Km

0.68xm

0.05xm

0.85/Km

0.65xra

0.08xm

^/Tm=1.0

0.65/Km

0.65xm

0.10xm

Tm/Tm=1.5

0.45/Km

0.55xm

0.14xm

x m /X r a =2.33

0-4/Km

0.50tm

0.21xm

*m/T m =4.0

Coefficients of K c , T ; , Xd estimatedfrom graphs; robust to ±25% process model parameter changes 6.558Kra 0.925Xm 0.061Xm x m /X r a =0.1 Minimum IAE Peng and Wu (2000). Model: Method 17

Minimum IAE Pessen (1994). Model: Method 1 Modified minimum IAE Cheng and Hung (1985).

2-31 lK m

0.677Xm

0.189Xm

im/Xm=0.2

1.479Km

0.633Xm

0.233Xm

x m /X m =0.3

1.117Km

0.647Xm

0.250Xm

x m /X m =0.4

0.947Km

0.705Xm

0.268Xm

tm/Xm=0.5

0.825Km

0.752Xm

0.289Xm

x m /T m =0.6

0.731Km

0.792Xm

0.312Xra

x m /X m =0.7

0.695Kra

0.871Xm

0.316Xm

x r a /X m =0.8

0.630Km

0.895Xm

0.326Xm

V/Tra=0.9

0.651Km

1.030Xm

0.306Xm

*m/T m =1.0

0.7KU

0.4XU

0.149XU

0.1<^2-<1.0 X m

3 'T ^ m K

m

f 5

T

N1137

(182)

T 6

TddS3)

=

0

X

4 g 2 T n

X V m

0.921

T

(

A 0 ' 749 6

0-878 \Tm )

T

(183)

Model: Method 12

158

Handbook of PI and PID Controller Tuning Rules Rule

T

Kc

Minimum ISE 1.495 f T j Murrill(1967)pages 358-363. ra m

K

0.945

x

lx J

,

m

T m

1.115

( V' XJ ,

T A

N

0.1<^s-
753 7

T

(184)

0.1<^-<1.0 T

(185)

1.1 < ^ 2 - < 2.0 Tm

id

0.641 7

m

T

id

1.130

Km

r

T

1.101

1.473 Minimum ISE K m V T nJ Zhuang and Atherton(1993). 0.735 Model: Method 1 1.524 (T ^

x 1.006

0.56Tm

m

0.970

N

,

\ 0.771

r

T

Comment

Td

,

A r a 6 [2,5], Minimum ISE Ho et al. (1998). Model: Method 1 Minimum ITAE -Murrill(1967) -pages 358-363.

8

(186)

K

T(l86)

ij

0.947

1.357

(TJ

Km

{zmj

T A


(186)

T

id

/•

0.1 < ^ 2 - < 1.0 T A m Model: Method 5

\ 0.738 9

m

0.842 V 1 m V

T X

(187)

0.1<^s-
d

m

1.468 f T j Minimum ISTSE

0.970

K m \xm) - Zhuang and Atherton(1993). 0.730 Model: Method I 1 . 5 1 5 f T j

Km W)

/ 7 ^(184)

8

K

=

0

_

T

Tm ( OA 0.957 ^Tm J

10 T (188) id

0.1 < - ^ < 1.0 T

10 rp (189) id

l.l<^2-<2.0 T

m

0598

\ 0.948

5 5 0 T n

T V 1m J (186) _ 1-0722 <|>n Km

Tm O 0-942 ( j m J

A.

, Td(185)=0.552Tn

T

T-

.86)

T l-2497T m ^ n 0.2099

(186)

T

_ 0.4763T m f Am 0.2035 /

-0.328

0.0961

V T*m J

/ T , \0.3693

Given A r a , ISE is minimised when m = 46.5489A m °' 2 0 3 5 (T m /T n ,)°

( H o e / al. (1999)); 2 < A m < 5 ; 0.1 < t m / T m < 1.0 . 9

10

(187)

T

T

(188)

= 0.381T m | ^ s _

=

Q.443T

T V im

, T d (189) = 0.444T;

T

Chapter 4: Tuning Rules for PID Controllers Rule

Ti

Td

Comment

•p (190)

0.144TU

0.1 < ^ 2 - < 2.0; T

Kc

Minimum ISTSE - Zhuang and Atherton(1993).

11

(190)

K

159

m

Model: Method 1 12

0.1 < -^s- < 2.0; T

T(191)

(191)

K

0.15TU

i

m

Model: Method 31 0.960

1.531 Tm Minimum ISTES 0.971 VXm J - Zhuang and Atherton(1993). 0.705 (T ^ T Model: Method 1 1.592 0.957 Km Minimum error step load change -Gerry (1998).

12

(194)

T

K

(194)

(194)

„(,9„

Ku,

'

15

T x

1.1 < ^ s - < 2.0 T

(193) d

m

T

is->5; T

0.471KU

Model: Method 1 6, < 2.4

m©u

(I90)

6.068K m K u -4.273*

r

16

N 0.597

Model: Method 35; Decay ratio = 0.15; 0.1 < i s - < 2.0 T

—

T< 192) =0.413T,

m ,

K

lj

5.758K m K u -1.058 15

0.1
(192)

T

T

4.434K m K u - 0.966 5.12K m K u +1.734

(191)

Jvc

15

0.5xm

Nearly minimum IAE, ISE, ITAE -Hwang (1995).

K (190)

x 0.746

0.3

16 ^

„

,

A

T

1.1622Km K „ - 0.748 « —

~

lu.

2.516K m K u -0.505 N 0.850

0.933

, T d (193) =0.414T n

T

V 1m J 0.0607fflHI2Tm2

0.674JL - 0.447coHltm + K m /(l + K H 1 K m ) ,(194) .

f

K.r>(l + K H I K m ) coH1Km0.0607(l + 1.05(DHItm-0.233coH,2Tm2)'

Handbook of PI and PID Controller Tuning Rules

160 Rule

Kc

Nearly minimum IAE, ISE, ITAE -Hwang (1995) - continued. Nearly minimum IAE, ITAE Hwang and Fang (1995). Model: Method 25

,7

K<195)=k,

17

K

T

(195)

Comment

Td

i

2.4 <e, <3

(195)

T

i 18

K

3<e, <20

T(196)

(196)

i 19

K

20 j ^

(197)

T

(198)

T(198) 1

e, >20

(197)

0.1<-^-<2.0; T decay ratio = 0.12

(198)

T

'd

i

0.778[l-0.467(oH,Tm + 0.0609coH2/xT HI ^m K m /(l + K H 1 K m ) j(195)

2h

Kc

(l + K H 1 K m )

» HI K m 0.0309(l + 2.84WH1xm - 0.532o>H12Tm2)" 18 ^

1.3l(0.519)m"|T" [1-1.Q3/S) +0.514/e l 2 f

(196)

K m /(l + K H ,K m )

J' 96)

KC" (1 + KH1K )

(196)

)H1Km0.0603(l + 0.9291n[coH1xm])(l + 2.01/s 1 -1.2/8 l 2 )' 19

K

(197)

Kt

izi

1.14|l-0.482coH,xm +0.068(DH,2Tm2]N K m /(l + K H 1 K m ) K c ( ""(l + K H 1 K j

• (197)

) H ,K m 0.0694(-l + 2.1co H ,T m -0.367 WHI 2 T m 2 )' 20

K

(198)

'x

0.802 -0.154 — + 0.0460 T i

^

K„

T

, V m J

K

,(198)

(198)

f K„0)u

^2

0.190 + 0.0532-52- - 0.00509 T T V 1m • (198) .

0.421 + 0.00915-21--0.00152P!! T I T

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

O'Connor and Denn(1972). Model: Method 1; Km=Tm;

Ti Minimum 0.5 0

ax m 2 + 2 X l

(2 + x , )

2T m a

Comment

Td QO

2i ax m 2 + 2x,

161

;'"K^ x

Recommended

lxm

2

x m a + 2x,

^2-<0.2 T

a e

0.6 2 '

1 2

m 22

(199)

K

T

(199)

T

(199)

X 1 2
2+22

K

(199)

=

_

x

0.31 + 0.6-

(199)

K„

_

T

0.777 + 0.45-2!T (199)

^COS 23

K

tan

1.57D„

(200)

K cos tan

0.16 DR J

6.28Tm

0.44 - 0.56

(~

Handbook ofPIandPID

162 Rule

Controller Tuning Rules

Kc

Ti

Comment

Td

Minimum performance index: servo tuning x,/Km

x x 2( T m + O 3(Tm + 0 Minimum IAE Representative coefficient values - deduced from graphs Gallier and Otto X x2 x3 x2 (1968). •tm/Tn, tm/Tm *i *1 3 Model: 0.95 12.3 0.0215 0.67 1.45 0.75 0.155 0.053 Method 1 0.85 0.21 0.93 0.041 1.50 0.63 6.8 0.11 0.57 0.88 0.083 4.0 0.53 0.19 3.4 0.25 / \ 0.869 Model: Method 5 Minimum IAE T 1086|Tm 1 24 T (201) Rovira et al. 0.1< — < 1 0.740-0.13— Km (1969).

Minimum IAE Marlin(1995)pages 301-307. Model: Method 1

l-3/K m

0.93xm

0.02x m

xm/Tm=0.25

l-0/K m

0.80xm

0.05x m

xm/Tm=0.43

0.8/K m

0.7h m

0.06x m

xm/Tm=0.67

0.7/K m

0.73xra

0.09x m

x m /T m =1.0

0.55/K m

0.64xm

0.12x m

tm/Tm=1.5

0.48/K m

0.56xm

0.15x m

xm/Tm=2.33

0-4/K m

0.52xm

0.21x m

tm/Tm=4.0

Coefficients of Cc ,Tj ,T d estima ted from graphs; robust to ± 2 5 % process model pa rameter Minimum IAE Sadeghi and Tych (2003).

25KC

Wang et al. (1995a). Model: Method 1

26 K

<202)

changes. Model: Method 1 1.45933xm

(202)

T

i

^ - + 3.27873 Tm

0.1<^-<1.0 T Minimum IAE

(203)

0.5T m x m Tm + 0.5xm 26

0.05 < ^ - < 6 "T

T m + 3.5xm

(204)

K

Minimum ISE

m

f 24

25

T

v

K

(201)

= 0.348T„

(202)

-,0.914

T

0.26266 + 0.82714-

T(202)

K„

0.7645 + 26 K (203)

0.6032

(Tm+0.5xm)

T

*m/ m

K m (T m + x m )

T

0.28743- S L +1.35955 . T 0.9155 +

0.7524

(T m +0.5x ra )

(204)

>KC

K r a (T m + x m )

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

1.048 Minimum ISE Zhuang and Atherton (1993). Model: Method 1 1.154

T,

Minimum ISE Sadeghi and Tych (2003).

Comment

Td

} 0.897

fry m

27 T (205)

T

(205)

28 j

T

(206)

0.1<^_<1.0 T

X f

\ 0.567 (206)

Minimum ISE Hoe/al. (1998). Model: Method 1

163

1.1 < ^s_ < 2.0

*d

A m e[2 J 5], 29 j r

(207)

(207)

T

(207)

T

i

0.1<^-<1.0. T 1.93576xm

(208)

T

3oKc(208»

^ + 3.83528 T

i\ Minimum ITAE 0.965 - Rovira et al. ,XmJ (1969).

0.1<^-<1.0; T Model: Method 1

0.85 31

T

(209)

T x

0.1
(209) d

m

Model: Method 5

27

T(205)

_

., Td(205) = 0.489Tn 1.195-0.368(206) _

28 ry (206)

0.490T„

1.047-0.22029 K

(207)

T(207)

'

T V

m J

'O T

V 1m J

1.8578 <|>n

,T,

0.4899Tmm

(207)

0.1457 (

\ 1.0264

T A„ Km A m ° 0.021 lTm[l + 0.3289Am +6.4572^)m +25.1914(x m /T m )], l + 0.0625A m -0.8079(), m +0.347(x m /T m )

Given A m , with 2 < Am < 5 and 0.1 < x m /T m < 1.0 , ISE is minimised when ^^29.7985 + ^ l ^ + 4 0 - 3 1 8 2 T - - ^ g ^ - ( H O e / a / . ( 1 9 9 9 ) ) . 0.40455 + 0.96441-

ry (208) _ ry

0.43770-2-+ 1.39588 T

K„ r

31 j

Td<209) = 0.308Tn

(209)

0.796-0.1465-

N

X„ V T my

0.929

164

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Minimum ITAE - Wang et al. (1995a). Model: Method 1 Minimum ITAE - Sadeghi and Tych (2003).

32

T m +0.5x m

(210)

K

i

3.855 Modified T 1 minimum ITAE - Cheng and Hung (1985). 0.855 Modified 0.965 (T N minimum ITAE m T - Smith (2003).

11

0.7303K

(210)

33 K (2")

=

_1

^ - + 4.00000 Tm

34T(212) i

0.308xm

1.26Tm

—1

35 j

(213)

T 1

0.05 < ^ - < 6 T m

0.1<-^-
Model: Method 1 £, = 0.707; Model: Method 12 Model: Method 1

(213) d

0.1 < ^2-< 1.0 T

(214) d

1.1 < i=-< 2.0 T

m 36 q-. (214)

T 1

0.5307 (T m +0.5x m ) tm/Tm Km(Tm+xm)

0.20360 + 0.80451-

T,(211) = T I 0 . 2 9 3 4 9 ^ + 1.29110

K„ , 34 ~ (212)

T <212) = 0.308Tn

\ 0.929

T

0.796-0.147/ 35 j

(212)

T

Comment

x 0.897

1.042 Minimum ISTSE - Zhuang and Atherton (1993). N 0.579 Model: Method 1 i.i42 rT m

32

1.55237Tm

T(211)

33KC<2>"

/

Td 0-5TmTm T m +0.5x m

T,

(213)

TH(213) = 0.385T„ 0.987-0.238-

,T V. m

/ 36 j ( 2 1 4 )

\ 0.906

-,0.839

, (214)

, T d l i "' = 0.384Tn 0.919-0.172-

T ,

Chapter 4: Tuning Rules for PID Controllers

165

Rule

Kc

T,

Td

Comment

Minimum ISTSE - Zhuang and Atherton(1993).

0.509KU

37 T (215)

0.125TU

0.1<^S-<2.0; T Model: Method 1

37

0.604 Ku

/•

0.130TU

38 T (217)

—l

38

T

_

T

39 T (218)

T 1

i

K

(220)

T

•

T

K

m

m»u

2.4<6,<3

T^2'6' =0.04| 4.972Kra Ku + 1 |T„.

(218)

f

\

0.892

T

:0.315T„

0.892-0.16540

l.l
i

, T d u l " =0.316T n

(218) _

(218) d

0.471KU

, T

0.1 <-^s_< 1.0 T

8, <2.4

0.977-0.25339

(217)

(219)

T

= 0 .05l(3.302K m K u +l)T u ,

T(217)

0.1<^-<2.0; T Model: Method 31

\ 0.904

Minimum ISTES 0.968 Tm) - Zhuang and Atherton(1993). / „ \ 0.583 Model: Method I 1.061 Km 40 ^ (219) Nearly minimum IAE, ISE, ITAE -Hwang (1995). 41 K (220) Model: Method 35

37 Tj(2i5)

T(216)

N 0.832

T V

m J

0.822[l-0.549co H1 T m +0.112a) H1 2 T m 2 ] N K m /(l + K H 1 K m )

(219)

K,<"»>(1 + K H 1 K B )

,(219)

2JT++ 6.96o)HITra-1.77coH1 T„ coH1Km0.0142^ 0.786[l-0.441fflH1Tm +Q.Q569coH1 2

(

41

K

(220)

KH1

KJ(I + K„,KJ T(220)

_

l\\

J

Kc'22°>(l + K H1 K m ) 2 21 u H1 coH1Km0.0172(l + 4.62co H l l nxTm -0.823a) H , T„

166

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Nearly minimum IAE, ISE, ITAE -Hwang (1995) - continued

42 K

(221)

T

43 j ,

(222)

T

Comment

Td

Ti

3 < e, < 20

(221) (222)

E,

i

>20

Decay ratio = 0.1; 0.1 < — < 2.0 m

Nearly minimum IAE, ITAE Hwang and Fang (1995).

42

K

44 K

(223)

T

(223)

Model: Method 25

(223)

T

0.1 < ^ - < 2.0 ; decay ratio = 0.03

1.28(0.542)'°H'X'"[I-0.986/E1+0.558/S12]

(221) H

'

K m /(l + K H 1 K m ) T

K

(221)

(1

c

+

K

H|Km)

raH1Km0.0476(l + 0.9961n[coHITm])(l + 2.13/s1 - 1 . 1 3 / e , 2 ) ' 43

K

(222)

1.14[l-0.466 mH1 T m+ 0.0647co H1 2 T m 2 l N K m /(l + K H 1 K m )

'K

.(222) ' 44

K

(223)

Kc

co H1 K ra 0.0609(-l + 1.97co H1 T m -0.323co H1 2 x m 2 )'

0.537 - 0.0165^2- + 0.00173 -^ T T T,

(l + K H 1 K m )

K„,

K

(223)

(223)

f

K

u»u

0.0503 + 0.163—m-- 0.0389 T T

\2

V 1m J • (221) _

0.350 - 0.0344-^- + 0.00644 - ^ T T

Chapter 4: Tuning Rules for PID Controllers Rule

Comment Tj Td Minimum performance index: other tuning

Kc

Simultaneous Servo/regulator tuning - nearly minimum IAE, ITAE - Hwang and Fang (1995).

167

0.1<^-<2.0; T 45 K

(224)

(224)

T

m

decay ratio = 0.05; Model: Method Model:25

Servo or _ regulator tuning 0.36 + 0 . 7 6 ^ - minimum IAE - Huang and Jeng (2003). 46 £

(224)

T

0.47x m +T m

(225)

T

0.47Tmtm 0-47Tm+Tm

(225)

Method 37; ^ 2 - < 0.33 T

(225)

T

1SL>O.33 x

Astrom and Hagglund (2004). Model: Method 1

45

K

(224)

l^ro+

X

2Tm

x

3Tm

+

x

4*m

XeTm^m tm+X7Tm

X

l

x2

Coefficient values X x3 x4 5

X

X

6

7

0.057 0.139 0.400 0.923 0.012

1.59

4.59

Mmax=l.l

0.103 0.261 0.389 0.930 0.040

1.62

4.44

M m a x =1.2

0.139 0.367 0.376 0.900 0.074

1.66

4.39

M m a x =1.3

0.168 0.460 0.363 0.871 0.111

1.70

4.37

M m a x =1.4

0.191 0.543 0.352 0.844 0.146

1.74

4.35

M m a x =1.5

f

0.713-0.176-2!-+ 0.0513 Is. T T

^

Ku, K

- (224)

(224)

21

Ku u 0.149 + 0.0556- - - 0.00566 m

( x. \2

T

T V

m J

0.371 - 0.0274 - 2 - + 0.00557 T IT 46

K

(225)

0.8194Tn+0.2773Tn. v

0.9738 T 0.0262

, T^">

co„K

(224)

=1.0297T m + 0 . 3 4 8 4 T „

T

(225) _

0.4575Tm+0.0302Tn. 1.0297Tm+0.3484t

my

Handbook of PI and PID Controller Tuning Rules

168 Rule

Kc

Ti Coefficient values Astrom and X x X x2 x4 l 3 5 Hagglund (2004) 0.211 0.616 0.342 0.820 0.179 - continued. 0.227 0.681 0.334 0.799 0.209

Comment

Td X

X

6

7

1.78

4.34

M m a x =1.6

1.81

4.33

M r a a x =1.7

0.241 0.740 0.326 0.781 0.238

1.84

4.32

M r a a x =1.8

0.254 0.793 0.320 0.764 0.264

1.87

4.31

M m a x =1.9

0.264 0.841 0.314 0.751 0.288

1.89

4.30

M m a x =2.0

Direct synthesis: time domain criteria Van der Grinten (1963). Model: Method 1

_L(o.5+^l

Regulator Gorecki et al. (1989). Model: Method 1

47

K

(226) _

Tra^m ^m+2Tm

T m +0.5t m

47 K

(226)

T

(226)

T

(226)

48 K

(227)

T

(227)

T

(227)

Step disturbance Stochastic disturbance Pole is real and has maximum attainable multiplicity; "tm/Tn, < 2

e_ K„

In

1 - 0.5e"°,',x"' +-la-e-<»d^

- (226)

(l - 0.5e""jT- )ln

(226)

T

x„

'K

(227) _

2

\

2T„

2Tm

-x,-9-

v2Tmy 1

T1

(227)

i

= TL

2T

2T

'O

2

x, - 9 - - ^ - 2T l2Tm;

n

21 + 3 ^ - + T

—1

X,

N2TJ_

A3

r -36-4.5-^T

(2T^

—

l2TJ A

T<227)=0.5x„

3+

6+-

2T

x,-9-

2T

2T

2T

A2

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Saito et al. (1990). Model: Method 18

T|

169 Comment

Td 0.315T m x m

0.215tm+Tm 1.37K m x m

2HL

0.315x r a +T m 0.315Tm+Tm

< 1

T m

49 x (228)

1<^-<10 T m

Suyama(1992). Model: Method 1

50 K (229)

Tm+0.309tm

T

OS=10%

(229)

0.56T m Fruehauf et al. (1993). Model: Method 2

T

5im

mK m

<0.5T m

^ 2 - < 0.33 T

<0.5x m

— T

m

0-5Tm T

Tm

mK m

Juang and Wang (1995). Model: Method I

51 K (230)

Abbas (1997) Model: Method 1

52 £ (231)

T

(230)

T

(230)

T

•M

>0.33

CL = « T m ,

0
Tm+0.5tra

2T

0< V<0.2 ;

m*m

m + tm

0.1 < -^2- < 5.0 T m

Camacho et al. (1997).

1

Tm+xm

4T m x m T

Tra+xm

Model: Method 1

m+*m

(228)_0.315xmTm+0.003T„

0.315Tra+Tm 50

v

K„

(229) _

1

0 . 7 2 3 6 - 2 - + 0.2236 . T,

K.

2-236T ra T m

(229)

7.236Tm+2.236xn ,2

51 K (230)

T Km

T

. (230)

a + IsL + o.5 U s T IT a+-

a

0.5 (230) _

a + ^-O.Sa^ T T

Vim J

V

a + is_ Tm

a +

2^

0.177 + 0.348 i " 52

K

(231)

K m (0.531-0.359V a 7 ' 3 )

Ii!L + o.5 Tm

Handbook of PI and PID Controller Tuning Rules

170 Rule

Kc 53

Cluett and Wang (1997), Wang and Cluett (2000). Model: Method 1;

K

Td

Ti

(232)

(232)

T

Comment

(232)

T l

d

TCL = 4xm , A m E [7.97 ,8.56], * m e [79.67° , 79.70°] (233)

54 ^

(233)

T

(233)

T l

d

T C L = 2 x m , A r a e [4.84,5.31], *„ e 73.9°,74.14° (234)

55 K

(234)

T

(234)

T l

d

i

0.01 < -^2- < 10 T

T CL =1.33T m ,A m E[3.81,4.16],
(235)

(235)

T

T

(235)

i

T CL = T m , A m G [3.30,3.56], <))m e [65.86°,68.50° (236)

57 K

T

(236)

T

(236)

T C L = 0 . 8 t m , A m 6 [3.00,3.19], * m e 60.60°,67.21°

53 „ (232) _0.019952x m +0.20042T m

0.099508xm + 0.99956Tm

(232)

0.99747x-8.7425.10 _ 5 T„ (232)

-0.0069905xm+0.029480T„ 0.029773xm +0.29907Tm

54 v

( 233)_0.055548T m +0.33639T m

0.16440xm +Q.99558Tm

(233)

0.98607x-1.5032.10~ 4 T„, m -0.01665 IT m + 0.09364 lTm (233) 0.093905xm +0.56867T„ 55 v

'K,

<234)

_0.092654x m +0.43620T m K

0.20926xm + 0.98518Tm

T (234)

0.96515x m +4.2550.10- 3 T„

T

m m

-0.024442xm +0.17669Tm 0.17150xm +0.80740T„ 0.24145xm+Q.9675 lTm

(234) .

56 K (235) _0.12786x m +0.51235T m ^ T<235) K

0.93566xm + 2.2988.10~2T„,

mTm

-0.030407xm +0.27480Tm 0.25285xra+1.0132Tm 0.26502xm +0.94291Tm 0.89868x m +6.9355.10" 2 T m m ' (235)

57 K„ (236) _0.1605lT m +Q,57109T m c

.

T(236) li

T d

(236) = -0.035204Tm+0.38823Tm 0.33303xm+1.1849Tm

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Cluett and Wang (1997), Wang and Cluett (2000) - continued. Valentine and Chidambaram (1997a). Morilla et al. (2000). Model: Method 28

58 v (237)

T|

(237)

58 K

Comment

Td

(237)

T

171

T

*i

(237)

*d

T CL = 0.67x m , A m e [2.80,2.95], ^ m e [52.35°,66.45° j

59 K

(238)

(238)

T

T

(238)

i

60 K

(239)

(239)

T

\ = 0.707 ; Model: Method I a = 0.1; 6 0 = [0.2,0.5]

al™

0.19067t m +0.61593T m T < 237) _ 0.28242Tm +0.91231Tm ^ n K m x„ ' ' 0.85491T+0.15937T

K„

(237)

-0.039589Tm +0.5194 IT,, 0.40950T„ + 1.3228Tm

59

K

rr\

(238) _

{£Jd)

1

rr\

0.746-1.242 In

T

T

r ^2 (238) 0.3106 +1.372-^1- _ 0.545 -T, T T V

T

16.015Tm -11.702x„

mJ

±1% T s = 2.5T m , 0 . 1 < - ^ - < 0 . 5 ; ±1% T s = 3.0Tm, - ^ - = 0.6; +1% T s 3.75Tm , ^=- = 0.7; ±1% T s = 4.3Tm ,^60

K

T^'V

(239)

+

= 0.9 .

, (239)

Kra[280+ron0(xra-Tj(239))l' • 5„Tm

= 0.8; ±1% T s = 5.0Tm , ^

'

1 + VT^4^'

V»02Tm»+Tm(TB,-Ti'ro>)-arr,<°!"]' T

(^-T^j-aP^]2

m (?m " T i

= desired closed loop response decay ratio.

with 8 0 1+

2TT

logc[b/a]

2

a

Handbook of PI and PID Controller Tuning Rules

172 Rule

61 j r

(240)

T

Comment

Td

Ti

Kc

Mann et al. (2001). Model: Method 31 or Method 33 Chen and Seborg (2002).

(240)

T

(240)

OS < 5%; 0<*m/Tm
(241)

x

(241)

OS < 5%; l^m/Tm<2

(242)

T

(242)

Model: Method 1

i 62 ^

(241)

T

i

«K C < 2 4 2 >

T

i

0.854"

0.770+ 0.245^261

K

Tm

lTnW

(240)

T(240)

'

i

1.262 + 0 . 1 4 7 ^ V

ir ,

N 0.854

0.262 + 0.147 (240)

_

V

m J

0.770 + 0 . 2 4 5 ^

0.603 + 0.275 62 j , (241)

r

T

,(241)

V m K

1.1618 + 0.165

m'Cni

N-2.4

T

T m '

V m

0.1618 + 0.165|-^ (241)

_

0.603 + 0.275 63

K

(242)

1 (2TmTm + 0.5xm2)(3TCL + 0 . 5 x J - 2 T C L 3 -3T C L 2 x m 2T C L 3 +3T C L 2 x m +0.5x m 2 (3T C L +0.5xJ

Km T(242,

_ (2T m t m +0.5Tm2)(3TCL + 0 . 5 x J - 2 T C L 3 -3T C L 2 x n tm(2Tm+Tmj Td(242)

2

. 3TcL xmTm + 0-5TmTm2 (3TCL + 0.5x m )- 2(Tm + xm )TC (2Tmxm +0.5xm2)(3TCL + 0 . 5 x J - 2 T C L 3 -3T C L 2 x m

T<242>s(l + 0 . 5 x m s ) r K ^ ' ^ s + l)3

Chapter 4: Tuning Rules for PID Controllers Rule

Kc 64 K

(243)

T

Comment

Td

T,

(243)

173

(243)

T l

i

6

Viteckova and Vitecek (2002). Model: Method 1 For 0.05 < ^ S - < 1.6 , T; = 1.2Tm ; overshoot is under 2% (Viteckova and Vitecek (2003)). (l-v)xmTm (l-v)rm+Tm

Gorez (2003). (l-v)xm+Tra Model: Method 1 XK T Sree et al. (2004). Model: Method 1

1 [T

1

-^2-+ 0.5 T

m ^

66T d < 2 4 4 )

T m +0.5x m

J

1.377 (T

0.8422 67

(245)

T

(245)

T

T

V

w

64 vK

65

0-v)x m +T m

m m

0.1<^_<1 T m

m J

^ l [ T m 2 T m X | 3 +(3TmTm

(243) =_ e

+ T m 2) X i 2 + T m X ]

_1];

K„

3 x"m m

x, =

0.5 | 3 2 + T A |T ^mm m

0.25 T 2 m

^ m ' ^ X , 3 + ( 3 T m T m + T m 2 ) x 1 2 +T X, - 1

• (243)

("c m 2 T m x, + 2 t m T m (243)

T/" J '=-0.5 T

'ForO^

+xj)

^ m 2 T m x , 2 +(4xmTm + x m 2 ) x 1 + 2 x m +2Tm T

"

2 2 1 m T+ T L m )x, v m X, - 1 m 2 T m X l 3 + ( 3 t'•m mT m m r-\ T+T m^l


iV=

>Tm+Tm2+0.5xm2 (^m+Tra)2 1 X=

F

o

r

T

m

< ^ ^ <

1

,

v

=

^

T

+

T

m

|

2M,(M,-l) ™

2

+

0

(^m+Tm) 66

x

(244)

5

2

( 1 . 3 M S 2 - 1 ) 1.56T m 2 (0.5x m +3T, 2M.(M.-l)

^

2

0.65M, M.-l

0.5im(Tm+0.1667tm) Tm+0.5xm

67

T<

245

>=i.085T„

T V mJ

,T,

(245) _ .

0 . 3 8 9 9 - ^ + 0.0195 T

(im+Tm)3

Handbook of PI and PID Controller Tuning Rules

174

Comment Td Direct synthesis: frequency domain criteria (246 68 j r (246) (246) Model: T T Method 1 Low frequency part of magnitude Bode diagram is flat.

Rule Regulator Gorecki et al. (1989).

69 K

Somani et al. (1992). Model: Method 1; Suggested A m : 1.75
68

K

^

Kc

(247)

X2Tm

*3*m

Coefficient values x2

^m/Tm 0.0375 0.1912 0.3693 0.5764 0.8183 1.105 1.449

X

1.869 2.392 3.067 3.968 5.263 7.246 10.870

0.71 0.67 0.63 0.59 0.56 0.53 0.50

3.57 3.33 3.13 2.94 2.78 2.63 2.50

x2

*m/Tm

3

2.38 2.27 2.17 2.08 2.00 1.92 1.85

3

0.48 0.45 0.43 0.42 0.40 0.38 0.37

(246) K

m

2

_

x„

-+1

- (246) V "m

/

rT

^

7 + 4 2 - ^ - + 135 T

(246)

-

T

T

T

+ 180

m

1 + 7 - ^ + 27 xm

(216) _

'N2

'T„°

+ 60

(T

+ 240

7+ 42-2!-+ 135 xm

-EL

T

< 1 , K (247)

\4

(y

(T + 240

/T T 15 2 - ^ - + l l + 3 ^ _ + 6

For

x

0.7809

+ 60 fy

\

A4

+ 180

+ 1 or

KmAm |0.803 + 4^s-+0.896

K

(247)

, v

0.7809 1 + 0.803 T KmAm1

K

(247)

0.7809 1 + 0.455 KmAm1

•>1.

Chapter 4: Tuning Rules for PID Controllers Rule

Kc 70 ^

Chidambaram (1995a). Model: Method 1

T

*i/K m A m

X

Iss. T

x

2tm

x

Alternative

3^m

ISL

x2

l

X

m

i

x2

0.389 0.543 1.03 2.92 3.69

5.104 3.781 2.262 1.209 1.105

2.19 2.14 2.00 1.72 1.67

x

3

2.34 0.37 2.33 0.37 2.29 0.37 2.27 0.36 2.23 0.36 2.21 0.35 2-4xm

1.03Tm

2-4xm

0.38xm

A ra = 1.5

0.38tm

A m == 1.75

• + 0.85

0-85

or

,

/T

A AC

,1+4.45

4.45 + 4 - S - - 2 . 1 1 5T„

5Tm (T

\

4.45

1.17

3.42

-0.15

+ 0.85

4.45 + 4 - 5 i - - 2 . i l Tm 0.8Tm

T <248> ~

0.8T

y

d

4.45 1.17

3

0.35 0.34 0.32 0.28 0.27

m m

KmAm

•(248)

X

T

3.42

(248)

(248)

T

Kmtm K

70 j r

(248)

Coefficient values

0.062 29.770 0.083 22.100 0.150 12.470 0.196 9.670 0.290 6.686 0.339 5.784 1.20Tm Chidambaram (1995a). Model: Method 1

Comment

Td

Ts

(248)

175

3.42

4.45 + 4-21--2.11

+ 0.85

/ T \2

0.15

\2

Handbook of PI and PID Controller Tuning Rules

176 Rule

Kc 0.90Tm

Chidambaram (1995a)continued.

Ti 2.4xm

71 K

(249)

(249)

T

T X

i

72 ^

(250)

T(250)

73 K

(251)

T(251)

v

K,

(249)

=

1

(249)

T x

(250)

mKucosfaJ

(251)

1.1 < ^ 2 - < 2.0 T

(252) i

T(252) 1

0.5 <-^S- < 1.1 T

d

T

74

0.1< — <0.5 T

d

T

i

71

Am=2.0

KmTm

Branica et al. (2002). Model: Method 1; M.=1.8 Minimum ISTSE - Zhuang and Atherton(1993). Model: Method 1

Comment

Td 0.38xm

a

i

* m = 60°

, Ti(249)=1.534Tn,T"

0.021 + 0.788-

/-

\ -0.055

(249)

T

V 1m J f

72

vK „ (250)

\ -0.554

_ _ J _ [ 0.292 + 0.671-^2- | , Ti(250) =1.239T„ v T V mJ r (250)

\ -0.069

: 0.230T„

T \-0.480 73

v

K„

(251) _

1

0.338 + 0.623^2-

T<

25

» = 1.228xm - ^

xm

T„

T d ( 2 5 1 ) =0.23kJ^H T

i = 0.614(1 - 0.233e-°'347K"'K»), <> | m = 33.8°(l - 0.97e"°'45K"'K"), 0.1 < ^ - < 2.0 ,

T/ a = 0.413(3.302KmKu +1), T;

252

tan(c|>J+ - + tan2(<|>J ' =a 1« 2co„

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum ISTSE - Zhuang and Atherton(1993) - continued.

Ti T

7S

a

4Td(253,

*m=60°

(253)

T

*d


Tan et al. (1996). Model: Method 30

Am=2,

(252)

T

(page 176)

(253)

Uet al. (1994). Model: Method 1

Comment

Td

(252)

i

mKucos(4)m)

76 j r

111

15°

n

A ra 2

2.8

4>m A m 30° 1.67

il 3.8

20° 1.67 3.2 25° 1.67 3.5

35° 1.43 4.0 40° 1.43 4.2

4h 7tApAm

0.3183T

-7—cosm

aTd

(254>

Am n 1.25 4.6 45° 50° 1.25 4.9

4>m

55° 1.25

4>m

Am 1.11 60° 65° 1.11

TI

5.4 5.5

5.2

0.0796T

simplified algorithm

77 x

a chosen arbitrarily

(254)

l

d

Arbitrary A m , 78 ry (255)

A ra

T

(255)

*d


'Model: Method31; m = 0 . 6 1 3 ( l - 0 . 2 6 2 e " u ^ ^ " ) , <(>m = 33.2U l-1.38e

-0.68K m K„

a = 1.687K m K u , 0.1 < - ^ < 2.0 . 76

K

4hcos0

(253) 7lA„

(253)

+ ^T*™

fiT^

tan(|)m - (b/VA.'-b

1

_

JJ

T1 + J T 1 2 + 471

with ±b = deadband of relay, T = limit cycle period.

}*[*,^ . • - " • • J . K 77

T

in

(254) _

T

— In

2

tancj>m + J — + tan i 47C

I recommended A m = 2 , d)m = 45

Handbook of PI and PID Controller Tuning Rules

178 Rule Friman and Waller (1997). Model: Method 1; Am>2

Kc

T;

Td

0.25K U

1.1879TU

0.2970TU

0.4830

2.6064

0.6516

Comment tm/Tm<0.25,

G

ra

pO .5o°]

$m > 60° 0.25 < -i=- < 2.0 Tm *m>45°

Robust Rivera et al. Tm+0.5tm (1986). K m ( ^ + 0.5x m ) Model: Method 1

X>0.1Tm,

Tn^m 2Tra+xm

Tm+0.5Tm

^>0.8xm.

0.5x m < X < 3x m (Zou et al. (1997)) Brambilla et al. 79 T m + 0-5xm (1990). K m (A. + i J Model: Method 1

Tm+0.5xra 2Tm+xm

0.1 < - ^ s - < 10 T m

No model uncertainty - X « 0.35x m T

Lee et al. (1998) Model: Method 1

80 y (256)

(256)

T

K r a (X + x m ) 81 K

Gerry (1999). ModelMethod 10

(256)

(257)

(257)

T

Tm K m ( ^ + 0.5x m )

T 1

T

T

CL

- ^

(257) d

0-5xm

X = 2x m (aggressive, less robust tuning); X = 2(T m + x m ) (more robust tuning)

X£[0.1Tm,0.5Tm],^<0.25; ^ = 1 . 5 ( x m + T m ) , 0 . 2 5 < ^ < 0.75 ; X = 3 ( x r a + T m ) , ^ > 0.75 (Leva (2001)).

80 y (256)

(256)

= Tm + 2(^ + t m )

2{X + Tm)

1-V

3T-i. (256) J

J 1

, (257) 81 K

(257)

T / ^ T Kra(2TCL+xra-a)'

+ a ^

li

+a

^-0.5xn 2TCL + xm - a

3

2 0 1fi7r m 3 -0.5ax 0.167x m

Tma(257)

2T rl + xm - a , (257)

Trl

2

+ ax m - 0.5x

2

2T r , +x m - a 2

a = T,m - T j l - ^ M e

_ISL

Chapter 4: Tuning Rules for PID

Rule

Comment

Td

T,

Kc

179

Controllers

Ultimate cycle McMillan (1984). Model: Method 2 or Method 44 Geng and Geary (1993). Model: Method 1 Wojsznis and Blevins(1995).

82 K

K

Tuning rules developed from

(258)

T

i

K

0.94T m 0.5x m

•^2- < 0.167 T

Tm+0.5tra

Tmtm 2Tm+xm

1.3
Tm+0.5xm aKra^m

K

83

aTd(259)

u«>S(|>m

84 K

u>Tu

2tm

Kmxra

(260)

(260)

T

Recommended ot = 4

(259)

0.1 < ^ s - < 1 . 0 T

(260) d

aT.(261)

85 j (261)

K u cos(|) m

T

x X

i

Am

1.415 Tm

(258) _

(258)

T 1

Peric et al. (1997). Model: Method 40 Matsuba et al. (1998). Model: Method 1 Wojsznis et al. (1999). Model: Method 1

82

(258)

.

T. (258) = x

1+

T

+x

1+

(258)

= 0.25TJ1 +

T +x V 83

x

/

(259)

tann a ,0.9

84

K

(260)

1.31 f T, Km Vtm

y

2.19 IT,

m

T 47t 0.9

Ti(260)=1.59x„

Kra^m T <260) = 0.40x„

= (tan<j)m + y 4 a + tan2<|>

I—J 4na

my

Handbook of PI and PID Controller Tuning Rules

180 Rule

0.5Kucosm Wojsznis et al. (1999)continued.

Td

Comment

0.15Ti(262)

Default values

T|

Kc

86ry

(262) i

0.38K U

1.2T

0.27K U

0.87TU

0.18TU

U

^m/Tm<0.2 Am=3,

0.13TU

4> m =33°; xm/Tm*0.25

Wojsznis et al. (1999). Model: Method 36

a, e [0.3,0.4]; 87

K

(263)

ry

(263) i

0.125Tj<263)

a2 E [0.25,0.4];

0.2<^-<0.7 T m

T i ( 2 M ) = 0 . 5 3 T „ ( t m ( | > m + 7 a 6 +• tan

87 ^

(263)

_

K

0.6

1.0-a,

r-p (263)

0.6

T-. -

1+e

-^-7.0

l + e^Tm

-j

Chapter 4: Tuning Rules for PID Controllers

181

4.1.2 Ideal controller in series with a first order lag f G c ( s ) = Kc T

V

U(s)

l

^\

is

j Tfs + 1

l + sT„

K

Table 45: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

Schaedel (1997). Model: Method 21

1K

1

Y(s)

Kme'""

•

1 + Tfs

1

me

l + sT„

Comment Td Ti Direct synthesis: frequency domain criteria

(264)

T,2 - T 2 2 T

T

T

(264)

Tf =T d ( 2 6 4 ) /N

(264)

5 < N < 20

Robust T m +0.5x m

Tra^m

Horn et al. (1996). Model: Method 1

T m +0.5x m

^ m

T

' 2^ + x J ' ^e[xm,Tra]

2T m +T m

X = max(0.25Tm,0.2Tm)(Luyben(2001)); X>0.25t m (Bequette, (2003)).

1^

0.375T

(264)

(264)

(264) K„

T, +-

_

^ ' , (264)

j

(264) _ T j d

- T,

T3

T22

N

- + Tm,T2=1.25TmT>

-T„,+0.5T

'

T,=0,

1 + 3.45-

1 + 3.45-

0<-^-< T

0.104:

0.7T

0.7T

T„-0.7x a m

- + 0.5t„ -+^ m - T 2 = T m C + "" ' Tm-0.7<

0.952T m 2 (C-0.l)

, 0-35T m 2 t m 2

T„,-0.7T*

Tm-0.7<

0.167xJ, — > 0.104.

Handbook of PI and PID Controller Tuning Rules

182 Rule

Tj

Kc

Td x T ^m+2T m

Comment

2T m +x r a

Model: Method 1

2 K (265) H^ optimal T m +0.5x ra Tan et al. (1998b). X = 2 : 'fast' response; X = 1: 'robust' tuning; X = 1.5 : recommended Model: Method I Lee and Edgar (266) i-p r-p ( 2 6 6 ) 3 K (266) (2002). T T (266) X lf - ld d M Model: Method 1 T m +0.5x m Tm Huang et al. T m +0.5x m 2T m +x m ' 2(X + xJ Kj^ + xJ (2002). X = 0.65x minimum ISE; X. = 0.5x overshoot to a step input m m Model: Method 1 < 5% , A m = 3 . In general, A ra = 2{X/xm +1).

Zhang et al. (2002).

4

K

(267)

T m +0.5x m

2 K (265)_0-265X + 0.307rT m K„ lxm 3

K

1

(266)

K m (^ + x m )

T m +-

. ( 2 6 6 ) _ry

. Tf

5.314X. +0.951 3Tm-xm

T

2(\ + T J

\6(*. + x Jr + -4(^ + x r a ) 2 3 T

X„

2(X + xm)

xn

m I—7"X„

m

m-'cm

^ +x J T

+

.

t„

4(^ + x r a ) 2 '

(266) _

[6(X + x J K

(267) _ _

1 Km

T m +0.5x„ 2X. + 0.5X.

2X. + 0.5x„

o r T f _AE^ m xm+2T

4(X + x J 2

Chapter 4: Tuning Rules for PID Controllers

183

4.1.3 Ideal controller in series with a second order filter r A 1 G c (s) = K£ V

T

is

J

l + a fl s + a f2 s

u(S;

E(s) R(s)

l + b fl s

l + bfls

K, l + — + Tds V TiS

l + a f ,s + a f 2 s

l + sT„

Table 46: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

Kristiansson (2003). Model: Method 11

5

K

Y(s)

K„e"

K„e" l + sT„

Comment Ti Td Direct synthesis: frequency domain criteria rp (268)

(268)

T

(268)

0.05<^S-<2; T m

1.6 <M S <1.9

' Equations continued into the footnote on page 184; f \2 T, 0.06 + 0 . 6 8 — 0.12 l.l^-O.l T T (268) _ K f~ \ K „ 0.68+ 0.84-=!--0.25 T T 2T„ 0.06 + 0.68 -5=- -0.12

T

. (268)

0.68 + 0.84 -^- -0.25 T T

T

Handbook of PI and PID Controller Tuning Rules

184 Rule

Ti

Kc

Kristiansson (2003) continued.

6

K

(269)

(269)

T

Comment

Td T

(269)

i

0.7 < — < 2 ; T m

1.6<M S <1.9 Robust Horn et al. (1996). Model: Method 1

T

7

K

T ro +0.5T m

(270)

(268) _

mTm

*• > ^

- *• < T m •

^m+2Tm

b l f = 0 , alf=0.8Tf, a2f=Tf 0.68 + 0 . 8 4 ^ - 0 . 2 5 ^ T Tra fx

\

with

)

2 -|2

0.06 + 0 . 6 8 ^ - -0.12 T T

1.1^-0.1

Tr=min-l 5+ 2 - ^ , 2 5 x„

0.6667(T m +T m 6

K

(269)

1.1 ^ - - 0 . 1 Xm

0.68+ 0.84-2!--0.25' O T T V . (269)

2

m /

0.6667(T m +x m ) n.T, ( ^2 0.68+ 0 . 8 4 ^ - - 0 . 2 5 T \LmJ m

0.1667(T m +x m )

(269)

(. \ 2 ] 0.68 + 0.84 ^~ T

-°ii) \

(T m + xJ 2 1 . 1 ^ - 0 . 1 xm

b l f = 0 , a,,- = 0.8T f , a2f = Tf with Tf = -

9T„ 5 + 2V 7

K

(270)

my

2Tm+xm _A. 2 x m +2T m x m (x m -A.), 2*,(2Tm->.) -,Dfl-, ; + T m (x m +2X) (x m +2X) ' 2(2A. + x m - b „ ) K n « ffll —

2XT m +2X 2 +b f ,T a , TT~. : \—i 2(2X + x r a - b f l ) '

^ x „ df2 f

2(2X + T m - b f i :

Chapter 4: Tuning Rules for PID Controllers 4.1.4 Ideal controller with weighted proportional

term

Table 47: PID controller tuning rules - FOLPD model G m (s): Rule

Kc

185

Ti

K m e" l + sT„ Comment

Td

Direct synthesis Astrom and Hagglund(1995) -pages 208-210.

3.8e-",+7JtITm K

5.2xme-—'

0.89T m e- 0 3 7 t - 4 w 2

or

or

T

m m

50t 4 8l! 0.46T m e 2 ' 8 T - 2 ' u 2 0.077T m e - '

b = 0.40e 0 ' 18t+2 ' 8x2 Model: Method 5 5T a93T2 or Method 14 8 . 4 e - * 6 t + M ' 2 T 0.86T m e-' 9 t - 0 4 4 1 2 m 3.2T m e-'- or

or

0.28T m e 1 8 T "'' 6 t 2

0.076T ra e 3 ' 4t - MT!

KmTm

M,=1.4; 0.14 < ^ 2 - < 5.5 Tm

Ms=2.0; 0.14 < ^ 2 - < 5.5 T

b = 0.22e a 6 5 , + 0 0 5 l T 2 0-v>m+Tm Gorez (2003). Model: Method 1

(l-v>mTm

0-v)tm+Tm

8

(l-v)xm+Tm

xKraTm b

_ (X-I)tm-Tm

:1

(l-v)tm+Tm

• < ' For 0 < ^ m +T m

T m T m +T m 2 +0.5T m 2 T

V =

(^ m +T m ) 2 1 2M,(M.-l)

Fori

<

V ^ m +T m

(1.3M S 2 -1)

{

2M.(M,-l)

<1;V_^Tm+Tm2+05Tm2_

frm+Tj

1.56T m 2 (0.5T m +3T m ),

0.65M„ M -1

(T m +T m ) 3

Handbook of PI and PID Controller Tuning Rules

186

4.1.5 Ideal controller with first order filter and set-point weighting 1 \ f 1 1 D , , l + 0.4T r s U(s) = K 1+ — + T d s R(s) — ^ J - - Y(s) Tfs + 1 1 + sT, T V >S J U(s)

E(s)

K(s) l + 0.4T r s w

1^+ T r s -

( K„

^

\ 1 1 + — + Tds

V

T

is

J

1 1 + Tfs

Table 48: PID controller tuning rules - FOLPD model G m (s) =

Rule

Kc

Normey-Rico et al. (2000). 0.375(x m + 2Tj Model: K mTra Method I

Td Direct synthesis

Ti

Y(s)

l + sT„

Ke-

l + sTm Comment

m^ra

T f =0.13x ra

2T m +x m

T r =0.5x m

T

T m +0.5x m

K e~

Chapter 4: Tuning Rules for PID Controllers

4.1.6 Controller with filtered derivative G c (s) = K£

187

T;s

Table 49: PID controller tuning rules - FOLPD model G m (s) Rule 1K 2

1

N=10

Km

(271)

(272)

1 ~Km

\

(

l + sT„

T

m

i s - + 0.1 T

T<27')

i s - + 0.2 T m

-

0.3i°- + 0. 75 T >

)

T(272) x

i

0.06 x

Minimum ITAE

m

i s - + 0.04 T

0 . 3 - ^ + 1.2 T

0.06 m

I Tm f

)

i=- + 0.0; T

T ^ 0.3 +-siX l mJ

\

(

0.8

> V

K

K

1

1 "Kro

2

X,

Comment Td T( Minimum performance index: servo tuning (271) Minimum IAE T (271) 0.01 HT m (272) Minimum ISE T(272)

(

0.1
K

.

ds

Kc

Tavakoli and Tavakoli (2003). Model: Method 1;

1

T

l

,

i s - + 0.04 T

+ 0.08 J

"\ 1

^--"-m

T

4

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Davydov et al. (1995). Model: Method 6 Kuhn(1995). Model: Method 14; N=5

3

K

T, Td Direct synthesis

3

K

(273)

T (273)

4

K

(274)

T

(273)

T

(274)

T 1

i

(274)

0.66(T m+ T m )

0.167(x m + T m )

2/K ra

0.8(Tm+Tj

0.194(x m+ T m )

, (273)

K J 1.552-^ + 0.078

^=0.9; 0.2
d

VKm

(273)

Comment

"Normale Einstellung" "Schnelle Einstellung"

0.186-E- + 0.532 T

Td(273) = 0.25 0 . 1 8 6 - ^ + 0.532 |T„ •(274) K„

1.209^2-+ 0.103 T

0.382-^s- + 0.338 Tm T

(274)

0.4 0.382^2-+ 0.338 T

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Schaedel (1997)

5

Goodwin et al. (2001) -page 426.

K

6j,

T

(276)

T

Comment

Td

Ti

(275)

(275)

T

(275)

(276)

T

(276)

Model: Method 21

i

0.375T (275)

Model: Method 1

T

,(275)

(275)

T, + T

189

2 (275) '

T

r-p J

FT* £,

Z

X

~

T

•(275)

ryi

Z

1

T

(275)

*d

l

2

h

T

T

2

N

- + Tm,T/=1.25TnX +

1 + 3.45-

-Tm+0.5TmM3=0, 1 + 3.450 < ^ - < 0.104: T

0.7T,

2

?ra,V=Tmxam

T m -0.7x...

0.7T„

T -0.7x a

•+o.V,

3_0.952T m 2 (x a m -0.l) + TmTmTan + °- 3 5 T r o Tm +0.167T m 3 , ^ - > 0 . 1 0 4 TV m m m _ m ' r p T m -0.7x a A _ a T 0 7 T l m-

-

m

m

r (275)

N: .(275) A

. (275)— (275)

iii^l+o.3 7 5 - ^

Kv

0.05 < a < 0.2 , Kv = prescribed overshoot in the manipulated variable for a step response in the command variable. 6

v

(276) ._ ( 4 f ; T C L 2 + T m ) ( x m + 2 T J - 4 T C L 2 2 K m (l6^ 2 T C L 2 2 +8^T C L 2 x r a +T m 2 ) '

T

(276) _ (4^TCL2 +T m )(x m + 2 T m ) - 4 T C L 2

,

+i _

2(l6i; 2 T CL2 2 +8i;T CL2 T m +T m 2 )

^

^

T d ( 2 7 6 ) =(l6^ 2 T C L 2 2 +84T C L 2 x m +T r a 2 ) l(4^TCL2 + xm ) 2 i m T m - 2TCL22(4^TCL2 + xm Xxm + 2T m )+ 8TC (4^TCL2 + xm )3 [(4^TCL2 + xm Xxm + 2 T m ) - 4T CL2 2 ] N=44TCL2^mT(276)

2T r

w

,thGcL(s)

=

.

'

T C L 2 V+2!;T C L 2 s + l

Handbook of PI and PID Controller Tuning Rules

190 Rule

Kc

Comment

Td

Ti

Robust T ra +0.5x m Chien (1988). K (^ + 0.5x ) m m Model: Method 1 Morari and Zafiriou(1989). Model: Method 1 Gong et al. (1996). Model: Method 1 Maffezzoni and Rocco (1997).

Tmtm

T ra +0.5x m

2T m +x m

Tm^m 7jr

(277)

T m +0.5x m

2T m +x m

8

K

(278)

T m +0.3866x m

0.3866Tmxm T m +0.3866x m

9

K

(279)

T

(279)

T

(279)

10 K

(280)

T

(280)

T

(280)

(Chien and Fruehauf (1990)); N=10 ^.>0.25x m , \>0.2Tm N = [3,10] Model: Method 12 Robust to 20% change in plant parameters

Kasahara et al. (1999).

Model: Method 1; N = 10; 0 < ^ - < 1 T

K

(277) _

Tm+0.5x„

1 ... .

(279)

2T m (^ + x m ) ^(2Tm+xm)'

T m + 0.3866x m . (0.1388+ 0.1247N)Tm+0.0482NTn , *• = —r-r . . ._. -T„ K m (x + 1.0009xm) 0.3866(N - l)Tm +0.1495Nxn

8 j , (278)

K

N=

^ + Tm

2T

m(Tm+X!)+Tm

=

2Km(tm+x,)2 T (279)_

2

-

T

(279)

=T +

2(Tn+x,)'

2

x m [2T ( x m + x , ) - x m x 1 ]

M

_

2Tm(xm+x1)-xmx1

2

Xj = 0.25t m

2(tm+x,)|2Tm(xra+x1)+xm J' x,[2T m (x m +x 1 )+x m 2 ] x for uncertainty on xm only; xx >0.1Tm>—— <0.4 for uncertainty on the

minimum phase dynamics within the control band. 10

K

(280

> = 0.6K, 0.718-0.057-

, T^ 8U ' = 0.5TU 0.296 + 0.838Td(280) = 0.125Tu 0.635-1.091-^ T

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Kasahara et al. (1999)continued.

Leva and Colombo (2000).

11

191 Comment

Td

Ti

11 K

(281)

x

(281)

T

(281)

12

K

(282)

T

(282)

T

(282)

13

K

(283)

T

(283)

T

(283)

14

K

(284)

Robust to 30% change in plant parameters Robust to 40% change in plant parameters Robust to 50% change in plant parameters A. not specified; Model: Method 1

2 T

K< 2 8 1 ) =0.6K, 0.690-0.076-

m

I

T m T x

2(X + T J

(284) d

, Ti(281) = 0.5TU 0.274 + 0.788Td(281) = 0.125TU 0.526-0.057-!2T

12

(282)

K/"";=0.6K,

v

0.331 + 0.149-

• (282) _

0.5T„ 0.041 + 0.857-ET Td(282) =0.125TU 0.495-0.064-

13

K <283) = 0.6K, 0.195 + 0 . 2 6 6 - ^ T

X (283) = 0.5T„ -0.013 + 0.731(283)

14

v

(284) _

1 2(X+Tm)Tm+Tm2

Km

2(X + TmY

(284 )

_

:0.125T„

0.489-0.065-52T

Tm{l + tm)

[2(^ m )T m +T ra 2 ]' N=

2T m (X + T m ) 2

x[2(A. + T m )T m +T m 2 J'

Handbook of PI and PID Controller Tuning Rules

192 Rule

Kc

Leva (2001). Model: Method I Leva et al. (2003).

(285) _

v

K

17j,

2(* + TJ

m i

(287)

+Tn

=rJ

2

^ Y }

2

T 1

m

(286) d

Model: Method 1

2(x + xJ T

(287)

T x

Model: Method 1

(287) d

2Tm(X + t J 2

N:

2(X + X m ) 2

m

2S5)

V

m

T T

(285)

T

T

Tm i

1 2(X + Xm)Tm K

2

(285)

16 j , (286)

Zhang et al. (2002).

5

15 j r

Comment

Td

Ti

«.[2(X + T , ) T m + x „

^ ' ^

2 |A e [ 0.1T m ,0.5T m ],^0.25;

2(X + Tm)[2Tm(^ + T m )+T m 2 J

Tm

>. = 1 . 5 ( t m + T m ) , 0 . 2 5 < ^ - < 0 . 7 5 ; A. = 3 ( T m + T m ) , i s - > 0 . 7 5 . 16

v

K„

(286) _

1 2 ( ^ + T m )T m + T m K m

N=

r

2(X + T m )

T

(286)

__

T m T m (X + T m )

[2(^ + x l ) T m + T r a 2 ] ;

2

2T

™(X + X™> ^ . Xe[0.1T m ,0.5T m ], i 2 - < 0 . 3 3 ;

42(^ + x m ) T m + t m 2

Tra

X= 1.5(xm+Tm))0.33<^<3U = 3 ( x

m +

Tj,^>3.

(287) 17

K

(287)

d 0.5xmm +T m -~AN

(287) , (287)

K m (2* + 0.5xJ (287) _

x lmmTi „m 2T (287)

T,

(287) N

T

'

:

0.5x m +T m

N

(287) N

2?,+0.5x„

orN=10.

Chapter 4: Tuning Rules for PID Controllers f

4.1.7 Blending controller Gc (s) = Kc V

E(s) R(s)

®—

+ T

1

\

T S

J

i

U(s)

f

\ 1 K, 1 + — + T d s

193

Y(s)

K e"

1 + sT

Table 50: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc 18

Chen et al. (2001). Model: Method 1

K

l + sTm

Comment Ti Td Direct synthesis: frequency domain criteria

(288)

T

(288)

T

(288)

Representative X values ^ = 9.56t m) Mmax =1.20; ^ = 4.68x ra , Mmax =1.26; ^ = 2.19x m , M max =1.50;^ = 1.40tm, M max =2.00; ^ = l . l l T r a , M r a a x = 2 . 5 0 ; X = 0.95Tm, M r a a x =3.00.

K

(288)

1 T K„

+T+2A. (*• + *.?

,(288) _

Tm+Tra+2X,T

(288)

T„+T m +2X.

Handbook of PI and PID Controller Tuning Rules

194

4.1.8 Classical controller 1 G c (s) = K

Table 51: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

Ti

Kme" l + sT m Comment

Td

Process reaction Witt and Waggoner (1990). Model: Method 2 Witt and Waggoner (1990). Model: Method 2

1

K

m m

Equivalent to Ziegler and Nichols (1942); N e [10,20]

Xm

*m

T

X! 6 [0.6,1.0] 1K

(289)

T(289)

T

Preferred tuning; N e [10,20]

(289)

Tuning equivalent to Cohen and Coon (1953).

V2

1 . 3 5 0 ^ _ + 0.25+ ^ . 0 . 7 4 2 5 + 0 . 0 1 5 0 ^ 3 - + 0.0625

K

(289)

T

2K„ *(289)

1.350^2- + 0 . 2 5 - i s - 0.7425 + 0.0150-^=- + 0.0625 ^ tm Tm V Tm T (289)

!!

r x \2 n

1 . 3 5 0 - ^ + 0.25 + - E - 0 . 7 4 2 5 + 0.0150^ -+ 0.0625 T tm xm V Tm V 1m J

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Witt and Waggoner (1990) (continued).

2

K

T,

(290)

195

Comment

Td

T(290)

x

(290)

Alternate tuning; N e [10,20]

Tm St. Clair (1997) page 21. Model: Method 2

Kmxm

5x ra

0.5x m

5xm

0.5x m

Tm

<0.25T m

0-5Tm Kmxm

Harrold(1999). Model: Method 10; N not specified

^m/T r a ^0.25

1/K m 0-5/K m 0.25/K m

Minimum IAE 0.817 Huang and Chao Km (1982).

2

K

l-75x ra

Kmxm

xm/Tm«0.5 xm/Tra>l

0.889T m Shinskey (2000). Model: Method 2

'aggressive' tuning; N>1 'conservative' tuning; N>1

0.70x m

^ 2 - = 0.167; T

N not specified Minimum performance index: regulator tuning 0.982 N = 8; 3T(291) (291) Model: Method 1 x X \Xm

d

J

f x „ \2 1 . 3 5 0 ^ - + 0.25 - ^ i 0.7425 + 0 . 0 1 5 0 ^ - + 0.0625 xm xm V Tm VTmV

(290)

2K„

X

(290)

f x„ \ 2 1 . 3 5 0 ^ + 0.25 + — . 0 . 7 4 2 5 + 0 . 0 1 5 0 - ^ - + 0.0625 xm xm V Tm T (290)

1.350^2- + 0.25 - — . 0 . 7 4 2 5 + 0 . 0 1 5 0 ^ + 0.0625 x„ xm V Tm T s 0.954

3 T (291)

=

Q.903T

, T d ( z y " = 0.602T,, T

, T ,

Handbook of PI and PID Controller Tuning Rules

196 Rule

T

Kc

Minimum IAE Kaya and Scheib (1988). Model: Method 5 Minimum IAE Witt and Waggoner (1990). Model: Method 2

4 ^

(292)

T 1

Comment

Td

i

(292)

T 1

i

0
(292) d

T

(293)

T

(293)

T

(293)

N=10 Preferred tuning

6 j r (294)

T

(294)

x

(294)

Alternate tuning

5K

0.1<-^2- < 0.258; N e [ l 0 , 2 0 ] m

0.95Tm/Kmira Minimum IAE Shinskey(1988) 0 . 9 5 T m / K m x m -page 143. 1.14T m /K m T m Model: Method 1 1.39T r a /K r a T m

4 K

(292)

(^

_ 0.98089

1.43Tm

0.52xm

^/Tra=0.2

l-17T m

0.48t m

xm/Tm=0.5

1.03xm

0.40xm

tm/T m =1

0.77x m

0.35xm

Tm/Tm=2

^ 0.76167

,

T

, (292)

T

0.91032

K„

N 1.05211

V m J

Trf<292) = 0.59974T

5 £ (293)

f ,. A

0.718 | T„ K

m

Is

1 + , 1 - 1.693

T

V my

T

V 1m J

r A1137 0.964T„

0.964T„ VTm V

, (293)

(293)

1 - , 1-1.693 IlIL ,T

l + J l + 1.693

T \

6 ^

(294)

0.718 K„

m J

-0.921

1 - , 1-1.693

T

T V

m J

f

r (294)

\1.137

0.964T„ Im. T /•

l - J l + 1.693 V

\1137

( 0.964T n • (294)

T

-,1.886

l+Jl-1.693

T m J

T

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IAE Shinskey (1988) -page 148. ModelMethod 1 Minimum IAE Shinskey (1994) — page 167. Minimum ISE Huang and Chao (1982). Minimum ISE Kaya and Scheib (1988). Model: Method 5

i

Td

Comment

l-43tm

0.52xm

Model: Method 1 xm/Tm=0.2

1.5Tm

0.55xra

l-8xm

0.70xm

T

Minimum IAE Shinskey (1996) 0.96T m /K m i m -page 119. Minimum IAE Edgar et al. 0.94T m /K m x m (1997) - pages 814. 8-15. Minimum IAE Edgar et al. 0.88T m /K m x m (1997) -page 815. Minimum IAE Smith and Tm Corripio(1997)K mTm page 345.

197

Model: Method 1 Time constant dominant; N=10 Model: Method 2; N=10 Model: Method 1;

Tm

0-5xm

0.56KU

0.39TU

0.14TU

xm/Tm=0.2

0.49KU

0.34TU

0.14TU

x m /T r a =0.5

0.1 < ^ 2 - < 1.5

0.51KU

0.33TU

0.13TU

Tm/Tra=l

0.46KU

0.28TU

0.13TU

Kuxm 3x m -0.32T u

7 T (295) i

0.14TU

^m/Tm=2 Model: Method 1; N not specified Model: Method 1; N=8

/-m

\°-881

I.IOI r - o 9

K

(297)

8 T (296) i

T

j (297)

T

(296)

(297) i

0<-^-
N=10

0.15—^-0.05 0.883 T(296)

=

U

3

4

(296)

T

= 0.563T„

>T

,

K

(297)

1.11907 TTm

Km

{xm

v T

• (297)

T 0.7987 T • (297)

= 0.54766T,

T V '• J

Handbook of PI and PID Controller Tuning Rules

198 Rule

Kc

Minimum ITAE 0-745 - Huang and Chao (1982). Minimum ITAE - Kaya and Scheib (1988).

T

Tm^

10 T (298)

T

„

(298)

= a 7 7

K (299)

Model: Method I; N=8

(299)

0
'd

11 K

(299)

12

(300)

T

(299)

T

K

(300)

T

T l

(300)

Preferred tuning

(301)

Alternate tuning

d

13

K

(301)

(301)

T

T

Model: Method 2; N e [l0,20]

f T

(298)

^=10; Model: Method 5

Minimum ITAE - Witt and Waggoner (1990).

>0

Comment

Td

i

I t„

0.77902 ( X

TT\?

\ 1.006

, T d (298) =0.597T rt I™. ,T •

>T;

T t. 1.14311 T

(299)

A

0.70949

TH(299) = 0 . 5 7 1 3 7 T j i

12 K

(300) _ 0-679 K„

, 0.1 < -**- < 0.379; T

1 + , 1-1.283

T

f

0.762T, .(300)

/•

N0.995

0.762T„

T

(300)

-v 0.995

T V 1m J

r

1-Jl-1.283

l + J l + 1.283

T V m y

13

K

(301) _

T

V mJ -1.733

0.679 K„

V- 733

1-Jl-1.283

T V 1m J

,

0.762T„ ,(301)

T

1 - J l + 1.283

0.762T„ • (301)

N0.995

>T

( 1 + , 1-1.283

A1'733 T

V 'my

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Ti

(302) _

Kc Minimum ITAE f Ou and Chen 1.357 (1995). Km

T

N 0.947

Is]

Minimum ITSE 0.994 (T Huang and Chao (1982). Km

Td

199 Comment

(302) _ 14 T

(302)

Tm O 0-842 [Tm J

v 0.907 15 j

(303)

T

(303)

'd

Model: Method 46 Model: Method 1; N=8

Minimum performance index: servo tuning f N-1.00 Model: Minimum IAE 16 T (304) (304) Method 1; Huang and Chao 0.673 U \ T N=8 (1982). Km I j J Minimum IAE Kaya and Scheib (1988). Model: Method 5

.04432 17 T (305)

X (3"5) l A

Km ^xm j

0<^-
<302>

14 x

(302)x (302)

N=±

15

T(303)

Is. |

= 1 - 0 3 2 T

V 16

T

T.(

305

Ti«302>+KmKc^(Td(3°2)+Ti^-Tm-Tj

, Td<303> = 0.570T„

m

<304)

•>T,

0.148

17

0.75K r a K c l ^'T d

'm

(304)

+ 0.979

>

= 0.502T, V

-.T, 0.9895 + 0.09539-

T

(305)

T mJ

= 0.50814T v T

N>0.

Handbook of PI and PID Controller Tuning Rules

200

Rule

Kc

Comment

Td

Ti

Preferred tuning; Minimum IAE Witt and Waggoner (1990). Model: Method 2

(306)

K

(306)

j (306)

T

(306)

19

K

(307)

T

(307)

T

(307)

Ne[l0,20] Alternate tuning 0.1 < ^ - S 1.5; T N not specified

0.83Tm

T

Kmxm

1.086 K„

0.1<^_<1; T

i

Minimum IAE Smith and Corripio(1997)page 345. Model: Method 1

18 ^

18

T

0.5xm

m

1 + Jl-1.392

0.74-0.13-

V'm/ ,

0.696T ,(306)

T V m J \u.»oy r

/

1-Jl-1.392

N0.914

' T„ 1 T

I ^0.74-0.13-

0.696T„ (306) ,

1 + Jl-1.392

x 0.869 T

, 19 j ,

(307)

1 - J l - 1.392

N

T

T

0.74-0.13-

T

V

mJ

0.914

0.74-0.13-

V 1m J

0.696T . (307)

1 + Jl-1.392

0.74-0.13/

0.696T„ (307) f

l-Jl-1.392

\ 0.914

T

\ 0.869

T

0.74-0.13-

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Td

Ti

Minimum ISE - 0.787 fT ^ m Huang and Chao (1982).

20 ry (308)

Minimum ISE 21 K (309) Kaya and Scheib (1988). Model: Method 5 / \ 0.995 Minimum ITAE 0.684 - Huang and K m V X nJ Chao (1982). Minimum ITAE 23 K (311) Kaya and Scheib (1988). Model: Method 5

T

T x

(309)

22-p (310)

(311)

T

(308)

0
T

(310)

Model: Method 1; N=8

(311)

0<^2-
d

T N=10

m

N=10

\ 0.971

TH(308) =0.594T„

1.678T,

Model: Method 1; N=8

(309)

T

f 20

d

Comment

T

1

T

(308)

201

T V 'm J

2,

K

(309) _ 0.71959

T

K„

(309) _ _

1.12666-0.18145,0.86411

T <309) =0.54568Tm U * 22

T

(310)

(310)

0.114| -!=-| + 0.' 23

v

K

(311)

1.12762 fT m

= 0.491T„

vT \

m

0.80368 ,(311)

0.99783 +0.02860-^ T .(311)

Handbook of PI and PID Controller Tuning Rules

202

Rule

Kc

T;

Comment

Td

Preferred tuning; Minimum ITAE Witt and Waggoner (1990). Model: Method 2

24

K

(312) = 0-965

Km

24 K (312)

T(312)

T

(312)

25 K

T

(313)

T

(313)

0.1<-^-
fxm

(313)

1 + , 1 -1.232

I T„

Alternate tuning

i 0.796 -0.1465-JHT

0.616T, •(312) ^0.85

r

1 - , 1-1.232

X

0.796-0.1465-

T V 1m J

0.616T„ (312) ,

N0.85

1+ Jl-1.232

0.796-0.1465-^T

T V 1m J 25jc

f

,3,3) _ 0.965 K„

T

N 0.929 r

1 - J l -1.232 ^ 2 -

\ 0.796 - 0 . 1 4 6 5 ^

vT m J 0.616T„

.(313)

1

Tra

T V m

N0.85

f

1 + ,(1-1.232

T

0.796-0.1465-

0.616T„ (313)

(. \ V

1 - , 1-1.232

Tm 7 0.796-0.1465-

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

T

Kc(314,

203

Comment

Td

i

=

Minimum ITAE 0.85 - Ou and Chen 0.965 ( l ^ (1995).

26 T (314)

1.00 Minimum USE 0.718 f T 1 Huang and Chao X Km „ mJ (1982).

T x

27 T (315)

(314) d

(315)

T

Model: Method 46

Model: Method 1; N=8

Direct synthesis Tsang et al. (1993). Model: Method 15

N = 2.5

0.25xra

Coefficient values *i

1.6818 1.3829 1.1610 Tsang and Rad (1995). Model: Method 15 Smith and Corripio(1997)page 346. Model: Method 1

26

T

Kratm \ 0.0 0.1 0.2

*i

0.9916 0.8594 0.7542

0.809Tm

\ 0.3 0.4 0.5

x

\ 0.6 0.7 0.8

i

0.6693 0.6000 0.5429 0-5tm

Overshoot = 16%;N=5

T

0-5xra

Servo - 5% overshoot; N not specified

K ro ^ m

T(314)

-. T,

(314)

£ 0.9 1.0

x

T

Kmtm 0-5Tra

i

0.4957 0.4569

•• 0 . 3 0 8 T „

0.796-0.1465T N = ±-

0.75KmK (314)

(314) |

( 3 1 4 ) T (314) (314)

T

028 27

T(315)

T

0.063 ^

,(315)

+ 1.061

(314)

+ K m K c l J ""lV"+T i : 0.596T„.

, T

+T

N>0, '^m-TrnJ

Handbook of PI and PID Controller Tuning Rules

204 Rule Schaedel(1997) Model: Method 21

28 K

(316)

X

O'Dwyer (2001a). Model: Method 1

K

Comment

Ti

Td

T2 T,--i2T,

-y/T, 2 -2T 2 2

N

T

Kc

N not defined

Am =

lTm X

m .r

w/2x,N

;

K= 0.57c-x,N

Representative result 0.079T m K

m

T

Am=2.0; 0.1Tm

m

1 Wang et al. (2002). Model: Method 1

1.04432

29 j

(318)

T

(317)

(318)

x

( 0.375 T,

0.5Tm

4 ^

4 2r]

0-65 f - Q

28 K

K

= 45°

Tm

+

K i+

30 K

Tm

T

(317)

T

(318)

Labeled IMC; N=10 Labeled IAE setpoint; N=10 Labeled IAE load; N = 10

2A

(316)

Kn

N

T,

v

'

2T, J

T

T,

2

a

l - + LTmm>,T, 2 =1.25T m x m +

1 + 3.45T.=

°'7Tm

T m -0.7C

I i L _ T m + o . 5 T r a \ 0 < ^ < 0.104; 1 + 3.45^-

+Tm,T22=TmTam+_°-7Tm_ Tm-0.7C

+0.5xm

, 29^(317)

2

, j ^ > 0.104.

s 1.08433

(317)

, T d ,J " =0.50814 0.9895 + 0.09539^2T

30 K (3.8) _ 0,98089 K„

T /•

— I

, Ti

(318)

=1.09851Trt

>, 1.05211

, T , (318)

: 0.59974Tm

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Wang et al. (2002) continued.

T,

Td

31 K

(319)

-p (319)

T

32 K

(320)

ry (320) i

T 1

T

33

K

(321)

T

(321)

34

K

(322)

T

(322)

T 1

M

Huang et al. (2005). Model: Method 42

31

K

(319)

0.65Tm Kmxm

T

(320) d

(321)

(322) d

0.4xm

Comment Labeled ISE setpoint; N=10 Labeled ISE load; N = 10 Labeled ITAE setpoint; N=10 Labeled ITAE load; N = 10 N=20 Am=2.7,

m

*m=65° The above is a representative, default, tuning rule; other such rules may be deduced from a graph provided by the authors for other A m ,4)m values.

0.71959 ( T,

:

(319)

205

^•m

,(319)

_

v ' m ;

1.12666-0.18145f (319)

T

'd

: 0.54568

Kr

-

(320)

-

= ]

252QT '

T

V

N0.95480

32 „ (320) _ 1.H907

\ 0.86411

m J

Xn

K„ f

Td<320) = 0.54766T,,

\ 0.87798

T V

33

K

(321) m 1-12762

, T=

K„

(321)

0.9978 + 0.02860T x

, 34 ^

(322)

;

0.77902 T ^

x 1.06401

= 0.42844

0.70949

f (322) = 0 8 7 4 8 1 T

(321) d

N

Td(322) =0.57137Tm

mJ

Handbook of PI and PID Controller Tuning Rules

206 Rule

Kc

Harris and Tyreus(1987). Model: Method 1

Tm Km(xm+^)

Td

Comment

T

0-5xm

N = (xm+^)A; X > max imum [0.45xm)0.1Tm]

Tm

0.5x m

0.5x m

Tm

0.5xra

Tm

T

0.5x m

Ti

Robust

Tm Chien (1988). K ( X + 0.5T ) n n Model: Method 1 0.5xra K m ( ^ + 0.5x m ) Zhang et al. (1996). Model: Method 38 Zhang et al. (2002). Shi and Lee (2002). Model: Method 1 Lee and Shi (2002). Model: Method 1

35

K

(323) _ _

35 K

(323)

36 K

(324)

K

(324)

^e[0.2x m ,

m 37 K

T

(325)

Model: Method 1

0.5xra m

1 38 K

(326)

Tm

tan

m

»g

2

I

J

(327)

0-5^

40 j

N_Tm(2X

=

Km(0.5xm+2^)

0. " m

(328)

+ 0.5x m )

tf

, N=

0.5x m (2^ + 0.5x m )

Tm K N=10or N = K m ( 2 ^ + 0.5x ra ) 0.5x m (2X + 0.5x m ) "bvyTm

38 ^ (326)

zr, N = l +

2co

^gO v

l+^tanfe^

K„

-tan

2

< » „

39

,N =

J°bwTm

K (327)

v

1+ ^ t a n V

K„

40T(328)=JJ!L

^

1 +

TnL T

m

+

2

,

_ l _ m

bwtra

1 +

^

t a n

[^

Suggested 0.6 Suggested 0.6

T 39 g

K m (o.5x m+ 2\)' 36

(Chien and Fruehauf (1990)); N=10

j

Chapter 4: Tuning Rules for PID Controllers Rule

Td

Ti

Kc

207 Comment

Ultimate cycle Huang et al. (2005). Model: Method 42

41

K

(329)

(329)

T

(329)

N=20 Am=2.7,

The above is a representative, default, tuning rule; other such rules may be deduced from a graph provided by the authors for other A m ,<|>m values.

K„ 41 ^

T

-1

(329) _ 0-65

, T i l " ! " = 0 . 1 5 9 T u J Ku TI

-tan"

-1

K,

Td(329) = 0.064 T„ 7t-tan ' J K„

Handbook of PI and PID Controller Tuning Rules

208

1 + NT d s

4.1.9 Classical controller 2 G c (s) = K c 1

i yV

T s

E(s) R(s)

-K

1 + Tds j

U(s) 1 Yl + NTHs 1+XsA 1 + Tds

Y(s)

K„e'

l + sT„

K„,e~

Table 52: PID controller tuning rules - FOLPD model Gra (s) = Rule Hougen(1979)~ page 335. Model: Method 1

1

l + sT„

Comment T; Td Direct synthesis: frequency domain criteria Maximise 0.45^E(330) crossover T N frequency; N e [10,30]

Kc

1 jr

Values deduced from graph. v (330) (330) _ IT (330) n A/T^ Kc =9.0/K =4.2/K m ,x m /T m = 0.2; r a ,x m /T m =0.1;K C K c ( 3 3 0 ) =2.8/K m ,x m /T m =0.3;K c ( 3 3 0 » = 2.1/K m ,x m /T m =( (330) K/ 3 3 0 ) =1.7/K .,l^m,.mml) x,mr -^,^ =1.45/K m ,T m /T r a =0.6 a /T cm = 0.5;KC K

(330)

= 1.2/K r a ,x m /T m =0.7 ; K c ( 3 3 0 ) =l.l/K m ,x: m / T m = 0 . 8 ;

£

(330)

(330) = 0.95/K m ,T m /T m =0.9;K = 0.9;Kc c ( 3 3 0 ) =0.85/K m ,T m /T m =1.0.

Chapter 4: Tuning Rules for PID Controllers

209

4.1.10 Series controller (classical controller 3) (l + sTd)

Table 53: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

Kraus(1986). Model: Method 19 Tan et al. (\999a) - page 25. O'Dwyer (2001b).

0.833Tm

T;

Td Process reaction

K ra x m

l-5xm

0.25xm

*m

^m

2Tm

X3tm

K m e" l + sT„.

Comment Foxboro EXACT controller pretuning

0.6Tra K

mXm

xjm Kmxm

x

Model: Method 2

Model: Method 2

Representative results Chien et al. X x 2 (1952) equivalent * 1 3 - regulator 0.7236 1.8353 0.5447 0% overshoot; 0.1 < ^ m / T m < l 0.84 1.4 0.6 20% overshoot; 0.1 < i m / T m < 1 O'Dwyer (2001b). Model: Method 2

K

(331)

Representatii/e results - Chien et al. (1952) equh alent - servo (331) (331) 0% overshoot; T (331) K T T m /T m <0.5

2

0.3T„

1-J1

2x„

.(331)

1+ Jl-

2x„

Handbook of PI and PID Controller Tuning Rules

210 Rule

Kc

O'Dwyer (2001b)continued.

3K

T

(332)

Comment

Td

Ti

(332)

(332)

T 1

d

20% overshoot; * m / T m < 0.7234

Direct synthesis Tsang et al. (1993). Model: Method 15

xjra

T

0.25xm m

l

S

1.8194 1.5039 1.2690

0.0 0.1 0.2

x

Pessen (1994). Model: Method 1

x \ i 1.0894 0.3 0.7482 0.4 0.9492 0.6756 0.8378 0.5 0.6170 Ultimate cycle

0.35KU

x

i

0.25TU

% 0.6 0.7 0.8

0.25TU

i

I

0.5709 0.5413

0.9 1.0

x

0.1<^s-
3 j , (332) _ 0 . 4 7 5 T ,

Kmtm

1+ 1-1.3824-

T;(332) =0.68T„ 1 + . 1-1.3824-

Td(332) = 0.68Tn 1 - 1 - 1 . 3 8 2 4 -

Chapter 4: Tuning Rules for PID Controllers

1+

4.1.11 Classical controller 4 G c (s) = Kt

211

- ^ L

1+

^

;

N

Table 54: PID controller tuning rules - FOLPD model Rule

Kc

Hang et al. (1993b) -page 76. Model: Method 2

0.83Tm K

T

m m

Td Tf Process reaction 0.25xm

l-5Tm

K„e~

l + sTm Comment Foxboro EXACT controller 'pretune'; N not specified

Robust Chien (1988). Model: Method 1

^ 6 [ T m. T m]

Tm Km(X + 0.5Tm)

Tm

0.5tm

0.5xm Km(X + 0.5xm)

0.5xm

Tra

(Chien and Fruehauf (1990)); N=10

212

Handbook of PI and PID Controller Tuning Rules

4.1.12 Non-interacting controller 1 U(s) = K,

E(s)—^-Y(s) v

T s

i y

N R(s)

U(s)

E(s) K,

'.+-0 — • V

T

is/

Y(s) —•

Ke~ 1 + sT

Tds T 1 + ^-s N

Table 55: PID controller tuning rules - FOLPD model G m (s) =

Rule

Kc

Huang et al. (2000). Model: Method I

0.8(T m+ 0.4x m )

T

Kmxm 0.6(T m+ 0.4T m ) ""m

m

i

Td Direct synthesis

T m +0.4x m

0-4Tmxm T m +0.4T m

K m e^ l + sT„

Comment N= min[20,20Td]; regulator tuning N= min[20,20Td]; servo tuning

Chapter 4: Tuning Rules for PID Controllers

213

4.1.13 Non-interacting controller 2a

E(s)

U(s) = K T s

v

i y

%-Y(s) 1+N

U(s) R(s)

/ C >

Y(s)

E(s)

^+-

K,

1 + sT

Tds T N

Table 56: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc /

K„e"

l + sT„.

Comment T( Td Minimum performance index: servo tuning \ 0.887

1.260 Minimum ISE Km Zhuang and Atherton(1993). 0.619 Model: Method 1 1.295 f T j

—]

4T(333) i

T

(333)

5

T

(334)

T

(334)

0.1<^a-<1.0 T 1.1 < ^ 2 - < 2.0 T m

/ x 0.930 Minimum ISTSE - Zhuang and 1.053 f T r a j Atherton(1993). K [ x j m

4

T

(333) _ _

6^(335)

T X

T

(. 1 VT m

^

,N=10.

j

-,T d <334) =0.378T m ' T "

, N=10.

0.661-0.110/

6 j(335)

, Td<335) = 0.349T,, 0.736-0.126-

0.1 < ^ S - < 1.0 ; T m

, Td(333) = 0.375T„

(334) _ _

d

Model: Method 1

0.701-0.1475

(335)

N0.907

X,

T

,N=10.

Handbook of PI and PID Controller Tuning Rules

214

Rule

Kc

Td

Ti

Minimum ISTSE 1.120 - Zhuang and Atherton (1993) (continued)

Comment

0.625

1.1 < ^ 2 - < 2.0; T N=10

7 T (336)

T

(336)

8j(337) i

T

(337)

0.1 <-^=-< 1.0 T

9T(338) i

T l

(338)

1.1 < ^ 2 - < 2.0 T

0.933

0.942 (T ^ Minimum ISTES \Tm J - Zhuang and Atherton (1993). r N 0.624 Model: Method I 1.001

Minimum ISTSE - Zhuang and Atherton (1993).

10 £

d

0.112TU

•p(339)

(339)

0.1<^2-<2.0; T m

Model: Method 1 11

K

(340)

0.271K u TuK m

0.116TU

0.1 <-^2-< 2.0; T m

Model: Method 31

7 j

(336)

, Td<336) = 0.350T„ 0.720 -0.114^2T : 0.308T„

,

m

(338) T d l "°'

= 0.308Tm \—\

m

34o)

=

N=10.

T V ni J

0.754 -0.116^2T

llKc(

N 0.897 X„

0.813

T

(338)

m J

, (337)

0.770 -0.130^2T 9 j

T V

T

• (337)

^

'x.

2.354KmKu-0.696£u)N=10 3.363KmKu + 0.517

, N=10.

Chapter 4: Tuning Rules for PID Controllers

215

4.1.14 Non-interacting controller 2b U(s) = Kc +

E(s)-^!_Y(s) N Y(s)

R(s) E(s)

+ o

*®

Xs

U(s)

>

K„e" 1 + sT

T„s T N

Table 57: PID controller tuning rules - FOLPD model Rule

Kc

Minimum IAE Kaya and Scheib (1988). Minimum ISE Kaya and Scheib (1988).

12 K

12

13

Comment T( Td Minimum performance index: regulator tuning

(341)

K

(341)

T

(342)

K

l + sTm

T i

(342)

T

x(3"l) _

''

K„

Model: Method 5

X <342>

i

(341) _ 1-31509

(341) d

'm

1.2587 T ^ 0.4576

(341)

T / " ' =0.5655T m | —2-

0 < -SL < 1 ; N=10. T

13 K (342) _ 1.3466 [ Tra K

m

T

,0.9308

V mJ

.

T

(342) i

1.6585 T f . (342) Td^z'=

0.79715T„

x0.41941

T

1

V mJ

0<-!S-
Handbook of PI and PID Controller Tuning Rules

216 Rule

Kc

Minimum ITAE Kaya and Scheib (1988).

14

K

(343)

T 1

Comment

Td

T, (343)

(343)

T

i

Model: Method 5

Minimum performance index: servo tuning Minimum IAE Kaya and Scheib (1988). Minimum ISE Kaya and Scheib (1988). Minimum ITAE Kaya and Scheib (1988).

14

K

(343)

15

K

(344)

T(344)

16

K

(345)

T

(345)

17

K

(346)

1.3176

T

(346)

T x

(346) d

1.12499 [T m

1.13031 I Tm m

Model: Method 5

- (343)

Td(343) = 0.49547T„

K

(345)

T l

d

K„

15 K (344)

(344)

T

0<-^-
, T

• (344) _ _

T

l m/

5.7527-5.7241/•

\ 0.17707

(344)

, 0<-S-
T V

1.26239 (l„ .Tj

m /

(345)

6.0356-6.0191T d ( 3 4 5 ) =0.47617T m ;^ ,7 „ (346)

0.98384 (Tm~) K„

lTm,

T-

0<-^-
(346) _ _

2.71348-2.29778-

T

N

• (346)

0.21443T,

0.16768

0<-a-
Chapter 4: Tuning Rules for PID Controllers

217

4.1.15 Non-interacting controller based on the two degree of freedom structure 1 f

\ 1

T„s

U(s) = Kc 1+ — + -

N

v (

f

\

E(s)-K c a +

PT d s 1+ ^ s N

J

\ PTds K a+ T 1+ ^ s N ; Y(s)

U(s)

-V& +

1

T„s

Ts

l+ is

K, 1 + — +

'

E(s)

R(s)

R(s)

*g>—•

K^e" 1 + sT

N ;

Table 58: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

Hiroi and Terauchi (1986) - regulator. Model: Method 2; a = 0.6; (3 = 1

0.95T m

T,

Td

Kme" l + sT m Comment

Process reaction T

Km m

0% overshoot; 2.38xm

0.42t m

0.1<^-<1 T m

20% overshoot;

1.2Tm K

mTm

2tm

0.42t m

0.1<-^=-
Handbook of PI and PID Controller Tuning Rules

218 Rule Taguchi and Araki (2000).

1

Shen (2002). Model: Method 5; N=0, (3 = 0, Ms<2

Comment Kc Ti Td Minimum performance index: Servo/regulator tuning (347) £ (347) Overshoot (servo T T (347) 1 i d step) < 20% Minimum performance index: other tuning oo

oo

Minimum f|r(t) - y(t)|dt + f|d(t) - y(t)|dt 0 2

K

(348)

T

0

(348)

T

(348)

0
T

1.224

1 v (347) _ _J_ 0.1415 + K,

-0.001582

•(347)

0.01353+ 2.200-2--1.452 T

+ 0.4824 2~\

(347)

0.0002783 + 0.4119—E-- 0.04943 T

a = 0.6656 - 0.2786-2- + 0.03966 T

, N fixed (but not specified);

P = 0.6816 -0.2054-J2- + 0.03936 T 2

K

(348) K

.(348)

mTm

exp 2.94-11.63

T

+T

+ 11.15 V T m+ T r a f

: xm exp 1.88-3.63 V ^ m "•"

(348)

•< 1.0. Model: Method 1.

+ 0.86 T V m+T r a y ^m J

:x

m exp •0.25-0.06

V T m+T m y

a = 1 - exp -0.22-0.90 ^ +

T

-1.99 (,

1

m

+ 1.45 m ;

*m

+ T

m/

+

'ni;

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Kasahara et al. (1997). Model: Method 1 Huang et al. (2000). Model: Method 1 Srinivas and Chidambaram (2001). Kim et al. (2004). Model: Method 1

3

K

3

•p. (349)

^T m +0.4T t t Kmxm

(349)

0-4Tmtra T m +0.4x m

Closed loop transfer function is 4th order; N = 0 For X = 0.80 , A r a =2.13

1, P = 0 , N = min[20,20Tdl. (350)

T

T

Model: Method 1;

(350)

N = oo

x,/K m Tm/Tm

X]

0.1 0.2 0.3

12.5 6.1 4.1

0.2xm3 + 1.2x

(349)

^

(350)

K

T

T + 0.4xm

T

a= 4

Comment

Td Direct synthesis

Ti

(349)

K

219

2

x x2Tm 3Tm Coefficient values x2 x3 0.22 0.04 0.41 0.08 0.57 0.11

T + 5 . 4 T T 2 + 9.6T

a

P

0.68 0.63 0.62

0.75 0.70 0.70

3

T m (x m +3T m ) 2

K.

frm+3Tm)3

Ti(349)=1.667xmi!i+3T -1.389T„ x m + 2 T m - " •-

(xm+2Tm)4

Tm2Tm(Tm+3Tj

(349)

1.2(xm+2Tra)3-xm(xm+3Tm)2' ct = l-0.723x„

P = l-0.261x r 'Sample K c

<350)

^ m +3T m V^+2Tm

1.667x„

*m+3T m ) 1 „ „ n T m 2 ( x m + 3 T m ) 3 -1.3891%,+2T„ (x m +2T m ) 4 1

T „ +2T„

, j.*

350

)^*

X

350

( 3 4 9 ) T (349) •

' taken by the authors correspond to the process reaction

tuning rules of Zeigler and Nichols (1942); a =

and P = •

T„ (350)

K„

<350,

T „

Km

2T:

(350)

T(350)

—-4

= 0.5-0.83^

= -0.16 for this tuning rule

Handbook of PI and PID Controller Tuning Rules

220 Rule

Kc T

m/Tm

Comment

Td T, Coefficient values x x2 3

*1

a

P

0.4 3.1 0.71 0.15 0.59 0.69 0.5 2.5 0.18 0.58 0.83 0.69 Kim et al. (2004) 0.6 2.11 0.94 0.21 0.56 0.69 - continued 0.24 0.7 1.82 0.54 1.05 0.69 0.8 1.61 1.13 0.28 0.51 0.68 0.9 1.44 1.22 0.31 0.48 0.69 1.0 0.34 1.33 1.26 0.49 0.66 OS < 20% (servo step i; settling time < that of corresponding rule of (^hien et al. 1952); a cost function is minimized. Robust (351) 5 <35 ) Leva and Model: T (351) T K Colombo (2004). Method 41 Ultimate cycle 6 o.6K, 0.5TU 0.1251u Model: 7 ry (352) 0.6K i Method 2 0.125T u Hang and Astrom (1988a).

5

K

(351)

K m (x m +X) (351)

=

Tm +

2(Tm+X)

V . l ' , ^ ;

2T m (T r a +X)+T m

T- (351) = T m +

"» 2

3AT

a not specified; (3 = 1. A, >

2(xm+^)' N

_

2(Tm+X)'

2T

m(tm + ^f

}

x[2Tm{xm+X)+xm2}

— , Ai m = maximum variation in the time delay, with

only variation in the time delay considered (Leva and Colombo (2004)); X chosen so that nominal cut-off frequency = <x>u (Leva, 2005). a = l-2(x-0.l)-1,66Tm , is-<0.3; a = l - 2 x - ^ - ,

0.3<^-<0.6;

x = % overshoot. N=10, P = 1. 7

T(352)

0.5 1.5-0.83-

T„. a = - ^ - - 0 . 6 , 0.6 < ^ 2 - < 0.8; T T a = 0.2, 0 . 8 < - S - < 1 . 0 . T

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Hang and Astrom (1988a)continued Hang et al. (1991) - servo. Hang and Cao (1996). Model: Method 31

i

Td

Comment

0.6K U

0.335TU

0.125TU

-^2->1.0; T

0.6K„ 9 0.6K U

0.5TH

0.125TU

0.222KT U

0.125TU

a = 0.2 Model: Method 2; (5 = 1

T

8

0.6K U

10 y (353)

0.1<^L<0.5;

T

N=10; (5 = 1

r,

10% overshoot;

15 + K

5 i _ , 20% overshoot with K = 2\ 1 1 ^ - / ^ - " J + 1 3 27 + 5K l37[Tm/Tm]-4/

a =1

10

(353)

T 1

d

0 . 1 6 < ^ - < 0 . 5 7 ; N = 10. a = l Tm

9

221

0.57 < - ^ s - < 0.96 ; N =10. ce = l - — f - K ' + l 1, 20% overshoot, 10% undershoot. 1719 T.(353) =1 0.53-0.22

T

T U >

2

X

(

(353)

l T

0 . 5 3 - 0 . 2 2 — — . a takes on values of L 4

1.0, 0.2 and 0.40(x m /T m ) -0.05(x m /T m )+0.58 at different points on the open loop process step response.

Handbook ofPI and PID Controller Tuning Rules

222

4.1.16 Non-interacting structure 2

controller based on the two degree of freedom

f

\

f

E(s)-K£ U(s) = K c 1+ — + - a X s 1, +. T^d s V N ; (

\ a +

PTds

X

Td„ 1+ ^ s N

l + T iS

\

pTds

a +l+ ^ s v N

x__

K„

1+ T S

*

Y(s)

U(s) K„

+ R(s)

T

l

,

1+ — T8 +-

.

E(s)

l+

K e" l + sTm

ds

d

i8

+

N .

Table 59: PID tuning rules - FOLPD model Gm (s) = Rule

Kc

K

T

m m

l + sT„

Comment

Td Process reaction

0% overshoot; 2.38tm

0.42xm

2t m

0.42xm

l-2Tm Kmxm

K m e-

Ti

0.95Tm Hiroi and Terauchi (1986) regulator. Model: Method 2

R(s)

0.1<^H-<1 T m 20% overshoot;

a = 0.6; p = l or (3 = -1.48 ; x == 0.15

0.1<^-<1 T

Chapter 4: Tuning Rules for PID Controllers

223

4.1.17 Non-interacting controller based on the two degree of freedom structure 3 U(s) = K c [ ( b - l ) + (c-l)T d s]R(s) + K (

1

R(s)-

Xs

(l + s T f ) :

-Y(s)

K c [(b-l) + (c-l)T d s] +

ir

Y(s)

U(s) K e" 1 + sT

(l + sT f ) 2 Table 60: PID controller tuning rules - FOLPD model G m (s): Rule 11

Astrom and Hagglund (2004).

12 j r

11

K

(354)

=

Comment Td Minimum performance index: other tuning (354) Model: T (354) 0-5xmTm i Method 1 °-3*m +T m No precisely defined performance specification (355) (355) Model: T (355) T Method 45 No precisely defined performance specification

Kc

Astrom and Hagglund (2004).

_

K

K m e" l + sT„

Ti

0.2 + 0.45-

Xm+0.1Tm

K„

m

T

b = l , - 2 L > l ; c = 0; Tf = 0.1xn 12

K

(355)

=

_ K„

0.2 + 0.45 - T - + 0 - 5 r ' T m +0.5T f

, b = l , c = 0. Tf =-

355) =

0.4x m + 0.8T m + 0.6T f x m +0.1T m +0.55Tf 355) _ 0.5(T T< m +0.5T f )(T m +0.5Tf) . xm 0.3x m +T m +0.65T f ' Tra

? }

, (355) d

N

,Ne[4,10];

224

Handbook of PI and PID Controller Tuning Rules

4.1.18 Non-interacting controller 4 1

E(s)-K c T d sY(s) U(s) = Kc 1 + v T>sv R(s)

U(s)

E(s)

+ X

K, v

T s

. y

—+®~2—-

Y(s)

K„e~

1 + sT

KJds

Table 61: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

VanDoren (1998).

l-5Tm Kmxm

ABB (2001). Model: Method 2

1.25Tm

T, Td Process reaction 2.5xra 0-4xm

K m e" l + sT„

Comment

Model: Method 2

0-5Tm 2^m K ra x m Minimum performance index: regulator tuning Minimum IAE Tm/Tm=0.1; Shinskey (1996) 1.32Tm/Kraxra 1.80xm 0.44xm Model: Method 1 - page 117. l-32T ra /K ra t ra l-77t m 0.41xm xm/Tm=0.2 Minimum IAE 0.41xm im/Tm=0.5 l-43xm Shinskey (1988) 1.35Tm/KmTm -page 143. 1.17x 0.37xm 1.49T m /K m i n m ^m/Tm=l Model: Method 1 l-82T m /K ra t m 0.92xm 0.32xm ^ra/Tm=2 Minimum IAE - 1.30Tm/KmTm Model: Method 1 0.45xra 1.8Tm Edgar et al. Model: Method 2 l-33T m /K m t m 0.63xm 2.1xm (1997) -pages 8-14, 8-15. Minimum performance index: other tuning 0.77KU 0.48T„ ^m/Tm=0.2 0.11T„ Minimum IAE 0.70KU 0.42TU 0.12TU x m /T m =0.5 Shinskey (1988) -page 148. 0.66KU 0.38TU 0.12TU xm/Tm=l Model: Method 1 0.60KU 0.34TU 0.12TU xm/Tm=2

Chapter 4: Tuning Rules for PID

Rule

Kc

T;

Shinskey (1994) -page 167.

1K

(356)

ABB (2001).

2j.

(357)

T

(357)

Haeri (2002)

3K

(358)

T

(358)

225

Controllers

Comment

Td 0.12TU

T2 0.125-^-

T

Minimum IAE; Model: Method I Model: Method 7

(357)

Direct s vnthesis

1

K„

(356)

K

T

Model: Method 1

(357)

i

_

>

T„

K„

K c <356)

i < 2 . 7 ; T „

3.73- 0.69-i

±->2.1.

2.62-0.35-^

T

X

m

*m 0.738

2

v

Kr

(357)

1.3570 (Tm

=

T/ 3 5 7 ' =1.1760T„ ^

T N 0.995

(357)

3

(358)

K

6.84

_J_

=

1 + 2.33

:0.3810TJ-^-

,^->0.5.

- + 0.64 •7.82j

T

,(358) _ ,

TT

n

0.29 < — T

<4;

- 4.65

+ 3.94

0.29 ^m """ T m

(358) __

m J

2.15 - 0 . 7 6 ^ 2 - + 0 . 3 3 - ^ Tm T„.

(358)

d

0<-^-<0.29, T

0.95+ 2 . 5 8 - S - +3.57 T T V

T1

\2

(

(358) _

T

T

VTm+^

0.87| -^2- - 0 . 4 9 ^ - 1

+0.09

m

>T

+tm

,0<-2L<1.79; T

1.79<-2-<4 . T

226

Handbook of PI and PID Controller Tuning Rules

4.1.19 Non-interacting controller 5 b + — E(s)-(c + Tds)Y(s)

U(s) = K

T s

v

i y

*-(b-l)Kc

U(s) R(s)

E(s)

>®

+

K,

+ T

v

T s

i v

Y(s)

K„e"

l + sT„

(c + T d s)

I

K„,e~

Table 62: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

T|

l + sT„

Comment

Td

Direct synthesis Chidambaram (2000a). Model: Method I

4K

K

(359)

mTra

(359)

0.15xm2 K ra T m

0-5Tm

c=K c ( 3 5 9 > c-

r \2 0.311 + 1.372^2--0.545 T T

b = 0.793-3.149^L + 8 . 4 0 5 ^ TA T m

K

16.016-11.7

-8.431 ,T

•<0.5,

V mJ

b = 0.0584 + 0.44^2. T

o.5 < ^ - < o.9 . T

b = 0.3238 + 0.4332^- Im. < 0 .8 ; b = 0.8507 , ^ - = 0.9 , T T T m

m

(359)T(359)

(359)

m J

J

m

1.2Tra K

4.114-6.575-J2- + 3.342 T T

V

5

T

'O*

=

K„

,(359)

T (359)

5 l-2xm K

4

(359)

mTm

Chapter 4: Tuning Rules for PID Controllers

227

4.1.20 Non-interacting controller 6 (I-PD controller) U(s) = ^ E ( s ) - K c ( l + Tds)Y(s) T:S R(s)

-J®

E(s)

U(s)

K.

Y(s)

K m e"

Xs

1 + sT K c ( l + T d s)

Z3Z Table 63: PID controller tuning rules - FOLPD model G m (s) = Rule

Comment Td Direct synthesis: time domain criteria

Kc 6

R

Ti

(360)

rj, (360)

(360)

T

*d

i

Suyama(1993).

K m e" l + sT„

Closed loop transfer function is 4lh order

Model: Method 1

(361) 7 j r (361) Shigemasa et al. T (361) T *d (1987), Model: Method 1 th Suyama (1993). Closed loop transfer function is 4 order Butterworth

6

K

0.2xm3 + 1.2xm2Tra + 5.4xmTm2 +9.6T m 3

(360)

^m(^ m +3T r a ) 2

K„

, (360)

: 1 . 6 6 7mt m T m + 3 T m - 1 . 3 8 9 x m 2 j T m + 3 T m j 4' xm+2Tm (xm+2Tm) Tm2tm(tm+3Tm)

(360)

1.2(x m +2T m ) 3 -x m (x m +3T r a ) 2 7

K

(361)

1.3319(x m+ 2T m ) 3 K m x m (x m +3T m )

2

T(36l)=12532

1 Km ' xm+3Tm _ L 6 9 ^ T..+2T,

(361)

:

tm(xra+3Tm)

2(xm+3Tm)

™ (Tm+2Tmr

1.5095(xm +2T m ) 2 -(x r a +T m Xt m +3T m ) 1.3319(x r a +2T m ) 3 -x m (x m + 3T„

Handbook of PI and PID Controller Tuning Rules

228 Rule

Kc 8

K

Ti

(362)

(362)

T

Comment

Td T

(362)

*d

Shigemasa et al. (1987), Suyama (1993) (continued). Model: Method 1

Closed loop transfer function is 4th order ITAE 9

K

(363)

(363)

T

T

(363)

i

Toshiba AdTune TOSDIC211D8 product

Closed loop transfer function is 4th order Bessel 10 K

(364)

T(364)

T

(364)

th

Closed loop transfer function is 4 order Binomial

K

(362)

2.6242(x m +2T m ) 3

1

2

Kr

Kmtra(Tm+3Tra)

T i ( 3 6 2 ) = .8898x 1 . 8 8 9m8Txmm+i3^T^m -0.720x m ^ rTa + 2J-9T Tm ' T X

K

(363)

(362)

_T

(z

+ Tr

^.3130(x m

d

2

ym+yTm]t , (xm+2Tm)4

+2Tm)

2

-(xm+TmXxm+3Tm)

2.6242(xm+2Tra)3-xm(Tm+3Tm)2

0.9450(x m + 2T m ) 3 Kmxro(xm+3Tj

1

2

Km

Tm+3Tm 2 T1(363) = 3.3333x 333xm 1X12M._ 3.527xm

T

T(363)

/

+2T

,_

.(Tm+fm|]

,

U . 3 4 4 4 ( x m +2 2TTmm) )2 2- -( (xxmm++T1m X x m + 3 T m ) 1.9450(x m +2T m ) 3 -x mm(x O m +3T m ) 2

K

(364)

_0.5622(xm+2Tj3 Kmxra(xm+3Tj

1

2

Km r

T/ 364 ' = 5.3337xm

Tm+3Tm - 9.488xm2 ( T " + 3 T " ) 3 Ill / x m + 2 ."IT-. Tm ( x m + 2_ T m ) 4 ' ^m+2Tm (x m +2T, (T + 3T ^ • 1 2 4 4 ( x m + 2 T m ) 2 - ( x m + T m X T m + 3 T r 0.5622(x r a +2T m ) 3 -x m (x m + 3T m ) 2 m 111

1

T

(364)

T

Chapter 4: Tuning Rules for PID Controllers

229

Rule

Kc

T,

Chien et al. (1999). Kasahara et al. (2001). Model: Method 1

11 K (365)

T(365)

T

(365)

Model: Method 1

12 j r (366)

ry (366) i

T

(366)

OS = 0%

(367)

OS = 10%

Ozawa et al. (2003). Minimum ISE Argelaguet et al. (2000).

13

K

(367)

(367)

T

Comment

Td

T 1

i

d

Model: Method I OS e[0%,10%] Minimum performance index: servo tuning 2T TX m + *m Model: Method 1 T m +0.5x m 2Tm+tm 2K t T(368)

14 K (368)

T

(368)

i

ra m

Underdamped system response - Z, = 0.707 . xm > 0.2T,,, K

1.414T C L 2 T m +T m T m + 0.25T m 2 -T c

(365)

" ^

2

2 2 X.,, CL2 +0.707T " ' Cr ,,T L2 l m m+0.25T m

1.414T +xl m- mT1mm+0.25x m z ^ Cri L,T 2 T m "T

(365)

T m +0.5x m (365)

_ 0.707TroTCL2Tro +0.25T m x m 2 -0.5x m T CL2 2 Tmxm +0.25x m 2 +1.414TCL2Tm - T C L 2 2

1 2 K ( 3 6 6 )

_

0.28l(x m +2T m ) 3

1

Kmxm(xm+3Tm)2

Kn

, (366)

:5.33x„

(366)

13

K

(367)

:

tm+3T„

-18.98x n -'

tm+2Tm

,0.562(xm + 2 T m ) 2 - ( x m + T m X x m + 3 T m ) tm(^m+3T m )0.28l(x m +2T r a ) 3 -x m (x r a +3T n l

1.066Tm -0.467x n

•(367) _

4.695x m (1.066T m -0.467x m ) x m +2T m T i

14

v

K„

(x m +3T m ) 3 (xm+2Tm)4'

d

(367) _ ~~ ln

0.331T m -0.334x n 1.066T„ -0.467x„

(368) .

1.6193(x m +2T m ) 3

1

2

K m x m (x m +3T m ) Km ' (x m +3T m , (368) = 2.7051x„ xm + 3Tm •1.6706x„ xm+2Tm -•" ( x m + 2 T m ) 4 ' T <368> _ T

IT

J-TT )

1.7793(xm + 2 T m ) 2 - ( x m + T m Xx r o +3T m ) id

"TmlTm

mj

1.6193(x m+ 2Tj 3 -x m (x m+ 3Tj 2

'

230

Handbook of PI and PID Controller Tuning Rules

4.1.21 Non-interacting controller 7 T

U(s) = - ^ E ( s ) - K t 1 + T,s

N

R(s) E(s)

K

U(S)

+ ^

l + sTm

Table 64: PID controller tuning rules - FOLPD model Rule 15

Ti

(369)

K

l + sT„

Comment Td Direct synthesis: time domain criteria (369) T (369) T N=10

Kc

Ogawa and Katayama (2001).

Y(s)

K m e"

TiS

i

Minimum ISE - servo. Model: Method 1

e

15

Desired closed loop response =

(l + sTCL)

2

K

1

T„

~K„

X

(369)

T CL

.

T V


Y(s)

-

T

X

T CL

f

2

V T*m

T

CL

v

T

^CL

T ra

x

m

J

lck_2^k +4

Tm A l

CL

+4

V (

f

T

Tm

(369) _ .

• T

T

V

m J

^ e

[0.01,1.00];

IcL__2lck + 4 T T

- ^ = -0.1902 T T

- 0.6974 -^SL + 0.007393 , - ^ e fo.05,1.00l. 1 J Tm T '

Chapter 4: Tuning Rules for PID Controllers

231

4.1.22 Non-interacting controller 11 >

U(s) = Kc

E(s)-K 1 (l + Tdls)Y(s) v

R(s)

E(s)

Kc(l+ —1 + ! »

T

is

j

U(s)

+ x-> _±*(g)

„

Y(s)

Ke" l + sTm

KiG + T^s)

Table 65: PID controller tuning rules - FOLPD model G m (s) = Rule

Kc

T,

K m e" l + sT„ Comment

Td Robust

Lee and Edgar (2002). Model: Method I

Ki =0.25K U , 16 jr (370)

T

(370)

l + 0.25K u K m T m -0.25K u K m x m l + 0.25KuKra K ra (^ + t m )

16 K (370) T(370)

T

|

(370)

Tdi=0

xn 2{X + xn

Tm-0.25KuKmxm | x — uKm + 2(X + xm)' l + 0.25K

1;

=

T^10)

_ l + 0.25K u K m 0.25K u K m x m 2 l Km(X + Tm) 2(l + 0.25K u K m )

3

6(^ + T J

4

4{X + xm)2 , xm2(Tm-0.25KuKmim)

2(l + 0.25KuKmX^ + O '

232

Handbook of PI and PID Controller Tuning Rules

4.1.23 Non-interacting controller 12 f

U(s) = K{ R(s) +

1 E(s)-K,Y(s) Tds + 1

T,s

E(s) ii

Table 66: PID tuning rules - FOLPD model G m (s) = Rule

Kc

Td

T,

K m e" l + sT„ Comment

Robust Lee and Edgar (2002). Model: Method 1

17 K

(371)

T

(371)

T x

n £ (371) _ l + O^KyKn, f T m -0.25K ll K ra T m K m (^ + Tm) l + 0.25KuKm 0.25KuKm 2(l + 0.25KuKra T

im 6(^ + Tm)

|

(371) d

K, =0.25KU

2(^ + x m )

xm2 4(^ + T m ) 2

|

(T m -0.25K u K m T m )T m 2 2(l + 0.25KuKm)(X + x m )

2

(371) _ Tm ~ 0 - 2 5 K u K m T m

l + 0.25KuKm

2(X + Tm)

0.25KuKm 2(l + 0.25KuKm

6(X + xJ

4{X + xm)2

0.25KuKm _2(l + 0.25K„Km

6(X + xm)

4(X + xm)2

Tm

T„

(T m -0.25K u K m T r a )i n 2(1+ 0.25K u K m )(^ + x m )

(371

(T r a -0.25K l l K m T m )x n 2(1+ 0.25K u K ra )(^ + Tm)

Chapter 4: Tuning Rules for PID Controllers

4.1.24 Industrial

controller

1 +T s

1+

U(s) = K

233

T,s

R(s)--^^Y(s) 1+ ^ N

U(s)

R(s)

Y(s)

K e~

K, 1 + XS

l + sT„

1 + Tds 1+ ^ s N Table 67: PID controller tuning rules - FOLPD model G m (s) = Rule

Kaya and Scheib (1988). Model: Method 5

Kc

x

i

Kaya and Scheib (1988). Model: Method 5

Minimum IAE Minimum ISE Minimum ITAE

l + sT„

Comment Td Minimum performance index: regulator tuning Ti

M1

T f

,0

X

3

\

x

x6

4

x

T

sT m

T

0<^-
N=10

Coefficient values X X X x4 i 3 5 6 0.91 1.00403 0.7938 1.01495 0.5414 0.7848 1.1147 0.8992 0.9324 0.8753 0.91107 0.56508 0.7058 0.99138 0.8872 1.03326 0.971 0.60006 Minimum performance index: servo tuning 2 "6 Tm x, TT V 0<-^-
Minimum IAE Minimum ISE Minimum ITAE

K m e^

x2

(O

0.81699 1.1427 0.8326

1.004 0.9365 0.7607

1.09112 0.99223 1.00268

0.22387 0.35269 0.00854

0.44278 0.35308 0.44243

0.97186 0.78088 1.11499

Handbook of PI and PID Controller Tuning Rules

234 Rule

Kc

Td

Comment

T

0.5tm

X > max imum [0.8i m ,T m ];N not specified

T

0.5xm

T

0.5xra

T, Robust

Harris and Tm Tyreus (1987). K ( 0 . 5 t ^ ) m m+ Model: Method 1 18

Hoetal. (2001). Model: Method 1

K

(372)

N not specified 1.551Tm K

-mAmTm

cog = frequency at which the Nyquist curve has a magnitude of unity.

Chapter 4: Tuning Rules for PID Controllers

235

4.2 Non-Model Specific

4.2.1 Ideal controller G c (s) = K£

R(s)

+

0

E(s)

A

K. 1 + — + THs T,s V )

U(s)

Y(s)

Non-model specific

Table 68: PID controller tuning rules - non-model specific Rule

Kc

Ziegler and Nichols (1942).

[0.6KU,KJ

Farrington [0.33K U ,0.5KJ (1950). McAvoy and 0.54KU Johnson (1967). Atkinson and 0.25KU Davey(1968). Carr (1986); Pettit and Carr (1987). Parr (1989)pages 190,191,193.

Ku

T,

Ta Ultimate cycle 0.5TU 0.125TU Tu

[0.1T U ,0.25TJ

T

0.2TU

0.75T„

0.25TU

20% overshoot - servo response 0.5T, 0.125TU

Comment Quarter decay ratio

Underdamped

0.6667 Ku

Tu

0.167TU

0.5KU

1.5TU

0.167TU

Critically damped Overdamped

0.5K„

T

0.2TU

£ =0.45

0.5KU

T

0.25TU

4

U

0.5KU

0.34TU

0.08TU

Tinham (1989).

0.4444KU

0.6TU

0.19TU

Blickley(1990).

0.5KU

Tu

[0.125TU, 0.167TJ

Less than quarter decay ratio response Quarter decay ratio

Handbook of PI and PID Controller Tuning Rules

236 Rule

Kc

Corripio(1990)page 27.

0.75KU Ku cos<|>m

Astrom (1982).

T: 0.63TU 1

T

(373)

Td

Comment

0.1T„

Quarter decay ratio

aT, (373)

0.87KU

Representative results 0.55TU 0.14TU

*m=30°

0.71KU

0.77TU

0.19TU

lm=45°

0.50KU

1.19TU

0.30TU

K = 60°

T Astrom and Hagglund (1984).

Ara Kucos<j>m

arbitrary

47l2T; 2

aT d <374)

T

Specify A m

(374)

Specify <j>m

Hang and Astrom (1988b).

Kusin<|)m

T u (l-cos<|) m )

T u (l-cos<|) ra ) 47tsin(j>m

Astrom and Hagglund (1988) - page 60.

0.35KU

0.77TU

0.19TU

0.4698KU

0.4546TU

0.1136TU

0.1988KU

1.2308TU

0.3077TU

0.2015K„

0.7878TU

0.1970TU

0.906KU

0.5TU

0.125TU

4>m = 25°

0.866KU

0.5TU

0.125TU

4>m = 30°

0.5KU

0.5TU

0.125TU

0.3536KU

0.1592TU

0.0398T;

Am>2, *m^450 A m = 2,

Astrom and Hagglund (1995) -page 141-142.


DePaor(1993). McMillan (1994) - page 90. Calcev and Gorez(1995).

, (373)

V"

2

T

(374)

4>m = 45° , small

xm
tm

= | tan<|)m + .J4a + t a n 2 c t > m ] — • a = 0.25 (Astrom, 1982); N J 4cac a = 0.5 (Seki et al., 2000); 0.1 < a < 0.3 (Seki et al., 2000). \T . tan
Chapter 4: Tuning Rules for PID Controllers Rule ABB Commander 300/310(1996). Luo etal. (1996). Karaboga and Kalinli (1996). Luyben and Luyben ( 1 9 9 7 ) page 97. Tan et al. (1999a)-page 26. Tan et al. (1999a) -page 27. Wojsznis et al. (1999). Yu (1999) -page 11. Tan et al. (2001).

237

Kc

T,

Td

Comment

0.625K U

0.5TU

0.083TU

'P+D' tuning rule

0.48K U

0.5TU

0.125TU

|p.32K u ,0.6K u ]

[0.213TU,

[0.133T U ,

1.406TJ

0.469TJ

0.46K U

2.20TU

0.16TU

0.4K U

0.5TU

0.125TU

0.5KU

Tu

0.125TU

Based on work by Tyreus and Luyben (1992)

Tighter damping than quarter decay ratio tuning 0.4K U

0.333TU

0.083TU

0.33K U

0.5TU

0.125TU

Some overshoot

0.2K„

No overshoot

0.5TU

0.125TU

Kc(375,

3 T (375)

Q25T_(375)

0.33K U

0.5TU

0.333TU

Chau (2002) page 115.

0.2K U

0.5TU

0.333TU

Robbins (2002).

0.45K U

4 lj

0.45K U

4

Smith (2003).

0.75K„

0.625TU

Rutherford (1950).

1.25K 37%

T37%

°-125T37%

!-12K37%

T37%

0.25T37%

T(376)

(377)

T

i

0.25TU

0.25TU

"Just a bit o f overshoot No overshoot Minimum IAE step change in setpoint "Good" response - step change in load

0.1T„

Other rules

(375) c

„ „ „ „ , (375), 2

2

, '

0.25[Ti ( 3 7 5 ) l 2 Q)„ 2 +l

i

= 0.3183T„tan

37% decay ratio

0.571 + Z G ; Q ( D U )

Gc(j ffl u) = desired frequency response of controller Gc(jffl) at cou - (376)

= (0.24 + 0.11 l K m K u )TU

T / 3 7 7 ' = (0.27 + 0.054K m K„ )TU

Handbook of PI and PID Controller Tuning Rules

238 Rule MacLellan (1950).

Ti

Td

1.77K 37%

1-29T37%

0.16T37./o

1.67K 37%

1.29T37„/o

0.16T37„/o

1.49K 37%

°-98T37%

0.24T 37%

25%

0.167T25./o

0.667T 25%

K

25%

0.667T 25%

0-167T 25%

j(378)

0.25Ti (378)

0.488T25„/o

0.122T 25%

25%

0.67T 25%

0-17T 25%

0-83K 25%

0.5T 25%

0.1T25o/o

Quarter decay ratio Quarter decay ratio x = 100(decay ratio) Example: x = 25 i.e. quarter decay ratio Quarter decay ratio 'Fast' tuning

0.67K25„/o

0.5T 25%

0-lT 25%

'Slow' tuning

0.45T 25%

0.1125T 25%

25%

0.667T 25%

0.167T25„/o

0.8497K 5 0 %

0.475T 50%

0.1188T 50%

0.6667TU

0.1667TU

5

Parr ( 1 9 8 9 ) page 191. McMillan (1994) -page 43

37% decay ratio

K

Harriott ( 1 9 6 4 ) pages 179-180. Liptak(1970)pages 846-847. Chesmond (1982)-page 400.

Comment

Kc

(378)

K

0.999K25o/o K

Wade ( 1 9 9 4 ) - [1.002K 2 5 % > page 114.

1.671^]

Hay ( 1 9 9 8 ) page 189. ECOSSE team (1996b). Bateson (2002) page 616.

K

K

6

25%

K

(379)

T(379)

Rotach(1994).

5

K

_0.5

(378)

6

K

(379)

Tx%

, (378)

0.5 + 0.3611nx M„

2Vl +0.025 In2 (379)

• G . ( j < D M _ ) , Tj'

A/M^ dco.

^=-

'modified' ultimate cycle

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Ti

Rotach(1994)continued. Garcia and Castelo (2000).

7 j , (380)

ry (380)

8

T

Lloyd (1994). Model: Method 3

(381)

K

Comment

Td T

(381)

(380)

0.25Ti(381)

0.4KU

0.3333TU

0.083TU

0.4 Ku

Tu

0.159TU

aTu,

0.125TU

0.7490

Jones et al. (1997).

239

CO, < C 0 U

Self-regulating processes Non selfregulating processes Model: Method 2

a e [0.2526,0.5053]

7

(380)

K

M„

•|G p (jCD r )|.

Mraax

-1

, (380) d

cor

^

,

.

-•n + tan

if

• -1

dcor (380)

0.5,

cor

r

dcor

1

jt-sin

7Ctann 7C-sin

+i

M„

Mmav 1 + tan'

7i - sin

+

M, tan 7t - sin

K

(381)

cos(l80°+
Mn

2[sin(l80° + +„ -ZG p (jm 1 ))+ 1] . co 1 cos(l80°+(|) m -ZG p (jco 1 ))

Handbook of PI and PID Controller Tuning Rules

240

Rule

T,

Kc

Shin et al. (1997).

9j£

(382)

(382)

T

i

NI Labview (2001). Model: Method 2

Dutton et al. (1997).

aT/

382

*

0.6 Ku

0.5TU

0.12TU

0.25 K„

0.5TU

0.12TU

0.15KU

0.5TU

0.12T„

P X 1 a,9(f

Model: Method 2 Quarter decay ratio 'Some' overshoot 'Little' overshoot

5^(383) ^

2.5465hsin<j>m Tang et al. (2002). Model: Method 4

Comment

Td

5 e [l.5,4]

^m

(383)

10 T

1

d

0.81 <x, <1

x, = 1 , 'small' xm ; x, = 0.91, 'large' Tm u

0.5K

e

T

u

1

e u

0.25Tue

pages 576-8.

y 2 = — s i n | ZG p (jco u ) |. Typical a : 0.1; typical i,: [0.3,0.7]. Ku 10

T

(383)

_ -cot4>m+Vcot2c|)m +4/5 2co 9 0 ,

Te = 1-

6.283 1110

with r = ratio of the height of the

55

first peak to the height of the first trough of the closed loop response, when controller gain Kc is applied; cod = damped natural frequency.

Chapter 4: Tuning Rules for PID Controllers

241

4.2.2 Ideal controller in series with a first order lag r \ 1 G c (s) = K£ 1 + — + Tds v

R(s) E(s)

-

T

1 j Tfs + 1

is

U(s), K

1+— + V TiS

T

«S

Y(s) Non-model specific

Tfs + 1

Table 69: PID controller tuning rules - non-model specific Rule Kristiansson (2003).

12

K

Comment Td Ti Other tuning: minimum performance index

Kc 12

K

(384)

T

(384)

T

(384)

"Almost" optimal tuning rule; K)J0„ > 0.03 ; 1.6 < M s < 1.9

0.60

(384)

K„

K

K-, o 150" -0.07 0.32 + 1.6- 5n K„ K„

K

150°

1.5

,(384) K

0.32 + 1.6-

150°

-0.8

^150"

0.6667

(384)

. K-nnO

0.32 + 1.6-

K„

-0.8

K, V Km J 0.4

(384) _ _ Kl5

»°--0.07

K„

K-,<„0

-or - J

0.32 + 1 . 6 — ^ _ - 0 . 8 | K„

^l;n> K„

(384)

20-

150°

+ 0.5 0.32 + 1 . 6 — ^ - - 0 . 8 |

K, K.

K„

min-^ 6 K

I50°

,25

Handbook ofPIandPID

242

13

Rule

Kc

Wang et al (1995b).

13 jr (385)

Controller Tuning Rules Td T: Direct synthesis

T (385)

T

Comment

(385)

'd

For a stable process, Km is assumed known; for an integrating process, Km and T„ are assumed known. K

(385)

GPO^CL^-GCLQCOCL)}

JCOCL G C L (j ffl CL)

Im . (385)

JMCLGCLQCOCL)

-Im •^CL

GpGttcLXi-GcLGaa.)}

1 Rl

J«CLG C L(J®CL)

_G p (jffl CL Xl-G CL (jQ) CL )}

1 j2coCLGCL(j2co JMCLGCLCPCL) •Rl + — Rl G (j(o Jl-G (j« )jj 1 G 02(00. Xl-Go. 020)0.)] p c CL CL p 3 Rl f (385)

JMcLGcLptfcL) J2(»CLGCL(J2C0CL) 1 -Rl Gp(jco CL Il-G CL Ora CL )]J [G p 02co CL )[l-G CL 02co CL ). Im

JOJCLGCLO^CL)

GpO ffl CLl 1 -G C LO (B CL)]j

Chapter 4: Tuning Rules for PID Controllers

243

4.2.3 Ideal controller in series with a second order filter f

1

A

l + b fI s J l + a fl s + a f2 s

G c ( s ) = Kc T

V

is

U(s)

R(s) E(s) K, -+®-+ + x

l + — + Tds

Non-model specific

l + a f ,s + a f 2 s

T s

V

Y(s)

l + bfls

i

Table 70: PID controller tuning rules - non-model specific Rule

Comment Td Minimum performance index: other tuning (386) rj, (386) (386) T M.*l.7 Ti

Kc

Kristiansson et al. (2000).

l

K

"Almost" optimal tuning rule

Gp(j(Du) G p (jco u ) 1.6 1.6-L^ ^ - L - 2 . 3 J — L + l.l K„

K„

l ^

(386)

, b l f - 0 , a lf - T f , a2f - Tf' Km 0.37 + 1.6

. (386)

G p (jco u )| K„

=r,T,

0.625

(386)

Gp(jcoJ 0.37 +

0.37 +

GpOu

K„

K„

,,M1. 2 . 3 W + 1 , K„

K„

0.37 +

G p (jco u )

|Gp(jcoj|

13-20

K„

Km

Gp(ja>„)

< 0.5 or

Kn

!,Ml_,3ra+1, K„

K„

Gp(j(Bu) 3co„ 0.37-

K~

13-20

|Gp(jfflj| GP(K) Km

Km

>0.5.

Handbook of PI and PID Controller Tuning Rules

244 Rule

Kc

Kristiansson (2003).

2

K

Td

Ti

(387)

(387)

T

T x

i

(387) d

Comment "Almost" optimal tuning rule; K, 5 0 „>0.03; 1.6 <M S <1.9

2

K

0.60

(387) K

K„

150°

-0.07

K, 0.32 + 1.6K„

-0.8 V

K

m

J

1.5

,(387)

0.32 + 1.6—^--0.8 Kn. V

K, K

m J

0.6667

(387)

K,,„o 150"

0.32 + 1.6-

blf = 0 , -0.8

^150°

V Km J

K„

0.8 2 0 ^

+ 0.5 0.32 + 1 . 6 ^ ^ - 0 . ^ K l 5 0 ' ^ K„

2 0 ^ 1 ^ + 0.5

K„

0.32 + 1 . 6 ^ L - O.gf K ' s ° u K„

K„

Chapter 4: Tuning Rules for PID Controllers

245

4.2.4 Ideal controller with weighted proportional term G c (s) = Kc T

v

is

Kc(b-1) R(s)

+ T

E(s)

Y(s)

A 1 > K r 1 + — + Tds T s v i J

+ T U(s)

Y(s) Non-model specific

+

Table 71: PID controller tuning rules - non-model specific Rule

Kc

Streeter et al. (2003).

0
K

(388)

ry (388)

T

(388)

4jr

(389)

T(389)

T

(389)

0
T

(390)

M max =2.0 0
3

Astrom and Hagglund (1995) -page 217.

Comment

Ti Td Direct synthesis

M max = 1.4 i

5

K

(390)

(390)

T 1

i

Mmax =2.0

Kc(388) =(p.33e-° J l M 2 )c u , T,(388) = (o.76e-L6K-036K2 )r u , (o^e" 031 *-* 2 )K U

T^^O.ne^-^Ji^b^.SSe-' Kc

<389)

= (o.72e-

L6K+1 2K2

)K U , Ti'

389

3K+0 38K2 ' = (o.59e-'-1.3K+0.38 )r u ,

Td(389) =(o.l5e

" % » = 0.25eu

1.4K+0.56I

K c (390) =(o.72e-''6K+L2l<2)K:u -0.0012340TU -6.1173.10"6 (390)

_ (o.59e-'-3K+a38K2 ](o.72e-'6K+'-2lc2 )K„ - 0.0012340T - 6.1173.10"6 \ru 0.72e~

(390)

0.108e"

38K

0.040430e"1JK ^•

]K U

log(l.6342'

w°

KUTU - 0.00266401

0.72K,1e"L6K+,'2l<2 -0.0012340T,, -6.1173.10~ 6 K„

246

Handbook of PI and PID Controller Tuning Rules

4.2.5 Controller with filtered derivative

G c (s) = K t 1 + — + — — N U(s)

E(s) R(s)

<8M

K„

+ T

,

T

1

ds

Y(s) Non-model specific

d

1 + T—s + * l+ ^ s N .

Table 72: PID controller tuning rules - non-model specific Rule

Kc

Leva (1993). N not specified Yi and De Moor (1994).

6

K

8

6

R

(391)

7

K

(392)

K

T

aT d (392) T

(393)

G

p(j<4

T

(393)

T X

*i

0)T:

(391)

Comment

Td Direct synthesis {391) 6.283/Pco

Ti

1-coT,

(3 = 10

(392)

6
(393)

N not defined

d

(391)

(391) 2TC

2

pj X

+ W

[T i <391) ] 2

(391)

. > < L " * • Y + ytan(m-o>-0-57c)

7

K

COT:

(392)

(392)

|G p (ja,)|V(l-a, 2 T i ^T d ^) 2 + f l ) 2 [T i « M 2 >] 2 • aco + y a 2 © 2 + 4aco2 tan2 (<(>m - <|>a - 0.5TC)

(392)

2aa>2 tan((j)m - <j>m - 0.57t) K

l(393) J„,

0.5cos[225°-ZG p (ja)^J

T<393) =

^

(393)

Gp(jW^) (393)

tan[225 0 -ZG p (j ffl ^)]+ A /4 + tan 2 [225°-ZG p (jffl^) 2®A

Chapter 4: Tuning Rules for PID Controllers Rule

T|

Kc

Vranfiic(1996)page 130.

2(A,-T, ( 3 9 4 ) )

9

(394)

T

N<10

(394)

T

A3A4-A2A5 10 J ,

(395)

T

Vrancic (1996) page 134.

11

(396)

T

Vrancic et al. (1999).

12

9

Comment

Td

(394)

T

Vrancic (1996)pages 120-121.

247

K

(395)

_

A3

(396)

N = 10; 1 = [0.2,0.25]

^(3%)

i

K

T(397)

(397)

T

N>10

A,A5

(397)

8 < N < 20

(394)

T

(394)-.2

A2-Td^>A,-

-(A3

2

N

- A5A,) + J ( A 3 2 - A 5 A J

-1(A3A2

- A5XA5A2 - A 4 A 3 )

(394)

N 10

*r (395) \."*> _

0.5A3\A3

v

C

(A3A2-A5)

A,A 5 j

~ (A32-A1A5)(A1A2-A3Kra)-(A3A4-A2A5)A12 ' A3(A32-A|A5)

T(395) = 2

' 11

K

0.5T;

(396)

(396)

A.-K,,,^ 12

K

T

(396)

(396)

(\3 -A,A5)A2-(A3A4-A2A5)A1

A2 - V A 2 2

-4XA,A3

2XA,

(397)

A1A2-A0A3-Td<397,A12-IIi- (397) .

A

V M T

W^

-T(397)A

[T, I

(397)-, 2 J__

N

(397) M , A , A 3 L (397)}

J+

1^

[d

J

, T6

N

(397)

-A 0 A, J

is obtained by solving:

K„

A 2 A 3 [ Td (39 7 )f

ApA5 N

+ (A 3 2 - A, A 5 lT d <397) ]+ (A 2 A 5 - A 3 A 4 ) = 0 .

Handbook of PI and PID Controller Tuning Rules

248

Rule

Kc

Lennartson and Kristiansson (1997).

13

T

T

14 j ,

c

v

K u K m >1.67

(399)

04T,(399,

(400)

04T(400)

(399) T •M

Kristiansson and Lennartson (1998b).

k

Comment

04T.(398)

i

Kristiansson and Lennartson (1998a).

13 v (398)

Td

(398)

Ti

(398)

K

N = 2.5

15

K

(401)

T

(401)

04T.(40„

16

K

(402)

T

(402)

0.333T;(402)

12KU Km - 3 5 K u K m +30

v

- N ^ n

T

;

;

1.

u m m +2.5|l2K„ ' ' — ' |_* "" * - u2 K " mm2 - 3 5 K u K m +30J K„K (398)

K

• (398) _ 3

-0.053co„ +0.47(0„2 -0.14co„ +0.11 2.5

N:

12

55_+

KmKu

30

K 22KKK m

14

v

(399) _

v

12K

v

2

2

u

Km2-35KuKm+30

KuKm+x,[l2Ku2Km2-35KuKm+3o]' K

, (399)

(399)

±— < 0.4;

-0.525CO, +0.473cou2 -0.143cou +0.113 K

, (400)

KUK

(400)

, - ^ > 0 . 4 ;

J

-0.185cou +1.052cou -0.854cou +0.309 N= -il15

K

(401)

12

K

7.71 , , 2 "-U K••m 2 K„

35 30 — + -K 2 K 2 K m u m u 9.14 " i K mmK„

KUK

x , = 3 , KuKm>10;x,=2.5, KuKm<10.

- + 3.14, K u K ra > 1.67 , ii—>0.45; m K„K„ ,(401)

(401)

K, • 0.63co„ + 0.39co„2 + 0.15co„ + 0.0082 3

16

v

12K„3K

(402)

2

-35K„2Km+30K„

" K m 3 K u 3 +2.5[l2K u 2 K m 2 - 3 5 K u K m + 3 o ] ' Kc(4°2)Km3Ku2

,(402)

j

2

2

2.5 —1,N =

co u [0.95K m K u -2K r K +1.4j u

KmKu

1 2 —K 1 * - + mKu

30

Km2Ku2

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Rristiansson and Lennartson (2000).

Comment

Td

17

K

(403)

j (403)

18

K

(404)

T

T

(404)

T l

(403)

0.1
(404)

<0.5 K„K r a >0.5

d

(l.lKu2Km2-2.3KuKm+1.6)2

(403)

'K

T;

249

2

K r a K u (-20 + 13K m KJ(l + 0.37K m K u ) 2 "l.6(-20 + 13K m K u )(l + Q.37KmKu) l.lKm2Ku2-2.3KmKu+1.6

-(403) _

KmKu(l.lKu2Kro2-2.3KuKm+1.6) cou(-20 + 13K m KJ(l + 0.37K m K u ) 2 1.6(-20 + 13K m KJ(l + 0.37K m K u ) 2

^

2

UKm Ku -2.3KmKu+1.6 (403) _ K m K u (l.lK u Km -2.3K u K m +1.6) o) u (-20 + 13K m K u )(l + 0.37K m K u ) 2 (-20 + 13K m K u ) 2 (l + 0.37K m K u ) 2 (l.lKm2Ku2-2.3KmKu+1.6)2 1.6(-20 + 13K m K u )(l + 0.37K m KJ 2

(403)

--1

N:

2

UKm Ku -2.3KraKu+1.6 r f

(403) _ K m K u (l.lK u Km -2.3K u K m +1.6) cou(-20 + 13K m KJ(l + 0.37K m K u ) 2 (l.lKu2Km2-2.3KuKm+1.6)2

(404)

K

2

2

3G) u K m K u (l + 0.37K m K u )

, (404) _ ( U K u 2 K m 2 - 2 . 3 K u K m + 1 . 6 ) 3cou(l +

x2 0.37K m K u r

2

4.8K m K u (l + 0.37K m KJ UK

2

K

2

•2.3K ra K u +1.6

4.8K m K u (l + 0.37K m KJ l.lKm2Ku2-2.3KraKu+1.6 3K m 2 K u 2 (l + 0.37K m KJ 2

2

(404) _

2

(l.lKu Km -2.3KuKm+1.6) 3a u (l + 0.37K m K u )^

(l.lKm2Ku2-2.3KmKu+1.6)2 4.8K m K u (l + 0.37K m K u ) l.lKm2Ku2-2.3KmKu+1.6

(404)

N: Tf

(404)

.Tf

(404)

l.lKu2Km2-2.3KuKm+1.6 3cou(l + 0 . 3 7 K m K J 2

(403)

Handbook of PI and PID Controller Tuning Rules

250 Rule Kristiansson and Lennartson (2000) continued. Kristiansson and Lennartson (2002).

19

19 K

(405)

20 K

(406)

T

T

(406)

T X

(405)

KuKm<0.1

(406)

KuKm<10

d

20.8(1.8+ 0.3K m K..,„)

13Kra(1.8 + 0.3KmK135o) • (405)

Comment

Td

j(405)

(-6 + 3.7K n ,„K m ) 2

(405)

K

T;

Kc

2

(-6 + 3.7KmK135o)

(-6 + 3.7K135„Km)K135oKm

20.8(1.8+ 0.3KmK135o)

13co135o(1.8 + 0.3K m K 1350 ) 2

(-6 + 3.7K ra K 1350 ) ' 169(1.8+ 0.3K m K 135 ,) 2

(405) _ ( _ 6

mK135°)KmK-i35» 13o3135.(1.8 + 0.3K m K 135 „) 2

"

+

(-6 + 3.7K m K m „) 2

3 7K

-

20.8(1.8 + 0 . 3 K m K „ ) (-6 + 3.7KmK135„)

(405)

N=l

(405)

.T f

(405)

f

-1

(-6 + 3 .7K m K 135 o)K m K 135 „ 13co135„(1.8 + 0.3K m K |35 „) 2

'

(406) = Ku [l.5(K m K u +4)(0.4K m K u + 0 . 7 5 ) - K m K u X l ] x , (0.4K m K u +0.75) 2 (4 + K m K u )

Km

(406) _ K m K u [l.5(K m K u +4)(0.4K m K u + 0.75)-K m K u x,] (0.4K m K u +0.75) 2 (4 + K m K u )

CO (406)

K

mKu

u (406)

N=-

(406)

(4 + K m K J [l.5(4 + K ra K u )(0.4K m K u + 0.75) - K m K u x, \l + N " 1 ) '

j (406) _ K m K u

(0.4K m K u +0.75) 2 (4 + K m K J

x, = 0.13 + 0.16K m K u -0.007K m 2 K u 2 .

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Kristiansson and Lennartson (2002) continued.

21

K

21 K

(407)

T

(407)

K u K r a >10

K n5 o [l.S(0.44KmK,35o +1.4)x 2 -KmK,35oX3Jx3

(407)

(0.44K m K 1 3 5 „+l.4)'x 2

K„

, (407)

KmK135o [l.5(0.44KmK,35o +1.4)x 2 - K m K u x 3 j (0.44K m K, 35 o+1.4) 2 x 2

co135„ (407)

K

mKi35°
(407)

N:

T

Comment

Td

Ti

(407)

251

d (407)

.Tf

[l.5x 2 (0.44K m K 1 3 5 „+1.4)-K ro K 1 3 5 „x 3 ll + N - 1 ) ' 2 K 2K (407) _ ^ m ^ 1 3 5 °

(0,

(0.44K m K, 1 ,„+1.4) i x 2

x 2 =min(14 + 0.5K m K 1 3 5 „,25),x 3 =1.35 + 0.35KmK,35„ - 0 . 0 0 6 K m 2 K | 3 /

Handbook of PI and PID Controller Tuning Rules

252

4.2.6 Ideal controller with set-point weighting 1 U(s) = K c (F p R(s)-Y(s))+^(F i R(s)-Y(s)) + K c T d s(F d R(s)-Y(s)) 1 :b

Table 73: PID controller tuning rules - non-model specific Rule

Kc

Mantz and Tacconi(1989).

0.6KU ; F p =0.17

T

i

Td

Ultimate cycle 0.5TU; 0.125T,; Fd = 0.654

Comment Quarter decay ratio

Chapter 4: Tuning Rules for PID Controllers

1

4.2.7 Classical controller 1 G c (s) = K

253

1 + sT,

T;S

N U(s)

R(s)

E(s)

Y(s)

K, 1 + — Xs

T

Non-model specific

N; Table 74: PID controller tuning rules - non-model specific Rule

Kc

Comment Td Minimum performance index: regulator tuning

Minimum IAE Edgar et al. (1997) -page 815.

0.56KU

0.39TU

Kinney (1983).

0.25KU

Ultimate cycle 0.5TU 0.12T„

Corripio(1990)page 27. St. Clair (1997)page 17. N>1

0.6KU

0.5TU

0.125TU

10 < N < 20

0.5KU

1.2TU

0.125TU

0.25KU

1.2TU

0.125TU

Harrold(1999).

0.67KU

0.5TU

0.125TU

'aggressive' tuning 'conservative' tuning N not defined

T;

0.14TU

N=10

N not defined

254

Handbook of PI and PID Controller Tuning Rules

4.2.8 Series controller (classical controller 3) U(s) = K

v

U(s) E(s)

R(s)

K,

(l + sT d ) — • T s

v

i y

'i + JL

(l + s T d )

T;sy Y(s)

Non-model specific

Ta ble 75: PID controller tuning rules -- non-model specific Rule

Kc

Td

Tj

0.25KU

Ultimate cycle 0.33TU 0.5TU

0.2KU

0.25TU

Pessen(1953).

Tan et al. (1999a) -page 26. O'Dwyer (2001b)representative results - Astrom and Hagglund (1995) equivalent -page 142. Tan et al. (2001).

0.3KU

0.15TU

0.25TU

0.2349KU

0.2273TU

0.2273TU

0.0944KU

0.6154TU

0.6154TU

0.1008KU

0.3939TU

0.3939TU

y (408) i

025T_(408)

K

(408)

=

Ziegler-Nichols ultimate cycle equivalent Am=2, * m = 20°
1K

(408)

>u|Gc(jttu)|

_

V[T i

'Some'overshoot

A r a =2.44,

,408)^ 1

'optimum' servo response

0.5TU

'optimum' regulator response - step changes 0.5TU 0.33KU 0.33TU

Pessen (1954).

Comment

<408) 2

2

] co u +lVo.0625[T i (408) ] 2 Q) u 2 + 1

6.25cou2 +4cou2 tan2 .(408)

co„ tan

2ZG;(jco u )+7t" -2.5co u 2

2ZG;(JCOU)+7I

G c (jco u ) = desired controller frequency response at oo = cou

+m=46°

Chapter 4: Tuning Rules for PID Controllers

255

/

4.2.9 Classical controller 4 G c (s) = Kc v

R(s)

1+- sTd i y l+ ^L V N

T s

U(s)

E(s)

K,

4- T

1+V

T s

i 7

sT„

Non-model specific

1 + s-

Y(s)

N

Table 76: PID controller tuning rules - non-model specific Rule

Kc

Td

Comment

0.20TU

N not specified

Ti

Ultimate cycle Hang et al. (1993b) - page 58.

0.35KU

1.13TU

Handbook of PI and PID Controller Tuning Rules

256

4.2.10 Non-interacting structure 1

controller based on the two degree of freedom f

\ a

U(s) = Kc 1 + — + T s

.

E(s)-Kc

>

PT d s

a +

R(s)

1+ ^ s

1 + ^-s N j

V

(

f

N

;

\ PTds a +

K_

1+^s N Y(s)

U(s)

R(s)

><%>-+

d K, 1+ — + T s . 1 + ^-s N .

+ E(s)

Non-model specific

Table 77: PID controller tuning rules - non-model specific Rule Shen (2002).

!

Kc 2

K

Comment Td Minimum performance index: other tuning Ti

T (409)

(409)

(409)

T

Minimise Jr(t) - y(t)|dt + f|d(t) - y(t)|dt; N=0, (3 = 0, M s < 2 : (

. 2 ^

K c ( 4 0 9 ' = K u e x p 0.17-2.62/ K m K„ +1.79,

T, ( 4 0 9 ) =T u exp - 0 . 0 2 - 2 . 6 2 / K m K u

Km Ku v

j

f

/> 2\

+1.34,

Km K u J

V

f

T d ( 4 0 9 ) =T u exp

*2\

1.70-0.59/ K K „ -0.25,

K m Ku V

If a = 1 - exp 0.30-0.48/ K m K u

+0.93,

Km

-2 Ku

J

Ku > 1/Km

Chapter 4: Tuning Rules for PID Controllers

257

4.2.11 Non-interacting controller 4 U(s) = K

1 +V

R(s)

E(s)

+ X

E(s)-K c T d sY(s) iS/

Non-model specific

+

Kr

T

XS

Y(s)

KJds

I

Table 78: PID controller tuning rules - non-model specific Rule

Kc

Comment Td Minimum performance index: regulator tuning

Minimum IAE Edgar et al. (1997)page 8-15.

0.77KU

0.48TU

VanDoren (1998).

0.75KU

Ultimate cycle 0.1TU 0.625TU

Ti

0.11TU

258

Handbook of PI and PID Controller Tuning Rules

4.2.12 Non-interacting controller 9 U(s) = Kc E s

R(s)

(>

3

1+ — + T d s

K„

+ X

E(s)-^-sY(s) N

T:S U(s)

Non-model specific

•

Y(s)

N

3: Table 79: PID controller tuning rules - non-model specific Rule

Kc

Comment

Td

Ti

Other tuning Bateson (2002) pages 629-637. Model: Method 1

3

K

(410)

=

10

»

3

K

(410)

{{-K^Do^iGp^^H

dB;T i (410) =min[ mg20 ,0.203,^. ]

2

T(410)

05

' ppO^HO 0 ]

N=10 180°

and

| G P U ° W ) | are given in

Chapter 4: Tuning Rules for PID Controllers

4.3 I P D M o d e l G m (s)

K

e"ST™

f

4.3.1 Ideal controller Gc(s) = Kc T

v

1

A

is

j

Table 80: PID controller tuning rules - IPD model G m (s) = Rule

Kc

Ford (1953). Model: Method 3

259

K m e" Comment

Td Process reaction

Ti

1.48 mxm

2xm

0.37xm

Decay ratio 2.7:1

Astrom and Hagglund(1995) -page 139.

0.94 Kmtm

2xm

0.5xra

Model: Method 1

Hay (1998)page 188.

0.4

K

Ultimate cycle Ziegler-Nichols equivalent

Hay (1998)page 199. Model: Method i; K C! T d deduced from graphs

Model: Method 3

0.55xm

Kmxm=0.1

0.30tm

K r a x m =0.2

0.25tm

Kraxm=0.3

2.0

0.25Tra

K r a x m =0.4

1.8

0.25xm

K m x m =0.5,0.6

3.2xra

10.0 4.0 2.5

X

Visioli (2001). Model: Method 1

0-8xm

Km 1 ™

3-2KmTm

2

Minimum performance index: regulator x x i/Kraxm 2Xm 3xm Coefficient values 1.49 0.59 1.37 1.66 0.53 1.36 1.34 1.83 0.49

tuning

Minimum ISE Minimum ITSE Minimum ISTSE

Handbook of PI and PID Controller Tuning Rules

260 Rule

Comment Td ^ Minimum performance index: servo tuning x x 0 3Tm l/KmTm Kc

Visioli (2001). Model: Method 1

Coefficient values 0.49 Minimum ISE 0.45 Minimum ITSE 0.45 Minimum ISTSE Minimum performance index: other tuning

1.03 0.96 0.90

Xl/KmTm

Astrom and Hagglund (2004). Model: Method 1

l

x2

x3

0.139 0.261 0.367 0.460 0.543

76.9 23.3 12.2 7.85 5.78

0.346 0.365 0.378 0.389 0.400

X

x

X2tm

3Tm

Coefficient values x Mmax i 0.616 1.1 1.2 0.681 0.740 1.3 1.4 0.793 0.841 1.5 Direct synthesis

x2 4.58 3.82 3.28 2.89 2.61

K

m

T

m

0.47KU Wang and Cluett (1997).

12.2tm

0.4lT ra

3.05TU

0.10TU

1

K

(411)

T(411)

2

K

(412)

T

T

(411)

T

(412)

M max 1.6 1.7 1.8 1.9 2.0

3

OS (step input) < 10%; Minimum IAE (disturbance ramp).

0.74 Leonard (1994). Model: Method 1

X

0.410 0.418 0.426 0.434 0.440

Model: Method 1 (412)

K u , Tu deduced from graph X

Cluett and Wang (1997). Model: Method 1

X

l

i/Kraxm

X

x2

X

3

K

(411)

K m T m (0.7138T CL +0.3904)

1 2 3

K

(412)

=

1 K m i m (0.5080T CL +0.6208)

X

3

4 m

x4 4 5 6

T/ 4 1 " = (l.4020TCL +1.2076)rm ,

1.4167TCL+1.6999 2

x2

= X T

0.3752 9.1925 0.1702 0.3144 11.1637 0.1453 0.2709 13.1416 0.1269

X

T (411) _

To.

3tm

Coefficient values X x4 l

0.9588 3.0425 0.3912 0.6232 5.2586 0.2632 0.4668 7.2291 0.2058 1

X

2tm

T C L e[T r a ) 16T m ], ^ = 0.707.

Ti(412) =(l.9885T CL +1.2235)r m ,

• (412) .

.0043TCL +1.8194

T C L e k , 1 6 T m ] , %=\.

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Ti

Td

1.60xm

0.48xm

261 Comment

1.21 Rotach(1995). Model: Method 5 Chen et al. (1999a).

K

m

T

m

Damping factor for oscillations to a disturbance input = 0.75. 3 T (413) 1.661 0 Model: Method 1 K

Am

rrJm

4

T

(414)

'd

Chidambaram and Sree (2003).

1.2346

Sree and Chidambaram (2005b). Chen and Seborg (2002).

0.896 Kmxm

K

5

m

K

T

m

(415)

4-5xm

0.45xm

Model: Method 1

2.5xm

0.55xm

Model: Method I

(415)

Model: Method 1

(415)

T

T

i

Robust Zou etal. (1997), Zou and Brigham 2 (1998). Km(X + 0.5Tm) Model: Method 7 or Method 8

X + 0.25xm —x 2X + xm

2 ^ + Tm

0.5x m <^ ^3xm

Td<413) = 1.273-cJl- 0.6002Am (l.571 • 4

T / 4 , 4 ) = T Jl.2732

i(A

riA m 1.6661

_. T,(415)S(l + 0.5T m s)e-^

'v ^•'/ desired " .(415)

Kc"""(TCL S + " 1)' = 3T CL +0.5T m , Td

(4i5)

' ' (415)

1.5T

_Tm

(3T C L + Q.5t m )

~K m(TcL+ 0.5x m r 2

x m + 0 . 7 5 T r , x 2 +0.125x m 3 - T x m (3T C L +0.5x m )

3

Handbook of PI and PID Controller Tuning Rules

262

4.3.2 Ideal controller in series with a first order lag r \ 1 G c (s) = Kc 1 + — + Tds T

V

R(

V «•>. K f + c

1

k

I

U(s)

1

^

1

T s

)

1 + Tfs

i

is

Kc

T

Tfs + 1

Y(s)

Kme"^

s

Table 81: PID controller tuning rules - IPD model G m (s) = Rule

j

K m e"

Td

Comment

0.5x m 2 +Xx m

Suggested X = 3.162tm

i

Robust Rice and Cooper (2002). Model: Method 1

6 j , (416)

=

_

6

K

(416)

2X + 1.5tn K, 1 ^ + 2 X x m + 0 . 5 t m 2 )

2X + 1.5xm

. Tf =

2X + 1.5Tm

0.5X 2 x„

^+2^xm+0.5xm2

Chapter 4: Tuning Rules for PID Controllers

263

4.3.3 Ideal controller with first order filter and set-point weighting 2 1

, l + 0.4T r s R(s) — - Y(s) 1 + sT j Tfs + 1

U(s) = Kc v

T

is

R(s) f

1 + 0.4Ts 1 + Ts

Y(s)

U(s)

+ E(s) —•

•K0Y(s)

A

i 1+ — + Tds

V

T

i

S

J

1 Tfs + 1

+ +&»

K m e"

Kn

Table 82: PID controller tuning rules - IPD model G m (s) = Rule

Kc

Normey-Rico et al. (2000). Model: Method 1

0.563 Kmtra

K m eComment

Td Direct synthesis

Ti

T f =0.13x m l-5xm

0.667tm

l ~ ° 2K ra x m T r =0.75x m

Y

Handbook of PI and PID Controller Tuning Rules

264

d

4.3.4 Controller with filtered derivative G c (s) = K( 1 + — +

N R(s)

?>

E(s)

»

Kc 1 ,

l

,

T s

>

L

T

«s l+ ^ s

V

Y(s)

U(s)

iN J

Table 83: PID controller tuning rules - IPD model G m (s) = Rule

Kc

Kristiansson and Lennartson (2002).

7 K (417)

Chien (1988). Model: Method 1

8 K (418)

7 K (417)

2.5(0,,

(417)

T

T

. 0 .08 + ^

K,

2X + xm

(417)

+

Robust Tm(X + 0.25xra) 2X + xm

J

(o.osVK^+fo.oss/Kj-o.os'

2.3 - 2 K

( O . Q 5 9 / K I 2 ) + (0.055/K,)- 0.08

_ (417) _ 2.5

2.3-2K, j (417)

T, 8

K

(418)

(417)

>Tf

(417)

0.4(10 + 1/K,)

(0.059/K) 2 )+ (0.055/K!)- 0.08" 0.4(10 + 1/K,)

2

0.4(10+ 1/K,)

—+ 1 N

(0.059/K! 2 )+ (0.055/K))- 0.08 , K, e [0.5,1]. 0.16cou(lO + l/K!)

2>. + T„ K m (0.5^ + T ra ) 2

N=10

^ "i

2

N=-

Comment

T, Td Direct synthesis

2.3 - 2 K , 2.5

Kme"

Model: Method 1

K„

•(417)

—P-

s

•; X e [l/K m ,T m ] (Chien and Fruehauf (1990)).

Chapter 4: Tuning Rules for PID Controllers

265

4.3.5 Controller with filtered derivative and dynamics on the controlled variable G c (s) = Kc 1 + — +

5-

E(s)-K 0 Y(s)

N R(s)

+

E(s)

U(s)

Y(s)

*®^

K,

+ T N

K„

Table 84: PID controller tuning rules - IPD model G m (s) = Rule

Comment Ti Td Direct synthesis - time domain criteria

Kc

Normey-Rico et al. (2001). Model: Method I

9

K

Kme^

(419)

(1.5-yk,

T

Y

(419)

X

2K

"°

m*ra

N = l; 0.1716 K

mTm

Y

0.5xra

*m

-

"°

l 2

Kmxm

Recommended y = 0.5 ; non-oscillatory response

•5-Y 4y + 1 - 4^/y2 +0.5y .T,

(419)

1.5-y

"Y^m.

N•

l-r(l•5-Y) (1.5 -y)y

Handbook of PI and PID Controller Tuning Rules

266

(

1 + Tds

4.3.6 Classical controller 1 Gc (s) = Kc V

T s

i V

NJ R(s)

E(s)

K„

&

+ T

v

T s

i y

Y(s)

l + sTd l+ s ^ N

U(s)

Table 85: PID controller tuning rules - IPD model G m (s) = Rule

Tf

Kc

Y(s)

Kme"

Kme" Comment

Td

Process reaction Bunzemeier (1998). Model: Method 4; G

P = K

x

X 2 tra

K mmTmm

X

3tra

10% overshoot (servo)

x4 Kmxm Coefficie nt values

P

s(l + sT p )"

0% overshoot (servo)

i

x2

x,

x3

x4

N

n

0.280 0.24 0.730 5.60 0 0.796 0.94 1.570 5.99 2 0.62 1.200 0.669 6.03 3 0.981 0.587 0.46 5.99 4 0.530 0.866 0.38 6.02 5 0.34 0.828 0.487 6.01 6 0.452 0.801 0.32 6.03 7 0.424 0.782 0.30 5.97 8 0.767 0.401 0.29 5.99 9 0.756 0.381 0.27 5.95 10 ^linimum p<;rformance index: regiilator tuning Model: 0.93 Method 2; 1•57xm 0.58x n i Kraxm N not specified

0.21 0.68 0.47 0.35 0.30 0.27 0.25 0.23 0.22 0.22 Minimum IAE Shinskey (1988) -page 143. Minimum IAE Shinskey (1994) -page 74. Model: Method 1

0.93 K

mTm

•60xra

0.58T n i

N=10

1• 4 8 t m

0.63x n i

N=20

0.93 Kmxm

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IAE Shinskey(1996) -page 117. Minimum IAE Shinskey (1996) -page 121.

Ti

Td

1.57xm

0.56xm

0.39TU

0.15TU

0.93 K

m

X

m

0.56KU

10

Chien (1988). Model: Method 1

10

Rice and Cooper (2002). Model: Method 1

11

K

0.5xm

(421)

2X. + 0.5xm

0.5xm

(422)

2X + im

0.5xm

K

K

Robust 2>. + 0.5xm

(420)

267 Comment Model: Method 2; N not specified Model: Method 1; N not specified

^[l/Km,xm] (Chien and Fruehauf (1990)); N=10 Suggested k = 3.162xm

Ultimate cycle Luyben (1996). Model: Method 9 Belanger and Luyben (1997). Model: Method 1

10

K

(420) _

2.2TU

0.16TU

Maximum closed loop log modulus of +2dB ; N=10

3.11KU

2.2TU

3.64TU

N=0.1

0.5x„

1

Km

0.46KU

2

^ + 0.5xJ J 2

K

2X + 0.5xm

(421) K„

2

Km^ +2^xm+0.5xm )

N=

[^ + 0.5x m ] 2

X.2+2A.xm+0.5x,

Handbook of PI and PID Controller Tuning Rules

268

l + NTds

4.3.7 Classical controller 2 G c (s) = K v

E(s) R(s)

T s

i y

U(s)

f 1 "\ l V l + NT s K i + T- s l + T sd

v

i A

Y(s) K„e"

a J

Table 86: PID controller tuning rules - IPD model Gra (s) = Rule

Comment Td Direct synthesis - freq uency domain criteria Maximise 0.45^200 (423) crossover N frequency Ne[l0,30] (424) oo ]QD°Sro i m +0-65] Five criteria are fulfilled; N = 10.

Hougen(1988). Model: Method 1

12

K

13

K

' Values deduced from graph: 0.1 0.2 Kmxm K

(423)

KraTm K

(423)

^

Kc

Hougen(l979)page 333-334. Model: Method 1

K m e"

0.3

0.4

0.5

9

4.2

2.8

2.1

1.7

0.6

0.7

0.8

0.9

1.0

1.45

1.2

1.1

0.95

0.85

Equations deduced from graph; Kc

' «—— 10

K„

T'

Chapter 4: Tuning Rules for PID Controllers

f 4.3.8 Classical controller 4 G c (s) = Kc v

P +—

l+

^

269

s

1-3*

T s

i y

N

Table 87: PID controller tuning rules - IPD model G m (s) = Rule

Kc 1

Chien (1988). Model: Method 1

K K

(426)

(125)

2 ^

(426)

1 I

[X + 0.5Tm j 0.5l„

Km{[x + 0.5xJ

Td

Robust 2>. + 0.5Tm 0.5xm 0.5xm

2?I + 0 . 5 T „

(425) K„

2

K

T(

J

2X + 0.5xm

K m e" Comment ^ e [l/K m ,t m J (Chien and Fruehauf (1990)); N=10

Handbook of PI and PID Controller Tuning Rules

270

4.3.9 Non-interacting controller based on the two degree of freedom structure 1 f

\

f

^

R(s)

U(s) = K,

Table 88: PID controller tuning rules - IPD model G m (s) = Rule

Minimum ITAE - Pecharroman and Pagola (2000). Model: Method 10

Comment Kc Td Tf Minimum performance index: servo/regulator tuning x,K u x2Tu *3Tu Coefficient values a x X x2 i 3 4»c 1.672 0.601 0.366 0.136 -164° 1.236

0.427

0.149

0.607

-160°

0.994

0.486

0.155

0.610

-155°

0.842

0.538

0.154

0.616

-150°

0.752

0.567

0.157

0.605

-145°

0.679

0.610

0.149

0.610

-140°

0.635

0.637

0.142

0.612

-135°

0.590

0.669

0.133

0.610

-130°

0.551

0.690

0.114

0.616

-125°

0.520

0.776

0.087

0.609

-120°

0.509

0.810

0.068

0.611

-118°

Chapter 4: Tuning Rules for PID Controllers Rule Taguchi and Araki (2000). Model; Method 1

Hansen (2000). Model: Method 1 Chidambaram (2000b). Model: Method 1 Chidambaram and Sree (2003). Model: Method 1

Kc

T,

Td

Comment

2.388xm

0.4137xm

^2-<1.0 T

1 f 1.253^ Km

271

m

a = 0.6642 , p = 0.6797 , Overshoot (servo step) < 20% ; N fixed but not specified Direct synthesis N = 0; P = l ; 0.938K m t m 0.313tm 2.7xm a = 0.833 N = oo; p = l; 0.9425 0.5xm K 2xm a = 0.707 or mTm 1.2346 K m x ro

4-5Tm

0.45Tm

a = 0.65 N = 0; p = 0 ; a = 0.6

Handbook of PI and PID Controller Tuning Rules

272

4.3.10 Non-interacting controller based on the two degree of freedom structure 3 /

U(s) = K c [(b - 1 ) + (c - l)T d s]R(s) + K c 1 +

\ + Tds R ( s ) -

1 Ts

1 (l + s T f ) :

•Y(s)

K c [(b-l) + (c-l)T d s] \ 1 1 + — + Tds V Tis J

(

+

-+<&->K

R(s)

+

Y(s)

U(s) K e"

>

+ (l + sT f ) 2

Table 89: PID controller tuning rules - IPD model G m (s): Rule

Astrom and Hagglund (2004). Model: Method 1

K m e"

Comment T; Td Minimum performance index: other tuning No precisely defined 0.5xra 0.45 8xm performance Km specification Kc

b = 0, - ^ - < l ; b = l , ^ - > l . c = 0. Tf = 0 . k m m

m

Chapter 4: Tuning Rules for PID Controllers

273

4.3.11 Non-interacting controller 4 E(s)-K c T d sY(s)

U(s) = K T s

v R(s)

f

E(s)

K, 1 + -

+

v

P

>®

+

i Xs i°y

i y Y(s)

U(s)

K„e"

KcTds

T Table 90: PID controller tuning rules - IPD model Rule Minimum IAE Shinskey (1988) -page 143. Model: Method 2 Minimum IAE Shinskey (1994) -page 74. Minimum IAE Shinskey (1988) - page 148. Minimum IAE Shinskey (1996) -page 121.

Kme"

Comment Td i Minimum performance index: regulator tuning Also defined by Shinskey (1996), 0.48xm 1.28 1.90xm page 117. K T Kc

T

m m

1.28 1.90xm

0.46t m

Model: Method 1

0.82KU

0.48TU

0.12TU

Model: Method 1

0.77KU

0.48TU

0.12TU

Model: Method 1

K

mTm

Handbook of PI and PID Controller Tuning Rules

274

4.3.12 Non-interacting controller 6 (I-PD controller) U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s) R(s)

>M

+

1 k

E(s)

U(s)

Kc TiS

r

ryy

4

Y(s)

K.e-'s

— •

K c ( l + T d s)

t Table 91: PID controller tuning rules - IPD model G m (s) = Rule Minimum ISEArvanitis et al. (2003a).

K m e"

Comment Td Minimum performance index: regulator tuning 1.4394 Model: Method I 2.4569xm 0.3982xm K T Kc

Ti

m m

1.2986 Arvanitis et al. (2003a).

Kmxm

3.2616Tm

0.4234xra

Model: Method 1

oo

je2 (t) + K m 2 u 2 (t)Jdt

Minimise performance index 0

Arvanitis et al. (2003a).

1.1259 Kmxm

6.7092xm

0.4627xm

Minimise performance index 0

Model: Method 1

;'«^'(T]" dt

Minimum performance index: servo tuning Minimum ISEArvanitis et al. (2003a). Arvanitis et al. (2003a).

1.5521 Kmxm

2.1084tm

0.3814xra

Model: Method 1

1.4057 Kmtm

2.5986xra

0.4038xm

Model: Method 1

oo

[e2 (t) + K m 2 u 2 (t)Jdt

Minimise performance index 0

Chapter 4: Tuning Rules for PID Controllers Rule

Kc 1.3332 Kmxm

Arvanitis et al. (2003a).

275

Ti

Td

Comment

3.0010xm

0.4167xm

Model: Method 1

Minimise performance index 1 e 2 (t) + K m 2 |

'duf~

dt

0

Direct synthesis Chien et al. (1999). Model: Method 2 Arvanitis et al. (2003a).

3

K

(427)

3

K

(427)

1.414T CL2 +t m

16(2^ 2 +l)

N2+lk,

2

(32^ +4JK m t m

1.414TCL2+xm

^ ( W + O J O T T c ^ + O ^ x , , 2r' ' . ±d

T

(427)

16^+4 _ 16 ( 2 ^ 2 + l j " m

(427)

Underdamped system response \ = 0.707 Model: Method 1

0.25Tm^+0.707TCL2Tn 1.414TCL2+xm

276

Handbook of PI and PID Controller Tuning Rules

4.3.13 Non-interacting

controller

8

\ f 1 U(s) = Kt 1 + — + Tds E(s)-K 1 (l + Tdis)Y(s) V R(s)

T

is

J

E(s)

Table 92: PID controller tuning rules - IPD model G m (s) Rule

T|

Kc

Comment

Td Robust

Lee and Edgar (2002). Model: Method I

4 j , (428) _

K; =0.25K U , 4

^

=

(428)

T

(428)

0.25KU

(XT

K

0 2 S ^

fr + O

(«8)

Tdi=0

,(428)

2{X + xn

K;„K„

{ Q

T

2_

T

3

6(X + t r a )

K„Km 2

4{X + xmf

'

2(X + i J

x m (l-0.25K u K m T m ), 0.5KuKm(X + x J

Chapter 4: Tuning Rules for PID Controllers

277

4.3.14 Non-interacting controller 10 U(s) = Kc 11++ ^f- E(s)-K f -^-s + 1 Y(s) I Tisy vTm y R(s) ^^y.

E(s)

+ /cx

K V

Y(s)

U(s)

K„e"

T s

, 7

iis +l T

K,

I

V m

J

Table 93: PID tuning rules - IPD model G m (s) = Rule

Kc

Ti

Kaya and Atherton(1999), Kaya (2003).

0.313Ku

0.733TU

(

0.753

A

Td Direct synthesis

A 0.333

Km Ku

^nJm K„

5

x

VTmT"

(429)

(

) A

A

0.567 Km Ku K„

T

V m T"

K„e"

•0.313Ku J

5

T X

Comment

(429) d

Model: Method 9

278

Handbook of PI and PID Controller Tuning Rules

4.3.15 Non-interacting controller 12 f T

V

R(s)

A

1

U(s) = Kc

i

s

j

1 E(s)-K,Y(s) Tds + 1

E(s)

—*®-* + r

Kc(l + -J- + Td

Table 94: PID controller tuning rules - IPD model G m (s) = Rule

T;

Kc

K m e- S t ™

Comment

Td Robust

Lee and Edgar (2002). Model: Method 1

6

K

(430)

6

K

(430)

0.25K,, (X + xJ'KuKm

(430)

T

•i.

"'

+-

K, =0.25KU

(430)

T

2{X + zm) 0.5

xm 0.5 , (430)

4

*m 6ft+ 0

,

V , (l-0.25K„K n ,T ni ) 4(X + t m ) 2 0.5K u K m (X + Tm)

x,

2{x+imy ^

0.5

6(^ + x ra )

+-

T

"

4(X + zm)

2

-+

(l-0.25K u K m x m ) 0.5K u K m (^ + x m ) -i0.5

T

(430) _

0.5

^

6(X + xm)

+_

T

"

4(X + xm)2

•

(l-0.25K u K m x m ) + 0.5KuKm(X + t m )

Chapter 4: Tuning Rules for PID Controllers

4.4 FOLIPD Model Gm(s) =

279

K e"


f A 1 4.4.1 Ideal controller Gc (s) = Kc 1+ — + Tds

TiS

l(!L^o E ( s )

J

U(s)

K e-sx"

KJ1+ — + Tds

+ T

s(l + sT m )

T:S

Table 95: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

Minimum ISE Haalman(1965).

2 3K m x ra

Y(s)

K m e
Comment Td Minimum performance index: regulator tuning Tj

0

T

Model: Method 1

Am=2.36;*m=500;M1=1.9. Direct synthesis 1 Van der Grinten (1963). Model: Method 1

K e

mTm

(431)

X K m (l + Xxm)

l-O-Se-^^+^-e-"'- tme

T m +0.5x m 1

-2«)d^

Kmtm

Tachibana (1984).

0

x

(431)

0 0

T

Step disturbance Stochastic disturbance Model: Method 5

Handbook of PI and PID Controller Tuning Rules

280 Rule

Kc

Ti

Comment

Td

ra T

Chen et al. (1999a).

p m

K A T <432)

0

2T

0

2

(432)

Model: Method 1

(433)

Model: Method 1

ri w rT m v

(433)

T

T l

d

0

Viteckova x l/KmXra (1999), Viteckova et al. OS X, (2000a). 0% Model: Method 1 0.368 0.514 5% 0.581 10%

T

Coefficient values OS OS x. 0.801 0.641 15% 30% 0.696 0.853 20% 35% 0.906 0.748 25% 40% x

l

l m

K

x

0

mTm

l

0.785 Chen and Seborg (2002).

3

K

(434)

n

X not specified

(433)

4m r 2 x m

^p^m 1 v + — P

T

Rot)USt T m (x m +2X) Tra+Tm+2^ xm+Tm+2^

1 . 2co

45% 50%

<> tm = 0.571-X, Representative coefficient values x Am Am l *m *m ) 0.524 s 45° 60° (434) Model: Method 1 T 3T CL +T ra

Rivera and Jun T m +T r a +2?t (2000). K (T +X) 2 Model: Method I ra m

(432)

OS

Ara = 0.5/x, ;

X T

O'Dwyer (2001a). Model: Method I

x. 0.957 1.008

[

1

T _ 0 25n(2r,A - 1 )

3(1

_(3TCL+xm>e-""

~K c <->(T CL s + i r

(434).. ( 3 T C L + T m X T m + T m )

'

~ X

(434)

(T CL+ xJ 3

•

2

3TCL Tm+3TCLTmTm-TCL3+TmTm2 (3T C L+x m XT m +T m )

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

281

Td

Comment

(«5)

0.25Ti(435)

Tuning rules developed from

<436)

5 T (436)

T| Ultimate cycle

McMillan (1984). Model: Method 1 Peric et al. (1997). Model: Method 8

K

(435)

5 T (436)

4

K

(435)

T

Ku>Tu K

aT d

uCOS(|) m

1-111 Tm

tand)m +,|— + tan a

• (435)

T 471

2lm

1+

Recommended a =4

Handbook of PI and PID Controller Tuning Rules

282

4.4.2 Ideal controller in series with a first order lag \ r 1 G c (s) = Kc 1 + — + Tds T

V

R(s)

E(s)

+

\ r 1 K, 1 + — + Tds T s v i J

1

i

s

j

1 + Tfs

Y(s)

K „e"

Tfs + 1

1

s(l + sTJ

U(s) Table 96: PID controller tuning rules - FOLIPD model Gra(s) = Rule

Comment

Td

Ti

Kc

K m e"
Robust H„ optimalTan et al. (1998b).

6

K

T(437)

(437)

T

Model: Method I

(437)

A. = 0.5 (performance), A, = 0.1 (robustness), A. = 0.25 (acceptable)

7 j r (438) 3^- + T m + x m 1.5xm
6 v (437)

(0.463X + 0.277)([0.238X + 0.123]Tm + T ) 5.750A. +0.590

K m x„/ . (437)

- Td

(437)

0.238A. +0.123' 7

K

(438)

3^ + T r a +x m

K ^ l3k'+3\xm+xm< K

(439)

0.0337T„ Kmx„

i.Tf=-

1+0.1225T„,

, K

Tmtm (0.238A,+ 0.123)T m +x n 3k'+3\xm+xm' (440)

0.0754T„

K mmx m

V

0.1863T„

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Tan et al. (1998a)continued. Rivera and Jun (2000). Model: Method 1

,

K (44i)

K

(442) _

Td

Comment

Tmtm

0.1344Tm KmT„

10

Ti

283

9

K

10

K

(441)

T m +3.9635x m

0.2523T ra +x m

Tf = 0.2863xm

(442)

2(T m +V) + Tm

2Tm(tm+^) 2(xm+X) + Tm

X not specified

1+-

0.2523Tm

2 T

( m+^)+Tm 2

j 2

K r a K + 4 T m X + X )'

T

_ f

m^

2

2T m +4T m ^ + ^

284

Handbook of PI and PID Controller Tuning Rules

4.4.3 Ideal controller with weighted proportional term

Table 97: PID controller tuning rules - FOLIPD model Gm (s) = Rule

Kc

5.6e-8'8T+68,! Astrom and Hagglund(1995) K r a (T m +T m ) -pages 212-213. Model: Method 1 ; 8 6 e -7.1x + 5 .4x or Method 3 Km(Tm+tm)

11 j

(443) _ Q 3 C

T

0.056T-0.60t2

Td Direct synthesis

Ti

l.hrae—'!

1.7tme-6-4T+20T2

K m e" s(l + sT m ) Comment

Mmax =1.4

b = 0.12e 6 ' 9t - 6W 1.0x m e" T - 2 ^

11

T

(443)

M max = 2.0

Chapter 4: Tuning Rules for PID Controllers

4.4.4 Controller

with filtered

derivative

285

Tds

G c ( s ) = Kc 1 + T S

.

/ R(s)

U(s)

E(s)

K, 1 + ^ +

7^

Kme"CT s(l + sT B )

- ^ N

Table 98: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

Kristiansson and Lennartson (2002).

12 ^

T (444)

T

Y(s)

Kme-" s(l + sT m ) Comment

T, Td Direct synthesis

(444)

1+ s ^ N

Model: Method 1; K, e [0.5,1]

(444)

Robust Chien (1988). Model: Method I

12 ^

(444)

2*. + ^ ra +T m 2^ + T m + x m

2.5co„ K„

•0.08 + ^ + K,

° ^ Ki J 2

(O.059/KI2)+(0.055/KI)-0.08

2.3-2K

2.3-2K, (444)

2.5 (

(O.059/K|2)+(0.055/KI)-Q.08A

.2

2.3-2K, (444)

N=l

d

(444) (444)

f

0.4(10+ 1/K,)

(0.059/K,2)+ (0.055/K,) - 0.08' 0.4(10 + 1/K,)

2.5

,(444)

^e[T m ,T m ] (Chien and Fruehauf (1990)); N=10

T m (2^ + x m ) 2^ + T m + x m

0.4(10 +1/K,)

(o.osg/K^j+fo.oss/Kj-o.os 0.16cou(lO + l/K,)

N

+

Handbook of PI and PID Controller Tuning Rules

286

4.4.5 Controller with filtered derivative with set-point weighting 1 G c (s) = K£

,

T

1

T iS

1+

l + bns

K,

1 + a,, s

^R(s)-Y(s)

. l + afls

/

E(s)

+

1+ b

Td

N

v R(s)

ds

U(s)

-U-^

1+

Y(s) K m e"

sd + sTJ

N ;

Table 99: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

^

K m e-

'(1 + s T j Comment

Td Robust

A, = x

n

K

T (445)

(445)

13 j r

Liu et al. (2003). Model: Method 1; recommended

(445)

T

Representative A values deduced from a graph OS Rise time OS Rise time A

X 0-3Tm

= 57%

= 3Tm

2T m

-0%

= 8Tm

0.5x m

= 33%

= 3.5xm

2-5Tm

= 0%

= 10.5x m

tra

= 7%

= 5x m

3Tm

= 0%

= 12.5x m

l-5t m

= 0%

- 8.5r m

3-5Tra

= 0%

= 14t m

(445) _{3X + xm+Tmi3X2+3Xim+zm2)-X3 2

^

2

Km(3X +3Xxm+xm } T

(445) = (3X + xm+TmJ3X2

+3Xxm+ij)-X3

(3X2+3^ra+Tm2) T

(445,

2

' 2

A3

Tm{3X + xmJ3X +3Xxm+xm ) (3A + T m +T m )(3A 2 +3Ax m +T m 2 )-A 3

Tm{3X + xm) N=^t(3A + T m + T j ( 3 A 2 + 3 A T m + T m 2 ) - A 3 ]

3A 2 +3AT m +T m 2 ' _

Chapter 4: Tuning Rules for PID Controllers

287

4.4.6 Controller with filtered derivative with set-point weighting 3 1

R(s)

1

T

ri + b fl s ± b il s 2 + b f3 s 3 _ R(s) _ Y(s) ^

ds

G c (s) = Kc 1 + — + — = 2 3 Ts - X 1 + ^ - s 1 + af ,s + a f 2 s + a f 3 s V N ;

l+bfls + bf2s2 +bf3s3 l + a n s + af2s2 +af3s3

Table 100: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

K m e" *(l + sTm Comment

Td

Tj

Robust r (446)

(446)

1^

rj- (446) X

Liu et al. (2003). Model: Method 1

K

d

a f 2 = X(\6X + 4xm),

af

= 4X2(3X + xm)

X

OS

Rise time

X

OS

Rise time

0-3Tm

= 43%

= 3.5x m

2Tm

= 0%

= 16.5t m

0-5x m

= 15%

= 4.5Tm

2.5Tm

= 0%

= 20x m

tra

= 0%

= 8Tra

3Tm

= 0%

= 24tm

l-5Tm

= 0%

= 12-5Tra

3.5Tra

= 0%

= 28Tm

(446, _ (3*. + t m + T m J3X2 + 3Xxm

+xm2)-X3

Representative X values (deduced from a graph)

•

a f l = 7X + xm;

K ^ + S ^ + T , , , Ti(446) =

2

{3X + xm+TmJ3X

2

)

2

+3Xxm+xm2)-Xi

b f [ = 3X , b f 2 = 3X , b f 3 = A.

l3X2+3^m+Tm2) (446)

Tm{3X + xmJ3X2+3Xxm+xm2)

*3

(3X + T m + T m ) ( 3 X 2 + 3 ^ m + T m 2 ) - X 3 N =

T m (3X + T m ) X3 [(3X + x m + T m )(3X2 + 3Xxm + xm2)- X3 ]

3X2+3Xxm+Tm2' t

288

Handbook of PI and PID Controller Tuning Rules Rule

Liu et al. (2003) - continued.

Kc

Comment

Td

Ti

a f , = 4X + x m ; a f 2 = X(3.25X + x m ) , a f 3 = 0.25A2 (3A. + x m ) A

Representative A. values (deduced from a graph) OS OS Rise time Rise time X

0.3x m

*61%

-3tm

2x m

*4%

-6.5xm

0.5t m

» 40%

-3im

2.5x m

*4%

*8xm

V

-15%

-3.5xm

3x m

*1%

"10Tm

l-5x m

*7%

-5xm

3-5xm

»0%

*12x„

Chapter 4: Tuning Rules for PID Controllers

289

4.4.7 Ideal controller with set-point weighting 1 U(s) = K c (F p R(s)-Y(s))+|^(F i R(s)-Y(s))+K c T d s(F d R(s)-Y(s))

Table 101: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

Oubrahim and Leonard (1998). Model: Method 1

0.6KU, F p =0.1

T,

Td

Ultimate cycle 0.5TU, 0.125TU, Fi=l

F d =0.01

K m es(l + sT m ) Comment

0.05 < — < 0.8 T 20% overshoot

Handbook of PI and PID Controller Tuning Rules

290

1 + Tds

4.4.8 Classical controller 1 Gc (s) = Kc v

E(s)

R(s)

K,

+

V

T s

i y

N ;

U(s)

l + sTd T T s i 7 1 + s-ii

s(l + sT m )

K m e"

Table 102: PID controller tuning rules - FOLIPD model G m (s) = Rule

Minimum IAE Shinskey (1994) pages 158-159. Model: Method 1; N not specified

2 jr

(447)

2

K

v

K

(448)

Ti

(447) ~ (447)

3

K

(448)

100 108KraTm 1.22-0.03-

3

S (l+sT m )

Comment Td Minimum performance index: regulator tuning 0.78 1.38(x m +T m ) 0-66(x m +TJ Km(tm+TJ N=10. Model: Method 1 Kc

Minimum IAE Shinskey (1994) -page 75.

Y(s)

K m e"

100 108Kmxm 1 + 0.4-

^-<2 T

0.56t ra+ 0.75T m

IHL>2 T

-

T(447)=157x

1 + 1.2 1 - e

T

™

Chapter 4: Tuning Rules for PID Controllers Rule Minimum ITAE - Poulin and Pomerleau (1996), (1997). Model: Method 1

Process output step load disturbance

Process input step load disturbance

Poulin et al. (1996). Model: Method 1

Kc 4

Ti

(449)

R

X

2^m+Tm)

291

Td

Comment

Tm

0< , T") , < 2 (T m /N)

3.33
m

tm

T m /N 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

x

x2 0.5231 0.5237 0.5241 0.5245 0.5249 0.5320 0.5315 0.5311 0.5311 0.5311 Direct

T m /N 1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0 synthesis

2l(0.2T m +T m )

T

i

5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397

S K (450)

Xl

x2

4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099

0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314

N = 5. Mmax = 1 dB

Robust Chien (1988). Model: Method 1

Tm

Kra(^ + 0

2

T

2X + xm

2X + xm

Km(X + t J 2

' *- + *m

Tm

^e[T m ,T m ] (Chien and Fruehauf (1990Y>-N=10

Handbook of PI and PID Controller Tuning Rules

292

v

R(s)

E(s)

®+

K

f

U(s)

, ~\

1+-

V

TiSy

1

l + NTds T s i y 1 + Tds j

1+-

4.4.9 Classical controller 2 G c (s) = K

1 + NTds V

1 +

T

d«

s(l + sTm)

j

Table 103: PID controller tuning rules - FOLIPD model G m (s) = Rule

Ti

x, values deduced from graph: 1 2 3 ^m/T m x

l

^m/Tra X

l

K m e
Comment Td Direct synthesis: frequency domain criteria Maximise 6 0.106x! 00 a7 0 3 crossover K 0.135Tm tm mTm frequency; N e [10,30] Kc

Hougen(1979)page 338. Model: Method 1

Y(s)

K„e-"»

4

5

6

6

3

2.1

1.8

1.6

1.3

7

8

9

10

11

12

1.1

1.0

0.9

0.8

0.7

0.7

Chapter 4: Tuning Rules for PID Controllers

293

4.4.10 Series controller (classical controller 3)

R(s) ~

E(s) K

&

v

T s

(l + sTd)

i y

G c (s) = Kc 1+ (l + sTd) v Tisy U(s) Y(s) Kme" s(l + sT m )

Table 104: PID tuning rules - FOLIPD model G m (s) = Rule Skogestad (2003). Model: Method I

Kc

T; Td Direct synthesis

0.5 K

mXm

8t m

T

Comment

294

Handbook of PI and PID Controller Tuning Rules

4.4.11 Classical controller 4 G c (s) = K c

'. + -L' V

R(s) •

E(s)

Kr

1+T

V

&

i

s

/

T

sTri

•?*

1 +

is/

1+-

U(s)

N Y(s)

K m es(l + s T J

1 + sN

K m eTable 105: PID controller tuning rules - FOLIPD model G m (s) = -


Rule

Kc

Td

Comment

2X + Tm

Tm

Tm

2X + xm

(Chien and Fruehauf (1990)); N=10

Ti

Robust 2X + xm Chien (1988). K m (^ + x m ) 2 Model: Method 1 Tra

Kmfr + 0

2

Chapter 4: Tuning Rules for PID Controllers

295

4.4.12 Non-interacting controller based on the two degree of freedom structure 1 THS PT d s U(s) = Kc 1 + - U R(s) E(s)-K c a + TiS l + (T d /N> l + (T d /N>

a+

PTds T

K„

N ;

U(s) Y(s) K e"ST"

R(s) +

K, T S

'

E(s)

+

l+ ^ s N

s(l + s T n )

Table 106: PID controller tuning rules - FOLIPD model G m (s) = Rule

Minimum ITAE - Pecharroman and Pagola (2000). Model: Method 4; Km = l; Tm=i; P = 1,N=10; 0.05 <x m <0.8

K m e
Comment Kc Ti Td Minimum performance index: servo/regulator tuning x,K u x2Tu x3Tu Coefficient values a X x2 *i 3 *. 1.672 0.136 0.366 0.601 -164° 1.236

0.427

0.149

0.607

-160°

0.994

0.486

0.155

0.610

-155°

0.842

0.538

0.154

0.616

-150°

0.752

0.567

0.157

0.605

-145°

0.679

0.610

0.149

0.610

-140°

0.635

0.637

0.142

0.612

-135°

0.590

0.669

0.133

0.610

-130°

0.551

0.690

0.114

0.616

-125°

0.520

0.776

0.087

0.609

-120°

0.509

0.810

0.068

0.611

-118°

Handbook of PI and PID Controller Tuning Rules

296 Rule

Kc

Taguchi and Araki (2000). Model: Method 1

1K

(451)

T

Comment

Td

Tj

(451)

T x

(451)

^s-<1.0 T

(452)

N=0; p = 0.

d

Direct s vnthesis Wang and Cai (2001). Model: Method 1

2

K

(452)

T

(452)

T

*d

5.9784xm

3T

(453)

5.9112xm

4

(454)

0.6068 K ra x m

Wang and Cai (2002).

0.6189

Model: Method 1

Xu and Shao (2003a).

5

Xu and Shao (2003c).

K„

,(451)

T

K

2(T m +1.5x m )

(455)

1.5xm+0.5Tra 2K m x m

Tm+tm

2

(T m +0.5x m ) 2 (T m +1.5T m ) l-5Tmxm 1.5x ra+ 0.5T m

Model: Method 1 6

Model: Method I

0.5184 0.7608+ [-2- + 0.01308]z

0.03330 + 3.997-

-0.5517

0.03432 + 2.058-

-1.774

3A (451)

= T

-0.6878

a = 0.6647 , P = 0.8653 - 0.1277(xm /T m )+ 0.03330(xm /T m f, N fixed but not specified; OS < 20% . K

(452)

1.3608-

1.2064

T

M„ (452)

,452) 1

(T m +0.1x m Xl.450M s -1.508) 1.3608MS-1.2064

=

x m (l.3608M s -1.2064) 0.29M,-0.3016 0.2M, , 1.3<M , a =' 1.3608MS-1.2064

Td(453) = 0.8363(Tra + 0.1x m ), a = 0.3296 , M s = 1.6 ( A m > 2.66 , <|)m > 36.4°) Td(454) = 0.8460(Tm + 0.1x m ), N=0, P = 0, a = 0.3232 , A m = 3 , <> | m = 60°. 0.5(Tm+1.5x„ (455) 0 a = K :T m +1.5x N=co,p = 0 , A m > 2 , 4 m > 6 0 K m x m ( T m + x m );> ' .5T+0.5X

, P = 0.6667 , N= oo , Am > 2 , $m > 60°

<2.

Chapter 4: Tuning Rules for PID Controllers

297

4.4.13 Non-interacting controller 4 U(s) = Kc 1 + T-

,sy

R(s)

U(s)

E(s) + X

K, 1 + — V

E(s)-K c T d sY(s)

Kme-

Y(s)

s(l + sTJ

T S

i 7

KcTds

I

Table 107: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

Comment Td Minimum performance index: regulator tuning

Minimum IAE 1.16 Shinskey(1994) K m (x r a +T m ) -page 75. Minimum IAE 7 ^ (456) Shinskey(1994) -page 159.

7

K

(456)

K m e" s(l + sT m )

T

i

l-38(t m +T m )

0.55(T m +T m ) Model: Method 1

-p (456) i

0.48x m+ 0.7T m Model: Method 1

1.28 K m x m (l + 0 . 2 4 - ^ - 0 . 1 4

T

(456,

=1.9x„

l + 0.75[l-e

Tm "\ Tm

]

Handbook of PI and PID Controller Tuning Rules

298

4.4.14 Non-interacting controller 6 (I-PD controller) U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s) T,s R(s)

>6*

+

E(s)

U(s)

Kc

' W

T iS

1

L

Y(s)

K m e-™

— •

" s(l + sT m )

K c (l + Tds) * Table 108: PID tuning rules -FOLIPD model Gra (s) = Rule

Kmes(l + sTJ

Comment T( Td Minimum performance index: regulator tuning 16 4 (8-P)T Arvanitis et al. (16-3P)K T Model: Method 1 (3 Tra 16 m m m (2003b). Minimum ISE 8 Minimum j[e 2 (t) + K m V ( t ) ] d t 9 Kc

0

P = 1.5841 + 0.3101| —

'O* + 0.219181 — ^

+0.38841

- 0.4622

T

'O

-0.060691

6

X,

•0.0012466 - ^ H +9.1201.10"5 T T

+ 0.010771

T

T V m jf

•3.8302.10" i*, T

+7.0316.10"8

\10

V1m J

V m

T ,

( \4 (- \ '^^ + 1.0972 M -2.7957 + 1.9019 T T T V mJ

P = 0.066485+ 2.0757

f

T

f

^

+ 0.14763 1M.

-0.68418

,T

+ 0.0016016

T

-7.2368.10" ^ - | +1.397.10"6

0<-S-<4.0;

T P = 2.128638-0.005608121-^-4.0

V mJ

^

•0.019733

T„

•>4.0.

t

T

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Arvanitis et al. (2003b). Model: Method 1

10 j ,

T;

Comment

Td

T (457)

(457)

299

(457)

T

Minimum ISE '' Minimum performance index 12,13

10

K

(457)

12.5664(x m + T m )-4 K m x m [25.1328(T m +T m )-8-(x m +2T m )a] r^™„ 3.4674x m +3.1416T - 1 , with a = [0.5708 + J ^ ^ ]% , ._

T (457)

'i

11

12.5664(x m +T m )-4

-

't

-0.7679 I™.

-0.1458

T

_

d

T

T 1

m

ra

2

12.5664(x m +T m )-4

(~

'o

\

V m

7

-0.0013561 H 2 -

+ 0.017833

T

> T , V my

-1.0823.10"

T

-2.61031

V m

V m

+ 5.83.10"

1

+ 5.5784 -^-\

-7.0371

T V 'my 5

\

x

A

x = 1.178 + 4.5663 f

T <457> _ T '

> T

0<-2=-<3.5, T

T

X = 2.2036 - 0.0033077| ^ - - 6.5 , ^ - > 3.5 . 00

:

Performance index = |je 2 (t) + K m V ( t ) ] d t f

13

x =-0.1185 + 4.781^2-]-3.9977 /

\ X„

+ 0.057708

m y

T

-1.733

-0.42245

>T

T m y

V. 1 r a y

0.0038421

+ 2.6386.10"

T X V

m y

-4.2486.10"

T V.

m

\4

/

X v

5

T V

x \

V

m y

T -i2-| +1.078.10"5' ^ s vTm

Handbook of PI and PID Controller Tuning Rules

300

Rule

Kc

Comment Td T, Minimum performance index: servo tuning

16 (16-3B)K m x m

Arvanitis et al. (2003b).

(8-P)

4

(3Tm

16

m

Model: Method 1

Minimum ISE M Minimum performance index 15, 16

14

p = 1.2171 + 0.7339Us- +0.083449\—\ - 0 . 1 4 9 0 5 ^ 2 -

T

l J

ITJ

^5

/ + 0.0045567

T

-0.0029943p

T

\Jm)

-0.00056821 1

-5.6642.10" M 2 -

|e 2 (t) + K m 2 u 2 (t)Jit

Performance index =

'P = 0.083709+ 2.28751

.18739

+0.067859-^2-

-3.5473.10" 5

•1.0059PB

T V

lTmJ m

IT.

m J

-0.0093104

lTmy L

m J 8

+ 2.7505.lO" !^ -

-0.3036^2-

lTmJ -5.4957.10"

T V

6

+ 0.030257^

+ 6.9487.10" I™.

VTJ

(-

\

T

6 = 2.444547-0.0146274714 •Is.-4.5 1 ^ 2 _ > 4 . 5 . T I T

v J

'T,

+ 0.00077161

, o<^2L<4.5

Tm

-0.79718 T

T

A

Chapter 4: Tuning Rules for PID Controllers Rule Arvanitis et al. (2003b). Model: Method 1

Kc

T;

1 7 K (458)

T(458)

301 Comment

Td T 1

(458) d

Minimum ISE 18 Minimum performance index 19,20

17

K

(458)

12.5664(x m +T m )-4 KmTm[25.1328(xm + T m ) - 8 - ( x m +2T m )a] 3.4674xm +3.1416Tm - 1

with a = [0.5708- (458)

12.5664(T m +T m )-4

(458)

= Tm - T *-m

12.5664(x r a +T m )-4 f

X = 1.1571 + 5 . 0 7 7 5 ^ . - 8 . 2 0 2 1 ^ vTm

ITJ

l

+ 7.0439.10"

m

-3.1355

UJ

ITJ

+ 0.92683-2!- -0.17635 T V my V m 5

+6.6353^-

+ 0.02158

T Xr

-0.0016402

J

^

X„

T V

m y

0<-in-<4,5 T

-1.3056.10"

X = 2.124307 - 0.004844182| i = - - 4.5 , — > 4.5 .

' Performance index = |je 2 (t) + K m 2 u 2 (t)Jdt

20

x =-0.70861+ 7.4212| ^2-1-8.1214 'O' T

V

•0.056975 —

+ 0.38998 ^

+ 9.3969.10"6 - a v

+ 4.8168

x.

-1.1861.10"

X = 2.1069751341 - 0 . 0 0 1 7 0 9 8 - ^ - - 4 . 5

-1.7217

m y

+0.0053141^-

111/

A3

T V

m y

V.T m J

V 'm y

r

-0.00030233^

V m y

0<_E-<4.5 T •>4.5.

V m y

]X

Handbook of PI and PID Controller Tuning Rules

302

Rule

Kc

T|

Arvanitis et al. (2003b) (continued).

Comment

Td

Minimum performance index 21,22 Direct synthesis

Arvanitis et al. (2003b). Model: Method 1

(459)

k +lk

24 K

(460)

y (460)

Performance index =

22

2

K

23

J

e 2 (t) + K m

% = 0.05486 + 0.816621 X

2

-1.4989pa V

-0.013386-^-

^

lTmJ

-5.307.10' V

m J

T

(460)

-1.1955 p V

-0.48464p-

m j

-0.00059665 ' x .

+ 1.6437.10"7

T

16(2i; 2 +l) T m

dt

l dt

V mJ

+ 0.11011

—

16^2 + 4

T

N?

+ 3.8633.10"5 - ^ L

0<-2L<6.0; T

m J

X = 2.1111 -0.001025 -^S--6 T, 2 16(2^ +l) 23 JJ- (459)

->6.

(3242 +4)K m T m ' 24

v

K

(460)

^+Tm+4Tm^2 2

2

K m t m (l+85 7

Tr)^m+Tm+4Tra^(Tr0,=

T

mTm

^+T

m

(l + 4^) +4t

m

^

Chapter 4: Tuning Rules for PID

303

Controllers

4.4.15 Non-interacting controller 8 /

U(s) = Kt

V

T

1

\

is

J

EC^-KiCl + T^^YCs)

Table 109: PID controller tuning rules - FOLIPD model G m (s) =

Rule

Kc

Wang et al. (2001b). Model: Method 7

0.4189 K

Kme" 5 (l

+ sT r a )

Comment

T( Td Direct synthesis 1.25(Tm+0.lTm) 4-t

Km^m

mTm

T d i =0; A m =3;
i „ (461)

Ki = 0.25KU , 1K

(461)

T

T

(461)

Tdi=0

i

0.25K,,

•xm

ft + O (461) _ 0-25K u

(461)

, (461)

+

tm +

2(A. + T „ )

4T m

2

Tm

|

2(^ + t r a ) :

Tm

Tm2(l-0.25KuKmTm),

0.5KuKm(^ + x J

J

Handbook of PI and PID Controller Tuning Rules

304

4.4.16 Non-interacting

controller 11 U(s) = K(

(l + T d s)E(s)-K i (l + T dl s)Y(s) T s

v

i y

E(s) R(s)

+

U(s)

f

1^

v

iy

< 8 ^ K C 1 + -L(l +Tds)^0 T s

>

Y(s)

K„e"

s(l + sTm)

Ki(l + Tdis)

I

Kme" " ' &

Rule

Kc

Majhi and Mahanta(2001). Model: Method 6

0.5236

" " V U

1

"

"

"

mv /

T| Td Direct synthesis

s(l + sT m ) Comment Tdi=Tm;

Kmtm

2xme^

T

KmTm

Am=3; 4»m=60°

Chapter 4: Tuning Rules for PID Controllers

305

4.4.17 Non-interacting controller 12 \ f 1 U(s) = Kc 1 + — + Tds T

V

R(s)

is

j

E(s)

1 E(s)-K,Y(s) Tds + 1

+ U(s)

1

•w § -

Tds + 1

K e"

Y(s)

(l + s T j

K,

I Table 111: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

Td

Ti

K.e" s(l + sT m ) Comment

Robust Lee and Edgar (2002). Model: Method 1

l K

^,

=

2

T

(462)

K, = 0.25KU

K

u

K

m

4T -- + K u K. m

-

C , Tm4 6(^ + x m ) 4(X + x m ) 2

- + 0.5xJ.

4 --Tm KuKm

^

2(^ + x m )

KuKm

4Tm

(462)

T (462)

0J5KJL_J__Xm+_L ^+tm

-(462)

(462)

K

+

+

x 2(X +

2

-

(l-0.25K u K m x m )x m 0.5K u K m (^ + x J

Tm)

0.5xJ-

0.5x

f

V , C , (l-0.25KuKmQTm 6(^ + x ra ) 4(^ + x r a ) 2 0.5KuKm(X + x m ) (l-0.25K u K m x m )x„

X

" 6(^ + x m )

4(^ + x m )

2

0.5KuKm(A. + Tm)

Handbook of PI and PID Controller Tuning Rules

306

4.4.18 Industrial controller U(s) = K

1+ — R ( s ) - ^ Y ( s ) N U(s)

R(s) K, 1+-

+

K e" TiSy

j Y(s)

s(l + sT m )

1 + Tds l+

^ s N

Table 112: PID controller tuning rules - FOLIPD model G m (s) = Rule Skogestad (2004b). Model: Method 1

K„e"

s(l + sT m )

Comment Ti Td Direct synthesis: time domain criteria 'good' 1 robustness T Km(TCL+xra) ^2(T + TJ Kc

CL

TcL = Tm ' 4 =0.7 or 1

N e [5,10]; N = 2 if measurement noise is a serious problem

Chapter 4: Tuning Rules for PID Controllers

4.4.19 Alternative controller 1 Gc (s) = Kc

307

1 + Tds

1 + TjS

N

N j

Table 113: PID controller tuning rules - FOLIPD model G m (s) = Rule

x

Tsang et al. (1993). Model: Method 2

T

N = 2.5

0.25xm

Coefficient values *i

Tsang and Rad (1995). Model: Method 2

i

Kmtm

K m e"
Td T, Direct synthesis

Kc

;

%

1.6818 0.0 1.3829 0.1 1.1610 0.2 0.809

x

%

l

0.9916 0.8594 0.7542 T

0.3 0.4 0.5

%

Xl

0.6693 0.6000 0.5429 0.5xm

0.6 0.7 0.8

Xl

\

0.4957 0.4569

0.9 1.0

Maximum overshoot = 16%; N = 8.33

Handbook of PI and PID Controller Tuning Rules

308

4.4.20 Alternative

controller 2 \C 1 + TjS

G c (s) = K£ V f

2„2

l + 0.5Tms + 0.0833T„ s N

N .

\/ 2

1 + TjS

K,

1+

„ 2

l + 0.5xms + Q.0833xm s

^

Kme"

l+ ^ s N

N

Y(s)

U(s) s(l + sTm)

Table 114: PID controller tuning rules - FOLIPD model G m (s) = Rule

Kc

T

i

Kme" >(l + sT m ) Comment

Td Direct synthesis

*i

Tsang et al. (1993). Model: Method 2

K m

T

T

m

N = 2.5

0.25xm

Coefficient values *1

4

1.8512 1.5520 1.3293

0.0 0.1 0.2

x

l

1.1595 1.0280 0.9246

% 0.3 0.4 0.5

x

i

0.8411 0.7680 0.6953

\ 0.6 0.7 0.8

x

i

0.6219 0.5527

\ 0.9 1.0

Chapter 4: Tuning Rules for PID

309

Controllers

4.5 SOSPD Model G m (s) =

K

fi-

ll+sTmlXl+sTm2)

orG m (s) =

Kme~ 2 2

Tml s +2^Tmls + l

4.5.1 Ideal controller Gc(s) = Kj 1 + — + Tds T:S

R(s)^E(s)

+S

{ I

1 T s i

U(s)

1 J

i

Table 115: PID tuning rules - SOSPD model G m (s) =

Gm(s) = Rule

Vis)

SOSPD model

Kc 1 + — + T d s

w

Kme"

(l + sT m ,Xl + sT m2 )

Kme" Tml2s2+2^mTmlS + l

Kc

Ti

Comment

Td

Process reaction Auslander et al. (1975).

1K

(463)

(463)

T

Model: Method 3

0.25Tj (463)

i

Note: equations continued into the footnote on page 310.

K

1.2

(463) . K

m(Tnil

~Tm2)

, +

T 111

TmlTm2 rr^

l

m\

~

, Tjni In

ry^

r-r->

X

X

ra2

(Tm/

Tm2 Tml-Tm;

T ml 1

m2

T

in2

T

, 1T„„-T m ! m2 '

5n2.1T--T-

(Tml-Tm

| !

T m2

j T m2

Tmi ~ T m 2 ^ T m ,

Trai Tmi

I T m2

Tm2 ^ T m l j

T m l -T m 2

1

Handbook of PI and PID Controller Tuning Rules

310 Rule

Kc

T

Comment

Td

i

Minimum performance index: regulator tuning Representative 5.8 tuning values; 0.36TU 0.21TU Km

Minimum IAE Wills (1962b) - deduced from graph. Model: Method 1

5 Km

0.29TU

0.29TU

0.38TU

0.15TU

= 0.1T n l

T Minimum I A E Shinskey (1988) -page 151. Model: Method 2

0.62K U

T

m2

m

Tm 2 0.68K U

0.33TU

0.19TU

0.79K U

0.26TU

0.21TU

x,/Km

x2Tml

x3Tml

T

T

- 05

m 2 + *m

T

Minimum ITAE - Lopez et al. (1969). Model: Method 17

T

+

m2

- 0

m2 +

1m

Tm,

1

Representative coefficient values (from graphs) x

i

25 0.7 0.35

25 1.8 9.0

x2

$« 0.5 0.5 0.5 1.0 1.0 4.0

*3

0.5 1.3 5 0.5 1.7 2

0.25

1.2 1.0 0.2 0.7 0.45

^m/Tmi

0.1 1 10 0.1 1 1

• (463) T„, 2

2

x

(Tm,-Tm2)

VTml

Tm,-Tm2

m2

(T

~\ T m , - T m 2 1

2T m l T m 2 T-ral

Tm2

Tml

)

m2

T

V ml J

+ 2x„

Tm2 ,

1+-

m2

Tml _ T m 2

J

X

m2

T V 'ml )

Tmi

Tm2

T

m2

V ml

)

7S

Chapter 4: Tuning Rules for PID Rule

Kc

Minimum ITAE Bohl and McAvoy (1976b).

2

2

T

Comment

Td

T,

(464)

K

311

Controllers

(464)

T

(464)

Model: Method I

T _E!l = 0.12,0.3,0.5,0.7,0.9; - ^ - = 0.1,0.2,0.3,0.4,0.5

o^y

> -1.2096+0.1760 In 1

K,

(464)

10-9507 K„

T

V

ml J 0.1044+0.1806 In 10^=2- -0.2071 In 10-^=-

l Tm,J

10-

N 0.7750-0.1026 In

T i < 4 6 4 ) =0.2979T n l

{ TmlJ

10-

10-2 T

V

ml ) J 1 (&L > 0.1701+0.0092 In

+0.0081 In [ 1 0 - ^ -

10-2 T V ml / v 0.6025-0.0624 In 10-

T d < 4 6 4 ) =0.1075T n ,

10- 2 V

T

T„,

ml J 0.4531-0.0479lnl 1 0 ^ - |+0.0128In| 10T„,

10'ml

)

Tmi;

Handbook of PI and PID Controller Tuning Rules

312 Rule

Kc

Minimum ITAE Hassan (1993).

J

3

R

T,

(465)

Comment

Td

(465)

T

T

Model: Method 7

(465)

i

0.5 < ^ m < 2 ; 0.1 < — ^ < 4 . K c ,40: " is obtained as follows: Tmi

log[K m K c (465) ]= 1.9763370-0.6436825^m - 5.1887660-^2- + 0.43755584„ + 2.9005550^2-]

+3.14680104 m ^!_-0.16972214 n V^ml

-0.81618084n

T

1.2048220^m2 -5!- - 0.08103734m3 f

-0.4444091-^-1 +0.03194314n

\3

VTml J

+ 0.1054399^

VTmiy

-0.16528074 m 3 ^i_ + 0.11759914 m 3 [^2-| -0.03752454 n 3 '

T

" ml J

T:

(4 5)

is obtained as follows: . (465)

-0.7865873 + 0.6796885^m + 2.1891540-^2-- 0.3471095^m2 Tmi

lot

-1.9003610 - ^ M -0.70078014 m -^- + 0.3077857^n ^Tmi J Tml -0.8566974^ 'ml y

- 0.2535062^m2 ^ - + 0.0412943^m3 Tmi

( \ x -0.1626562^ ^ = + 0.3484161 VTml J

+ 0.2247806£m3 - $ - - 0.2470783^n ^ Td IOE

(465)

T,

V T m1 )

-0.06618994 m 2

v 1mi y

+ 0.049301 l ^ m 3 U =

is obtained as follows (continued into footnote on page 313): (465)

= -0.6726798 - 0.20724774m + 2.6826330^2- + 0.0807474^m2

-1.7707830 -^s- -1.6685140^ m -^- + 0.0845958^m2 Tmi ^Tml J Tml

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum ITAE - Sung et al. (1996). Model: Method 9

4 j,

T

(466)

(, + 0.7159307^n

Comment

Td

i

rj, (466)

\

T

0.05 < i s - < 2

(466)

+ 0.56314474m2 i * - - 0.0225269£m3 Tmi

T

V miy

-0.0616288^ i s - | -0.0626626^

+ 0.2821874

313

2

V^ml

V^rol

m] /

- 0.0372784^m3 i s - - 0.0948097^m3 i s - | +0.0272541^

3

V

ml .

Formulae continued into footnote on page 314. r (466)

K

-0.67 + 0.297

T

K„ V

K

(466) _ __1_

+ 2.189

T

-1.4

+2.189

,

->0.9;

T

V1ml

V i ml . (466)

•<0.9 or Tm,

V T ml ,

ml J

-0.365 + 0.260

\ -0.766 1„

N 0.520

2.212

-0.3

•<0.4 or

V T ml /

Ti(466) =T m l {-0.975 + 0.910 •is--1.845 + a} , - i s - > 0.4 with T VTml J ml

a = 1-e

T

5.25-0.1

(466) _

1-e

1.45 + 0.969

-1.9 + 1.576 V ^ml

Handbook of PI and PID Controller Tuning Rules

314 Rule

Kc

Nearly minimum IAE, ISE, ITAE -Hwang (1995). Model: Method 10

5

(467)

K

f

K

5

R

T

s 2 < 2.4

(467) (467)

T

6

Comment

Td

Ti

(467)

2.4 < e2 < 3

(468)

K

(468)

T

7 ^

(469)

T(469)

3 < e2 < 20

+0.052a)H, 2 T m 2 P K m /(l + K H 2 K m )

„ , - ^ ;l-0.435
K c ' 467) (l + K H 2 K j

, (467)

Km0.0697(l + 0.752co H2 t m -0.14503 H2 2 T n , 2 j' T 6

K

d

-=^^fl-^l(l-0,12co K £ m (»u I n (468)

u

x

m +

0,0

3

co

u

S

2 m

).

0.724[l-0.469(0 H2 T m +0.0609co H2 2 T m 2 l N K m /(l + K H 2 K m )

'K

Kc«68>(l

.(468)

+

KH2Km)

o)H2Km0.0405^1 + 1.93oiH2Tm-0.363cDH2iTm/ 7

K

(469) _

1.26(0.506)'°"2T'"[l-1.07/e2+0,616/e22]N K m /(l + K H 2 K m ) T

l«"l .(469)

K

c

(1 +

K

H2Km)

) H2 K m 0.066l(l+ 0.824 ln[coH2xm])(l + l . 7 l / E 2 _ l . l 7 / e 2 2 ) '

Chapter 4: Tuning Rules for PID Controllers Rule

Kc 8

~

(470)

T

(467)

i

Hwang (1995)continued

Comment e2 >20

(page 314) Decay ratio = 0.2; 0.2<x m /T m l <2.0 and 0.6
Minimum IAE Wills (1962b) - deduced from graph. Model: Method 1 Minimum IAE Gallier and Otto (1968). Model: Method 1

(470)

K

Td

Ti

315

Tm2

7/K m

0.95TU

x,/Km

9T(471)

0.22TU T x

(471) d

=

T

m

= 0.1Tml Tmi

=

Tm2

Represent itive coefficient values - deduced from graphs x X x2 x2 x3 i 3 V *1 2Tml 2Tml 0.24 0.64 0.053 6.7 0.89 0.67 0.24 1.05 0.24 4.15 0.84 1.50 0.70 0.55 0.26 0.11 0.24 2.35 0.77 4.0 0.52 0.50 0.20 0.25 Minimum ITAE 18 Tmi = T m 2 ; Wills (1962a)«T *0.2TU Kra ^ m =0.1T m l ; deducedfrom Model: Method J graph.

K

(470) _

' H2

fm

1.09[l-0.497(o H2 T m +0.0724a) H2 2 T m 2 l N K m /(l + K H 2 K j T (470)

' "if 7 " =x 2 (T m l + T m 2

+

Kc

fflH2Km0.054(-1

x j , Td<471) =x 3 (T m l + T m 2

(l + K H 2 K m )

+ 2.54a H2 x m - 0.457a> H2 2 x m 2 )' +

xj.

Handbook of PI and PID Controller Tuning Rules

316 Rule

Kc

Minimum ITAE - Sung el al. (1996). Model: Method 9 Nearly minimum IAE, ISE, ITAE -Hwang (1995). Model: Method 10

10 £

11 j r

(472)

j

j

(473)

/ 10

K

(472) _

1

(472)

x 1

(472)

0.471KU Kmcou

(473)

(472) _

1

0.05<-^2-<2

d

Decay ratio = 0.1; 0.2 < -^2- < 2.0; Tml 0.6<^m<4.212

N-0.983

- 0.04 + 0.333 + 0.949

K

Comment

Td

Tj

5 m < 0 . 9 or

-0.544 + 0.308—i2-+ 1.408 *

.Til

if'

5

"

5m>0-9.

Ti(472) =T m l [2.055 + 0.072(x m /T m l )J m , T m /T ml <1 or T/ 472 ' =T ml [l.768 + 0.329(T m /T ml )£ m , x m /T m , > 1 , (472)

T,„i

J _

11

K

(473)

K

0.870

e

0.55 + 1.683

0.822t-0.549(o H2 T m +0.112co H2 2 T m 2 ] N K m /(l + K H 2 K m )

K ( ^ ( 1 + KH2K,J

- (473)

(oH2Km0.0142(l + 6.96(D H2 T m -1.77o) H2 2 x m 2 )' r 12

^

^< 0.613+ 0.613^21-+ 0.117 T

'ml

T v

m,y

2

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Hwang (1995)continued.

Hwang (1995)continued.

13 v (474) _. „ '

Td

Comment

0.471KU

Decay ratio = 0.1;

Ti

13 j ,

(474)

T

(474)

15

K

(475)

T

(475)

16

K

(476)

T(476)

17 K

(477)

T

K

m

m

K

mfflu

0-786]! - 0.441coH2Tm + 0.0569coH22Tm2 K m /(l + K H 2 K j T(474)

_

u

0.471KU

(477)

Kc 2 K m 0.0172(l

317

0.2 < ^-2-< 2.0 ; Tm, 0.6<^m<4.214 Decay ratio = 0.1; 0.2<^-<2.0; Tral 0.6<4m<4.218

h

(l + K H 2 K m ) + 4.62 WH2 x m -0.823co H 2 2 t m 2 )'

E<0.613 + 0.613-^- + 0.117 U ^ Tml ^T m 15 K

(475)

Kt

1.28(0.542) c ° H;tm [l-0.986/s 2 +0.558/e 2 2 } Kc'475>(l + K H 2 K m )

, (475) 2 K m 0.0476(l

16 j . (476)

Kt

2 2 .H[I 1.1411-0.466co H2xm + 0.0647co H2 T m ^ K m /(l + K H 2 K m )

(476)

Kc

r-

T;

17

K

(477)

+ 0.9961n[coH2Tm])(l + 2.13/s2 - 1 . 1 3 / e 2 2 ) '

(l + K H2 K m ) —2—51T m J

coH,Km0.0609 - 1 +1.97co„,xm - 0.323co H2

0.794|1 - 0.541coH2Tm + 0.126(oH22Tm2 " *[l-0.: K m /(l + K H 2 K m ) .(477)

K< 477 '(l + K H 2 K j a>„2Km0.007811 + 8.38coH2Tm - 1.97coH2zxn

£> 0.649+ 0 . 5 8 ^ - - 0 . 0 0 5 Tm, V T ml

)

Handbook of PI and PID Controller Tuning Rules

318 Rule

19 j r

Hwang (1995)continued.

(478)

T

Comment

0.471KU

Decay ratio = 0.1;

(478)

T(479)

21 j r (479)

Hwang (1995)continued.

Td

T,

Kc

22 K

(480)

T

(480)

23 K

(481)

T

(481)

K

mwu

0.2 < -^5- < 2.0 ; Tn, 0.6 < ^ m < 4.2 20 Decay ratio = 0.1;

0.471KU

0.2 <-^2-< 2.0; 0.6<^m<4.224

19

K

(478) _

j ^

2 2 0.738[l - 0.415oo.H2xm + 0.0575co H2 xm \> , K m /(l + K H 2 K m )

H 2

(478)

T

K

1

20

X

( T

^

£,> 0.649 + 0 . 5 8 - ^ - -0.005 —s 'ml

21

(l + K H 2 K m )

a H2 K m 0.0124(l + 4.05co H2 x m -0.63co H2 2 x m 2 )'

K

(479)

V^miy

1.15(0.564) m " iT "[l-0.959/e 2 +0.773/e 2 2 ]"

Kt

K m /(l + K H 2 K m ) K

T.(479)

c

J' (1

+ K

H2Km)

coH2Km0.0335(l + 0.9471n[coH2xm])(l + 1.9/8 2 -1.07/e 2 2 )' 22

K

i

1.07|l-0.466co H2 x ro +0.0667 mH2 2 x m 2 ] N , J K m /(l + K H 2 K j

(480) _ K

H 2

v

Kc'^(l + K H 2 K j

r (480)

mH2Km0.0328(-1 + 2.21C0H2X,,, - 0.338coH22Tm2)' 23

K

2 2 0.789|l-0.527(D ?[i - o.: H2 x m +0.11 C o H2 T m l^ —-• = K m /(l + K H 2 K m )

(481) K

H 2

.(481)

K™(1

+

KH2KB)

) H2 K m 0.009(l + 9.7co H2 xlmm -2.4co H , 2 T m 2 )' I < 0.649 + 0.58—— - 0.005 T

T

V ml 7

,^>0.613 + 0.613^a- + 0.117

X„

T

V ml J

Chapter 4: Tuning Rules for PID Controllers Rule

Kc 25 K

Hwang (1995)continued.

T

Comment

Td

Ti

(482)

319

Decay ratio = 0.1;

(482)

27 K

(483)

rp (483)

28 K

(484)

T

0.471KU Kmwu

Q.2<^-<2.0;

(484)

Tmi

0.6
"[e2(t) + x 2 2 K m 2 (y(t)-yJ 2 ]dt; Tm, >T m2

Minimum

Wilton (1999). Model: Method 1

•0

Tm2 1 + T m2 /T ml Coefficient values (obtained from graphs) T r a 2 /T m l =0.25 T m 2 /T m l =0.5 T m2 /T m , =0.75 x

i

T m l +T m 2

x

i

x2

x

i

1250.1 0.0001 22.946 0.04 20.396 4.8990 0.2 4.8990 2.1213 1 2.1213

25

K

(482) _

'

x2

i

x2

22.946 4.8990 2.1213

0.04 0.2 1

x

0.04 0.2 1

xm/Tml=0.1

0.76[l-0.426o) H2 T m +0.0551 mH2 2 T m 2 l A K m /(l + K H 2 K m ) Kc«82>(l + K H 2 K m )

• (482)

+ 4.37co H2 T ra -0.743» H2 i T n

raH2Km0.01531 2

26

27

( \ 'x >' ,4>0.613 + 0 . 6 1 3 ^ - + 0.117 £,< 0.649 + 0.58-^—0.005 T T T V miy Tmi V ml V K

1.22(o.55)"'H'T" [l-0.978/s 2 +0.659/e 2 2 j K m /(l + K H 2 K m )

(483)

K

j t, i(483) oj;

c

(1

+

K

H2Km)

coH2Km0.042l(l + 0.9691n[coH2xm])(l + 2.02/e2 - l . l l / s 2 2 ) ' 28

K

(484)

'

l.ll[l-0.467co H2 T m +0.0657coH22Tm2lN K m /(l + K H 2 K m ) T;

(484)

Kc'^(l + K H 2 K j > H2 K ra 0.0477(-l + 2.07co H2 T nl -0.333 fflH2 2 T m 2 )'

Handbook of PI and PID Controller Tuning Rules

320

Rule

Kc x

Wilton (1999) continued.

i +

Tmi

Comment

Td

Ti Tm2

Tm2 1 + T m2 /T ml

Coefficient values (obtained from graphs) T m 2 /T m l =0.25 T m 2 /T m l =0.5 T ra2 /T Ml =0.75 x

i

x2

x

x2

i

x

x2

i

180.01 0.0001 230.01 0.0001 300.01 0.0001 8.1584 0.04 9.1782 0.04 10.708 0.04 0.2 3.4293 0.2 3.1843 0.2 2.9394 1 1.4142 1 1.3435 1 1.2728 90.004 0.0001 100.00 0.0001 112.01 0.0001 5.354 0.04 4.5891 0.04 4.6911 0.04 0.2 2.0821 2.1311 0.2 2.0331 0.2 1 1.0748 1.0324 1 0.8768 1

T m /T m l =0.5

xm/Tml=l

29 K (485) (485) Keane et al. T (485) T 'd (2000). Model: Method 1 Minimum ITAE servo plus ITAE regulator, with minimum Mmax and with good noise attenuation; Tml = Tm2 1

Huang and Jeng (2003). Model: Method 13

29

K

(485)

30 j -

(486)

I

2^ m T ml

( l n K j T u + T m , ( l + lnK u )]

T

(485) _

T

(486)

Sn.Sl-1

T T 'u 'ml

T u +T m l (l + l n K u ) ' (lnK u )[T u +T m l (l + lnK u )]

-(485) _

(lnK u )(l + lnK u ) + [lnK u +ln(l + e T »- K -)][ln(K u +1.33419T m )-l] 30 K (486) _ 1.2858Tm^m [ T„ KmTm (486) _ [(-0.0349^ m +1.QQ64X +(0.4196^m -0.1100> m P 2T,^ m

Chapter 4: Tuning Rules for PID

Rule

Ti

Kc

Huang and Jeng (2003) continued.

31

K

Controllers

(487)

Comment

Td

(487)

T

321

(487)

T

4 m > 1-1

i

Servo or regulator tuning - Minimum IAE Direct synthesis: Frequency domain criteria

Hang et al. (1993a). Model: Method 8;

^ m l

0-5T ml

2T m l

AmKroTm

Tml ~~ Tm2

Sample results Ara=2.0,

1.5708Tml ^2->0.3 Tml (Yang and Clarke (1996)).

KmTm

K

(487)

T/

2T ral

0-5T ml

Am=3.0,

Kmxm 32 K

T(488)

(488)

T

*m=60°

(488)

Tm/Tm<2^m

i 33 j

MpTml

(489)

T m2

Tm, > T m 2 .

A n ,K i n

T 0 . 0 0 5 2 - 2 ^ - + 0.8980T m2 + 0 . 4 8 7 7 t m + T m l

0.5890 K m t m y ^ra

487

0-5T ml

1.0472T ml

H o e ; al. (1994). Model: Method 1 Hoet al. (1995a). Model: Method 1

31

2T m l

V

' = 0 . 0 0 5 2 - 2 ^ - + 0.8980T m2 +0.4877x m + T m l , T 0.0052 - = ^ - + 0.8980T m2 + 0 . 4 8 7 7 i n

T

(487) _

T

0.0052 - = = - + 0.8980T m2 +0.4877x m + T m l J 32 T , (488) _

2co Tm,l 2 f ' " " pn 1 m 7tA„

„5 ,7t-20)T

m

T (488) _ 2 > M

V«pT m , (488) _

2 'ml

f

,

^ i s ! - + t - 2«pxm V T ml«p TtT. • + t T m l -2Tm]copTm

1 2co„

4©D2Tm

1

Handbook of PI and PID Controller Tuning Rules

322 Rule

Kc

Ho et al. (1995a) - continued.

0.7854Tm,

Ti

Comment

Td Sample results

K

m

T

Am=2.0,

m

Tml

Tm2

Tm,

Tm2


0.5236T ml Kmxm

Ho etal. (1997). Model: Method 1 Leva et al. (1994). Model: Method 15

34

34 K

36 K

(490)

K

(490)

tAmKm

T 1

(491)

T

(490)

(490)

i

35

(491)

*m=70° (at least).

O^TA 491 '

,Tr0,=-Tmlfem+,-2,m),

**>m+K-2l

7t

ml J

TtT,ml

(490) _

T

T


_ Tm

2 ( ^ m T m l + 7 t T m l - 2 T m ) ' T, 8™m^

< 1 , 2£ > ^ 2 - .

- (Am2-l)-27tAm(Am-l) 4A„

(491) 36

v

K

(491) _

2 2 2 l + »c„ 2 T m , 2 ) + 4 ^ m c o c n T m l

Wcnji

Km

, (491)™ (4911

^

2

r

, ( 4 9 1]) 0w^

1-T^'V X" r+[Ti

4.07 - <|>ra + tan"

, (491)

-tan 0-5 c o c n T m + < | > m - 0 . 5 7 t - t a n "

m

^

i[2^xmTml(0.57I-ctm) (0.57t-<^ r a ) 2 T m l 2 -x m 2

2^ m co c n T m l V^cn

l

2

T

ml

- ! y

o r ^ m > l (with 0 . 5 7 1 - ^ >

3T„

^m+V^2-1

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Leva et al. (1994)continued. Wang et al. (1999).

37 K

Wang and Shao (1999). Model: Method 12

1

T 1

(492)

(492)

T

i

Model: Method 12.

0-5Tml

SmT r a l K

Comment

Td

Ti

(492)

323

2^raTm,

T

m m

4m

38

T K

m

X

m

Some coefficient values Comment *i

*i

1.571

Am=2,
1.047

T 37 ^

2^ m T ml

Am=3,
Comment

0.785

Am = 4 , ^ = 6 7 . 5 °

0.628

A r a =5,<j> m =72°

2 |C0cn2+-V(2^V^ • 1 -1

(492) _ _ ^ c n j _ m l _ (492>

1

ml

V

coj+x2

ll

(0.5TI -(|>m) Tml

-xm

co„

ffl

cnTml

tan

^m+V^n Tralz + U

m

+^

2 m

-l

(492) ,

. (492)

2

Sm + ^ m - 0 with

. 5m>l

Tmlz + Um +

3x„

< 0.571 - < | > m <

£m "V^ir

^ m > 0.7071 or 0.05 <

°-7°7h<0.15, T m l A /24 m 2 -l

QJQ7lT

T

m l

">U„*1 ^ - 1

or 0.05 < % ^ s - < 0.15 , - ^ 2 - > l,^m < 1. Tml

kJ-l Tml

Sm+A^

Handbook of PI and PID Controller Tuning Rules

324

Rule Wang and Shao (2000b). Model: Method 12

Kc

^

Td

39 ^ (493)

2^ m T ml

2^ m

Tml

X

Chen et ah (1999b). Model: Method 1

39

40

m l j£ (493) _ 2" ^»m»Tm

K

(494)

2^m *i

A m = 3 . 1 4 ,
1.22

A m = 2 . 5 8 , ^ = 5 5 . 0 ° , M.=1.10 A m =2.34,
1.40

A m =2.24, 4>m = 50.0° , M , =1.26

1.44

A m = 2 . 1 8 , * m = 48.7° , M,=1.30

1.52

A m =2.07, < L = 4 6 . 5 ° , M S =1.40

1.60

Am =1.96, 4)m =44.1°, M,=1.50

40 K (494)

16(T m 2 +t m )

41 K (495)

5T m2 +4T m

.451-

m

l(5Tm2

T x

(495) d

1.508

4(Tm + 4 T

4(T m 2 +x m )

M„

8K m (T, T

2^ m Comment

1.00

:

41 rr (495) _

Suggested Mmax = 1.6

lSmTmi

1.34

Leva et ah (2003). Model: Method 1

Comment

m)

8K m (T m 2 + T m ) 2

T (495)

,

)

•

4T m2 (T m2 +x m ) 5T m2 +4T m

'

TA , > 8(Tm ml —

+ x„

Chapter 4: Tuning Rules for PID Controllers Rule

Comment T, Td Direct synthesis: time domain criteria

K X

Wills (1962a). Model: Method 1 Servo response;

i/Kra

X T x2Tu 3 u Representative coefficient values - deduced from graphs x2 x2 x2 *3 *1 *3 *1

*i

2.5 2.75 3.2 20 12 1.4 1.5

4 = 1•Cm = ° - 1 T m l

2.9 2.9 2.9 2.9 2.9 1.4 1.4

42 £

Van der Grinten (1963). Model: Method 1

0.035 1.6 0.073 16 0.143 10 0.285 0.42 0.57 0.4 0.035 10 0.073 2 T m l +T m 2

(496)

43 j r

(497)

Pemberton (1972a), (1972b). Model: Method 1

(T m l +T m 2 ) K ra x m

(496)

T m l +T m 2

T r a l +T m 2

K

(497) _

T 1

0.2 8 0.09 0.1 0.11

0.48 0.48 0.28 0.28 0.28

*3

0.143 0.36 0.073 0.143 0.28

(496)

Step disturbance

(497)

Stochastic disturbance

d

TmiTm2 Tml +T m2

Model: Method I

TmiTm2

0.1<^-<1.0 Tm2 0.2<^-<1.0 Tm2

f

Q 5

i

T

ml +

T

(T m ,+T m 2 )x m +2T m l T 1 1lm 2 ^+2(Tml+Tm2)

(496)

m2 I

K. 43

0.143 0.28 0.72 0.073 0.143 0.36 0.70

T

3Kmxra

1

=

(497)

T

2(T m l +T m 2 )

K

1.4 1.4 1.4 0.70 0.70 0.70 0.70

+0.5Tm

Pemberton (1972b).

42

325

T„, + T„

e

1 - Q.5e-a'x- +K„

T ml + T „

• (497)

X

l-0.5e" (Tml+Tm2)T m

• (497)

l-0.5e~

*ml

+

'm2

(T ml +T m2 ) / e~

M T

J m

326

Handbook of PI and PID Controller Tuning Rules

Rule

Kc

Ti

Td

Comment

Pemberton (1972b). Model: Method 1

2(Tral+Tm2)

Tmi +T m 2

Tra,+Tm2

0.1<^k.<1.0 Tm2

Chiu et al. (1973). Model: Method 1

4

3Kmxm

0.2 < ^ 2 - < 1.0 T m2 ^ T m l + Tm2

Tmi + T m 2

K n ( l + XT m )

TmlT,n2 Tml+Tm2

A. variable; suggested values are 0.2, 0.4, 0.6 and 1.0. Appropriate for "small xm "

Mollenkamp et 2s m T m i 2s m T m i Tml al. (1973). Kra(TCL+xra) 2^m Model: Method 1 Smith et al. Wml T m , > T m2 (1975). Tml Tm2 Km(l + ^ t m ) Model: Method 1 X = pol e of specified FO] J D closed loop response. Suyama(1992). T m iT m 2 Model: Method 1 OS=10% T m i +T m 2 Tml+Tm2 2Kmxm Wang and Clements (1995).

2^mTml

24m

K r a (l + ^ m )

Gorez and Klan (2000). Model: Method 1

44 K

(498)

2s m T m i

Tml 2^m

x,(Tml + T m 2 ) Viteckova (1999), Viteckova et al. (2000a). Model: Method 1

K

(498) _

Model: Method 16; Underdamped response Non-dominant time delay

2s m T r a i

T m iT m 2 T m i + T ra2

Kmtm i

OS (%)

x,

OS (%)

0.368 0.514 0.581 0.641

0 5 10 15

0.696 0.748 0.801 0.853

20 25 30 35

x

2^mTml Km(2^mTml+!„,)•

Tm,+Tm2 i

OS (%)

0.906 0.957 1.008

40 45 50

x

Closed loop overshoot (OS) defined Overdamped process; T„i > T m 2

Chapter 4: Tuning Rules for PID Rule

Kc

Ti

iS m T m i KraXm OS (%) x, 0.736 1.028 1.162 1.282

0 5 10 15

Skogestad 2^ m T ml (2004b). Km(TCL+xm) Model: Method 1

Tml 2^m OS (%)

24mTml i

OS (%)

1.392 1.496 1.602 1.706

20 25 30 35

x

Comment

Td

x

Viteckova (1999), Viteckova et al. (2000a) continued.

327

Controllers

*1

1.812 1.914 2.016

40 45 50

Tml 2^m

2^mTml

Underdamped process; 0-5 < ^ m < 1

Recommended = T TCL m

A ra = 3 . 1 4 , <|>m = 6 1 . 4 ° , M m a x = 1 . 5 9 Vitefikova et al. (2000b). Model: Method 4 Arvanitis et al. (2000).

0-74Tml KmTm 45 jj- (499)

Bietal. (2000). Model: Method 8

l-0128^ m T m l

Chen and Seborg (2002).

46 Tj (500)

45

0-5T ml

Tml - Tm2

2^dmesTml

Tml ^edes

Model: Method 1

0.5064T m l 2

l-9747K m x m T

Kmtm

(500)

x

(500)

Model: Method 1

2^sTm

(499)

K

Kmxm

2T m l

6 K (500) ^ 1 [(T m | + T m 2 )x m + T m |T m 2 ](3T C L + T m ) - T C L 3 - 3T C L 2 x m C

"Km

T(500) =

(TCL+^)

3

[(T m[ + T m 2 ) x m +T m l T m 2 ](3T C L + T m ) - T C L 3 - 3 T C L 2 t m T miTm2+(Tmi+Tm2+Tm)Tm

(5oo) = 3T C L 2 T m l T r o 2 + T m l T m 2 x m ( 3 T C L + x m ) - ( T m l + T m 2 + x m ) T C L 3

T

[(T m , + T m 2 )x m + T m ,T m 2 ](3T C L + x m ) - T C L 3 - 3 T C L 2 x m

"

T(500)se-sx„

_ 'desired . d Vdesi

Ti Kc

se -

(TCLS + 1 /

328

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Chen and Seborg (2002) continued.

47 K

(501)

(501)

T

Comment

Td

Ti

(501)

T

i

Robust 48 y-

Brambilla et al. (1990). Model: Method 1

(502)

T

Representative coefficient values (deduced from graph) ^m T

Lee etal. (1998). Model: Method 1

..)

K„(5° ^

Tml + Tm2 1.0 2.0 5.0

0.50 0.47 0.39

T(503)

49 j

X

T m l +T m 2

X

0.29 0.22 0.16

10.0

0.14

(503)

T J

(503) d

Km(2A. + Tn,)

(2^Tm,xm+Tm,2)[3TCL+Tm)-TCL3-3TCL2T

1 _K

•tm

1m

X

ml+Tm2

0.1 0.2 0.5

47

ry, (502) »d

(502)

(TCL+XJ3

m

j (5oi) _ j2^ m T m l T m +T m l J/K~"-CL j 3 T C L'+vtm -3T C L m )/ - TJ CL C + T m l 2 +(2f T . + Trm)/ TT im + l 2 S m 'ml (^^m^ml ++ T m

(501) _ 3 T g L 2 T m l 2 + T m l

T

+ T ml z <3T CL

IdJ, . /desired 48

(502)

K

=

Kc Tml

+T

T m JTcL -3T CL ^

+

(T C L S + 1 / m2 +°-5Tm

K m T m (2^ + l)

(502)

T

'

'

= Tml +T m,2+0.5x m

j (502) _ T m |Tm2 + Q - A T m l d

49 T (503) _

'

7?

~1^'^

T

T 2X 2 ^ ~-Z m 2(2X ihx ++Tir a ) '

+

Tm2/tm

T m ,+T m 2 +0.5T m T d

(503) _ "

T

'

Q| <

'

\

' ~ TT m l^+T m 2

<10.

T (503) _ , , T ml _ ^ml"»+T](503)

x s x 3 e- " —,.-,. .m r ; desired closed loop response = — 6Ti(503)(2>. + Tm) (Xs + l) 2

Chapter 4: Tuning Rules for PID

Rule

Kc

T;

(504)

T

329

Comment

Td

50 y (504) *i

i

Lee et al. (1998). Model: Method 1

Controllers

(504)

T

'd

Km(2A. + T m ) X = max imum[0.25Tm ,0.2Tml ] (Panda et al., 2004) 51

(505)

T

T

(505)

Desired closed loop response =

(505)

T

*d

i

K ra (^ + Tm) rj, (506) 1i

52

(506) *i

(506)

T

T

(X.s +1)

'd

Kro(^ + 0 Huang et al. 24mTral Tml (1998), Rivera 2^ m T ml K m (x ro +A.) 2^ m and Jun (2000). Model: Method I A, specified graphically for different OS and TR values (Huang et al. (1998)) j (507) (507) 53 ^ (507) Model: Marchetti and x 'd Method I Scali (2000).

50

T(504)

_

T

T

2^2-xm2 2(2A + x m )

T

(504) _

T

T

(504)

T

T

mlTm2 • (504)

-rrrrr r ; desired closed loop response = — 6T i (504) (2A + T m ) (Xs + l) 2 (

51 x (505)

Ti^'=2^mTml +

52

T

(506)

:

T

2(^ + T m )

> 'd

(505) _ T (505) ~ li

»). sm

lC

T

3

^

6(A + T J

T

'ml

.(505)

1 m 1 + * m ~> "*"

2(X-

T 1

53

K

(507) _ 2^ m T m l +0-5T m K m (X-

d

(506) _ T (506) ~ M

T(507) = 2 £

J

+ 0 5t

,

( T

I'ml

T

+

T

-v 'ni2'

(5 7)

°

6(X + ,(506)

- Tml(Tml + ^ m O 2^mTml+0.5Tn

Handbook of PI and PID Controller Tuning Rules

330 Rule

Kc

Ti

Td

Comment

Smith (2003). Model: Method 1

Tml

Tml

Tm2

^eK,.Tm2], Tml>Tm2

Kj^ + O

Kristiansson (2003).

54 K

(508)

x,/K m Wills (1962a). Model: Method 1 Quarter decay ratio tuning. Tmi = T m 2 ; ^ m =0.1T m l

i

x2

10 8 14 33 18

2.9 1.4 2.9 2.9 1.4

x

Bohl and McAvoy (1976b). Landau and Voda (1992). Model: Method 1

55 K

(509)

56 £

(510)

2^raTral

TmI 2^ m Ultimate cycle

x2Tu x3Tu Coefficient values (deduced from graph) x x x x2 x2 i *3 i 3 0.036 0.036 0.068 0.143 0.068

55

v

K

K

(508) _

1.4 0.67 0.48 0.28 0.67

4 + (3 1

2^mTml

0.143 0.068 0.068 0.068 0.143

18 0.9 18 18 13

T

(509)

4

1

4 + M,35» 0.13TU

0.51T„

0.48 0.28 0.67 0.48 0.28

X

Model: Method 1

1 < (3 < 2 (» u T m <0.16

M„

T

(509) _ 2 . 2 6 8 9

2.0302+0.9553 In

I

IT.

(V

K

\1.3135+0.18091n(K„K„)+0.52191np!!.

T

K„

r

\-0.0517-0.0643 In

—

(K K ) a 5 9 1 4 - 0 0 8 3 2 1 n ( K « K m ) + 0 0 6 8 7 1 n — 1

T -4.2613-1.58551n

Td(509) = 0.0068TU

56

K

(510)

(K K

4+P

P

4

2V2|Gp(ja>135„)|

3

0.143 0.143 0.24 0.24 0.28

. Recommended values of Mraax : 1.4,1.7,2.0.

Kmxm

T:(509)=0.2693T„

28 6 2 0.6 30 T(509)

3KU

54

Model: Method 1

V 1 1 4 3 6 -°- 2 6 5 8 l n ( K uK m )-lll001n| ^2-1

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

T;

Td

Landau and Voda (1992)continued.

1.9KU

0.64TU

0.16TU

331 Comment 0.16
332

Handbook of PI and PID Controller Tuning Rules

4.5.2 Ideal controller in series with a first order lag G c (s) = Kc 1 + R(s) ms;

^1

Kc|l + T - J s- + Tds - •

,

I

i

J

U(s)

1 Tfs + 1

K

Ti

Comment

Td

(1985).

Model: Method 1

Tmi + T ral

Arvanitis et al. (2000). Model: Method 1 Panda et al. (2004). Model: Method 1

T ra , 2

(512)

K

e^

m

(l + sTm,Xl + sT m2 )

Direct synthesis: time domain criteria Tml+Tm2 lmlT m 2

(511)

^w

T m l 2 s 2 + 2 4 r a T mlS+1

Kc 1K

V(s)

K m e"

G mm(s) v /

Umetal.

Tfs + 1

SOSPD model

Table 116: PID controller tuning rules - SOSPD model G m (s) =

Rule

"Tds

T

-^tdes

2 ^ % ,

T

Tm,

Sm*ml

2^ m T m l

T

Km m

%m

^

T

2^m

ml f

" 2N^m '

N not defined Direct synthesis: frequency domain cri teria Huang et al. (2005). Model: Method 14

Tm, 2^ m T r a l

KmTm

Am*3,
Hang et al. (1993a). Model: Method 8

Robust T m , +0.5x m

2T m l + Tm 2{X + xm)Km

K

(5U) =

2Tml+xm

2 ^ (512)

T,CL2

T m l +T m •.T

K m (2^T c

2CTm, CL2 '•ra/ KJSdm e %,,+0' l

T

f

2

= =

f

T l 4T*CL2 m C L 2+tn

2, ^ ^CL2 T r i , 2~-' lir „n 2

es

2^?m T»CL2 r ,,+Tm T

l

1

^>0.25xm.

^mlTm

Model has a repeated pole ( T m l ) .

1

°-025T ml

T

24 m

^ m

T

'

2(X + i m )

333

Chapter 4: Tuning Rules for PID Controllers

Rule Rivera and Jun (2000). Model: Method 1; X not specified Lee and Edgar (2002). Model: Method 1

Kc

T,

24mTml X

2^ m T r a l

T

Tml 2^ m T r a l

Km(2Tm+X) K

= - c

f

m

2^m

2^mTml 3

Comment

Td

(513)

(513)

X

Tf=

^ 2xm+X

T

*' • 2A. + t m

2^m (513)

T

i

T m iT m 2

T m i + Tm2

Zhang et al. (2002). Model: Method 1

Tmi

Kmk+2^)

+

Tm2

+

Tml

Tm2

H ^ method 4

K

(514)

Tml+Tm2

I ml I m 2

5% overshoot: X = 0.5x m

T

=

f

Tmi + T m 2

Xx

™

X + 2xm

H 2 method T(515)

5T(515)

(515)

T

Tf = 0 ; Maclaurin method

K m ( 2 ^ + Tm)

3

K

(513)

=

_

Km(2X

K

xm)

+

0.5x

(2f; T S mm mml l

2

-^2

2X + x 2^Tml +

6{2X + xJ J (513) l i

.it T - z S m 'ml

0-5xm2-X2^ 0.5xJ-X2 2X + xm 2X + x„

,

+

2A. + T m

6(2^+ xm)

2^ m T m l +

0.5x

2

- ^ 0.5x„/-^

2X + x m

(513)

6(2^+ xm) 4

5

v

K

Ami l

(514)

T(515)

m

2^ m T m l +

0-5x m 2 -X 2

0.5xm2-X2

2A, + x m

2A. + T„

;

2X + x„

9

Km(Tml+Tm2X^ + 2 i m ) '

:

Tml+T„.

2X2-xJ 2(2X + xm)'

t m l ^rtl

• (515)

12A. + 6x m -(515)

2^-x„ 2(2X + x m ) '

;

334

Handbook of PI and PID Controller Tuning Rules Rule

Ti

Td

^Sm^ml

2^ m

Kc

Comment

Kristiansson (2003). Model: Method 1

6

K

(516)

7

K

(517)

Panda et al. (2004). Model: Method 1

8j£

(518)

T(518)

T

(518)

9

(519)

-p (519)

T

(519)

K

7

2^ m T m

(516)

Kmira 2^raTm

K „

K

a-

a -1 + -

1—1

M„

T T

Mn

f

—

ml

• "T'

1 + 0 . 0 1 9 — - JO.346 + 0.038-!-211- + 0.00036 Suggested M m a x = 1 . 7 , p = 10.

8 K (518) _ 2 4 m X 1 + 0 - 2 5 T m l + 0 . 5 ^ m T Km(X + 2^ m T ml ) T(518)

_ 4^ m Tml +0.ST ml +4 m T m

'

2^„

Tf =

9

K

(Tml + 2^ m T m JT m |^ m

(518) _

T

4^ m 2 T m , + ^ m t m + 0 . 5 T m

^(T m l +2^ m x m ) -, X = max imum 0.25 44ma. + 44 m x m +2T m 2^ n 0.828 +0.812-2^-+ 0.172Tmle Jm]

(519)

-0.4^ m T,

-(6.9T„,/Tml) . 0.558Tm2T„, -+ T ml +1.208T ra2

0.5T„

K„

A, + 0.828 + 0 . 8 1 2 ^ - + 0.172Tmle-(6-9T™;/T"") Tmi 0.558T ,T s 9> T/ ' =0.828+0.812ii^+0.172T . e ^ " " ' ^ ^ u ' 3 3 8 1 ^ 1 " " + 0 .5x n Tml T ml +1.208T m2

(519)

0.828 + 0 . 8 1 2 ^ + 0.172T ,e-( 6 ' 9T - /T ">T LU6T^T^ +T Tml »T m ,+1.208T ra2 1.656 + 1 . 6 2 4 - ^ + 0.344Tm,e-(6-9T"'/T-')+

Tf =

L116T T ^ "" T ml +1.208T ro2

+T

4l-'16T m ,T m2 +T ro (T m ,+1.208T ro2 )] 2(^ + xmXTm, +1.208Tm2)+2.232Tm2Tml '

X = max imum

°-279Tm2Tml T ml +1.208T m2

+Q2^

0 - 1 6 5 6 + 0 - 1 6 2 4 l!°2. + 0 - 0344 T

e-MTm2/Tml)

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

335

Comment

Td

T, Ultimate cycle

Huang et al. (2005). Model: Method 14

10

10 K

T (520)

(520)

T

(520)

T

Tf=

(520)

d

20

A m «3,
/

6.283xn

K„ T„

K <520) = 0.080-

T. (520) =0.159K„ Tu

T

T V

6.283t„

u

J

f

Td(520) = 0.159-?K„

\

6.283xr

1 + K„ cos

T V

6.283T„ T

u

J

Handbook of PI and PID Controller Tuning Rules

336

4.5.3 Ideal controller in series with a first order filter f G c (s) = K c T

V R s

( )

E(s) K, 1+ — + Tds

l + bf]s l + a fl s

Kc

T

,s

j afls + l

K„e"

TmlV+2^Tmls + l

Table 117: PID controller tuning rules - SOSPD model G m (s) = Rule

^b s +l fl

Y(s)

U(s) H

-

K m eTmlV+2^mTmlS + l Comment

Td

i

Robust b f l =0.5T m ,

2sm'mi

Jahanmiri and Fallahi (1997). Model: Method 6

Km(tra+^)

2s m T m i

24m

^ m

"" X = 0.25T m+ 0.1^ m T ml

2(T m +^)

Chapter 4: Tuning Rules for PID Controllers

337

4.5.4 Ideal controller in series with a second order filter ( 1 G c (s) = Kc 1 + — + Tds T

V

l + b fl s ) l + a fl s + a f2 s

is

U(s

R(s) E(s),— T:S

l + afls + af2s

T ml 2 s 2 + 2^ m T ml s + 1

Table 118: PID controller tuning rules - SOSPD model G m (s) = Rule

Kc

Ti

Y(s)

K m e"

l + bfls

K m e" TmlV+24mTmls + l

Td

Comment

Tml 2^ m

Mmax=1.7,

Robust Kristiansson (2003). Model: Method 1

11

K

(521)

2^mTmi

b If = 0 , a I f =0.08T m I ,a 2 f =0.01T ml

II ^

(521) _ 2 ^ m T m l

P = 10. 2

1 + 0 . 0 1 9 ^ - - .0.346 + 0.038-^- + 0.00036-=^

Handbook of PI and PID Controller Tuning Rules

338

r ,

4.5.5 Controller with filtered derivative Gc(s) = K,

R(s)

,

& - K

T

1

+

T s

i

U(s)

ds

Y(s) SOSPD model

l+ ^ s N

Table 119: PID controller tuning rules - SOSPD model G m ( s ) = G m (s) =

Rule

Kme" (l + sTmlXl + sTm2) or

Kme" Tml2s2+24mTmls + l

Kc

Comment

Td Direct synthesis

Ti 12

WpTml

Hang et al. (1994). Model: Method 1

ds

NJ

c

E(s)

T

!

T

(522)

N = 20.

Tm2 T

AmKm

ml > T m 2

Sample result 0.7854T m l

Tml

Am=2.0,

Tm2

Kmxm

<J)m = 45° Robust

Hang et al. Tml (1994). Km(TCL+xm) Model: Method 1 13 K (523) Zhong and Li (2002).

N = 20. Tmi T

Tm2

(523)

T

T m l > ™m2

Model: Method 1

(523)

1 2co„

n

v

(523)

-

4co„ t m

i

2^mTml(2a + T m ) - a K m ( 2 a + xm)^ j

(523) _ rj, (523)

(523)

2 a + x„ 2^ ra T ml (2a + T m ) - a 2

2^ m T m l a z (2a + T m ) - a 4 (2a + x m ) 2

N =

2 0 ^ ^ 5 2 3 ,

Chapter 4: Tuning Rules for PID Controllers 4.5.6 Controller

filter

with filtered derivative in series with a second f \

Gc(s) = Kc

1 +— +—^— T S

i

l+

^s

l + bfls + bf2s

order

A

\ 1 ~r SfiS T Sf^S /

N

V R(s)+

339

U(s)

E(s) 1 1+ T ; s

T

,

K,

+

l + bf,s + bf2s l + afls + af2s

dS

£— 1+N

(l + sTml)(l + sTm2)

Table 120: PID controller tuning rules - SOSPD model G m (s) = Rule

Kc

|

Y(s)

Ke"

^

Td

K m e" (l + sTmlXl + sTm2) Comment

Robust Shi and Lee (2004). Model: Method 1

14 K

(524)

(524)

T

0.5 }

14 j , (524) m

br, ="

•m

(524)

,b f l =-

-6dB ,

2co_

1

4co

T(524)=T

T

i

2co_ + T

0.5

' >af2

T

_

(524)

2co„6dB(l + co_ 6dB x m )' =

TmlTm2(p_6dB2 - 0.5[(Tml + Tm2 )co_6dB - 0.5] [(Tml+Tm2)ffl-6dB-0-5]

N=

0.5[(Tml+Tra2)«_6dB-0.5] T m ,T m2 a_ 6dB 2 -0.5[(T ml +T m2 >o_ 6dB -0.5] '

Handbook of PI and PID Controller Tuning Rules

340

4.5.7 Ideal controller with set-point weighting 1 U(s) = Kc (FpR(s) - Y ( s ) ) + ^ ( F i R ( s ) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S

Table 121: PID tuning rules - SOSPD model G m (s) = Rule

Kc

K„e"

(l + sTmlXl + sT m2 )

Td

Comment

0.125TU

10% overshoot

0.125TU

20% overshoot

Ti Ultimate cycle

Oubrahim and Leonard (1998). Model: Method I; Tmi

=

Tm2 ;

0.5TU

' 0.6KU

Fi=l 2

0.6KU

0.5TU Fi=l

0.1<^-<3

' F p = 1 . 3 1 6 - K " K " , F dd = 1 . 6 9 f 1 6 - K " * ° 17 + K K 17 + K m K u y F„ = -

38

1.717 (l + 0.121K m Kj

Chapter 4: Tuning Rules for PID Controllers

341

4.5.8 Ideal controller with set-point weighting 2 f

1

Af

iS

J

U(s) = K c

R(s)-Y(s) T

V R(s) + _E(s)[— T W + T i S +l

f

\ \ l + — + Tds T

V

i

s

TiV+TiS + l

wx

j

,Y(s) K„e~

Tml2s2+2^mTmls + l

Table 122: PID controller tuning rules - SOSPD model Gm (s) = Rule

Kc

Arvanitis et al. (2000).

3

K

(525)

T

i

K„e" 2 2

T m l s +2^ m T m l s + l Comment

Td Direct synthesis

3 ^

(525)

T(525)

4

(526)

T

(525)

T

Model: Method 1

X^ K ra x m

K

2^ m +

,(525) =2m_ a + 2£' a2

u

T1

ml

(526)

x X

(526) d

, 0 < % < 1, with suggested x = 0.4 ;

+ 2yl

1 • n'l -! 7-',

. with

2

1+

T-^ r + E2 - - 1 >0andwith a = l + j — ^ r: XT m] (24 m T m +T ral ) U J X T m l (24 m T m +T r a l )

(525) _ 1

TmlTm

X 2 L T » +Tml 4

K

1

(526) K

m*m

.(526)

(526)

(2i;TCL2 + Tm p^mxm

+ Tm, )Tml - TmTCL22

2

T C L 2 +T r a (2^T C L 2 +T m )

_ (24TCL2 H ^ m X ^ - T m + ' m l m^CL2 ^2+^m2Tml(2^ra+Tml) Tml PcL2 + T m (2^T CL2 +x m )] (2^TCL2 +T ra X2£ m T m +T ml )T m , - x m T c

342

Handbook of PI and PID Controller Tuning Rules

1 + Tds

4.5.9 Classical controller 1 G c (s) = Kc 1+ T s

v

i y

E(s) R(s)

+

U(s)

®—

K, v

Y(s)

1 + Tds T T s i y l + -^s N

SOSPD model

Table 123: PID controller tuning rules - SOSPD model G m (s) =

K m e" (l + sTmlXl + sT m2 )

K m e" m \ J

Rule

Comment Td Minim um performance index: regulator tuning Tm2 0.80Tral n ""\ l-5(T m2 +T m ) 0.60(T m2 +x m ) T KmTm Tj

Kc

Minimum IAE Shinskey (1988) -page 149. Model: Method 1; N not specified

m2+tm

0.77Tml Kmtm

5

K

(527)

l-2(T m 2 +x m )

T m2

0.70(T m2 +x m ) T

0.83Tml KmTm

Minimum IAE Chao et al. (1989). Model: Method 1

T m l 2 s 2 +2^ m T mlS + 1

5

K

(527)

0.75(T ra2 +x ra ) T (527)

0.60(T m2 +T ra ) T

(527)

T

=0.5

m2 + x m

T 1 m2

m2+tm

- 0 75

N=8

0.8 < £m < 3; 0.05 < — ^ — < 0.5 2^ ra T ml

-1.890+0.7902^-0.1554^'

-0.01133 + 0.5017^ m -0.07813^ n K„

2s m T m i , -0.03793+0.3975$„, -0.053545 m

Ti<527) = 2 ^ m T m ,(l.235-0.41264 m +0.09873^ n

2s m T m i, 0.5696-0.8484^m +0.05055m!

Td

(527)

=2^ m T m ,(l.214-0.6250^ ra +0.1358^ ^Sm^ml ,

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IAE Shinskey (1994) -page 159. Model: Method 1 N not specified

6

(528)

R

T(528)

T JjH2_<3

(528)

T

xra 2-5Tm, K

i m +0.2T m 2

mTm

i^>3

x m +0.2T m2

0.36TU

0.59KU

xm -^2- = 0.2, Tml

0.26TU

2 k = o.2 Tml 0.85TmI 1.98xm

0.86xm

2.30xm

1.65xm

Kmxm l-25Tml

2.50xm

2-00xm

Kmxm

2.75xm

2-75xm

K

m^m

2 k = o.i Tml T - ^ . = 0.2 Tml

0.87Tm, Kmtm

- ^ s - - 0.2; Tm, N not specified

l-00Tml

x

Minimum ISE McAvoy and Johnson (1967).

i x

^m

K

Comment

Td

M

Minimum IAE Shinskey (1996) -page 121. Model: Method 1 N not specified Minimum IAE Shinskey (1996) -page 119. Model: Method 1

6

Ti

343

x

2Tm

3

2 k = o.5 2 k = i.o Tml Model: Method 1

tm

Representative coefficient v alues (deducec [from p-aph) X x X x2 x2 l i 3 S-

2k

N = 20

1 1 1

xm 0.5 4.0 10.0

N=10

1 1 1

0.5 4.0 10.0

0.7 7.6 34.3

0.97 3.33 5.00

0.75 2.03 2.7

0.9 8.0 33.5

1.10 4.00 6.25

0.64 1.83 2.43

4 4 7 7 4 4 7 7

3.0 22.7 5.4 40.0 3.2 23.9 5.9 42.9

0.5 4.0 0.5 4.0 0.5 4.0 0.5 4.0

1.16 1.89 1.19 1.64 1.33 2.17 1.39 1.89

X

3

0.85 1.28 0.85 1.14 0.78 1.17 0.78 1.04

100

(528)

2^

48 + 57

1-e

x

™

K

mTm

1 +0.342k _o.2 xm 1.2Tml \

T„, "\ ,(528)

_,

1.5-e

1 + 0.9 1-e

x

-

,T d <528) =0.56t„

1-e

T

™

0-6Tm2.

Handbook of PI and PID Controller Tuning Rules

344 Rule

Kc

Minimum ISE Chao et al. (1989). Model: Method 1 Minimum ITAE - Chao et al. (1989). Model: Method 1

7

(529)

7

K

(529)

T,

Td

T (529)

(529)

T l

d

Comment N=8

0.8 < S < 3; 0.05 < — ^ — < 0 .5 2i;mTml 8

K

(530)

T(530)

0.8<4m<3

-1.652+0.54165m-0.095445m

0.2538 + 0.3875^m -0.04283^, K„

(530)

T

V^Sm 'ml 0.1363+0.23424m -0.01445^ ra 2

T;1529* =2^ m T n , 1 (l.8283-0.8083^ m +0.1852§ m 2lf

X

2s m T m i j 0.4772+0.04424m+0.0l694^m

Td<529) =2^ ra T ml (l.077-0.4952^ m + 0 . 1 0 6 2 ^ 8 K (530)

_ - 0 . 1 0 3 3 + 0.53474m -0.07995);,

V

T

2^raTml N-1.798+0.71914 m -0.1416^ m 2

2 /

K„

^ S n i 'ml J N -0.1277+0.52204m -0.086294 m

Ti(530) =2^ m T ml (o.9096-0.1299^ m + 0.04179i;ra2 J — I 2^mTmi , 0.4780-0.03653^m +0.04523^'

T d <530) =24 m T ml (l.2226-0.6456^ m +0.1373^

2!

X

2£ m Tml

J

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum ITAE - Chao et al. (1989)continued. Minimum ITSE Chao et al. (1989). Model: Method I

9

K

(531)

T(531)

T1

T 1

i

0.3<4 m <0.8

(531) d

N = 8 ; 0 . 0 5 < — ^ — <0.5 2smTmi 10

K

(532)

(532)

T

T 1

i

0.8<^m<3

(532) d

-1.6159-0.066t6^ m +0.53515 m 2

2^ m T ml

K„ (53,)

Comment

Td

Ti

1.3292-2.85274 m +2.03654„

(531)

K

9

345

=2^ m T m l exp(x 1 ), x, =(3.3368-7.7919^ m +4.3556^ m 2 ) + (l .4094-4.458 l^m + 3.4857E,m2 )lnj

2^ramT,ml

+ (o.H60-0.8142^ m +0.74024 m 2 ; In Td

(53,)

= 2^ m T ml exp(x 2 ), x 2 = (2.3054- 6.4328^m + 3.9743^m ) + (l.8243-5.6458^ m + 4.6238^m2)ln

V^SmTml J

(o.2866-1.4727^ m +1.2893£ m 2 ; In

10

K

2s m T m | ,

2

(532)

-1.713+0.6238^-0.1178!; m

0.1034 + 0.4857i;m - 0.0709^n K„

24«,T„

2^mTmi j 0.1008+0.26255m -0.022035 m

Ti(532, = 2 ^ m T m ,(l.5144-0.6060^ m +0.1414^

2^ m T n 0.4316+0.08163^m +0.009 l73E,m2

Td

(532)

=2^ m T ml (l.0783-0.4886^ m + 0.1038(;„ ZtuJu

346

Handbook of PI and PID Controller Tuning Rules Rule

Comment

K„

Minimum ITSE Chao et al. (1989)continued.

X

0.3<^ m <0.8

(533)

(533)

N = 8; 0.05 <

2^ m T ml

<0.5

Minimum performance index: servo tuning Minimum IAE Chao et al. (1989). Model: Method 1

K

12

K

(534)

0.8 <^ m <3; 0.05 <

(533) _ 1.8342-3.8984$ m +2.7313;

2

N=8

(534)

(534)

T;

24 m T ml

<0.5

\-1.5641+O.16285 n ,+O.093375 m !

/

<2smTml , T,' 5 ") =2^ m T m l exp(x 1 ), x, =(3.2026-7.4812^ m +4.3686^ m 2 ) + (0.9475 - 3.6318^ra + 2 . 9 9 6 9 C

W r ^ -

+ (- 0.06927 - 0.3875^m + 0.4220£m2 y In

1 mT, v '2^ Sm ml J

Td(533) =24 m T m l exp(x 2 ), x 2 =(l.7393-3.7971^ m +1.7608^ m 2 ) + (l .0983 -1.5794^ m +1.0744£m2 )ln| — 2$ m T n + (o.01553-0.01414^ra -0.009083£ m 2 | In

12 K

(534, _ - 0 . 1 7 7 9 + 0.67634 m -0.1264^ K.

f

T„

^

v2smTmi j

\-0.9941+0.16695„,-0.04381£ m 2

2s m T m i ,

T/ 534 ' =2^ m T ral (o.2857 + 0.4671^ - 0 . 0 8 3 5 3 ^ m 2 ( - ^ x, = 0.1665 - 0.3324£m + 0.1776^m2 - 0.02897£m3; Td(534) =2^ m T m l exp(x 2 ), x 2 = (-0.8776 + 0.4353^ -0.1237^ m 2 ) + (- 2.0552 + 2.786^m - 0.5687£m2 )ln

,2^ m T m l

+ (- 0.5008 + 0.6232^m - 0.1430£ra21 In v2SmTmi

,

Chapter 4: Tuning Rules for PID Controllers Rule

Minimum ITAE - Chao et al. (1989). Model: Method 1

13 K

T,

T,

Kc

Minimum ISE Chao et al. (1989). Model: Method 1

13

K

(535)

T

Comment

Td

(535)

N=8

(535)

T

0.8 < £m < 3; 0.05 < — ^ — < 0.5

14

K

(536)

T(536)

(535) _ 0-1446 + 0.4302^ - 0 . 0 7 5 0 1 ^ K, (535)

347

T X

f

x_

>

^•%m*-m\

0.8<^m<3

(536) d

-1.214+0.3123?m - 0 . 0 6 8 8 9 5 „ 2

/

=2^ m T m l exp(x 1 ), x, =(-0.1256 + 0.1434^ m -0.03277^ m 2 ) + (- 2.2502 + 3.7015£m -1.7699^ m 2 + 0.2680£,m3)lnj

2£nJ m ,

-(-0.4823+ 0.9479^m -0.4913^ m 2 +0.07792^m3^ lnf

— \

V''Sm1ml J

Td(535) =2^ m T ml (0.9105-0.3874^ m +0.08662^ m 2 J — ^ _ x, = 0.0779 + 0.28554m - 0.01985^ m 2 . ,4 „ (536) K„

- 0.2641+ 0.7736^ m -0.1486^ 2 K.

-0.7225-0.04344< m -0.0012474 ra i

f

2^ m Ti m l 21

Ti(536) =2^ m T ml (o.22257 + 0.7452^ -0.1451$ m 2 ]

J t„

v^SmTml J

x, = 0.03138 -0.04430^ m +0.01120^ m Td(536) = 2£ m T mI exp(x 2 ), x 2 =(o.05515-0.7031^ +0.1433^ m 2 ) + (-1.2256 +1.8544^m - 0.3366^m2 )ln| — -

24raT,m l

+ (-0.2315 + 0.3402^ - 0 . 0 6 7 5 7 ^ 2 ] In

>

%

2s m T m i,

2 . »

Handbook of PI and PID Controller Tuning Rules

348 Rule

Kc

Minimum ITAE - Chao et al. (1989). Model: Method 1 Minimum ITSE Chao et al. (1989). Model: Method 1

5

K

15

K

T,

(537)

T

T

0.3<^ m <0.8

(537)

N = 8; 0.05 < — ^ — <0.5

16

K

(538)

T

(538)

x

0.8<^m<3

(538)

-0.9904-2.531 lS„,+2.

1.1295-2.8584^ m +2.1176£ n

(537)

Comment

Td

(537)

K„ • (537)

Ti WJ " = 2^ m T ml 18.6286-20.70^m +13.2034

2£ T x, = 0.7962-2.4809^ m +1.7611^

(537)

:

2^ m T ml exp(x 2 ), x 2 = (3.9285 - 12.874^m + 8.6434^m2) + (3.2228-12.0344m +9.2547^ m 2 )ln

•(o.6366-3.0208^ m +2.4603^ m 2 In ^ S m ^ml j

16

K

(538)

0.0820+ 0.4471^m -0.07797£,n K„

1™

-0.9246+0.07955^ ro -0.02436^„

v2s m T m i

= 2i;raTral (o.5930 + 0.1457^m -0.02062!; n

2

T„

x, =-0.2201 + 0 . 1 5 7 6 ^ - 0 . 0 3 2 0 2 ^ (538)

^ m T r a l e x p ( x 2 ) , x 2 =(-0.6534-0.1770^ m -0.0240^ m 2 ) + (-1.0018 + 1 . 5 9 1 4 ^ - 0 . 2 8 3 1 ^ r a 2 ) l n f - l ! B _ V^Smiml J

+ (- 0.2291 + 0.3064^m - 0.06344,/ In 24mTra

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum ITSE Chao et al. (1989)continued.

17 £

(539)

(539)

T

i

0.3<^m<0.8

N = 8; 0.05 < — ^ — <0.5

^T ml

Direct synthesis Tml Tm2

Kn.K.+l)

Smith et al. (1975). Model: Method 1

x

l *ml

K

m

T

TCL N not specified Tral

Tm2

T m

D -

'

Trai + Tm2 N=10

Coefficient values (deduced from graph) x x P i 4m i ^ra

P

x

i

0.50 2 >0.06 0.45 0.42 1.75 0.09 0.39 0.44 0.17 0.38 1.5 0.28 1.0 0.50 0.40 0.48 0.49 1.0 0.13 18 K (540) TmiTm2 Tmi + Tm2 Also given by Astrom and Hagglund (1995) -page 250-251. Model: Method 7. N = 8 - Honeywell UDC6000 controller.

6 1.75 1.75 1.5 1.0 Sklaroff(1992).

m

P

4m

K

Comment

Td

(539)

T 1

Smith et al. (1975). Model: Method 1

„

T:

349

>0.02 0.27 0.07 0.11 0.25

0.51 3 0.46 1.75 0.36 1.5 0.33 1.5 0.46 1.0 T m i+T m 2

>0.04 0.13 0.33 0.08 0.17

,539, ^ 0.9666-2.3853^ m + 2 . 4 0 9 6 ^ f xn .24 m T m l . K„

-1.533-0.054425ra+10.9165,,

T,(539) =2^ m T m l (6.3891-15.773^ + 1 0 . 9 1 6 ^ m 2 | - ^ x, = 0.04175-1.38605 m +1.40534 n Td

(539)

= 24 m T ml ex P (x 2 ), x 2 = (s.2620- 10.7894m + 7.6420!;m2)

)lnf + 2.277-8.94024 m +7.7176^,/lln

T

- 1

V^SmTml

(o.3465-1.92744 m +1.80304 ro 2 , In 18 j r (540) 3x„

Kjl + Tml

+

Tm2 J

f

\ In,

24 T, V^^mm 'ml J

Handbook of PI and PID Controller Tuning Rules

350 Rule

Kc

Ti

Poulin et al. (1996). Tml Model: Method 1 K ( T + x ) m ml m Schaedel (1997). Model: Method 1 Panda et al. (2004). Model: Method 7 Panda et al. (2004). Model: Method 1

19

K

20 K

(541)

T (541)

(542)

2^ m T ml

T

Comment

Tm2

N = 5; Tml ^ 5 T m 2 ; Minimum

(541)

N not defined

N=8 2^ m

TB,

2K m t m

Tm2

N not defined

4^m2Tml2+2^TmlTro+0.5Tm2-Tml2

19 jr (541) _ 0 3 7 5 K„

Tml

Td

(2?mTml + T J 2 +1 i-lj(2? m T ml +i m )T ml V2(24 ra 2 -l)-x * = 44 m 2 T ml 2 +2^ m T m l x m + 0.5xm2 - T m l 2 ;

• (541) _

4^m2Tml2+2^mTmlTm+0.5Tm2-Tml2 2^Tm,+Tm

Td 20

K

(54

" = V t e n J m l + x m ) 2 -2(T m l 2 + 2 ^ T m l x r a +0.5x m 2 ) .

(542)

Km 1

1^„

Chapter 4: Tuning Rules for PID Controllers

351

l + NT d s

4.5.10 Classical controller 2 G c (s) = Kc 1+ — T s v i y

1+ Tdsy

E(s) Y(s)

K m e"

(l + T ml sXl + T m2 s)

Table 124: PID controller tuning rules - SOSPD model G m (s) = Rule

Kc

Td Direct synthesis

Ti

0.8T ral a7 T ra2 a3 Hougen(1979)Km page 348-349. Model: Method 1 0.84Tm,°-8Tra20-2

0-5T m l + T m 2 21 j

0.lV^ T m.T m2

(543)

T J

tra

Hougen (1988). Model: Method 1

22 K

K m e" (l + sTmlXl + sTm2)

0.5T ml +T ra2

(544)

T

(543) d

(544)

Comment

N=10 N=30 N=10;x m
2, Tj(543) = a 5 3 T m i + 1.3T m2 ,T d (543> =0.08(x m T ral T m2 ) a28 . X|-x 2 log,

-10 K„ T

m

. Coefficients of Kc<544) deduced from graphs:

m/Tm2

0.1

0.3

0.5

1

2

*1

0.9

0.42

0.24

-0.10

-0.39

0.7

0.7

0.68

0.70

0.69

x

2

0.331ogl0| ^

|-1.40

T d ( 5 4 4 , =(T m ,+T m 2 >0

= 0.1

m-

0.3 l l o g , 0 - J ! ! - -1.63

Td(544) = (T ml+ T ra2 )l0l

, 0.2<-2^-
Handbook of PI and PID Controller Tuning Rules

352

4.5.11 Series controller (classical controller 3) G c (s) = Kc

0 + Tds) T s

v

i y

U(s) E(s)

R(s)

K, T s

v

+

ll+T ml sXl+T m2 s)

i y

Table 125: PID controller tuning rules - SOSPD model Gm (s) = Rule

Kc

K m e" (l + sTm,Xl + sT m2 )

Comment Td Minimum performance index: regulator tuning Ti

2Tml 3x m K m

Minimum ISE Haalman(1965).

Y(s)

K fi-

(l + sTd)

Tm2

Tral

Model: Method I

Tral>Tra2.Ms=1.9, Am=2.36, ^ = 5 0 ° . Bohl and McAvoy (1976b).

1

K

(545)

T

(545)

T

(545)

Minimum ITAE; Model: Method 1

M T

(545)

x

- ^ - = 0.1,0.2, 0.3,0.4,0.5; - ^ = 0.12,0.3 ,0.5,0.7,0.9.

K

(545)

5.5030 K„

f

N-1.1445+0.1100

10V T1 ml J

iiot) (

^0.1648+0.17091n| 1 0 - ^ |-0.21421n| 1

T

V

ml J

0.6212-0.0760 In 10-

T<545) =1.8681TTr

10

f

l 0 2k

V

\0.3144-0.00621nl 10-=^ 1+0.0085 Inl 10Tm,

T 'ml J

(545)

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Ti

353 Comment

Td

Direct synthesis Yang and Clarke (1996). Model: Method 8

ffl T

p ml

KmAm x

O'Dwyer (2001a). Model: Method 1

2T(546)

Tml

Tml

Tm2

Tmi

iTmi

Kmxm

A

=

Tm2 %

-

•

2x, *m =0.57C-X 1 .

Sample result 0.785T ml

Skogestad (2003).

Kmxm

Tm2

0-5T ral

min(T m l ,8T m )

Tm2

Model: Method 18

T mI

Tmi

K

Majhi and Litz (2003). Model: Method 8

A m = 2.0; Tml

3

T

m

m

(547)

K

* m = 45°

T

(547)

=

Tm2

Ultimate cycle Bohl and McAvoy (1976b).

2

T

T

4

T

(548)

T

(548)

T

4ffl

P

P

(547) _

T

m

+

71

Model: Method 1

1 T,ml

2$ ^ m +it(Am-l) 2(Am2-l)

Tm Km

, (547)

1 + T.i .

A

1» - ^1[ 2 ^ t , ( A „ - l ) | l -

K ^)

4

K

(548)

N

0.5458

, A " .1(2^„+»(AJ,-1)) t(Am2-l)

1.2264+0.4568 In — /„

„

y.2190-0.09581n(K„K|T,)-0.10071n

~

K„

-1.9314-0.62211n

T;<548) = 0.0393T,

v Tu

(548)

(548)

(546) _

2C0_ 3 v

(548)

K

(K u K m )

-0.0703-0.1125ln(K o K m )-0.29441nP !!!

354

Handbook of PI and PID Controller Tuning Rules

4.5.12 Non-interacting

controller 1

E(s)

K m e(l + sTm,Xl + sT m2 ) or

Table 126: PID controller tuning rules - SOSPD model Gra(s) = G m (s) = Rule

Huang et al. (2000). Model: Method I

%-Y(s)

K„e--

Tml2s2+2^Trals + l

Kc

Ti

*KC<*»>

T(549)

Comment

Td Direct synthesis

1

T

(549)

i

ral

i

m\

'

i

'

m\

'

Servo tuning X = 0.6^m<\

A. = 0.7,

K ^

r a

< 2

^ = 0-8»-z;— £m > 2 > Regulator tuning 1

5 Kc (549)

=x

2Tml^m+0.4Tm

Tj(5 49)

^

ml

^

+0.4TlI

' <549> = T ml +0-8T m l 5 m T l t 0.8T m l ^ m T m

Chapter 4: Tuning Rules for PID Controllers Rule

Comment T| Td Minimum performance index: regulator tuning (550) (550) Model: Method T (550) T i 7;N=10

Kc

Minimum IAE Huang et al. (1996).

6

355

6 ^

Note: equations continued into the footnote on page 356; 0 < Tm2 /T m l < 1 ; 0.1<xm/Tml
1

(550)

1

,

1.1863 IHL. T

N

0.6642

-23.1098

,

+ 20.3519

-52.0778

N2.1482

X,

VTmiy

m> ,

K„

xmT„

-3.3397

0.1098 - 8.6290-2- + 76.6760-

N 0.8405

• 12.1033 M 2 2 -

T V ml J

K„

*m

-13.6581-^2-

9.4709-

Tmi

Trai j

\11427

AT

K„

•19.4944Tm,

X

-28.2766-

m2

T

V T ml 7

v 1mi y

C!k -19.1463eT"" + 8.8420eT™' +7.4298e T"" - H . 4 7 5 3 ^ 2 K„ ,(550)

=T

- 0.0145 + 2.0555^2- + 0 . 7 4 3 5 ^ _ 4.4805 ^ 2 _

x T

m m +1.2069-^-^1

V^ml

+ T„

L

0.2584

m2

+ 7.7916

rmi j

^-\

+ Tm, 3.9585l

+ Tm

ml /

-3.0626

-6.0330Tmi l^Tml J

\3

/T

- 7.0263 V T mi y

V T ml 7

m l V ^ml

-1.5769

-2.7755

7.0004V T ml 7

V^ml

V T mi y

VTml 1

+ T„

-0.9439

+ 2.4311

3.1663 T

V mi y

1

mi y

m2

T

V ml J

'mi y

Handbook of PI and PID Controller Tuning Rules

356

2/

+ Tm

-2.4506-

-0.2227 T

I. Jml 1.9228

+ T„

V ^ml (550)

-0.5494-^S- p 2 2 Trai V.Tmi

ml J

- 0.0206 + 0.9385-^2- + 0 . 7 7 5 9 ^ _ 2.3820 — Tmi T ml vT m l

+ T„

-3.2336|^-

2.7095-

T

+7.27741-^5-

( T

'ml V ml + Tm

+ 2.9230

TmTm2

-9.9017Tmi vT m l

+ 6.1539 l m 2 ' — 11.10181 -^s l V ^ml ml J

10.6303-

'm2

+ 5.7105

- 2.274

T

VTml J

^ml V ml

V^-ml

V 'ml J 4

f N5 - 7.9490 - i s - p n ? . - 6 . 6 5 9 7 + 8.0849 V^ml Tml V Tml ) VTml ) + T„

-4.4897-

• 7.6469 VTml J V^ml

'ml V ml J (T + T„

2.2155

T

\

L

-5.06941-^2-

ml / V 'ml J 0.519-

-1.1295

'm2

V T ml J

m2 + 4.1225 f-^r lTml. 'ml 7

•1.6307

^m2

T

VTml J

V 'ml \3 -0.9321-Tm I VTml J

V^ + T„

2.2875

+ 0.9524 ml /

V ml

Tm2

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IAE Huang et al. (1996).

7

7

Comment

Td

Ti T(551)

(551)

K

357

Model: Method /;N=10.

(551)

T

Note: equations continued into the footnote on page 358. 0.4 < £,m < 1, 0.05<x m /T m , < 1 .

K

(551)

- 35.7307 + 1 4 . 1 9 ^ _ + 1.4023^m + 6.86184m Tmi \ T,ml

K„

,

-0.9773 ^ -

+55.5898£,n

K„

53.8651^=H m 2 +11.491 \\J

X,

- 3.3093^n

T

V^ml

+0.8778| •

K„

V -0.6951

/

ml j

N -0.4762

+ 53.535 ^ j

-29.8822 K„

J

\ 0.086

-16.9807^,

u197

^Tmi ,

N -2.1208

-25.4293^ m 14622 -0.1671^ m 58981 +0.0034^ n V T ml )

K„

. 25.0355^K m 3 ' 0 8 3 6 - 5 4 . 9 6 1 7 - ^ H j ' 2 1 0 3 -0.1398e T K„

-8.2721e^ +6.3542e T™' + 1.0479IE-JSL K„ A2

/ .(551)

- 9.7061£

0.2563 + 11.8737-21- - 1.6547^m -16.1913 1

+ T„

3.5927^m2 + 19.5201 ^ - \ - 14.5581£,n

+ T.ml -0.4592^ m -34.6273

+ T„

i.9259CP=X

V ml J

xn

/

VTral7

V T ml )

+ 8.6966^-^m3-6.9436^m4 iml

A tm

+ 2.939^

+ 50.5163^n

VTml J

iml

ml J

V

'ml /

358

Handbook of PI and PID Controller Tuning Rules

"\5

f + T„

20.0697^n

27.2386 V T mlV X

-T ml 8.5019

1

+ Tm

-42.2833^„ V T ml J

Em3 -12.2957 ±S- \m* + 8 . 0 6 9 4 ^ - 2 . 7 6 9 1 ^ V 1 ml .

ral J

ft V •7.7887 - E -

-2.3012£n

V ml,

/ 8.8984 VTmiy

+ 4.6594 - ^ K m 5

Tml

T

(551)

^Tmi

V

£„/+10.2494 ^ U

V lmlJ

m

- 5.4906 V 'ml

-0.021 + 3.3385—2- +-0.185^m -0.5164 ^ M

-0.96434m^

V 1 ml )

+ Tm

-0.8815^ m 2 +0.584 - ^

\2

f 3

'ml

- 1 2 . 5 1 3 4 m | ^ L | + 1.3468^ V ml .

+ T„

2.3181^ m 3 -5.2368-^2-J

+ 15.3014£n

-3.1988£,n V T ml 7

V*ml

f + T„

^=-

H-9607^

- 2.041 \ ^ J

+3.4675

^

x,

VTmiy

+ Tm

•15.07184n

T

V^ml J

V 1 ml J + T„

\2

f

-0.8219£ n

0.484 if

•1.8859 ±m_] f 3 V 'ml J

^ C+2.28214m5-0.9315|^

Un.y

+ 0.529^m|-^

2 -0.6772^,-1.4212 - ^ U r a +7.1176 ^ L .

-2.3636 ^ V'miy

1ml.

ml J

b

4 m

+0.5497

^ 'ml

J

^

Chapter 4: Tuning Rules for PID Controllers Rule

359

Comment T; Td Minimum performance index: servo tuning (552) (552) (552) Model: Method T T 1 d 1. N=10

Kc

Minimum IAE Huang et al. (1996).

8

K

' Note: equations continued into the footnote on page 360; 0 < T m2 /T ml < 1, 0.1<xm/Tml
(552) _ _

. TmT„

7.0636 + 66.6512—25- -137.8937K„

-122.7832-

'ml /

0.0865

Kn

26.1928

V ml J

V^ml

\

+ 141.51l| VTml J \

V ml J

+ 29.4068-

— j

'ml

( . 34.3156^521 Tmi T1

-70.1035-2^ T,ml

V

T

ml J

-47.9791-

152.6392K„

(T

2M5

/T K„

+ 3.0098

T

V *ml

-10.9347

K„

N 2.6405 X

+ 33.6578

Tml V

V^ml

ml

f

\ -0.4062

-57.9370eTml + 10.4002eT«" +6.7646e T""2 +7.3453 V 1m J

K„

• (552)

_,

0.6424p5Sl +2.504^555- + 2.5324- m2 ^ml V T mlJ VTml, 2.3641-

T

Tml V

+ T„

| V

•1.8759

+ 2.0500

*m

, T X

-0.6712-

VTml J

Ami

+ T„

A2

f T.

1.4510

0.9923 + 0.2819—52- - 0.2679:

^m ml J

'x

^

VTml J

v'ml

ml J

-1.3496-

0.8519Tmi j

- 3.4972 VTmiy

•3.1142 - l m l

+ T m -2.4216V^ml

V^ml J

+ 0.5862

VTml J

VTmlV

L

ml /

360

Handbook of PI and PID Controller Tuning Rules

2

+ Tm 0.0797 ^ W ^ M

+0.985 i ^ T

V^ml

+ T„

V

( - ^VT

1.2892

V T ral J

m

+ 1.2108

V T m i y V '"ml (552)

^x„V

V T ml J

0.0075 + 0.3449^2- + 0.3924^mL - 0.0793 -

+ 2.7495 ^ V

T

\2

f-r + T „ 0.6485

-9.7483^21

+ 0.8089^2V T ml J

V T ml 1

3.4679^-f^( + T„

12.0049-

10.0045

/T

- 5 . 8 1 9 4 | - ^ - ] - 1 ..0884 """"' T

\

T V 'ml J

V^ml

f + 0.3520

\5

.6.3603^

T

ml V ml )

+ 7.0404 V T ml 7 V 'ml

+ T„

'm2

1.4294

V 'ml V T 'ml J

1.1839-

V 'ml

\

-0.0471

T

V T ml J

V 'ml J / \6

+ 1.7444| JEs

+ 1.7087

• 1.28171 ^ 2 _ P ^ - l

V^ml

$

fry

- 6.9064

T m l V ml 7

+ Tm

i a i - J -3.7055^ffl-|in2_ V ml J Tm, ^ Tml j

-1.4056

-3.2980

+ T„

J

3

T V 1 ml J

\

X

VT'ml

V T ml 7

-2.128l|

ml J T

"

+ 1.5121^IS-LlmL T m l V ml ,

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IAE Huang et al. (1996).

9

K

(553)

T

Comment

Td

Ti

(553)

361

Model: Method 1. N = 1 0 .

(553)

T

i

Note: equations continued into the footnote on page 362. 0.4 < ^ m < 1; 0.05<xm/Tml
(553)

- 8.1727 - 32.9042-2- + 31.9179£m + 38.3405£n K„

VTml J

lmt ,

xO.1571

+ 29.3215

+ 0.2079 K„

+ 35.9456

VTml J

^ [ - 2 1 . 4 0 4 5 ^ m 0 1 3 U +5.1159^m'-9814 -21.9381^ m '- 737 ] ,

K„

N

1.2008

-26.8655£n

-17.7448^m - s u

V^ml

ml J

T„

1

52.9156—s-^ m 11207 -22.4297-S_^ ra °' 3
I T

8.5175e^-1.5312e T™' +0.8906

[Huang (2002)];

K„

T

(553) _

T

1.1731 + 6.3082—^- - 0.6937^m +8.5271 ^

-24.7291^^ l

V * ml J

+ Tm

+7.9559^ m 2 -32.3937 ^ s - | + 5.5268!;,

•6.7123^ m |-^-

+ T m -27.1372 - ^ -

+ 166.9272^

m

\ +36.3954^„ 2

V^ml

V^ml

-^2_

-94.8879^n

m\

f

-22.6065^ m 3 -1.6084^2_^ m 3 + 29.9159^

-93.8912^

-84.3776^n l

ml /

\

VTml J

V^ml

+ T^ 49.6314'

6

L

ml J

ml /

Handbook of PI and PID Controller Tuning Rules

362

+ T„

V

(

V 1ral

K ral J

+ T„

ro

•11.75894n

-12.4348 VTml

+ T„

1™ | ^ m 4 - 1 9 . 7 5 6 9 ^ 5

110.1706 -^2£,m3 -25.1896| T

+ 68.3097

U,ml

VTml7

-17.8663 -^M ^m3 - 22.5926Jr^r-) C + 9 . 5 0 6 1 ^ m 5 V ml J m\ ;Tmi J X

(553)

_r

0.0904 + 0.8637—— - 0.1301^m + 4.9601

2 + Tml 0.7170^m -12.53 l l | * "

-21.4907!;,

+ T m 102.9447^m

v T miy ' T ^

-42.5012^

( -

\

+ 7.5855^,

•17.2130^n

T

50.6455U=V

+ Tm

+ T„

21.6061

+

^

„

T

V miy <'

T „

A

-110.0342£n

V miy

VTml7

^ m 3 - 1 6 . 7 0 7 3 U ^ I C -16.2013^ m 5 +5.4409^ n V

ml ,

+ 29.4445 VTmiy

^

19.i257^-^ m 3

T

-10.9260^ V^ml

+ 17.0952^

V T ml

V ml7

ml J

-0.0979

+ 14.3899^n

2

T

V miy

+ T„

^ "

-6.95554m -12.3016

V ml7

10.8688

TmJ VTmiy

T

+ T„

l

TmJ

C+6.2798^C

5 m 3 -24.1917' vTmiy

Chapter 4: Tuning Rules for PID Controllers

363

4.5.13 Non-interacting controller based on the two degree of freedom structure 1 r

\

f

\

s PT d s 1 + T d— U(s) = Kc ,1 + — E(s)-K ( ot + R(s) T:S 1- +X^ s 1 + ^-s V N ; N ;

pTds T N

a+

K„

Y(s)

R(s)

*U(s)

-Mg>

K, 1 +

+ T

1

+-

2 2

Tml s +2^mTmls + l

+

N ;

E(s)

K e"

THs

Table 127: PID controller tuning rules - SOSPD model Rule

Xl=exp[a

T Comment Td i Minimum performance index: regulator tuning Model: (554) j (554) (554) Method 1; T 2 d Km=l

1K

+e % ^ + Tm,

+d

+b | ^ | +c VTml7

V^ml

+h

Sm

m

T

+i

VTmiy (554) _ T m l

• (554)

(554)

fi;m+g^2

\3

r

K

T m l 2 s 2 +2^ m T m l s + l

Kc

Minimum IE Shen (1999).

1

K^e-1-

:

Sm + j

V lm\ )

x x., a = 1-x, ; coefficients of x, are

given below (and continued into the footnote on page 364). 1.4 < M s < 2 ; |3 = 0; N = 0. a (554) Kc(554, T (554) T a b c

1.40 -11.60 14.17

2.43 -2.50 -4.16

1.32 -0.51 0.70

-1.44 1.27 1.34

364

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Minimum IE Shen (2000).

2

(555)

(continued)

Kc(554,

d e f g h i

-6.44 8.28 -0.07 0.13 -4.66 -0.99 -3.03 2.89

T

(555)

J

U m +T m l J l.Tm+Tm,J ( \ (

a

(554)

T

'd

2.85 -0.30 0.1 -1.72 15.51 -7.98 -8.45 4.66

x,=exp[a + b f — ^ — ] + c f — ^ — ]

Model: Method 1; Km=l

(555)

T

(554)

T

Comment

Td

i

k 2

K

Ti

-0.37 1 -5.97 -0.78 -10.63 18.85 -10.97 6.44

-1.06 0.02 -0.09 -7.43 1.77 2.64 4.77 -3.74 + df—^—)

U m +T m l J V c

+e^ ra +f^ ra 2

V

+ g ^ h — L + h - — ^ — 5m+i—Jbr-Um ltm+ T mlJ

Kc

(555)

= _ n i . X | j Xf x

=T m Xi,T d

^m+TmlJ

{lm

given below; M s = 2 ; |3 = 0 ; N = 0; 0 < £m < 1; 0 < -^2- < 1. ml

a b c d

e f g h i

J

k 1

,555,

1.73 -17.51 30.73 -24.69 0.15 -0.12 3.30 33.07 -37.64 2.63 -34.57 36.88

T

(555)

2.28 0.87 -24.00 15.09 -0.01 -0.02 -8.03 45.21 -22.48 3.89 -22.39 8.32

J

= T m x,, a = 1-x, ; coefficients of x, are

m

Kc

+Tm\

T

(555)

1.43 -1.94 14.82 -21.44 -0.25 0.16 -1.06 -43.39 51.16 -3.25 36.04 -34.95

a -1.60 4.30 -12.16 26.54 1.33 -1.08 -17.56 38.13 -56.45 14.80 -39.33 51.18

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IE Shen (2000). Model: Method 1

3

(556)

T

Comment

Td

Ti

(556)

K

365

(556)

T

i

Km=l

*d

Minimum performance index: servo/reguiator tuning N=10; Model: 0.7236K, 0.5247TU 0.1650TU Method 11

Minimum ITAE - Pecharroman and Pagola (2000).

a = 0.5840, (3 = 1, 4>c =-139.65° ,K m = 1; Tml = 1; ^ m = 1 x,K u

Minimum ITAE - Pecharroman (2000). Model: Method 11

x

"3TU x2Tu Coefficient values and other data a x x2 3

i

0.509

0.167

0.585

-146°

0.727

0.524

0.165

0.584

-140°

x, = exp[a + b|

+ d

x m +T,ml

/

+g

VTm+Tra,

+J

K (556)=JJHLXIJ

4>c

0.803

VTm+Tml

^m+h

T

m

+

^m+Tmiy

VTm

Tmi j

+

T

m

^m+i|

l J

Sm + k

+ e4 m +f^„

T

m

+

X

Tm] J

m +Tm] J

k T d (556) "°'=T r a x,, a = l - x , ; coefficientsof x,

, (556) Ti»o^=TmX

are given below; M s = 2 ; p = 0 ; N = 0; 0.4 < ^ m < 1; 1 < -&- < 15 . Tmi K

a b c d e f g h i j k 1

(556)

128.9 -560.3 769.2 -338.9 -324.8 194.3 1421 -1981.6 895.0 -848.1 1186.1 -538.1

4m 1 ;

T

(556)

92.8 -391 521.6 -225.5 -224.5 137.5 963.9 -1310.6 574.4 -590.8 808.0 -356.7

T x

(556)

a

d

-3.1 46.3 -107.4 63.1 5.2 -2.5 -79.6 176.0 -103.6 36.2 -78.4 45.8

-119.9 539.5 -785.1 371.8 233.9 -115.7 -1068 1560.5 -741.5 529.2 -774.6 369.2

Handbook of PI and PID Controller Tuning Rules

366 Rule

Kc

Minimum ITAE - Pecharroman (2000) continued.

i

0.672

0.532

0.161

0.577

-134°

0.669

0.486

0.170

0.550

-125°

Km=l;

0.600

0.498

0.157

0.543

-115°

Tm, = l; P = 1;N=10;

0.578

0.481

0.154

0.528

-105°

0.557

0.467

0.149

0.504

-93°

0.544

0.466

0.141

0.495

-84°

0.537

0.444

0.144

0.484

-73°

0.527

0.450

0.131

0.477

-63°

0.521

0.440

0.129

0.454

-52°

0.515

0.429

0.126

0.445

-41°

0.509

0.399

0.132

0.433

-30°

0.496

0.374

0.123

0.385

-19°

0.480

0.315

0.112

0.286

-10°

0.430

0.242

0.084

0.158

-6°

0.1<x m <10

Taguchi and Araki (2000). Model: Method 1

4

x

Comment T, Td Coefficient values and other data (continued) a x x2 3 4>.

K

(557) _

1

Km

T

(557) _

4

K

1.389 + -

(557)

T

(557)

T

(557)

S» = i-0

0.6978 - ^ - + 0.02295]2

T

0.02453 + 4 . 1 0 4 - a - - 3.434 Tm,

+ 1.231

T

0.03459 + 1 . 8 5 2 - a - - 2.741

+ 2.359

-]4\ (557) _

T

LA

— ^rr

a = 0.6726-0.1285-2--0.1371

(3 = 0.8665 - 0.2679-E- + 0.02724 Tml TmI

-0.07345

-0.7962

Chapter 4: Tuning Rules for PID Controllers Rule

T;

K,

Taguchi and Araki (2000) continued.

5

R

(558)

T

(558)

T

367

Td

Comment

(558)

f

= 0.5

T m /T ml < 1.0; Overshoot (servo step) < 20% ; N not specified Direct synthesis 6K

(559)

(559)

• (559)

Huang et al. (2000) - servo tuning. Model: Method 1

X = 0.6,-2Km 2

Am=2.83. Am=2.43. Am=2.13

0.5T 2T ml ^ n - 1 . P=2T ml 4 m +0.4T m • • T m l '+0.8T m l 4 m x m N = min[20,20Td]

5K

(558) _ _

0.5013

0.3363-

K„

[-J2- + 0.01147]2 Tmi 3 "\

-(558)

= Tral

(558) _

+ 2.054

• 0.02337 + 4.858-S- - 5.522 Tm,

Tm, 0.03392+ 2.023-^--1.161

a = 0.6678 - 0.05413-^- - 0.5680 Tml

+ 0.2826 Tm, + 0.1699

P = 0.8646-0.1205^!_-0.1212 6 „ 3X(559)

K' "=X

.,

2 T m , ^ m + 0 . 4 Tm K

(559)

T-1

x

' = 2T ,E +0 4x

mTm

. (559) _ T m /+0.8T m l ^ m x n d 0.8Tml^mTm

368

Handbook of PI and PID Controller Tuning Rules

4.5.14 Non-interacting controller 4 E(s)-K c T d sY(s)

U(s) = K, 1 + R(s)

U(s)

E(s) K

1+ —

+

*g>

SOSPD model

Y(s)

KcTds

T Table 128: PID controller tuning rules - SOSPD model G ra (s) =

Kme"^ (l + sTmlXl + sT ra2 )

K„e"

G m (s)=TmlV+2^mTmls + l Rule

Minimum IAE Shinskey(1996) -page 119. Model: Method 1; - ^ - = 0.2 'ml

Comment Ti Td Minimum performance index: regulator tuning T 1.18Tml -2^-= 0.1 2.20xm 0.72xm Kmxm Tml T l-25Tml - ^ - = 0.2 2.20xm 1.10xm K Tm Kc

m

1.67Tml Kmxm

2.40xm

1.65xm

2.15xm

2.15xm

2-5Tml ^-m^m

• ^ = 0.5 'ml • ^

Tmi

= 1.0

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Minimum IAE Shinskey (1994) -page 159. Model: Method 1

1K

T

(560)

IEI2_<3

tm

3.33Tml

Minimum IAE Shinskey (1996) -page 121. Model: Method 1

R

Comment

Td

T(560)

'd

K

1

T|

(560)

369

^ m +0.2T m 2

^ m +0.2T r a 2

2k_ >3

m'Cni

^ - = 0.2, T 0.85KU

0.35TU

'ml

0.17TU

T - ^ - = 0.2 Tm,

100

(560)

^2^

/

x

38 + 40 1-e

1 + 0.34-^-0.2

»

v T„,

0.5 + 1.4[l-e

15tm

"\

]

1 + 0.48 1-e

T

J;

f

T d (560) =0.42x n 1-e

1.2Tm T

-

+ 0.6Tm

370

Handbook of PI and PID Controller Tuning Rules

4.5.15 Non-interacting controller 5 U(S):=KC b + — E(s)-(c + Tds)Y(s) T S

V

i V

>(b-l)Kc U(s) R(s)

E(s)

+

K,

I

v

*®

T s

i y

Y(s)

.+

+

Kme""™

(l + sT ml )(l + sT ra2 )

(c + T d s)

I

Table 129: PID controller tuning rules - SOSPD model G m (s) = Rule

Kc

K m e" (l + sTm,Xl + sT ra2 ) Comment

Td

Ti

Direct synthesis b = 0.198; nearly zero overshoot b = 0.289; nearly minimum IAE b = 0.143; conservative tuning

1.54T.(5«)

Hansen (1998). Model: Method 1

2

K

(561)

T

(561)

561

1.27T;*

'

1.75T;(561)

2

K

2kiTm2+(Tml+Tm2)rm+0.5Tm2f

(561)

9K m [T ml T ra2 i m +0.5(Tm, + T m2 )x m 2 + 0.167t m 3 f T(56.)

3[rm,Tm2Tn, +0.5(Tm, + T m 2 > m 2 +0.167T m 3 l [T m ,T m 2 +(T m l +T m 2 )x m + 0.5T m 2 ]

2k,T m 2 +(T m , + T m 2 )r m + 0.5T m 2 ] 2

(561)

3K m [T ml T m2 x m +0.5(Tml + T m 2 > m c = K,

(561)

1

Km '

2

+0.167xn

T

ral+

T

m2+*m

n—E;

Chapter 4: Tuning Rules for PID Controllers

controller 6 U(s) = —c- E(s) - K c (1 + T d s) Y(s) T:S

4.5.16 Non-interacting R(s)

U(s)

E(s)

>6*

+

371

Y(s) Kme-""

Kc T

1 L

—4

iS

T m l V+2^T r a l s + l

K c (l + Tds)

t Table 130: PID controller tuning rules - SOSPD model Gm (s) = Rule

Kc

Arvanitis et al. (2000). Model: Method 1

3 y- (562) _

sTm|

3

K

KmeT ml 2 s 2 +2^ m T ml s + l

Td Direct synthesis Tm,

Ti

T (562)

(562)

Comment 0<s
24m + — Tm J

ETr T,

2^„

(562)

[ | T m l (2^ m T m + T m l )(44

=- L

where

8(l- e )2k(2^ i n T i n

Jk.(24mxm+Tml)+K

+Tml)

2

(l- S ) + S )'

fe2(l-e)

with

+ e )iHL( 2 ^ T m

+T ml )+T n

e + 2ct 2 (l-s) 2 [5=L] (24 m x m+ T ml ) 2 > 0 . VKm/

^

Handbook of PI and PID Controller Tuning Rules

372

f 4.5.17 Alternative

controller

4 U(s) = K

sT,

E(s) +

Table 131: PID controller tuning rules - SOSPD model Gra (s) = Rule

R(s)

K m e" (l + sTm,Xl + sT m2 )

Comment Td Direct synthesis: time domain criteria (563) (563) N=10 T

Kc

Bohl and McAvoy (1976a). Model: Method 1

4

K

*d

Response reaches a new steady state in minimum time, to a servo input, with no observable overshoot. Upper and lower limits exist on the manipulated variable. -^2- = 0.1,0.2,0.3,0.4,0.5 ; i s ! = 0.12,0.3,0.5,0.7,0.9 .

K

(563)

4.486

-1.240+0.1409 In 10-

10-

K„

,-0.0252+0.1542 In l o i = i -0.16161n 1 0 - ^ -

. T„, 1

I T,,, j

{ Tml ,

,0.3664-0.0476 In 10(563)

10l

ml J

10 Im2

T

0.7797-0.07731n| l O ^ i |+0.02701n| 1 0 - ^ - 1 T

-

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Bohl and McAvoy (1976a). Model: Method 1

5

K

(564) _

0.2455 K„

5

K

Td Ultimate cycle

(564)

(564)

x

»d

,

373

Comment N=10

-.-0.4986-0.7510 In - a ,0.3875-0.2667 ln(K„K m )-l .1111 I n p s

(K„K m y T

v uy

N -6.1523-1.4285 i n - ^

(K

Td<564) =0.000726TU

T

K

y 1 - 4 4 9 4 +°- 0 4 0 2 | n ( K u K m )- 0 -' , 6 0 2 1 n —

Handbook of PI and PID Controller Tuning Rules

374

4.6 I 2 PD Model Gm(s) =

K„e~

4.6.1 Controller with filtered derivative with set-point weighting 2 A T l + b fl s + b f2 s 2 , 1 ds •R(s)-Y(s) G c ( s ) = Kt 1 + — + %r l + a fl s + a f2 s 2 Xs 1. +T,^ - s N V E(s) R(s)

U(s) Y(s)

l + bfls + bf2s

Kc 1 +

l + afls + af2s

l

^

+

— •

s2

I

N )

Table 132: PID tuning rules - I 2 PD model G m (s) = — 2 ^ Rule

Kc

T

Td

i

Comment

Robust Liu et al. (2003). Model: Method I

1 £

(565)

1

K

T (565)

(565)

{AX + zm jlX3 + 6X\n

x

(565)

+ 4Xzm2 + t m 3 ) - XA

Km(AX3+6X2zm+4Xzm2+zjj (565) _ (4k + T J4X3 + 6X2Tm + 4ATm2 + T 3 ) - X4 , Am\~'+zm2 , - 3T \4Xi+6X2x2_m+4Xz T

(565, _ (6X2 + 4Xzm + zj )AX3 + 6X2xm + 4Xz2 (4X + Tm ^

2

2

+ zj 3

)

4

+ 6X zm + 4Xxm + T m ) - A

4X 3 +6X 2 T m +4Xt m 2 +T m 3

a f , = 4X + zm ; af2 = 6X2 + 4Xxm + z„

6X2+4Xzm+zm ——. rf-— r—~ ~—; T\ '. X3[(4X + tm)(4X3 +6X 2 t m + 4Xx m 2 +T m 3 )-X 4 ]

1 » b n = 2X , b f 2 = X

Chapter 4: Tuning Rules for PID Controllers Rule Liu et al. (2003) - continued.

Kc X 0.8xm

375

Comment Td T, Sampi e data (deduced from a graph) Rise time OS OS Rise time X a 33% -4xm 2.5xra *0% «10t m

*m

« 20%

*5x m

3t m

«0%

-12.5x m

l-5xm

a 3%

«Vxra

3.5xm

*0%

*14x m

*0%

*8x m

4xm

*0%

*16x m

2xm

Handbook of PI and PID Controller Tuning Rules

376

4.6.2 Controller with filtered derivative with set-point weighting 4 \

f

G c (s) = Kc

1+ - U ^ T s

>

1+ ^ s N

R(s)

>

1 + b f jS + b f 2 s + b f 3 s 3 + b f 4 s 2

3

l + a fl s + a f2 s +a f 3 s + a f4 s

R(s)-Y(s)

4

J

U(s)

Y(s)

l + b n s + b f 9 s 2 + b f ,s 3 + b f 4 s 4 1 + a fl s + a f 2 s 2 + a f 3 s 3 + a f4 s

Table 133: PID tuning rules - I 2 PD model G m (s): Rule

T,

Kc 2 j^

T

(566)

Sampl e data (deduced from a graph)3 OS Rise time OS X = 14% 2.5xm = 0% = 3r m

X 0.8xm

K

Comment

Td

-,(566) i

<566>

Liu et al. (2003). Model: Method 1

| Robust

K m e"

Rise time = 7x m

tm

= 4%

= 3.5tm

3xm

= 0%

= 8.5T,„

1.5Tm

= 0%

-4 5x

35r m

= 0%

= 10.5xm

(566) _ (4X + Tm fr3 + 6X2Tm + 4XTJ

+ Tm3 ) - X4 ^

Km(4X3+6X2Tm+4Xrm2+xm3j .(566)

_ (4A. + T )(4X3 + 6A2xm + 4Axm2 + x 3

2

2

3

)-X 4

3

^ +6X xm+4Xxm +xm j (566)

_ (6X2 + 4XT

+ x m 2 &A3 + 6X2xln + 4Ax, 3

2

2

+ xm3)

(4X + x m X4X + 6A xm + 4Xxm + x m ) - X4 X4 4X 3 +6A 2 x m +4Xx m 2 +x m 3

2

3

, b f l = 4 A , b f 2 =6A 2 , b f 3 = 4 A 3 , b t 4 = A 4 ,

l 22 2 6A. '+4Ax " t m + T +x„ m

X'[(4X + x

m

^ +6X\^+4Xxm2

+xm3)-X4]

a f , = 8A + x m , a f2 = 26k2 + SXrm + x m 2 , a f3 = 4x(lOA2 + 20ktm + 4x m 2 ), a f 4 =4X 2 (6X 2 +4Ax m + x m 2 ).

Chapter 4: Tuning Rules for PID Controllers Rule Liu et al. (2003) - continued.

Kc 2Tm A 0.8xm T

4

m

377

Comment Td Ti Sample data (deduced from a graph) - continued «0% *6x m «0% «13Tm 4xm Sample data (deduced from a graph)4 Rise time OS OS A * 46% 2.5x « 4% ~6xm m * 34%

*20.5x m

«13.5xm

3xm

-2%

«24x m

3-5xm

*1%

-28im

4Tra

«0%

«32x m

l-5xm

* 16%

*13xra

2Tm

*8%

«16.5xm

a fl =5A + t m , a f 2 =10.25A 2 +5AT m +T m 2 , a f3 = A(7A2 + 4.25Atm + i m 2 ) , a f 4 =0.25A 2 (6A 2 +4Ax m +T m 2 ).

Rise time

378

Handbook of PI and PID Controller Tuning Rules

4.6.3 Series controller (classical controller 3) G c (s) = Kc l + J-](l + sTd) v

TiS

U(s)

R(s)

E(s) K,

~ ^ =

(l + sT d ) V

K„e"

T s

i 7

Table 134: PID tuning rules - I 2 P D model G m (s) = Rule

Kc

Skogestad (2003). Model: Method 1

0.0625 K

mtm

Y(s)

T; Td Direct synthesis 8xm

8t m

K„e-«-

Comment

Chapter 4: Tuning Rules for PID Controllers

379

4.6.4 Non-interacting controller based on the two degree of freedom structure 1

f U(s) = K

\

(

^ PTds

E(s)-K c a + V

R(s)

1+ ^ s N

Y(s)

*g)->* +

s U(s)

Table 135: PID tuning rules - I 2 PD model G m (s) = ^ ^ Rule

Kc

Hansen (2000). Model: Method 1

3.75KmTm2

T( Td Direct synthesis 5.4xm 2-5tm

Comment N = 0; p = l ; a = 0.833

Handbook of PI and PID Controller Tuning Rules

380

4.6.5 Industrial controller U(s) = K( v

T s

iy

R(s)-^Y(s) 1+N

Table 136: PID tuning rules - I 2 PD model Gm{s) = ^ L . Rule

Comment Tj Td Direct synthesis: time domain criteria 'good' Skogestad robustness 0.25 2 (2004b). 4(TcL+x m ) TcL = T m > K (T +T ) 2 4^ (T C L+x m ) Model: Method 1 m C L m £=0.7 or 1 N e [5,10]; N = 2 if measurement noise is a serious problem Kc

Chapter 4: Tuning Rules for PID Controllers

381

K e"st"'

4.7 SOSIPD Model (repeated pole) Gm(s)

sfl + T ^

4.7.1 Non-interacting controller based on the two degree of freedom structure 1 (

\

f

\

U(s) = K

Table 137: PID controller tuning rules - SOSIPD model (repeated pole)

Kme~ s(l+T f f l l s)'

Rule Minimum ITAE - Pecharroman and Pagola (2000). Model: Method 2 Km=i;

Comment Kc ^ Td Minimum performance index: servo/regulator tuning x x,K u x3Tu i \ *! 1.672

x2

Coefficient values x3

a

0.366

0.136

0.601

-164°

1.236

0.427

0.149

0.607

-160°

Tm=i;

0.994

0.486

0.155

0.610

-155°

0.1 < t m <10;

0.842

0.538

0.154

0.616

-150°

P = l , N = 10.

0.752

0.567

0.157

0.605

-145°

*c

Handbook of PI and PID Controller Tuning Rules

382 Rule

Kc

Pecharroman and Pagola (2000) continued.

Taguchi and Araki (2000). Model: Method 1

I

(567)

K

=

_

•.

*| 0.679

0.610

0.149

0.610

0.635

0.637

0.142

0.612

-135°

0.590

0.669

0.133

0.610

-130°

0.551

0.690

0.114

0.616

-125°

0.520

0.776

0.087

0.609

-120°

0.509

0.810

0.068

0.611

-118°

-140°

^-<1.0; 1K

(567)

(567)

T

T

*i

0.1778 + K„

Comment

Td Coefficient values (continued) a X *2 3 Ti

'ml

(567)

Overshoot (servo step) < 20% ; N fixed but not specified

0.5667 - + 0.002325 J n4>

T

(567) _

T

0.2011 + 11.16-^--14.98

•13.70

-4.835 'ml

V

(567)

=T m l 1.262 + 0.3620-^ml J

P = 0.8206 - 0 . 0 9 7 5 0 - ^ + 0.03845

a = 0.6666,

Chapter 4: Tuning Rules for PID

383

Controllers

4.8 SOSPD Model with a Positive Zero Km(l-sTm3)e-^ _Km(l-sTm3)e-"or G m (s) G m (s) = 2 2 (l + sTml)(l + sTm2) T ml s +2^ m T ml s + l

4.8.1 Ideal controller G c (s) = Kc

Xs U(s)

R(s)

*R\ +Y T 1

E(s)(

f

1

1

K c 1+ — + Tds T.s v *•" y

Y(s) — •

G m (s)

—w

Table 138: PID controller tuning rules - SOSPD model with a positive zero Gm (S)

Rule

= M ^ - s T ^ K ^ ^ Kn (1 - sTm3 y^ (l + sTm,Xl + sTra2) Tml¥+2i;mTmls + l Kc

Minimum IAE Wang et al. (1995a). Model: Method 1

Comment Td Ti Minimum performance index: servo tuning

i

i Poqi-Pir K m Po

p0q2-P2 PoQi-Pi

Po

Pi Po

: Po T m l + T m 2 + T r a 3 + T m , p , = T m I T m 2 + 0 . 5 T r a ( T m I + T m 2 ) - 0 . 5 T m T m 3 , p 2 = 0.5T m l T m 2 T m , qi = T m l + T m 2 + 0.5t m , q 2 = T m l T m 2 + 0.5x m (T ml + T m 2 ) ,

r = 8 0 + 8 1 - E - + 52

'n,2

V T ml J

+ 5,

S7 V T ml J

+8 V T ml J

+ 83

'1113

'O

Tml

V T ml J

V^ml

f

'T, V T ml J

+ 8C ml J

+ 8,

+ 5B VTmiy

Tn,

fT

\

V T ml J

S 0 = 3.5550 ,8, = -3.6167 ,8 2 = 2.1781,83 = -5.5203 ,5 4 = 1.4704 , 85 = -0.4918,8 6 = 2.5356 ,8 7 = -0.4685 ,8 8 = -0.3318 ,8 9 = 1.4746 .

Handbook of PI and PID Controller Tuning Rules

384 Rule

Kc

Minimum ISE Wang et al. (1995a). Model: Method 1 Minimum ITAE - Wang et al. (1995a). Model: Method 1

2

i K

Ti

p0q2-p2

Poqi-Pir Po

Po 2

m

Comment

Td

PoQi-Pi

Po

Poq2 - P 2

__P_L

PoQi - P i

Po

3

i K

p 0 qi - P i

m

Po

r

Po

2

Direct s ynthesis Wang et al. (2000b), (2001a). Model: Method 2

4 ^

(568)

2s m T m i

T

2

M,=1.5

"•ml

p 0 ,Pi,p 2 ,q,,q 2 ,r defined as on page 383. S0 = 3.9395 , 5 , = -3.2164 , 6 2 =1.6185 , 5 3 = -5.8240,8 4 =1.0933 , 65 = -0.6679 , 56 = 2.5648 , 5 7 = -0.2383, 5 8 = -0.0564, 5 9 = 1.3508. 3

Po>Pi>P2>qi>q2>r defined as on page 383. 8 0 = 3.2950 , 5 , = -3.4779 , 5 2 = 2.5336,8 3 = -5.5929,5 4 = 1.4407 , 8 5 = -0.3268 , 5 6 = 2.7129 , 8 7 = -0.5712 , 5 8 = -0.5790 , 8 9 = 1.5340 .

4

K

(568)

1

2

C0S9

^roTmlm

M s cos[anm - t a n - ' ( - T m 3 c o ) J K m ^

,V+i

co and 9 are determined by solving the equations: cos 9 tan '(-T ra3 co)-coT m -0.57t and tan M, - s i n 0 l = tan~

~ T m 3 T m t 0 2 ~ l)cOs(cOT m ) - C0Tm sin(cOT m )

r T m 3^ m co 2 -l)sin(coT m )+coT m sin(coT m )

Chapter 4: Tuning Rules for PID Controllers

385

4.8.2 Ideal controller in series with a first order lag f V R(s)

\

1

Gc(s) = K( T

i

1

J Tfs + 1

s

U(s)

E(s)

/ 1 K r 1 + — + Tds T s +1 T s v i J f

Y(s) jKm(l-sTm3)e-^ (l + sT ml Xl + sT m 2 )

—•

Table 139: PID controller tuning rules - SOSPD model with a positive zero Km(l-sTra3>-^ G m (s) = (l + sTmlXl + sT m2 ) Rule

Kc

Li et al. (2004). Model: Method 1

5

K

(569)

5

K

(569)

Comment Ti Td Direct synthesis: time domain criteria Tm, +T m2 TmiTm2 T m l +T m 2

T m l +T m 2 ;.Tf Km (2TCL2 + Tm3 + xra ) ' f

2TCL2 + Tm3 + xra '

desired closed loop transfer function =

(l-sTm3>-

Handbook of PI and PID Controller Tuning Rules

386

T 1 ds 4.8.3 Controller with filtered derivative G c (s) = Kc , 1 +— +-

R(s)

Table 140: PID controller tuning rules - SOSPD model with a positive zero K m ( l - s T m 3 ) e - o r G m ( s ) = JC m (l-_sT m 3 ) e G m (s): (l + sTmlXl + sT m2 ) T m l 2 s 2 +24 m T m l s + l Rule

Kc

T

Comment

Td

i

Robust 6 ^

Chien (1988). Model: Method I

7

K

j(570)

(570) (571)

x 1

x(571)

T

(570) d (571)

Tml > Tm2 > T m 3 ; N=10; X = [T mI ,x m J (Chien and Fruehauf (1990)).

6

(570)

K

Tmlx„ Tml + T „ , + ^• + T m 3 + T m

T

(570) _

T

Tm3Tm

T

K m (^ + T m 3 + x m ) T X

d

(570) _ -~

Tm3Tm + ^• + Tm3 +T ra J

X + T m , + T„ TmlTra2

+

x

,

T

m3 T n

^ + T m 3 +x m 7 ^ (571) K„

(571)

OE T _j m3 m ^>m ml ^ x A + 1 m3+Xm

Km(X + T m 3 + x m )

(571) _ / - j . 1T M - Sm ml +

T

T

Tm3 m ^ + Tm3+*m

oK

x , 2^ m T ml +

" ^ n . *- + T m 3 + 1 : m

X T m3 m

1

k+Tn>3+*m

'

Chapter 4: Tuning Rules for PID Controllers

387

1 + Tds

4.8.4 Classical controller 1 G c (s) = Kc 1+ — T s

v

i y

N R(s) E(s)

U(s),

1 + Tds

K, v

Y(s)

Km(l-sTm3)e-

T s

i y l+ ^ s V. N .

(l + sTmlXl + sTm2)

Table 141: PID controller tuning rules - SOSPD model with a positive zero Km(l-sTm3)e-^ G m (s) = (l + sTm,Xl + sT m2 ) Rule

Kc

Poulin et al. (1996). Model: Method I

O'Dwyer (2001a). Model: Method I

T; Td Direct synthesis

Comment Minimum

8

K

(572)

x

iTmi

K

mTm

Tml

Tm2

Tm,

Tm2

A m =0.57t/x, ; K = 0.57t-x,; N = T m2 /T m3

Sample result 0.785Tml KraTm

Am=2.0; Tm,

Tm2


Robust Chien (1988). Model: Method 1

Tm2 K m (^ + T m )

Tm2

Tml

Tml

Tn,

Tm2

N=10; ^e[T m l ,T m ] (Chien and Fruehauf(1990))

*•*(*•+ 0 Tml > Tm2 > Tm3

8

K

(572) _

Tral j ^ _ Tm2 (Tm, + 2Tm3 j Km(Tml+2Tm3+xJ: 1 m, = x Lm . = 56° when Tm3 m, = T ml m

388

Handbook of PI and PID Controller Tuning Rules

4.8.5 Series controller (classical controller 3) G c (s) = K t 1 + — (1 + T d s) U(s)

E(s)

R(s) ^

K, 1 + — (l + sTd) Ts

•

Y(s)

Km(l-sTm3)e (l + Tm]sXl + T m2 s)

Table 142: PID controller tuning rules - SOSPD model with a positive zero Km(l-sTm3)TSI™ Gm(s) = (l + sTmlXl + sTm2) Rule

Kc

Zhang (1994). Model: Method 1

Kmxm

T; Td Direct synthesis Tml

Tm2

Comment "Good" servo response; xm is

"small".

Chapter 4: Tuning Rules for PID Controllers

4.8.6 Classical controller 4 Gc (s) = Kc 1 + — 1 + v

389

TdS

T s

i y

N

;

U(s)

R(s) E(s)

-0—

1+ .

K, V

T

(l + sT ral Xl + sT m 2 )

iS/ V

N

Y(s) K m (l-sT m 3 )e-

^

;

Table 143: PID controller tuning rules - SOSPD model with a positive zero G m (s) = Rule

Kc

K m (l-sT m 3 )e- S T " (l + sTmlXl + sTm2) T,

Td

Comment

Robust Tm2 K m (^ + x m )

Chien (1988). Model: Method 1

Tm2

9

Tmi

10 T (574)

T 1

(573) d

Tml

Km(*- + 0

N=10; X e [T, Tm ] , T = dominant time constant (Chien and Fruehauf (1990)).

= T„,

+-

Tm,x„

^ + Tm3+xm 10

x

1^

(574) _

T

Tn^Tp.

— A m T -1

^ + T m 3 + Tra

390

Handbook of PI and PID Controller Tuning Rules

4.8.7 Non-interacting controller 1 U(s) = Kc

E(s) T

V

iS/

^-Y(s)

i + 3fi N

R(S)

v, ^

U(s) •

+ K T S

V

. /

Y(s)

K m (l-sT m 3 >TmlV+2^Tmls

+

l

Tds l+

^ s N

Table 144: PID controller tuning rules - SOSPD model with a positive zero K m (l-sT m 3 )e- S T " G m (s) = T m l 2 s 2 +2^ m T m l s + l Rule

Huang et al. (2000). Model: Method 1

Kc ..K e (5 ">

Td Direct synthesis (575) T 2T ml ^ m +0.4T m Ti

Comment

'd

^ =0.5,^m
N not specified

1

ml

' Tm

u0 7

- ' > T1

£

Ami ~> 9

Sn,^^

servo

'

tuning

ml

^ = 0 . 6 , ^ m < l.X = 0.7, 1 < - I ^ m < 2 . T

ral

'

TmI

T

T . _ un o

*•

ii

K

(575) = -^2T r o l ^ m +0.4x m K m (T m +2T m 3 )

T (575)

-°> T

l

'me

. 9

S m ^ ^ regulator tuning

ml

_ Tml + 0,8Tmli;mTn 0-8Tml^mTn

Chapter 4: Tuning Rules for PID Controllers

391

4.8.8 Non-interacting controller based on the two degree of freedom structure 1 f

1

T

1

U(s) = Kc V

1+-T" ijS + 1 + PTds

a+

T N

"\ dS d Tds/N

T

PT s

d E(s)-K c a + R(s) 1 + T d s/N

K„

U(s)

R(s)

Y(s)

-U-^

1+

+

T S

>

E(s)

Kra(l-sTm3)e-

l+

V

+

^ s N

(l + sT m l )(l + sT m 2 )

Table 145: PID controller tuning rules - SOSPD model with a positive zero K ra (l-sT ra3 )e- SI ™ G m (s) = (l + sTmlXl + sT m2 ) Rule

Comment

K„

Direct synthesis Huang et al. (2000). Model: Method I N = min[20, 20Td]

12 K

(576)

, (576)

X = 0.6,^m<\ Tml

'

(576)

Am=2.83. X = 0 . 7 , K ^ Tml

Am=2.43.X = 0 . 8 , ^ r a > 2

= 3 L

2Tm,^+0.4Tn,

Ti(576)

= 2 T m |

m

<2

Am=2.13.

0-5Tml2 --1 •1. P =T m / + 0.8T m l ^ m T n

2Tm,^ a =2T m l ^ m +0.4x n

,2 ^ ( 5 7 6 )

Servo tuning

^

+ Q 4 i n

Km(tm+2Tm3) (576)

T m l 2 + 0.8T mml^rri l ^ r a x"m 0-8Tra]^mTn

Handbook of PI and PID Controller Tuning Rules

392

4.9 SOSPD Model with a Negative Zero K m (l + sT m3 )e-"K m (l + sT m3 )e-"orGJs): G m (s) = (l+sT mI )(l + sTm2) Tmiy+2^Tmls +l

4.9.1 Ideal controller Gc (s) = K(

/

\

1 T

V

R(s)

i

S

J

iU(s;

*6?)

1f-

E

^ » K fl + - L + T s c d T s I

i

V(s)

K m (l + s T m 3 > - " TmlV+2^mTmls + l

J

—v-

Table 146: PID controller tuning rules - SOSPD model with a negative zero K m (l + sTm3)e-sx™ G m (s) = T m l V+2); m T r a l s + l Rule

Kc

Wang et al. (2000b), (2001a). Model: Method 2

1

K

(577)

1K

1

Comment

Ti Td Direct synthesis 2? m T ml

(577)

T 1

cos 6

2

^mTmlffl

,

2

M,=1.5

ml

•; co and G are determined by

M s cos[mTm-tan- (Tra3co)] K m ^ T m 3 V + solving the equations: tan

cos 9 = tan '(Tm3co) —coxm -0.5TC and M V S -sinO 6 = tan"1

T m 3 ^ m m 2 - 1 ) c O s ( ( 0 T m ) - C 0 T m SJn(c0T m ) lTm3'Cmffl2-1)sin(aiXm)+ff>'Cmsin((aXm)_

Chapter 4: Tuning Rules for PID Controllers

393

T 1 dS 4.9.2 Controller with filtered derivative Gc (s) = Kc , 1 +— +- d N

R(s)

E(s) +

K„

a

,

T

1

1 +T— s + • '

Y(s)

U(s)

ds

;

G m (s)

—

1+ ^ s

N . Table 147: PID controller tuning rules - SOSPD model with a negative zero K m (l + s T m 3 ) e - ^ _^ ^ _ K m (l + sTm3)e-STG m (s) = orGm(s): (l + sTmlXl + sT m2 ) T m l 2 s 2 +2^ m T m l s + l Rule

Kc

Ti

Td

Comment

Robust 2

K

(578)

(578)

T

*d

2s m T m i ~T m 3

2 v (578) _ Tml + Tm2 - Tm3 K

Kra(X + t J

(578)

.V

(578)

3

T

(579)

Tmi

(2i;mTml Tm3jTm3 2smTrai _ T m 3

:

Tml + Tm2

. TralTm2

(578)

T

i

Chien (1988). Model: Method 1 2s m T m i ~ Tm3

3

T

(579)

N=10; *-e[T ml ,T m ] (Chien and Fruehauf(1990)).

Tm3 .

(Tmi+Tm2

T^JT^

• > Tm| > Tm2 > T m3 .

394

Handbook of PI and PID Controller Tuning Rules

4.9.3 Classical controller 1 G c (s) - Kc

1

v

R(s)

E(s)

K, 1 + —

+ x

1 + Tds

+

^ T s i y

^ K

1 + Tds

I

m

N ( l

+

; sT

m 3

^

Y(s)

(l + sTralXl + sTm2)

1+ ^ s N

Table 148: PID controller tuning rules - SOSPD model with a negative zero G m (s) = Rule

Zhang (1994). Model: Method I

Kc

K m (l + sT m 3 >-"(l + sTmlXl + sT ra2 ) T( Td Direct synthesis

Comment

Trol K

mTm

Tm2

Tmi

N = T m2 /T m3

"small" xm ; "good" servo response.

Poulin et al. Tm, (1996). K m (T r a l +T m ) Model: Method 1

Tm2

Tmi

N=Tra2/Tm3; Minimum * m =90°;
Robust Chien (1988). Model: Method 1

Tm2 K m (^ + t m )

Tm2

Tml

Tm,

Tm2

Tml

Km(^ + 0

Trai >T r a 2 ;N=10; ^e[T m l ,x m J (Chien and Fruehauf( 1990)).

Chapter 4: Tuning Rules for PID Controllers

395

/

4.9.4 Classical controller 4 G c (s) = K v

T

l+-

T

's

iV N

U(s)

R(s)

E(s)

1+ -

K„

v

T

is/

T

N

Y(s) K m (l + sT m3 )e(l + sT ml Xl + sT m2 )

^

;

Table 149: PID controller tuning rules - SOSPD model with a negative zero Km(l + sTm3)e-^ Gm(s) = Rule

Kc

(l + sTmlXl + sT m2 ) Ti

Td Robust

Chien(1988). Model: Method 1

Tm2

Tm2

Tmi - T m 3

Tm,

Tm2 ~~ T ra3

K m ( * + Tm)

Tm,

Km(^ + 0 Tm, > T m 2 > T m 3 ; N = 1 0 a 6 [ T r a l , T m ] (Chien and Fruehauf (1990)).

Comment

396

Handbook of PI and PID Controller Tuning Rules

4.9.5 Non-interacting controller 1 r

U(s) = Kc

E(s)-

TdS

Y(s)

N Y(s)

Km(l + sT m3 >-"T m ,V + 2^ m T m l s + l

Table 150: PID controller tuning rules - SOSPD model with a negative zero K m (l + s T m 3 ) e - ^ G m (s) = TralV+2^Trals +l Rule

Kc

Huang et al. (2000). Model: Method 1

Td Direct synthesis (580) T 2T ral ^ m+ 0.4T m Ti

4KC(580)

servo tuning

N not specified l =0 . 6 , ^

m

< U = 0.7,K^

m

regulator tuning

4 K (580)

=?

Comment

2Tml£m + 0,4t m K m (T m +2T m 3 )

(580)

_ Tm, +0.8T ml £ m T m 0-8Tml^mTn

<2.^0.8,^

m

>2.

Chapter 4: Tuning Rules for PID Controllers

397

4.9.6 Non-interacting controller based on the two degree of freedom structure 1 f

\

U(s) = Kc

f

\

PTds T 1+ ^ s N

R(s)

1+ ^ s N

N

a+

(3Tds

E(s)-K c a +

K„

U(s)

R(s) K, 1 + E(s)

1

Km(l

THs

*g>> T

+T=s 1 + ^ s N .

+

Y(s)

sT m 3 )e-™

miy+2^mTmls

+l

Table 151: PID controller tuning rules - SOSPD model with a negative zero K m (l + sT m3 )e-"" G m (s) = TmlV+2^mTmIs + l Rule

Huang et al. (2000) - servo tuning. Model: Method 1

Comment

K, sK

Direct synthesis (581) 2T ral ^ m +0.4T n

(581)

2T ml $ n 2T m l ^ m + 0.4T n

N = min[20, 20Td]

0-5Tra,2 •l.p = T r a /+0.8T m l ^ m T n

X = 0 . 6 „ ^ K m < 1 A m = 2.83 . X = 0.7,1 < ^ Ami

'

Am =2.43. X = 0.'<

'

-lm>2

5 K (581) _.| ZTml^m +0-4l m ^ (581) _ Tml +0.8T ml ^ m T n Km(im+2Tm3). d 0.8T ml ^ m i m

m

Ami

Am=2.13

•1.

<2 '

Handbook of PI and PID Controller Tuning Rules

398

4.10 Third Order System plus Time Delay Model K e~ST|" G

m W

K e~STm

m

(s) =

m

or G (s) 2

l + a1s + a 2 s + a 3 s

3

m W

(l + sT ml Xl + sT m2 Xl + sT m 3 )

A / 1 4.10.1 Ideal controller G c (s) = Kc 1+ — + T d s V Tis U(s) R(s)

Y(s)

K„e"

(l + STmlXl + sTm2Xl + sTm3)

Table 152: PID controller tuning rules - third order system plus time delay model G m (s) = Rule Standard form optimisation binomial Polonyi (1989). Standard form optimisation minimum ITAE Polonyi (1989).

1

K

Kc

1K

(582)

K„e-"-

(l + sTm,Xl + sTm2Xl + sT m3 ) Td Ti Minimum performance index T(582)

i

2

K

(583)

(582) _

6t m

Tm3

T

(583)

Comment

,Xm

Model: Method I

tm

Tm, > 10(T m 2 +x m )

T/ 5821 = 4 V 6 T m 2 T r a + t r a .

Chapter 4: Tuning Rules for PID Controllers

399

4.10.2 Ideal controller in series with a first order lag /

G c (s) = Kc T

v

1

\

1

is

j

Tfs + 1

—JU(s) 1

K, V

T S

i

Y(s)

K„e"

Tfs + 1

l + a,s + a 2 s 2 + a 3 s 3

Table 153: PID controller tuning rules -Third order system plus time delay model Kme"^ G ra (s) = l + a,s + a 2 s + a 3 s Rule

Kc

Schaedel(1997). Model: Method 1

3

K

(584)

T

(584)

(584)

(584)

, (584) _ a l ~ a 2

(584)

Km U , + T m + (584) _ a 2 +a 1 x m +0.5T m 1J

T

Tf=Td(5S4)/N; 5 < N < 20

0.375T

(584)

K

3

Comment Td Ti Direct synthesis: frequency domain criteria

- (584)

+a T

a, + t m - T ,

N a 3 +a 2 T m +0.5a,T m +0.167xn

—

a 2 +a 1 T m +0.5T n

l m +0-5'cm (584)

Handbook of PI and PID Controller Tuning Rules

400

4.10.3 Controller

with filtered derivative

i 1 G c ( s ) = Kc 1 + +

T

ds

=r

N U(s)

R(s)

E(s)

1

K, 1 + — • + T S

'

THs

K„e"""

Y(s)

l + a,s + a 2 s 2 + a 3 s 3

1+ ^ s

N

Table 154: PID controller tuning rules - Third order system plus time delay model K m e~ G ra (s) = ija T « 2 a

Rule

a

Comment Ts Td Direct synthesis: frequency domain criteria (585) (585) Model: Method 1 T (585) T

4

K

0.375T,-

(585)

K

3

Kc

Schaedel(1997).

4

a

(585)

T(585) =

a , 2 - a 2 + a , T m +0.5T n

(585)

*i + t m - T d

• (585)

K„

(585)

N

(585) _ a 2 + a,x m +0.5x m a, + x „

a3+a2xm+0.5a,xm

+0.167Tn

a2+a,xm+0.5x (585)

N: a + Tm-Tj

(585)

Jki^flt

8 5 ) T (585) ry ( 5yjOJfy

0.375-L K„

0.05 < a < 0.2 , Kv = prescribed overshoot in the manipulated variable for a step response in the command variable.

Chapter 4: Tuning Rules for PID Controllers

401

4.10.4 Non-interacting controller based on the two degree of freedom structure 1

(

\

1 U(s) = Kc , 1+ — + T s i

T

r

\ PTds

dS

E(s)-K c a + R(s) 1+ ^ s 1+ ^ s d

N

a+

N

j

Fds

K, T N ;

U(s)

R(s) +

*g» d + sT

K,

n

Y(s)

K„e-"»

E(s)

ml)

3

N

Table 155: PID controller tuning rules - third order system plus time delay model Kme^ G m (s) = 0 + sT m l ) 3 Rule Taguchi and Araki (2000). Model: Method 1

5

K

Comment Td Kc Ti Minimum performance index: servo/regulator tuning Overshoot (servo ry (586) (586) (586) T step) < 20% ; N fixed but not specified

' Equations continued into the footnote on page 402; xm /T m l < 1.0 .

K

(586)

0.4020+ K„

1.275 - + 0.003273

) 2

Ti

(586)

rp " 1ml

a!

0.3572 + 7.467- --12.86

^m

Tml.

4

3

+ 11.77

"l

-4.146 _Tmi

T

. m!_

Handbook of PI and PID Controller Tuning Rules

402

2\ (586)

= T„

0.8335 + 0.2910-

a = 0.6661 - 0.2509- 12 - + 0.04773 Tm,

- 0.04000

, p = 0.8131 - 0.2303—S3- + 0.03621 T

Chapter 4: Tuning Rules for PID Controllers K„e""-

4.11 Unstable FOLPD Model G m (s)

Tj-1

f T

V r

is

U(s)

1

1 + — + Tds

K,

+

V

T s

i

A

1

4.11.1 Ideal controller G (s) = Kc

_RCs)_^E(s)

403

Ke"

Y(s)

Tms-1

J

Table 156: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Kc

Minimum ISE Visioli (2001). Minimum ITSE Visioli (2001). Minimum ISTSE -Visioli (2001).

1.37

Comment Td Minimum performance index: regulator tuning Ti

f

2.42Tra

T

1

(

Model: Method 1

0.55xm

Model: Method 1

0.50xm

Model: Method I

\ 0.90 m

4.12Tm

Km

0.60xm \ 1m J

\

1.37

1.70

K m e" Tms-1

I

T

f

1.13

T 1

4-52Tmf^ V

m

j

Minimum performance index: servo tuning Minimum ISE Visioli (2001).

/

NO.92

1.32(0

4.00T„^j

1

T

(587)

2

T

(588)

Model: Method 1

K m lT m J Minimum ITSE Visioli (2001).

1.38'

1 T^(587)

1 - 0.84

.90

T

3

78T

^

T \09S

f =

T V m

/

2

Td(588) =

3

62T

^

1-0.85

N 0.02

\ 0.90

f

4.12Tm

A093

f

T V mJ

Model: Method 1

Handbook of PI and PID Controller Tuning Rules

404

Rule

Kc

Minimum ISTSE -Visioli (2001).

,

1.35fO

Comment

Td

Ti 1.13

N0.95

3

Model: Method I

(589)

T

4.52Tmf^

KralTJ

Minimum performance index: servo/regulator tuning

Jhunjhunwala and Chidambaram (2001).

4

(590)

K

(590)

T

Model: Method 1

(590)

T

Direct synthesis De Paor and 0'Malley(1989). Model: Method 1 Chidambaram (1995b). Model: Method 1

3

x

(589)

4

K

6

5

(590) _ ~

K

T

(592)

25-27^2-

rr

•<0.2:

, 0.2<-^-<1.4; T T

- , 0 <-^- < 1 ; T: (590) = 0.3444T e

cosJU-^Ll^L+J1

Tm//Tm

K„

T

,(591)

l-t„/Tm T

f

6

v

K„

(592) _ _ 1 _

tan(0.75(f))

m/T m

\

(j) = tan v

^2_<0.6 T

0.46x m

0.5643-^2- + 0.0075 , 0 < ^ _ < 1 . 4 . T T

T X

<1

T

1-0.861

1.397

(591)

K

(591) d

m

Ti (590> = 0 . 8 5 6 T e

'd

T 1

m

K„

T

(591)

T

13.0-39.712—2 Tm

Kml (590)

(591)

K

i

: 3.70T„

(590)

K

5

T m

y

T

m m

1.3 + 0.3^2-

K/ \

m V

\ m /

sinjf 1 - - T "

T

-

, 1<^L<1.4:

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Chidambaram (1997). Model: Method 1

1.165

A

m

405

Comment

Ti

Td

7 T (593) 1 i

j(593)

a

Valentine and Chidambaram 1.165 (1997b)dominant pole placement. Model: Method 1

,

S T (594) ^i

\ 0.245

T T

lTmJ

(594)

^d

9T(595)

<0.6

0.6 <

<0.8 Tm m

10 j

0.8 <

(596)

<1 m

f

\2

7 ^(593) = 0 . 1 7 6 + 0.360^2- T m ; a = 0 . 1 7 9 - 0 . 3 2 4 ^ - + 0.161 2m. I \

* m

/

*• m

vT™

t n <0.6

a = 0.04, 0 . 6 < ^ s . < 0 . 8 ; a = 0 . 1 2 - 0 . 1 - ^ - , 0 . 8 < ^ - < 1 . 0 . T T T 0.176 + 0.36, (594)

m J

0 . 1 7 9 - 0 . 3 2 4 ^ + 0.161 — T T 9 j

(595)

10 j (596)

o.ne + o^-^psT^ 0.176Tm+0.36T„ 0.12-0.1^-

r-Td

(594)

0.176 + 0.36-

Handbook of PI and PID Controller Tuning Rules

406 Rule

Kc

Pramod and Chidambaram (2000). Model: Method I

T

11 K

Comment

Td

Ti (597)

(597)

T

12 j

(597)

T

- <0.6

(598)

0.6
13

K

14 v

(599)

T

(600)

T(600)

f 11

K

(597)

=

_

2.121-1.906-2-+ 0.945 K„

T„

rj, (599)

(599)

t, T

<0.8

0.1<^L<0.6 T

0.4917tm

0.01 < ^ L < 0.9 T

\2

0.176 + 0.36(597) _

,(597)

/" \ 2 ' 0 . 1 7 9 - 0 . 3 2 4 - ^ + 0.161 T T 12 j

(598)

(599)

V T

T„ m J

\ m y

16.327 VTm/

196

V

T

K„

y (600) _

(

4 . 1 0 4 ^ 2 . + 2 . 0 9 9T- ^ + 0.096 V mJ

+ 0.419-E- +0.019

T

1.4183 ('

= T„

\2

rt V

14 xj (600) _

• (599)

0.299 + 0.726-

0.822

0.176 + 0.36-

2.5T„

0.176 + 0.36-

Kn

d

+ 5.5778^2- + 0.8158 , 0.01 <-^< T T

- 247.28 -J2- + 87.72 , 0.6 <-^2-< 0.9. T T

0.6;

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Ti

Sree et al. (2004) - continued.

15 pr (601)

y (601)

Td T x

(601) d

407 Comment 0.01
Robust Tan et al. (1999b). Model: Method 1 Lee et al. (2000) Model: Method 1

(602)

v

rp (602)

(602)

T

-^-<1.0 T

•M

17

y (603)

(603)

K

x J

•M

(603) d

0<^2-<2 Tm m

15

v

K

1.2824

(601) _

T

K„ •. (601) _ ,

. (601) _

16 if

5.5734-iS-- 0.0063 , 0.01 < - a - < 0.5; T T T

™ , 0.5 < ^ - < 1.2 ; Td(601) = Tn 0.507-21-+ 0.0028 T

0.483Trae '

1

(602)

0.0373

Kn

X 10.5045

+ 0.1862 ^ - +[0.6508-0.1498 log?.] 3.2312

+ 0.8032

- 0.3445

, (602)

0.0373

(602)

+ 0.1862 ^ - + [0.6508- 0.1498 log*.]

l 0.9065X,0-"'"-12-Tm 0.2075X + 2.0335 00373

+0.1862 | ^ - + (0.6508-0.14981og?0 , X not specified.

, (603) 17 £

(603)

Km(2TCL+xm-a) . (603) _

•Tm+aT a

, (603)

.<* = Tn

ick + l T

e*.P._l

TCL2+aTm-0.5Tm2 2TCL+tm-a T m 2 (0.167x m -0.5a) 2TCL + T m - a ,(603)

T C L 2 +at m -0.5T m 2 2TCL+xm-a

Handbook of PI and PID Controller Tuning Rules

408

4.11.2 Ideal controller in series with a first order lag •>

f

\

1 Tfs + 1

G c (s) = Kc V

U(s) / 1 R(s) ^ '/v0 v" E ( s ) K, 1 + — + T d s + X

T

V

i

s

j

1

T

is

Kme"

Tfs + 1

Y(s)

T„s-1

Table 157: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Kc

Chandrashekar et al. (2002).

18

v

K„

(604) _

_J_

18 j r

4282

Ti Td Direct synthesis ry (604) 0-5Tm

(604)

-1334.6-21-+ 101 T

K„.

y (604) _ - ,

36.842

T

X

-10.3

Model: Method 1

1

m

N-0.9427

.1161 Km V

0.2 < -!=- < 1.0. T

m J

+ 0.8288 , 0 < — < 0 . 8 ; T

T V

(604)

Comment

0<-S-<0.2; T f

K

K m e" •o-i

m J

T: ( 6 0 4 ) = 76.241T„

, 0.8 < - 2 - < 1.0 T

Tr = 0 . 1 2 3 3 - ^ + 0.0033 T m , 0 <-$-< 1.0. /•

(TIS + l)e~STm x„ Desired closed loop transfer function = ^~- r ^ ; TCL2 = 4.5115Tn 2 T (TCL2s + l) f

0.8<^<1.0,TCL2

\i \

0.0326+ 0.4958-2L +2.03 T T V

\ 5.661

m J

T m , o < - s - < o.: T

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

T, Td Alternative tuning rules Tf Td

409 Comment

Chandrashekar et Kc al. (2002). Model: Method 1 101.0000

0.0404Tm

0.0

0.0

0.02Tm

0.0

Kra 10.3662

0.3874Tm

0.0435Tm

0.0134Tm

0.10Tm

0.1

5.3750

0.8600Tm

0.0884Tm

0.0250Tm

0.20Tm

0.2

3.2655

1.8018Tm

0.1375Tm

0.0435Tm

0.40Tm

0.3

2.4516

3.0400Tm

0.1868Tm

0.0581Tm

0.60Tm

0.4

Kra 2.0217

4.6500Tm

0.2366Tm

0.0696Tm

0.80Tm

0.5

1.7525

6.7286Tra 0.2866Tm

0.0784Tm

1.00Tm

0.6

1.5458

10.337Tm

0.3380Tm

0.0885Tm

1.30Tm

0.7

1.3723

17.693Tra

0.3910Tm

0.1005Tm

1.80Tm

0.8

1.2420

33.610Tm 0.4440Tm

0.1124Tm

2.60Tm

0.9

1.1400

80.620Tm 0.4969Tm

0.1247Tm

4.20Tm

1.0

Ti

TcL2

T

m/Tm

Handbook of PI and PID Controller Tuning Rules

410

4.11.3 Ideal controller with set-point weighting 1 U(s) = Kc (FpR(s) - Y(s))+^(F i R(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S

Table 158: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Kc

Prashanti and Chidambaram (2000). Model: Method 1

1

K

(605) _ _J_

1K

Td Direct synthesis

Ti

(605)

(605)

T

T

F p =0.17

Fd = 0.654

Fi=l

Tm//Tm —+.) sin T V x /T

COS J 1

v x

(605)

*i

(605) _

T 'my

(605)

V 1 V / 'm

tan(0.75(|») T

n,/Ti>

Tm

(j) = tan V

m/

m

xra \

m \

m/

Tms-1

Comment Modified method of De Paor and O'Malley (1989);

h./Tmj
Chapter 4: Tuning Rules for PID Controllers Rule Prashanti and Chidambaram (2000)Alternative tuning 1. Model: Method I; (605)

K

Td

T(605)

specified

on page 410.

Prashanti and Chidambaram (2000) Alternative tuning 2. Model: Method T, (605)

K

Td

(605>

T(605)

specified

on page 410.

Kc

T;

(605)

K

(605)

T

0-05<xm/Tm

*d

<0.90 F p = 0.309

Representative results 2 Fd = 2.834 Fi=l

xm/Tm=0.05

F p =0.281

F,=l

Fd = 2.605

Tm/Tm=0.10

F p = 0.244

Fi=l

Fd =2.413

tm/Tm=0.20

F p =0.213

F; = 1

Fd = 2.346

xm/Tm=0.30

F p =0.182

Fi=l

Fd = 2.336

tra/Tm=0.40

F p =0.150

Fj = 1

F d = 2.207

xm/Tm=0.50

F p =0.116

F,=l

Fd = 2.093

Tm/Tm=0.60

F p = 0.080

Fi=l

Fd =1.884

im/Tra=0.70

F p = 0.040

F;=l

Fd =1.508

xra/Tm=0.80

F p =0.0091

F|=l

Fd =1.159

tm/Tm=0.90

y {605)

(605)

K

T X

(605) d

-6.405T„,

F p = 0.9726e

Tm

, F; = 1, Fd = 0 ; 0.05 < ^ - < 0.90 . Representative results =0.05

F p =0.046 , T m / T m = 0 . 5 0

Fp = 0 . 4 5 7 , t m / T m = 0 . 1 0

Fp = 0 . 0 2 6 , x m / T m = 0 . 6 0

Fp = 0 . 2 2 7 , x m / T m = 0 . 2 0

Fp = 0 . 0 1 3 , T m / T m = 0 . 7 0

Fp = 0 . 1 2 1 , t m / T r a = 0 . 3 0

Fp = 0.0057, x m / T m = 0 . 8 0

Fp = 0 . 0 7 8 , x m / T m = 0 . 4 0

F p = 0.0022, x m / T m = 0 . 9 0

F P = 0 . 7 7 7 , xjTm

2 !

Comment

Td

(605)

T

411

F n = 0.3127-0.3369-^- + 0.0378 ^ _ m

\ *m J

V

f -0.045^ V

• F= = 1 .

m j

/ FH =2.779-2.44-2!-

+5

x„

^2

T V

m J

-4.81

T

Handbook of PI and PID Controller Tuning Rules

412 Rule

Kc 3

Prashanti and Chidambaram (2000). Model: Method 1

Prashanti and Chidambaram (2000) alternative tuning. Model: Method 1

K

T

^(606)

Representative results Fd =2.916

Fp = 0.542

3=1

Fp = 0.437

Fi=l

Fd = 2.878

tm/Tm=0.10

Fp = 0.366

F|=l

Fd = 2.525

^m/Tm=0.20

Fp = 0.303

Fj=l

Fd = 2.307

^m/T m =0.30

Fp = 0.263

Fi =1

Fd =1.948

x m /T m =0.40

Fp =0.214

F|=l

Fd =1.729

x r o /T m =0.50

Fp =0.164

F;=l

Fd =1.530

tra/Tm=0.60

F„ =0.151

F,=l

Fd =1.297

x m /T m =0.70

Fp =0.139

F,=l

Fd =1.093

tm/Tm=0.80

Fp =0.113

Fi=l

Fd = 0.962

x m /T m =0.90

x m /T m =0.05

2

Fp = 0.775 7-1.289^2-+ 0.6267 T T m V xm J 0.05 < ^ < 0.90 T

1, Fd = 0 ;

m

Representative results Fp = 0.605 , T.n l /T m =0.05 Fp = 0.322 , Tm / T m = 0 . 5 0

f 3

Comment

Td

i

rj, (606) i

(606)

K C ( 6 0 6 ) = — [ 2 . 1 2 1 - 1 . 9 0 6 ^ 2 - +0.945 Km T a = 0.179-0.324-^2. +0.161 \ l^-\ T T

t

m

T

\2 I ,

-,(606)

' "

•

T a

0.176 + 0.36-

, — < 0 . 6 a = 0.04 ,0.6 < 1^- < 0.8 ; T T a = 0.12 - 0.1-^2-, 0.8 < ^ - < 1.0; T T

Fn =0.5165-0.8482^2L +0.5133f T"

^

f

-0.071

1M.

;Fi=l:

T

f

A3

F d = 3 . 2 0 6 - 3 . 4 6 1 ^ L + 0 . 9 8 4 p 2 - | +0.089 m

V',

V

m J

Chapter 4: Tuning Rules for PID Controllers Rule

413

Comment Td Representative results (continued) Fp =0.664, T m /T m =0.10 Fp = 0.248, x m /T m =0.60 Kc

Prashanti and Chidambaram (2000) continued.

T

i

Fp = 0.550, T ra /T m =0.20

Fp =0.228, x m /T ra =0.70

FP =0.456, x ra /T ra =0.30

Fp = 0.209, x m /T m =0.80

Fp =0.395, i m / T m =0.40

Fp = 0.169, x ra /T m =0.90

Jhunjhunwala and Chidambaram (2001).

4

K

T(607)

(607)

T

Model: Method I

(607)

*i

Robust ry, (608) x d

y (608)

5 K (608)

Lee et al. (2000). Model: Method 1

2

1 as + 1

F p = Fi =

4 K (607)

=

_J_[ 13.0-39.712- T " 1.397

(607)

T (607) = 0.856Tme

2.044tm T

-

III

(607) _ .

etm/Tm

_lTm

,

J

, 0.2<-!2-<1.4; T

K„

|

T

-^-<0.2: T

K„

K

0<-^-<2 T

, 0 < ^ - < 1 ; T,'607' = 0.3444Tme '

rji

'

T

-

111

I

, 1 < - ^ - < 1.4 ; rji

'

0.5643— + 0.0075 , 0 < - 2 - < 1 . 4 : T T

F i = 0 ; Fd = 0 ; F p = 0 . 6 8 6 3 , - ^ - = 0.1 ;F p =0.4772e T™ , 0.2 < ^ - < 1. 1 m

* TTI

(608)

T; ". Km(2TCL+Tm-a) ,(608) _

-T„ + a •

, (608) _

=

T

n

(TX

CL

+1

e ^ - 1

T

T C L 2 +aT: m -0.5T m 2 2T CL +T m -CX 2

-T„a--

a

m

(0.167T m -0.5a) a +T m - a

2T

T (608)

T C L 2 +aT m -0.5T m 2

Handbook of PI and PID Controller Tuning Rules

414

4.11.4 Classical controller 1 G c (s) = K E(s) R(s)

®—K,

+ T

Table 159: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Shinskey(1988) - minimum IAE -page 381. Method: Model I N not specified

K„e'

T m s-1

Comment Ti Td Minim um performance index: regulator tuning 0.91Tra 0.60xm 1.70Tm Kn/Tra| = 0.1 Kmtra Kc

1.00Tm Kmxm

1.90xm

0.60im

k / T m | = 0.2

0.89Tm Kmxm

2.00xra

0.80xm

k / T m | = 0.5

0.86Tm KmmTmm

2.25xm

0.90xm

k / T m | = 0.67

0.83Tra Kmtm

2-40xm

1.00xm

k/T r a | = 0.8

Chapter 4: Tuning Rules for PID Controllers

415

4.11.5 Series controller (classical controller 3)

\ A"

G c (s) = K, +

Table 160: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Kc

Huang and Chen (1997), (1999).

6 ^

(609)

T m s-1

Comment Td Direct synthesis: time domain criteria (609) y (609) T Model: Method 4 X Ti

d

0 < —— < 1; 'Good' servo and regulator response.

K

(609)

0.5 1 + 1.61519 Km T

(609),

. ^ (609)

K =

c

K

ro T m [

2x

l

V < 609 >K1I T ^-,J^-m ~ 1 \ 1 r a

+

2m. T 'my

with recommended x, =T m -1.3735.10" 3 + 0.22091^2-+ 1.6653PST T T

(609) _

T

4

, n O'}c/in m , ^ 1 £mr\ _ 1.5006.10"4 +0.23549—E2+ 0.16970I

n

V T1 m J

Handbook of PI and PID Controller Tuning Rules

416

4.11.6 Non-interacting controller based on the two degree of freedom structure 1 f PTds ', 1 Tds ^ R(s) U(s) = Kc E(s)-K t a + 1+ — TiS + l + (T-d-/N> l + (T d/N>_ J v a +-

(3T d s

K_

T N

j

U(s) Y(s) K e"

1+^

*K,

+

T S

R(s)

'

E(s)

-

^

T

+

1+ ^ s N

s-1

Table 161: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Kc

Ti Td Direct synthesis 7j

0.245 Chidambaram 1.165 (T ~\ (2000c). K m lTmJ Model: Method I

Tms-1

Comment

(610)

T

(610)

4.4T m +9x ra

4 = 0.333; N = oo;(3 = l

K m e"

T 0.6 <

8T(6U) i

0.8 <

- <0.6

T T m

0.176 + 0.367 j

(610)

(610) f

A2

'

d

0.176 + 0 . 3 6 - ^ T . T

0.179-0.324^2-+ 0.161 T ,T 8 Ti(61 , ) =

0.176T m +0.36x m; 0.12-0.1^2-

a = 074 +

0 4 2

I^_ T

™

0 2 3

\2

/ T V'n/

<0.8 <0.9

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Srinivas and Chidambaram (2001). Model: Method 1 Chandrashekar et al. (2002). Model: Method 1

9

T,

Comment

Td

T(612)

(612)

K

T

417

(612)

N = oo

i

10

0.5Tra

T(613)

(613)

K

N = oo;|3 = 0

'Sample K c (612) , Tj (612) ,T d (612) taken by the authors correspond to the dominant pole placement tuning rules ofValentine and Chidambaram (1997b) (see page 405); •r

,(612)

10 ^

1 K

(613)

(6,2,

•; P =

K

4282 - r

(612)

K

2X ( 6 1 2 )

(612)K

-1334.6-^- +101 , 0<-!S-<0.2; T T

K„

K

(613)

1.1161 K„

, T V

f

36.842

, m J

, 0.2<-S-<1.0. T x

m

N2

, T

•10.3 t

T

+ 0.8288

0<JU2-
T:(613) = 76.24IT '

Tn

0.8<-^<1.0. T

a = 0.83, - ^ - < 0 . 1 ; a = 0.8053-0.1935-^-, 0.1
(rjs+_l)e^. (TCL2S + 1) 2 '

\2^ 0.0326 + 0.4958-

-2.03

Tm , 0 < - ^ < 0.8 ;

T r l , =4.5115T

T

, 0.8 < - 2 - < 1.0 T

418

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Td

Ti

1

Comment

Alternative tuning rules Chandrashekar et al. (2002). Model: Method I

Kc = — K

(614)

Coefficient values X x4 5

i

x2

x3

101.000 10.3662 5.3750 3.2655 2.4516 2.0217 1.7525 1.5458 1.3723 1.2420 1.1400

0.0404 0.3874 0.8600 1.8018 3.0400 4.6500 6.7286 10.337 17.693 33.610 80.620

0.0 0.0435 0.0884 0.1375 0.1868 0.2366 0.2866 0.3380 0.3910 0.4440 0.4969

x

Wang and Cai (2002). Model: Method I

, Tj = x 2 T m , Td = x3Tm , Tf = x 4 T m , TCL2 = x 5 T m

m

11

0.5236Tm

K

(614)

T

(614)

M

0.4764 Pf^ - I

T (6I4) )

A;

0.02 0.10 0.20 0.40 0.60 0.80 1.00 1.30 1.80 2.60 4.20

0.0 0.0134 0.0250 0.0435 0.0581 0.0696 0.0784 0.0885 0.1005 0.1124 0.1247 T 1

a

*m/Tm

0.505 0.742 0.767 0.778 0.803 0.828 0.851 0.874 0.898 0.923 0.948

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

N=0; p = 0

(614) d

_ 0.5236Tm - 0 . 4 7 6 4 ^ ,

-

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Xu and Shao (2003b). Model: Method 1 Sree and Chidambaram (2005a). Model: Method 1

12

(615)

K

Td

Ti T

(615)

T

(616)

K

j(616) i

T

12

K

(615) _

0-5

Km

(615)

13

T 1

T m

I

T

'if

0.5/L Tm

N=0; P = 0

(616)

N=oo; p = l

'd

/T

|T„

a

IT

(615)

2^ A'

Comment

(615)

*d

13

419

mTm

(Ara>2,4>m>60°).

=

T m +VT;

Sample K c (616) , Tj*6'6' ,T d <616) taken by the authors correspond to the tuning rules of Chandrashekar et al. (2002) and Huang and Chen (1999); the latter tuning rule is defined for an unstable SOSPD process model. Desired closed loop transfer function =

iQ-qfc^s + ilr"-

T CL 2 s 2 +2£T CL s + l The optimum value of a is chosen by minimizing the ISE (servo response). Then, 2TCL5 2TCL a = l - - .(616) , £,= 1; a = l- T (616) , S < 1 ;

^+^~i)(-^-^~i)2 a = l-

.(616)

with x, =-(-$-fi*~Tj

J-z+fiT^

+ x,

4>i

^+Vi^T -^-V^T +x + ^

+ A /^T7j

^ _ ^ T 7 j and

x^^-^T^-^V^J^^+V^J^-V^ If it is desired to minimize the overshoot (servo response), for £, > 1, then a = 1.

420

Handbook of PI and PID Controller Tuning Rules

4.11.7 Non-interacting controller 3 U(s) = Kc

E(S)-^£Y(S) T

V

s

i /

N R(s)

U(s)

E(s)

-*R>

K„ T s

v

>

K e"sx™

i y

Y(s)

T„s-1

KcTds 1+ ^ s N

Table 162: PID controller tuning rules - unstable FOLPD model Rule

Kc

Huang and Lin (1995) minimum IAE. Model: Method 2

Servo response: K

T: Td Minimum performance index

1K

(617)

-p (617)

2

(618)

rj, (618)

K

(617)

(617)

T

T X

(618) d

K m e" Tms-1 Comment

0.01 <

T

<0.8; r>[=10

- 0.433 + 0.2056-2-+ 0.3135 T

K„

6.6423 \ ,(617)

= T

• 0.0018 + 0.8193-^- + 7.7853 T v

(617)

T

= T

0.0312 + 1.6333- SL + 0.0399e T

7.6983-^ T

™

m J

• Regulator response: Kc

. (618)

(618)

=

-1.004 A

0.2675+ 0.1226-=i- +0.8781 T

0.0005 + 2.4631 -S- + 9.5795 T

i(618)

= 0.001 lT m +0.4759x„

Chapter 4: Tuning Rules for PID Controllers

421

4.11.8 Non-interacting controller 8 1

\

;s

j

f

U(s) = Kt v

R(s)

T

E(s)

-*8h* K

+ Tds)

+ x-x

E(s)-K i (l + Tdls)Y(s) U(S)

—•(X)

•

Y(s) -•

K e" T„s-1

+ f KiO + V )

I Table 163: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Kc

Comment

Td

Ti

Kme~ Tms-1

Robust Park et al. (1998). Model: Method 1

3

K

T (619)

(619)

T

(619)

K

_ i |T7 ' Km Vtm Tdi=0

3

Apply minimum ITAE - regulator tuning rule or minimum ITAE - servo tuning rule defined by Sung et al. (1996) for SOSPD model, ideal PID controller (pages 313, 316). For these formulae, Km , Tml and £m are replaced by the following parameters from the unstable FOLPD model: K„

• (Unstable FOLPD) -> Km (SOSPD) -1

A

7 1 X

0.25

0.75

0-71Tm xm V T A05 „

7

,L

0.5

(Unstable FOLPD) -» Tml (SOSPD)

0.5 L 0.25

rT

• (Unstable FOLPD) -> %m (SOSPD)

\0-5 -1

Handbook of PI and PID Controller Tuning Rules

422 Rule

V 4

(620) _

K

K

i

<620>l
4

K

(620) _

l

m

Comment

Td

y (620)

(620)

(620)

x l

6

1 _ 1

T

K

(620)

(620)

Km(^ + x J T

X

T;

Kc

Lee and Edgar (2002). Model: Method 1

Ki

K-

Tdi=0

T

Km-l

2(X + T J

(620) „ v -K-i K-mX„

(620)

Km(^O^K,<620)Km-l)

6(X + x J

4(X + T m ) 2

2(K/-»Km-lJ(X + xJ}' K,

(620)

1H.+

In

f

V

1-^ T ,

T

T

K„

\

m /

V

1

m J

V

my

Chapter 4: Tuning Rules for PID Controllers

423

4.11.9 Non-interacting controller 10 E(s)-K f

U(s) = K( v

R(s)

E(s)

+

xg>

K,

+ X

T s

V

T

iS/

i y

U(s)

fT

^

VTm

J

Y(s)

K„e-"-

Y(s)

T m s-1

+l Kf -^-S T Vm

J

T Table 164: PID tuning rules - unstable FOLPD model G m (s) = Rule

Kc

Majhi and Atherton (2000) - minimum ITSE Majhi and Atherton (2000) minimum ITSE. Model: Method 3

5

K

Comment T; Td Minimum performance index: servo tuning

(621)

K

(622)

f

T

ln(l + K) 4tanh-'K

(622)

2

1

0.1227+ 1.4550^2--1.271 l ^ r -

_ 0.801 l(l - 0.9358VRT^)) Kraln(l + K) (622)

'

m

0 < ^ - < 0.693 T

2T

J _ O Z , K = Ap/(Kmh), p/V KraVln(l + K)'

0.1227 + 1.4550 ln(l + K)-l.271 l[ln(l + K)f 2tanh-'K

Model: Method 1

0.5tm

(621)

T

i

6

K m e" Tms-1

Handbook of PI and PID Controller Tuning Rules

424

Rule

Kc

Majhi and Atherton (2000) - continued.

7

K

(623) K„

7

K

(623)

T (623)

0.801 l(K + 0.9946) K +0.0682 3

Td

T, T x

(623) d

0.7497VK + 0.9946 VK + 0.0682

2

, (623) _ 0.0145K + 0.5773K + 2.6554K + 0.3488 , K3 + 5.2158K2 + 4.4816K + 0.2817 , (623) = 0.0237(K +34.5338), " K + 4.1530

Kf

_1_ 12(K +0.9946) KV K +0.0682

Comment 0.693 <-^2-<1 T

Chapter 4: Tuning Rules for PID Controllers

425

4.11.10 Non-interacting controller 12 \ f 1 U(s) = Kc 1 + — + Tds T

v

R(s)

is

j

E(s)

Tds + 1

+ (1+

+T d s)

U(s)

Y(s)

K^eTms-1

Tds + 1

T;S

E(s)-K,Y(s)

K,

I Table 165: PID controller tuning rules - unstable FOLPD model G m (s) = Rule

Kc

T

T„s-1

Comment

Td

i

K m e~

Robust Lee and Edgar (2002). Model: Method I

8

K ^ 1,

(624) _

K

| a

8

K

(624)

T J

(624)

\ „ - l T 1 -m K , «1 K „ im „''tn

K,(624)Km-l

(624)

2(X + x m )

2M | ^ , (T m -K, ( 6 2 4 ) K m T m ) 6(X + xm) 4(X + T J 2 2(K 1 ( 6 2 4 , K m -l)(^ + Tm)

K

2(K,(624,Km-l) . (624) _

T

i

K m (^ + x J K

(624)

T

(624)f T K ft-1 mTm 624

V m ~

K,< »K

l

,

-1

m

2(^ + t n

K, (624, K

Tm

2(K,(624)Km-l)

(Tm-K,

T„

6(^ + x m )

4(X + T m ) 2

Kmxm)

2 ( K , ( 6 2 4 , K m - l ) ( ^ + Tm)

(624) (624),

(Tm-K,^Kmxm)

{

2(K 1 < 6 2 4 ) K m -l) K

(624) K„

cos J 1

m \

6(^ + x ra ) m

,

4(^ + T m ) 2

2(K 1 (624) K m -l)(A. + Tm) f

m

T

\ 1 —2HL

T

^m

T

Handbook of PI and PID Controller Tuning Rules

426 Rule

Kc

Lee and Edgar (2002). Model: Method 1

9

K

9 K (625) _ l + 0.25K u K m Km(X + xJ

(625)

(625)

T

T

(625)

Comment K, =0.25KU

( T m -0.25K u K m x m

l+0.25K u K r a

0-25KuKm _ xm 2(l + 0.25K u K m 6(X + xm) .(625)

Td

Ti

i _T nm - 0 . 2 5 K u K m x m +

l + 0.25K u K m

+

2(X + TJ

xj 4(X + xm)2

+

(Tm-0.25KuKmxJxm2 2(1+ 0.25K u K m )(k + x m )

2

x + 2(X + xm)

0.25KuKm 2(l + 0.25KuKm

6(X + xm)

4(X + xm)2

(T m -0.25K u K m x m )x n 2(1+ 0.25K u K m )(^ + x m )

0.25KuKm 2(l + 0.25KuKm

6(X + xm)

4(X + xm)2

(T m -0.25K u K m x m )T„ 2(1+ 0 . 2 5 K u K J ( ^ + x m )

(625)

Chapter 4: Tuning Rules for PID Controllers

All

4.12 Unstable SOSPD Model (one unstable pole) Gm(s) =

4.12.1 Ideal controller

1

G c (s) = Kc

+ ^ - + Tds

U(s)

R(s)>(C>E(s) K,

+

T

v

**e (T ml s-l)(l + sTm2)

is

K e~ (T ml s-lXl + sTm2)

Y(s)

Table 166: PID controller tuning rules - unstable SOSPD model (one unstable pole) Kme"sx» G m (s) = (T m l s-lXl + sT m2 ) Rule Wang and Jin (2004).

1

v

K„

(626) _

K

m

T

1K

i

(626)

(Tml ~

T

T

m XTm2

(TCL3+tm) .(626)

Comment

Td Direct synthesis

Ti

Kc

+

T

(626)

„

(626)

Model: Method 1

m )

3

1 TCL3 C L 3 + 3xmTCL3 + 3(TmlTm2 + x m T ml - x m T m2 JTCL3 + TmTm2 (Tml - xm j

. (626) _ (Tm2 +Tm ~T m iJT CL3 + 3TmlTm2TCL3(TCL3 + x m J + T ml T m2 x n

Handbook ofPIandPID

428 Rule

Controller Tuning Rules

Kc

Comment

Td

Ti

Robust 2

K

(627)

(627)

T

(627)

T

i

Rotstein and Lewin(1991). X determined graphically - sample values provided Model: Method 1 T m /T m =0.2,^e[0.5T m ,1.9T m ] Km uncertainty = 50% Tm/Tm=0.4Ae[1.3Tm,1.9Tj T m /T m =0.2^e[0.4T m ,4.3T m ] Km uncertainty = 30%

x m /T m =0.4,A.e[l.lT i n > 4.3T m ] im/Tm=0.6,X6[2.2Tm,4.3TJ Lee et al. (2000). Model: Method I

3 ^

(628)

(628)

T

(628)

T l

0<^-<2 T

d

i

m

(627) _ .

— + 2 | + T m2> V 'ml

(627)

VTml

M—

J

+ 2 +Tm

. (628) 3 £

(628)

•K m (2X + T m - a )

, X = desired closed loop dynamic parameter,

a = T.

T (628)__ T

_

1

, e

X1 + axm - 0 . 5 i m 2 2X + xm - a

- T mia + T m 2 a-T m ] T m 2 , (628) n(628)

x m 2 (0.167T m -0.5a) 2X + zm-a

X2 + axm-0.5xm2 2X + x„ - a

-/T-'-l

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

429

Td

Comment

~ (629)

Tuning rules developed from KU,TU

T| Ultimate cycle

McMillan (1984). Model: Method I

4 ^

(629)

4

(629)

K

T (629)

1.111 Tm,Tm Km

T

1+

T<629,=2xjl

lTm| + T m 2 / r m l T a _ vTml

Tm2 ATmi

(Tmi + T m2 JT ml T m2 +

T d ( 6 2 9 ) =0.5x m l

\Jm\

+

Tm2XTmi

^m^m.

(Jm! +Tm2jTmlTm2 \}m\

T m2 j(T ml

Tm/C„

tm/^mj

Handbook of PI and PID Controller Tuning Rules

430

4.12.2 Ideal controller in series with a first order lag r \ 1 G c (s) = Kc 1 + — + Tds

R(s)

- 1 <§>* +

A f 1 Kr 1 + — + Tds V Tis J

1 (Tfs + l)

T

V

U(s)

is

J

Tfs + 1 Y(s)

K„e~

(T ml s-lXl + sTm2)

Table 167: PID controller tuning rules - unstable SOSPD model (one unstable pole) Kme-"" G m (s) =

(Tmls-lXl + sTm2)

Rule

Kc

T,

T,

Comment

Robust Zhang and Xu (2002). Model: Method 1

5

K

(630)

T

(630)

d

Tm2Tmi(Tmi - x J + X3 +3X2Tm, +3XTJ+T

5 j , (630) .

(630)

T 1

2

Tm

2

K m ( ^ + 3 ^ T mml, +3XTmlm*,xmm +x m m T *m mll )j T

(630) _ 1T

M

- m2 +

(630)

v

2 2 r ^3+3X.2T ml, ~+3VT m , +T -^" T ml + T m TT, ml

^mllTmi

XraJ

Tm2(X, + 3X Tmt + 3XTml +T ro T m , ) T ml T m2 (T ml - x m )+ X3 + 3X2Tml + 3XTml2 + xmTra

T,

(630) _

^ TmlVml ~ T m ) 2

T m l (^ + 3X T m ,+3XT m l T m +T m 2 T m l

for^<0.5,A = 2-5(tn+Tm2).

r

*m <

T

ml

Chapter 4: Tuning Rules for PID Controllers

431

4.12.3 Ideal controller with set-point weighting 1 U(s) = Kc (FpR(s) - Y(s))+^(F,R(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S

Table 168: PID controller tuning rules - unstable SOSPD model (one unstable pole) „ u

Rule

K

., m V y

m

e ^

~(T ml s-lXl + sTm2 )

Kc

Comment

Td

Ti

Robust 6jr

(631)

-p(631)

T

(631)

0<^-<2 Tm

Lee et al. (2000). Model: Method 1

Fp=Fi

1 = fd -

, , '

as + 1

a

-/T-,_!

Tm, V 'ml

J

, (631) 6

K

(631)

K m (2^ + x m - a ) .(631)

•ir"=-Tml+Tn2+a-;

-

T

m l

a

+

T

m2

a

-

T

, X = desired closed loop dynamic parameter,

X +a,Tmm - 0 ". •5"T" m 2X, + xm - a ml

T

m2-

(631) - (631)

x m 2 (0.167x m -0.5a) 2>. + x m - a

X1 + a tv mm - 0 . 5 xmn 2 2X + xm - a

Handbook of PI and PID Controller Tuning Rules

432

(

1 + Tds

4.12.4 Classical controller 1 Gc (s) = Kc T s

v

i y

l + -*-s N

R(s)

E(s)

f K,

1+- P

Xs

1 + Tds

Y(s)

U(s) H

K„e-

(Tmls-lXl + sTm2)

l+ ^-s V N ,

Table 169: PID controller tuning rules - unstable SOSPD model (one unstable pole) G m (s)= Rule

K e ^ (T ml s-lXl + sT ra2 )

Comment ^ Td Minimum performance index: regulator tuning 7 K (632) Minimum ITAE Model: T (632) Tm2 - Poulin and Method 1 Pomerleau Coefficient values (deduced from graph) T (1996), (1997). X T m+T r a 2 m + m2 x x x x2 2 i i T T Kc

ml

Process output step load disturbance

Process input step load disturbance

ml

0.9479 1.0799 1.2013 1.3485 1.4905 1.1075 1.2013 1.3132 1.4384 1.5698

0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25

X

7

K

(632)

2.3546 2.4111 2.4646 2.5318 2.5992 2.4230 2.4646 2.5154 2.5742 2.6381

0.30 0.35 0.40 0.45 0.50 0.30 0.35 0.40 0.45 0.50

1.6163 1.7650 1.9139 2.0658 2.2080 1.6943 1.8161 1.9658 2.1022 2.2379

2.6612 2.7368 2.8161 2.9004 2.9826 2.7007 2.7637 2.8445 2.9210 3.0003

l '-ml

X

^\ m + ^ml}

K r a (x 1 T m l -4[T m +T,

T

(632)

4T m l (x m +T m 2 ) _ x,Tml-4(Tm+Tm2)! 0<

(T„/N)

^2;0.1Tm2<-^-<0.33Tm2

Chapter 4: Tuning Rules for PID Controllers

433

4.12.5 Series controller (classical controller 3) G c (s) = K( 1 + — (1 + Tds) T s

v R(s)

U(s)

E(s)

—i

K

1+ T s

v

(l + sT d )

i y

i y Y(s)

K„e"

(Tmls-lXl + sTm2)

Table 170: PID controller tuning rules - unstable SOSPD model (one unstable pole) Kme-"G ra (s) = (T ml s-lXl + sT m2 ) Rule

Comment Td Direct synthesis: time domain criteria Ti

Kc

Chidambaram and Kalyan (1996). Huang and Chen (1997), (1999). Model: Method 4

8 ^

(633)

(633)

T

i

9

Model: Method I

Tm2

'Good' servo and regulator Tm2 i response Direct synthesis: frequency domain criteria

(634)

K

T

(634)

ffl T

Ho and Xu (1998). Model: Method 1

p ml

AmKm

K

9

Tm2

Representative result 0.7854Tml

8

10 T (635)

"T ml

Am=2,

Tm2

4>m=45°

T

m m

Kc(633) and T;(633) are the proportional gain and integral time defined by De Paor and O'Malley (1989), Rotstein and Lewin (1991) or Chidambaram (1995), for the ideal PI control of an unstable FOLPD model [see Table 31]. K

(634)

0.5

(634),

1 + 1.11416^

(634) _ Kc K m T ro I 2x t ' ' ~Kc(634>Km-llTm T

Recommended x, =0.160367Tme

•57cop-cop xm-

T

T^ Tmy

-' ; 0 < - 2 - < 0.8 . Tml

434

Handbook of PI and PID Controller Tuning Rules

4.12.6 Non-interacting

controller

3

E(s)-i^Y(s)

U(s) = K v R(s)

T

i

s

i3

/

N Y(s)

U(s)

"^®—

E(s) Kr

K e" (T r a l s-lXl + sT m 2 )

KcTds T 1+ ^-s N

Table 171: PID controller tuning rules - unstable SOSPD model (one unstable pole) G m (s) = Rule

K„e-"»

(T m l s-lXl + sT m2 )

Comment T, Td Minimum performance index: regulator tuning Model: j (636) (636) (636) Method 2; T x d i N not specified

Kc

Huang and Lin (1995)minimum IAE.

1 jr

Note: equations continued onto page 435. 0.05 < T m /T ml < 0.4 ; Tm2 < T r a l . K

(636)

1 - 174.167 - 3 1 . 3 6 4 - 2 - + 0.4642 Km Tml VTmiy ST.

\

VTml

)

2 M

- 66.962I E I L S L + 59.496 i k

-83.916 K„

-103.069-^-

ml /

3n_

. 7 0 7 9 i.mT „2 _ +

isL

T im

2 3 121e T ml + 1 2 6 .924e

T

Tm2Tm

"" +26.944e T""'

Kn . (636)

\2

f

0.008+ 2.0718-2-+ 6.431

T 1

v miy 2.4484 T m 2 T m -18.686 T1 V^ml ral

+ 0.4556-E2- + 0.7503 VTml J

-2.9978

i

m2

V^ml

1

-21.135 V

m2xm

'ml

J

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

Huang and Lin (1995)continued.

+ T„

2

(637)

T

12.822

1

J

m2Tm

3^

4

T

V ml J

ml

J fT

A

T

m2

2 'm2

-0.527

m

T

T " = T

+ 22.848

T

3 x

cT

^

n

+ 39.001

3

T

•4.754

't

m'm2

rT

(636)

N not specified; Model: Method 2

(637)

T

i

V + T„

-

Comment

Td

Ti

(637)

K

435

T

2^ m

+ 1.64

"

V T ml 7

/ \ ,T, 1.1766- 2! --4.4623 + 0.5284-522--1.4281 Tml T„ VTmiy V T mi 6 T m2 T

+ Tm

+nA7J2js_)

lTml

Tm, /

T

-2.3542

'x

2

T

-0.0301-0.5039 v

-9.8564

3

1

mi

x„

f

2

T

- 5.0229

T

V/

I

T

ml

T

V 2 A

f

3

"

'mi \

y

4

m ^2 -0.1457 T T 4 v mi y v 1mi y V x mi y : Note: equations continued onto page 436. 0.05 < t r a /T m l < 0.25, Tml < Tm2 < 10Tml ;

+ T„

+ 9.3804

2T

m2

T

V^ml (

T

m

-7.528

y

3

T

Isl

+L0886

y

T

1

K

1

(637)

1750.08 +1637.76-=- + 1533.91

-7.917-

V T mi y

K„

6.187 K„

T

- 6.451 Tm2T,ro +0.002452

L 3729-sL-l739.77e

vLi

T

v 1mi y Tml

-0.000296eT"" +2.311e T""

K„

j (637)

_

T

+ Tm

Y

51.678 - 57.043- SL - +1337.29 - ^ H T . Tm,x„

7.7266-

, 2

-6011.57—2 I j

T + 0 . 1 7 4 2 - 2 ^ - 0.1524

+ 0.0213

V

-9135.52

ml

Handbook of PI and PID Controller Tuning Rules

436 Rule

Comment Td Minimum performance index: servo tuning N not specified; (638) (638) x Model: Method 2 T(638)

Kc

Huang and Lin (1995)minimum IAE.

3

K

T

i

i

f

-65.283

T

-0.0645 V

ml

'd

A

V

J

0.003926

•ml

"Cr.

T

4

I ,ml

(637) _

'd

_

Tml "

l ,ml J ^

+ 0.0117^ A3

/

+ Tm

-0.01291-^=^-] + 0.6874 Tm2\m T ^3

fT

+ T„

0.002455 -22-1 +1.4712 V

T

ml V

+0.6884

lTml f

+ T„

-0.1222

\

2

T

1

V

'm2 J

-0.1289

m2 m 4 'ml

-0.000135 -Tm2 V

J

i

fa

T m

T

J

V

^

m2 3

'ml

-0.007725

2

'ml

m

V

T

2

^xl T1 T

J

T

V

T

t, VTml J

'ml i

+ Tml -238.864-^2-

+140.135

( X 2T A m

T

J

- 0.0605 + 4.6998^2- - 29.478 — TmI lT m

T

'ml

J

Tm2

-0.001077

m

-49.007eT"" + 0.000026eT"" + 0.2977e

+ T„

T

x

1

2

x

'm2 V

J

T m 2 '•m T V 'ml

J

2

T

-2.0997

T

+ 274.851

ml

f

+ T„

2 \

T

m1m2

i m2 4

'ml

J

V

ml ,

J

Note: equations continued onto page 437. 0.05 < x m /T m l < 0.4 , Tm2 < T ml : -1.344

K

(638) .

1

10.741 - 1 3 . 3 6 3 — + 0.099 —

+ 727.914—s^-

0.995 K„

Tm, -708.4811-^21 T,

ml J

K„

1.031/

+ 9.915 Tm2T , m +84.273

'ml

T 'ml

1^ - 90.959-2^+ 9.034eT"" -2.386e Tml -16.304e T""2 xm

V^ml

N 0.997

Chapter 4: Tuning Rules for PID

.(638)

-149.685-141.418-

,717

' T

Controllers

I

17 29

T

437

T

T

V 'ml /

+ T m 20.5181 ^ 3 ^ -

Tm2T m

-12.82

lTm,J

,

Tml2

V^ml 7

Tm

+ T„

(638)

N0.286

+3.611 T„,;

V 'ml Tm;Tm

0 . 0 0 0 8 0 5 - 2 ^ +141.702e Tml -2.032e T "" +10.006e T "" 2

• 0.4144 + 1 5 . 8 0 5 ^ - - 142.327( - ^ - I + 0 . 7 2 8 7 ^ Tm, -18.317 T m 2 T , m +486.95 1SL) T Tml2 V 'ml

+ Tm 0.1123

(

+ T m 204.009

i

2 t

f

T

+ 41.26

T

T

_io.542ps2-

2

m *-ml

+ 396.349 T 3 T 3 VTml J V 'ml J V 'ml J (^ l 2 T 2 ^ fx 3 T >* 3^ T + Tm -138.038 xm 'm2 + 52.155 m'm2 - 646.848 m 'm2 T T V 'ml J V 'mi y V 'ml 7 5 T i m m2 + Tml 19.302|^22_] - 4 7 3 1 . 7 2 | ^ - | +425.789 Jml V ml J m '1112

f. T 2T 3A ^x 3 T 2 ^ mTm2 -841.807 'm 'm2 + 1313.72 m 'm2 5 T T V T'ml J V 'ml 7 V 'ml 7 T

+ Tm

-289.476

1 T 1 (, \ m2 + 6264.79 -161.469 mry m2 6 T T V ml ) V X ml J V 'ml / f.X 4_. 2 A f T 5^ rT 2T L 1 204.689 Tm ^m2 + 25.706 m 'm2 -791.857 m m2 V ^i J V Tml J

-3.7688

T

+ T„

648.217

m ^m2

V T' m l

-5.71: J

4

\

438

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Huang and Lin (1995)minimum IAE continued.

1

4

(639)

K

(639)

T

Comment

Td

Ti T

N not specified; Model: Method 2

(639)

Note: equations continued onto page 439. 0.05 < x m / T m l < 0.25 , T m l < T m 2 < 1 0 T m l ;

r 1

K (639)

s

-, -0.3055 X„

-130.2 + 8 5 . 9 1 4 - 2 - + 34.82

+ 10.442 ^ L

Tml (»

\°-5174 X

•22.547 K„

m2

-14.698

Tm2T m 2

V^ml J

(T

-51.47

'ml

\

(T

+52.4081^VTn

Tml J

s 2 - +53.378e T "« - 0.00000 le T "" +0.286e T™i

K„

, (639)

= T

72.806 - 268.146^-\2

{j + T„

4.922l[ ^ 2 . + 2468.191 - a . Trai J ml J

+151.351 Tm2\m

0.6724

-6914.46

V T ml J f

+ T„

- 795.465

T

^

1

1

m

3

x V

+ T„

2 l

T

^

ml

J

T^m Tx m 2i

1

V + 0.4057 V

V T ml J

+ 5580.17

Tml

ml

J

X1 T , 3 ^ ml J

fx

2T

V

T " 'ml

x + 55.536 '•m m2

2^

J

678778 VTml J

(639) _ ~ - ' m l 1 7 5 . 5 1 5 - 8 6 . 2 - ^ 2 - +348.727

Tm,

f

+ T„

"\3 m2

2\

T

( d

-0.0092

T 1

-0.001119J ^ina_ J _44.903e T "" + 0.000034e T"" - 1 5 . 6 9 4 e T""'

VTml ,

T X

/

VTml J

•14.27 m2 V 1 ml J

+ Tm, 1 4 1 7 . 6 5

+ T„

x

'O*

-,1.1798

-^-1

-0.008207^-

V^ml

NO.1064

- 5 5 . 6 1 9 _JH2_

V T mlJ

+ 0,0418 Tm2\m Tm,

+78.959 VTml J

VTml y

Chapter 4: Tuning Rules for PID Controllers

439

4.12.7 Non-interacting controller 8 \ / 1 U(s) = Kt 1 + — + Tds E ^ - K ^ l + VJYCs) V

E(s) R(s)

T

is

J

U(s) V

i

Y(s)

K^e""-

+

+ Tds K, 1+-JT S

(T ml s-lXl + sT m2 )

J Ki(l+Tdis)

Table 172: PID controller tuning rules - unstable SOSPD model (one unstable pole) G m (s) = Rule

Kc

Kwak et al. (2000). Model: Method 3

5 j , (640)

+ Tm 0.005048

5

'T.

(T m l s-lXl + sT m2 ) T

i

j , (640)

^

Comment

Td Direct synthesis T

(640)

Optimal gain margin: regulator response

-187.01eT"" + 0.000001eT"" -0.0149e T""2

T T Note: equations continued onto page 440. 0 < —— < 1 — — ,

K

(640)

-0.67 + 0.297 K„

K

(640) _

,(640) _ .

1

Kn

s-2.001

VTml J

,

+ 2.189

^-0.766

T

- 0.365 + 0.260 - i s - - 1 . 4 + 2.189 ^ V^ml VTml J ( - "\ 2.212

T

-0.3

<0.4 or

-<0.9 or

5» ' T •>0.9;

Handbook of PI and PID Controller Tuning Rules

440

Rule Kwak et al. (2000) continued.

T^u,

=Tmi

6

(641)

K

(641)

T

{_o.975+ 0.910

Comment

Td

Ti

Kc

Optimal gain margin: servo response

(641)

T

+ a} , -J2- > 0.4 with

-1.845

T1

V 'ml

ml

5.25-0.:

1-e

--2.8 V^ml

(640)

^

AT

K,

A1171

1.45 + 0.969

-1.9 + 1.576 l

ml J

2^

x,+x 2 -^-+x 3

Tml with /

\2

x

X, =-0.0030 + 0 . 6 4 8 2 - ^ ^ - 2 . 2 8 4 1 ^ +2.6221^ Tmi ^ Tml J ^ T,ml y X, = 0.2446 - 1 . 0 4 1 0 - ^ -13.6723

^

-0.9611 VTml

-16.76221-5 s2 -1 +5.1471 ^=2. V

X, = 0.1685+ 0.8289-^2--9.3630 ^22-) +2.9855 V^ml

+ 7.3803 ^=2. V

T

ml

j

K: = 6

+ jTdlcou)|Kn

T T Note: equations continued onto page 441. 0 < —— < 1 — . Tmi •> - 0 . 9 8 3

K

(641)

Kn

-0.04 + 0.333 + 0.949

T,m l

lm < 0.9 or J

ml

^3

f'r-

A/|Gm(jcou)(l

y

lTmlJ

j

Chapter 4: Tuning Rules for PID Controllers

441

/ K

(641)

-0.544 + 0.308 — + 1.408 K„

. lm > 0-9 .

Tmi

T^ 640 =T m] [2.055 + 0.072(T m /T ml )fe m , x m /T m l <1 or T^641' =T m l [1.768 + 0.329(xra/Tml )]^m , x m /T ral > 1 ; T

(641)

Tml

1-e

0.870

.

0.55 + 1.683

Tml

2^

X, + X 2 :

x.

T fT X, =-0.0030 + 0.6482-2^- -2.2841| ^2

-2.6221

'T,

^

-0.9611

Trai

^2

( -Y X, = 0.2446-1.0410-^-13.6723 Tml T X, =0.1685 + 0 . 8 2 8 9 ^ - -9.3630 \ —

l T .ml

16.7622

+ 5.1471 V^ml

\ + 2.9855-^/

V^ml

+ 7.3803-^. \

ml j

442

Handbook of PI and PID Controller Tuning Rules

4.13 Unstable SOSPD Model (two unstable poles) G ra (s) =

K„e"

(T ml s-lXT m2 s-l)

f \ 1 4.13.1 Ideal controller G (s) = Kc 1 + — + Tds

E(s)

-+6a

TS<

>

Kc

f 1 + —1 + T s

I

i

U(s) ^

1 Tds

Y(s)

Krae-»

(Tm,s-lXTm2s-l)

J

—w

Table 173: PID controller tuning rules - unstable SOSPD model (two unstable poles) G m (s) = Rule

(T r o l s-lXT m 2 s-l)

Kc

Wang and Jin (2004).

7 jr (642) _

K m e- sx ™

K

7

K

Td Direct synthesis

Ti

(642)

(642)

T

T

(642)

Comment Model: Method 1

i

mTi

( T ml ~ T ro X T m2 ~ T m )

(TCL3+^)3 • (642)

T =

CL3 3 +

3T

mTCL32 + 3(Tm,Tm2 + t m T m l - T m T m 2 ) T C U + T m T m 2 (Tml - Tm ) \ T m l ~ ^m A T ra ~ T m 2 )

. (642) =

( T m2 + Tm ~ T ml )^CLZ + 3 T ml T m2 T CL3 (TCL3 + Tm ) + T ml T m2 1 : m 2

Tr'^-tjT^-xJ

Chapter 4: Tuning Rules for PID Controllers Rule

Kc

T(

443

Comment

Td Robust

Lee et al. (2000). Model: Method 1

8 j r (643)

T

(643)

T X

(643)

0<^-<2 T

d

.(643)

K

(643)

Km(4X + x m - a , )

.(643)

X = desired closed loop dynamic parameter.

6X2 - a 2 + a!Tm -0.5x m 4X + tm - a . a2-(Tml+Tm2)a,+TmlT,

4X3+a2xm+0.167tm3-0.5a1tm2 4X + T m - a ,

(643) • (643)

6X2 - a 2 +a,T m -0.5x m

with cC|,a2 values determined by solving 1

(q 2 s 2 + oc1s + l); (Xs + l) 4

_j

L

=0.

Handbook of PI and PID Controller Tuning Rules

444

4.13.2 Ideal controller with set-point weighting 1 U(s) = Kc (FpR(s) - Y ( s ) ) + ^ f a R f s ) - Y(s)) + KcTds(FdR(s) - Y(s)) Y(s)

Table 174: PID controller tuning rules - unstable SOSPD model (two unstable poles) K m e- S ^ G m (s) =

(Tmls-lXTra2s-l)

Rule

Kc

Comment

Td

Ti

Robust 9j£

(644)

(644)

T

T

(644)

0< — < 2 T

Lee et al. (2000). Model: Method 1

m F

F

P= .

a2s + a , s + l 2

1

(a2s +a,s + l ) e - v i

frs + l)

4

, ls

,

0

=^^7

, (644) 9

K

(644)

Km(4A, + T m - a , ) , (644)

= -T m i - T r a 2 + a ,

X = desired closed loop dynamic parameter.

6X2 - a , + a , t m - 0 . 5 t 4X + x m - a ,

«2 - ( T m l + T m 2 ) a l +TmlT, (644)

2

4^+a2xm+0.167TmJ-0.5a,Tn 4A, + T m - a 1 , (644)

6X.2-a2+a|Tm-0.5Tm2 4^ + T r a - a ,

Chapter 4: Tuning Rules for PID Controllers

445

4.14 Unstable SOSPD Model with a Positive Zero Gm(s) =

Km(l-Tm3s)e-"" TmlV-2SmTmls + l

4.14.1 Ideal controller in series with a first order lag f G c (s) = K ( T s v i E(s) U(s) R(s) \ 1 K m (l-T m 3 s)e> <&*K 1 + — + Tds T „ 2 + v is j Tfs + 1 Tm,V-2^mTmls + l

\

1

J Tfs + 1 Y(s)

2

Table 175: PID controller tuning rules - unstable SOSPD model with a positive zero Km(l-Tra3s)r^ G ra (s) = T m l V-24 m T m ,s + l Rule Sree and Chidambaram (2004). 1

Comment Td Direct synthesis: time domain criteria

Kc

10

K

T

i

11 T (645) i

(645)

T x

l-sT m 3 )(l

' Desired closed loop transfer function =

+

(645) d

Model: Method 1

T, (645) s)e-^

l + sT^Jl + sTg'Jtl + sx,)'

, (645) 10

v

(645) ( o ^

=

)

(645)

)

iT& +T& +x2+Tin3+0.5Tin-Ti (645)

» If f

• (645)

11

)' 0.5TmQ<2L>x2

(645) 0.5lT 3.5(T ; -0.5T,

-

2 2 X 1 " T„, (T« - T I/ 6 4 5 ' ) ' ml nl V \ A CL ^+ T< CL> T+AX 2 ,T+T mm 3 , +0.5x m m

Equations continued into the footnote on page 446. j (645) _

x

l ~ T m l lTCL

+ T

CL + X 2

+ T

2

m3

0.5T m l T m 3 T m -T m l x, = T M x

2

+ 0.5x m T m l

2

tr^Tg>

+ T

m]

with

4

+

T&>T&>

+

T^T™)+^TralxmT<^x2

Handbook of PI and PID Controller Tuning Rules

446

_ Tm,4(0.5Tm3Tm - T j f a ' T f f

x

+

0.5xmfc'L> x4

+

TC2L> - T m 3 ] r f f

+ 2i; m T m 3 x 3 )

, T m , 4 ( - 2 ^ T m l +T m 3 + 0.5x Jo.5xmTc'L>T,l2L> - T r o 3 2 x 3 ^ x4 x

3

=

TCL

+ T CL + T m3 + x m , 2

x 4 =-T m l (o.5T m 3 x m - T m l 2 X r m l 2 [ r S + T<2> +0.5t m + 24 m Tj-0.5x m T< 1 L ) T< 2 L >) -(-2£Tral+Tm3+0.5xm) fc^M

2 2 + 0 . 5 x m ( T S +T< L»)]+4mTmlxmT<'L»T< )

-Tml4).

Chapter 4: Tuning Rules for PID Controllers

447

4.14.2 Non-interacting controller based on the two degree of freedom structure 1 f \ r \ T s PT d s d U(s) = Kc ,1+ —1 + R(s) ^— E(s)-K t a + T s * 1 + ^-s 1+ ^ s N N PTds

a +

K„

N

U(s)

R(s)

Y(s)

X0-

Km(l-sTm3)e-

K,

+ x

T s

'

E(s)

i

+

^ s N ,

+

TmlV-2^Tmls +l

Table 176: PID controller tuning rules — unstable SOSPD model with a positive zero Kra(l-Tm3s)e-^ Gm(s): TmlV - 2 ^ ^ , 8 + 1 Rule Sree and Chidambaram (2004). Model: Method 1 13

Comment Ti Td Direct synthesis: time domain criteria

Kc

12 j r

(646)

13

T

(646)

T

P = 0;

(646)

N = oo

( l - s T Vl + T(646)s)e~STra Desired closed loop transfer function = •) 2i£ '—-x . (l + sT&lJll + sT&'Xl + sx,)

. (646)

K^T^+T^+x.+T^+O.St,,,-^6461 (646)

T(646)

13

}'

O.SJT^-O.stj,

Equations continued into the footnote on page 448.

T m 3 + 0.5(T^ + Tg) '

T(646)

Handbook of PI and PID Controller Tuning Rules

448

(646) _

x

l ~ T ml \ T CL + T CL "•5Tmi

*. = ^ 0 2

+ X

im3tm

2 + T m3 + T m j

_

j ^

lml

2

+

w

0.5TmTm, (T<'L>T<2L>

_ T ,4(o.5Tm3Tm -lj)T&lg x2 - m

v

+

T&>T&> + T<3L>T<'>)+ ^ m T m ,x m T<'>T< 2 >x 2 ,

+0.5t m [T^ + Tff - T ^ j r g + 2^ m T m3 x 3 )

x4 , Tro,4(-2^mTm, +Tm3 + 0.5xmjo.5xmT^T^ -T m3 2 x 3 )^ x4 X3=T< 1 L ) +T< 2 L »+T m 3 +x m ,

x4 =-Tm,2(o.5Tm3Tm -T ml 2 )(T ml 2 [TS + T<2) + 0.5xm +2^mTml]-0.5tmT^T<2L)) -(-2!;T ml +T m3 +0.5x m ) ( ^ [ i M

2 2 + 0.5x ra fe'» +T< »)]HmTm,xmT^T< L>

-Tj).

Chapter 4: Tuning Rules for PID Controllers

449

4.15 Delay Model G m (s) = K m e" f \ 1 4.15.1 Ideal controller G c (s) = Kc 1+ — + Tds T s

V R

.

J

Y(s)

U(s)

( s > ^ E(s) K„

! + — + Tds

K

e-SI™

+ T

Table 177: PID controller tuning rules - Delay model G m (s) = K m e Rule

Kc

Td Process reaction

Ti

Comment

, 0.749 Callender et al. (1935/6). Model: Method 1

K

m'cm

2.734t m

0.303xm

1.31xra

0.303xm

1.24

2 K

raTm

1

Decay ratio = 0.015; Period of decaying oscillation = 10.5xn

2

Decay ratio = 0.12; Period of decaying oscillation = 6.28tm

Representative results deduced from graphs

Handbook of PI and PID Controller Tuning Rules

450

4.15.2 Ideal controller in series with a first order lag A r 1 G c (s) = K 1 + — + Tds T

V

j«%^

\ / 1 K, 1 + — + Tds V Tis J

1

is

J

U(s)

Tfs + 1

Y(s) Ke~

1 + Tfs

Table 178: PID controller tuning rules - Delay model G m (s) = K m e Rule

Kc

T(

Td

Comment

0.167xm

2^ 2 -0.167x m 2 4^ + Tm

Robust Bequette (2003) -page 300. Model: Method 1

Km(4X + x m )

0.5xra

Chapter 4: Tuning Rules for PID Controllers

4.15.3 Classical controller 1 Gc (s) = K

1 + Tds

'.+-L^ V

T

i

s

/

^ E(s) R(s)

1 + Tds

— K +

v

T s

451

N

U(s)

Y(s) * K„e-"»

i y l+

^ s N

Table 179: PID controller tuning rules - Delay model G ra (s) = K m e~ Rule

Kc 3

Hartree et al. (1937). Model: Method 1

4

Td T; Process reaction

Comment

0.70 KmTm

2.66xm

*m

0.78 K m x ra

2.97xm

0.5xm

3

N = 2. Decay ratio = 0.15; Period of decaying oscillation = 4.49t m

4

N = 2. Decay ratio = 0.042; Period of decaying oscillation = 5.03xn

Representative results

Handbook of PI and PID Controller Tuning Rules

452

4.16 General Model with a Repeated Pole G m (s) =

K m e" (l + sTra)n

/ \ 1 4.16.1 Ideal controller G c (s) = Kc 1 + — + Tds T

v

RQO

E s >(0v

( )

T

V

i

J

U(s)

\

1

K, 1+

is

+ Tds s

K„e"

Y(s)

0 + sTJ-

J

Table 180: PID controller tuning rules - general model with a repeated pole K e ^ d + sT ra ) n

G m (s)= Rule Skoczowski and Tarasiejski (1996).

5£

(647)

<

co„T w g * mm

Km ^ l

o „

=•

5

T

(647)

R

2 2 1 + co/T, (i+» g Tm r 2 + W8

[T d (647 >f "

(647)

T

(647)

T

Model: Method I

n-1 Tm , n > 2 with n+2

2n + 4 xm 4n + 2 n2-2n-2 ++
b = Tm

Comment

Td Direct synthesis

Ti

Kc

Tm

n 2 _ 2 n _ 2 + 2E±l^ +

c = 4(n + 2) 1

7t

Tm

71

4n +

2

+ c , with

7t

„ „ 4n + 2 ( T 2 n -4n + 3 + 71

T,

2n-2 +

< = specified phase margin.

Chapter 4: Tuning Rules for PID Controllers

453

4.16.2 Ideal controller in series with a first order lag f 1 \ 1 G c (s) = Kc 1 + — + Tds V

R(s)

7)

+

• Kc

L

f

1

I

T s

i

T

U(sj

^

1

K

m

e^

d + sTJ"

) Tfs + 1

is

j

1 Tfs + 1

Y(s) — •

Table 181: PID controller tuning rules - general model with a repeated pole K m e-"» G m (s) = (l + sT m ) n Rule Schaedel(1997). Model: Method 2

6

T

Comment T, Td Direct synthesis: frequency domain criteria (648) 6 j (648) T 0.563 "n + f Tf=Td(648)/N, Km . n - 2 . 5 < N < 20 Kc

(648) _. 3(n + 1) /

5n - 1

x T (648) _ n + 1 (_

6n

\

454

Handbook of PI and PID Controller Tuning Rules

4.11 General Stable Non-Oscillating Model with a Time Delay

4.17.1 Ideal controller Gc (s) = K I 1 + — + Tds T:S

R(s)

/0

+

f 1 K, 1 + — + T d s

vE(s)

<&

T

V

i

s

U(s)

j

General model

Y(s)

Table 182: PID controller tuning rules - general stable non-oscillating model with a time delay Rule Comment Kc Ti Td Direct synthesis Gorez and Klan (2000). Model: Method I

7

K

(649)

T(649) 1

K

l + Jl + 2 T

(649)

_ j

V

T.(649)+T

T

1 ar , (649)

T

(650)

025Tj(649)

(649)

(649) _

(649)

alternative tuning

i

. (649) 7

T

fr, (SW) +T a &t

!

"

alternative tuning

T ar J

2T„

1-K<649>^1 +

2T. (649)

3T,

(650) _ y (649) 2m_l j _ 2m_\ , T^ = average residence time of the process (which T

equals Tm + Tm for a FOLPD process, for example); T^ = additional apparent time constant; T = 1-K,

(649)

1+-

2T

(649)

455

Chapter 4: Tuning Rules for PID Controllers

4.18 Fifth Order System plus Delay Model ^ _ K m (l + b!S + b 2 s 2 +b 3 s 3 +b 4 s 4 +b s s 5 )e" m ' S ' ~~ ~

Ti

2

3

4

TV

^l + ajS + ajS +a 3 s +a 4 s +a 5 s J 4.18.1 Ideal controller Gc (s) = Kc 1+ — + Tds Xs E(s) U(s) R(s)

K, 1 + - - + Tds

+

Y(s)

Km (l + b,s + b2s2 + b3s3 + b4s4 + b5s5 p (l + a,s + a2s2 + a3s3 + a4s4 + a5s5

T

Table 183: PID controller tuning rules - fifth order model with delay K m (l + b1s + b 2 s z +b 3 s 3 +b 4 s + b s s )e G m (s) = (l + a,s + a 2 s 2 +a 3 s 3 + a 4 s 4 +a 5 s 5 ) Rule

Kc

Vrancic et al. (1999).

8

K

(651)

Td Direct synthesis

Ti T

(651)

T

(651)

Comment Model: Method I

' Note: equations continued into the footnote on page 456. (Ml)

K

a =

l3 Tal2bl +alb2 - 2 a l a 2 + a2bl

+ a

3 ~ b 3 +Yl

w i t h

2Kj^a72b,+a,a2+a;b|2-a3-b,b2+b3 tyj] Yi = t m ( a i 2 - a 1 b 1 - a 2 + b 2 ) + 0 . 5 ( a l - b l ) r m 2 + 0 . 1 6 7 x m 3 and Y2 =(a, - b , ) 2 T m +(a, - b , ) t m 2 +0.333x m 3 - ( a , - b , + x m ) 2 T d <651) ; (65i) =

T

'

a,3 -ai 2 bi + zxb2 -2a,a 2 +a 2 b 1 +a 3 - b 3 +y,

.

[a,2 - a . b , ~a 2 +b 2 +(B, - b , ) r m + 0.5xra2 -(a, - b , + x m )r d (651) J '

Td(651> is found by solving the following equation: T d (651) x, + 360x2 -360x m x 3 + 180x m 2 x 4 -60x m 3 x 5 - 15x m 4 x 6 -3xm5x7+7xm6(b1-a1)-xm7=0,with x, = 360[a, 4 b 2 - a t (a 2 b] +a 3 + b!b 2 +b 3 ) + a, 2 (a 2 2 + a 2 ( b 1 2 - 2 b 2 ) + 3a3b1 + 2a 4 H - b ^ - , + b 2 2 - b 4 ) - a , ( a 2 ( 2 a 3 - 3 b 3 ) + a3(2b, 2 - b 2 ) + 3a 4 b, +a 5 - b , b 4 +2b 2 b 3 - b 5 )

456

Handbook of PI and PID Controller Tuning Rules

- a 2 b , b 3 +a 3 2 +a 3 (b,b 2 - 2 b 3 ) + a 4 b, 2 +a 5 b, —b,b5 + b 3 ] -360T m [a, 4 b, - a , 3 ( a 2 + b, 2 +2b 2 ) + 3a, 2 (a 3 +b,b 2 ) + a,(3a 2 b 2 - 3 a 3 b , - 3 a 4 - b , b 3 - 2 b 2 +2b 4 ) - a 2 ( b , b 2 +b 3 ) + a 3 (b, - b 2 ) + 2a 4 b, +a 5 - b , b 4 +2b 2 b 3 - b 5 ] + 180x m 2 [a, 4 - 4 a , 3 b , + 3a, 2 (b, 2 +b 2 ) +a 1 (3a 2 b, - 3 a 3 - 5 b , b 2 +b 3 ) - a 2 ( b , 2 +2b 2 ) + a 3 b, +2a 4 +b,b 3 +2(b 2 2 - b 4 ) ] + 60Tm3[4a,3 - 9 a , 2 b , -a,(3a 2 - 5 b , 2 - 4 b 2 ) +4a 2 b, - a 3 - 5 b , b 2 +b 3 ] + 15xm4[ 9a, 2 -14a,b, - 4 a 2 +5b, 2 +4b 2 ] -42x m 5 (b, - a i ) + 7Tm6 , x2=a, b 3 - a , (a2b2+a,b3+a4+b,b3) + a, 2 (a 2 2 b, +a 2 (2a 3 +b,b 2 - 3 b 3 ) + a 3 (b, 2 + b 2 ) + 2a 4 b, + a s + b 2 b 3 - b 5 ) - a , ( a 2 3 + a 2 2 ( b , 2 - 2 b 2 ) + a 2 (2a 3 b, - 2 b , b 3 + b 2 2 - b 4 ) + 2a 3 2 +a 3 (2b,b 2 - 3 b 3 ) + a 4 (b, 2 + b 2 ) + a 5 b, - b , b 5 +b 3 2 ) + a 2 2 a 3 +a 2 [a 3 (b, - 2 b 2 ) - a 5 - b , b 4 +b 5 ] + a3 b, + a 3 (a 4 — b,b 3 + b 2 - b 4 ) + a 4 (b,b 2 - b 3 ) + a 5 b 2 - b 2 b 5 + b 3 b 4 , x3=a, b 2 - a , (a2b,+a3+b,b2+b3) + a, 2 (a 2 2 +a 2 (b, 2 - 2 b 2 ) + 3a3b, +2a 4 +b,b 3 + b 2 2 - b 4 ) -a,(a 2 (2a3-3b3) +a 3 (2b, 2 - b 2 ) + 3a 4 b, +a 5 - b , b 4 +2b 2 b 3 - b 5 ) - a 2 b , b 3 +a 3 2 + a 3 (b,b 2 - 2 b 3 ) + a 4 b, 2 +a 5 b, - b , b 5 + b 3 2 , x 4 = a , b, — a, ( a 2 + b , +2b 2 ) + 3a, 2 (a 3 +b,b 2 ) + a 1 (3a 2 b 2 -3a 3 b, - 3 a 4 — b,b 3 - 2 b 2 2 +2b 4 ) -a 2 (b,b 2 + b 3 ) +a 3 (b, - b 2 ) + 2a 4 b 1 +a 5 - b , b 4 +2b 2 b 3 - b 5 ) , x 5 = a , - 4 a , b, +3a, (b, +b 2 ) + a , ( 3 a 2 b , - 3 a 3 - 5 b , b 2 + b 3 ) - a 2 ( b , 2 +2b 2 ) +a 3 b, +2a 4 +b,b 3 +2(b 2 2 - b 4 ) , x 6 =4a, 3 - 9 a , 2 b , - a , ( 3 a 2 - 5 b , 2 - 4 b 2 ) +4a 2 b, - a 3 - 5 b , b 2 + b 3 , x 7 = 9a, 2 -14a,b, - 4 a 2 + 5b,2 +4b 2 .

Chapter 4: Tuning Rules for PID Controllers

457

4.18.2 Controller with filtered derivative G c (s) = Kt

i+-U T,s

Tds l + (Td/N>

E(s)

Y(s)

U(s)

R(s) ,

1

T

dS d

1 + T—s + . l+ ^ s N .

Km (l + b,s + b 2 s 2 + b3s3 + b 4 s 4 + b 5 s 5 p (l + a,s + a2s2 + a3s3 + a 4 s 4 + a5s5

Table 184: PID controller tuning rules - fifth order model with delay K m (l + b1s + b 2 s 2 +b 3 s 3 +b 4 s 4 + bss5)e~ST"' Gm(s) =

Rule

(l + a,s + a 2 s 2 +a 3 s 3 + a 4 s 4 + Kc

ry (652) Vrancic(1996)pages 118-119. <652) j Model: Method 1 2(A,-T i

(652)

-7)

Td T, Direct synthesis

[T,

9j

(652)

T

(652)

i

Comment

N<10

(652),2

N 2

-(A 3 -

A 5 A , ) + .J(A 3 2

- A ^ J - ^ ( A 3 A 2 -A 5 XA S A 2 -A 4 A 3

(652)

(A3A2-A5) N A, = K m ( a , - b , + x m ) , A 2 = K m ( b 2 - a 2 + A , a 1 - b I T m + 0 . 5 T m 2 ) , A 3 = K m ( a 3 - b 3 + A 2 a , - A,a 2 +b 2 x m -0.5b,x m 2 +0.167x m 3 ), A 4 =K m (b 4 - a 4 +A 3 a, - A 2 a 2 + A ,

b3Tra +0.5b 2 T m 2 + 0.167b,xm3 +0.042x m 4 )

A5 =K m (a 5 - b 5 + A 4 a, - A 3 a 2 + A 2 a 3 - A,a 4 +b 4 x m -0.5b 3 x m 2 ) + K m (o.l67b 2 x m 3 -0.042b 1 x m 4 +0.008x m 5 )

458

Handbook of PI and PID Controller Tuning Rules Rule

Kc

Vrancic et al. (1999). Model: Method 1

10

K

(653)

=

10

K

(653)

(653)

T

Comment

Td

Ti T

8 < N < 20

(653)

i

a, - a , b, +a,b 2 - 2 a , a 2 + a 2 b, +a 3 - b 3 +y,

_

2 2 2K, ^ - a / b j + a . a j + a ^ , - a 3 - b , b 2 +b 3 +(a, - b , ) x m +y 2

t, with

Yi = t m ( a i 2 - a , b 1 - a 2 + b 2 ) + 0 . 5 ( a 1 - b , ) T : m 2 + 0 . 1 6 7 T m 3 and y 2 =(a, - b , ) x m 2 +0.333x ra 3 - ( a , - b , +x m ) 2 T d

(653)

rT

S

< 6 5 3 >1 2

V J -(a,-b I + x B );

a, - a , b , + a , b 2 - 2 a , a 2 + a 2 b , + a 3 - b 3 + y .

• (653)

a,2 - a , b , - a 2 +b 2 +(a, - b , ) r m +0.5xm2 -(a, - b , +xm)Td (653) Td

[T,

(653)-,2

is found by solving the following equation:

[Td (653) ] 4 N -3 Z) + [ T d (

6

» ) ] 3 N - 2 z 2 +[T d ( 6 5 3 ) ] 2 N- , z 3 +

2 4 A TH<653)x, +360x, -360x m x, + 180x m 2x< 4 - 60xm x< - 15x'•m xA 6

-3tm5x7+7Tm6(b,-a1)-xra7=0 with x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 and x 7 as defined in Table 183; z,,z 2 and z 3 are defined as follows: z, = 360[a, - a t 2 b , +a,(b 2 - 2 a 2 ) + a 2 b, +a 3 - b 3 ] + 3 6 0 x m [ a 1 2 - a 1 b 1 - a 2 + b 2 ] + 180x m 2 (a,-b,) + 6 0 x m \ z 2 = 360(a, -b,)[a, - a , b , + a , ( b 2 - 2 a 2 ) + a 2 b , + a 3 - b 3 ] + 360t m [2a, 3 - 3 a , 2 b , -a,(3a 2 - b , 2 - 2 b 2 ) + 2a 2 b, +a 3 - b , b 2 - b 3 ] + 180x m 2 (3a, 2 - 4 a , b , - 2 a 2 +b, 2 +2b 2 ) + 240x m 3 (a, - b , ) + 60x m 4 , z 3 = z 4 + z 5 x m + z 6 x m 2 + z 7 x m 3 + 135(a, - b,)x m 4 + 27x ra 5 , with z 4 =-360[a, 4 b, - a , 3 ( a 2 + b , 2 + b 2 ) +a, 2 (2(a 3 + b 1 b 2 ) - a 2 b 1 ) + a,(a 2 2 +a 2 (b, 2 + b 2 ) - a 3 b , - 2 a 4 — b,b 3 - b 2 2 + b 4 ) -a 2 (a 3 +b,b 2 ) + a 4 b, +a 5 +b 2 b 3 - b 5 ] , 4

3

2

2

z 5 =360[a, -3a, b, +a, (2(b, + b 2 ) - a 2 ) +a,(3a 2 b, - a 3 -3b,b 2 ) - a 2 ( b , 2 + b 2 ) + a 4 +b,b 3 + b 2 2 - b 4 ] , z 6 =540[a,

- 2 a , b, - a , ( a 2 - b , - b 2 ) + b , ( a 2 - b 2 ) ] and

z 7 =180[ 2 a , 2 - 3 a , b , - a 2 + b , 2 + b 2 ] .

Chapter 4: Tuning Rules for PID Controllers Rule VranCic et al. (2004a), Vrancic and Lumbar (2004). Model: Method 1

Kc(

4>

11

K

Td

Ti

Kc (654)

T

(654)

T

(654)

459 Comment 8 < N < 20

and Td(654) are determined by solving the following equations:

, Tj

0.5^(A 4 A, - A 5 K j - 0 . 5 - ^ ( A , A 2 -A 3 K m ) K

(654)

K , 1(654) "" • (654) J r „ K„

x

(654) .

A

l

^m

2 2 A
2 L (654) I 2 K l + 2K m K c (654) + Km K

, - -

l ° ,. J and

2(A 1 + K r a 2 K c ( 6 5 4 , T d < 6 5 4 ) ) A 3 K m -A, 2 K m K c ( 6 5 4 ) T d ' 6 5 4 ) - A 2 A , 2A,A 9 K

x 2 =[(2A,KmKc<654>Td<«4>

+

+

A2Km2Kc<654)Td<654>

+

x

-A,3-A,Km2

A 2 )A 2 - A , A 3 - A 3 ( K m 2

+

with

A,2)Kc'«4»Td<654»] [A, + K r a 2 K c < 6 5 4 , T/ 5 4 >],

and with A

i =Km(ai-bl+xm) , A2=Km(b2-a2+A,a1-b1xm+0.5xm2),

A3 =K m (a 3 - b 3 + A 2 a, - A,a 2 +b 2 x m -0.5b,x m 2 +0.167T m 3 ), A 4 =K m (b 4 - a 4 + A 3 a, - A 2 a 2 +A,a 3 - b 3 T m +0.5b 2 x m 2 + 0.167b,xm3 + 0.042Tm4) A5 = K m (a 5 - b 5 + A 4 a, - A 3 a 2 + A 2 a 3 - A,a 4 +b 4 x m -0.5b 3 x m 2 ) -K m (0.167b : x m 3 -0.042b,x m 4 +0.008x m 5 ).

Handbook of PI and PID Controller Tuning Rules

460

4.18.3 Non-interacting controller 7 U(s) = - ^ E ( s ) - K ( 1+

R(s) E(s)

U(s)

K„

T S

Y(s) ' 1+ ^ s N ; Y(s)

K m (l + b,s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b 5 s 5 )s

Xs

(l + a,s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 J

Table 185: PID controller tuning rules - fifth order model with delay G m (s) = Rule

(l + a^ + a ^ 2 +a 3 s 3 + a 4 s 4 +a 5 s 5 J

Kc

Vrancic et al. (2004a), Vrancic and Lumbar (2004). 12

„-stm K m (l + b,s + b 2 s + b 3 s +b 4 s +b s s 5 \)e

12

(655)

K

Comment

Td Direct synthesis

Ti

(655)

T

Model: Method 1; 8 < N < 20

(655)

T

Note: equations continued into the footnote on page 461. Kc are determined by solving the following equations:

' , T;<655) and Td

0.s42-(A4A1 - A 5 K n )-0.5^-(A 1 A 2 -A 3 K m ) K

(655)

T

K„ A, 2 2 A,A,K m + A,A,A, A.A |rv2j-t3 n4*-i, - A K„

(655)

Kc<655)_l + 2KmKc , (655)

K

(655)

(655)

+ Km

2V

A

3

K

m

x'=U2A,K m K c

~

A

l

K

m

2

[KH :|2

( 6 5 5 ) ^ (655)

K

( 6 5 5 ) T (655) c

2A,A,K ( 6 5 5 ) T (655) . ld - t - / \ >)A 2 i n 22

and 2

•A,A, + A , K m zVK 3

-A, -A,K

( 6 5 5 ) x (655)

+ X

2

2 2 A n ], nAj , - A n1 (3 iK i vmm + A 1 ) K l

, with

( 6 5 5 ) T (655)

[A,+Km2K( (655)y

(655)1

Chapter 4: Tuning Rules for PID Controllers

461

and with A, =K m (a, - b , + x m ) , A 2 = K m ( b 2 - a 2 + A , a , - b , t r a + 0 . 5 x m 2 ) , A3 = K m(a 3 - b 3 +A 2 a, - A,a 2 +b 2 t r a - O ^ t , , , 2 +0.167-cm3), A 4 =K m (b 4 - a 4 + A 3 a, - A 2 a 2 + A,a 3 - b 3 t r a + 0.5b2xm2 +0.167b,T m 3 +0.042x m 4 ) A5 = K.m(a5 - b 5 + A 4 a, - A 3 a 2 + A 2 a 3 - A,a 4 +b 4 x m -0.5b 3 T m 2 ) + K m (o.l67b 2 T m 3 -0.042b,x m 4 +0.008T m 5 ).

Chapter 5

Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers

5.1 Introduction This chapter will discuss the compensation of FOLPD processes using PI and PID controllers whose parameters are specified using appropriate tuning rules. The gain margin, phase margin and maximum sensitivity of the compensated system as the ratio of time delay to time constant of the process varies, are used as ways of judging the performance and robustness of the system. This work follows the method of Ho et al. (1995b), (1996), in which good approximations for the gain and phase margins of the PI or PID controlled system are analytically calculated. The method will be extended to determine analytically the maximum sensitivity of the compensated system. Insight will be obtained into the range of time delay to time constant ratios over which it is sensible to apply various tuning rules, to compensate a FOLPD process. The chapter is organised as follows. Formulae for calculating analytically the gain margin, phase margin and maximum sensitivity are outlined in Sections 5.2 and 5.3. The performance and robustness of FOLPD processes compensated with a variety of PI and PID tuning rules are evaluated in Section 5.4. In Section 5.5, tuning rules are designed to achieve constant gain and phase margins for all values of delay, for a number of process models and controller structures. Conclusions of the

462

Chapter 5: Performance and Robustness Issues...

463

work are drawn in Section 5.6. This work was previously published by O'Dwyer (1998), (2001a). 5.2 The Analytical Determination of Gain and Phase Margin

5.2.1 PI tuning formulae The controller and process model are respectively given by G c (s) = Kc K

G m (s) =

T s

(5.1)

i ;

K_e~ 1 + sT

(5.2)

Then rw> = KmG~>aZm K c ( J » T i + l ) (5.3) l + j©Tm jooT; From the definition of gain and phase margin, the following sets of equations are obtained: m =arg|G c (j© g )G m (ja) g )j+7i (5.4) (j r

cu*c

m (jco)G c (jco)

Am =

1 |Gc(jcop)Gm(jcop)

(5.5)

where co and co are given by |Gc(jcog)Gm(jcog)| = l

(5.6)

arg[Gc(ja> )Gm(ja> ) J = - T I

(5.7)

From Equation (5.3), K m K c J l + G>2Ti2 Gm(jco)Gc(]co)-^^ !22 ^ 2 coTj -y/l + co Tn Z - 0.57T + tan -1 coTj - tan -1 ©Tm - coxm (5.8) Therefore, from Equation (5.4) (|>m = 7t-0.5n + tan~1cogTi -tan _1 co g T m -cogTm (5.9)

Handbook of PI and PID Controller Tuning Rules

464

with co given by the solution of Equation (5.6) i.e.

K m K c J l + co 2T,2 s

y

= =i

(5.io)

cogTiA/l + co g 2 T m 2 Also, from Equations (5.5) and (5.8), 1 A = . m

|G m (j(D p )G c (jco p )|

conT; P '

Jl + co 2 T 2 E EL.

K ^ ^ l + co/Ti

(5 11)

2

with co given by the solution of Equation (5.7) i.e. 0.57t + tan ' c o T . - t a n ' co Tm -co x m =-n pi

p m

(5.12)

p m

From Equation (5.10), co may be determined analytically to be T,(K c 2 K m 2 - l ) + A / ( K c 2 K m 2 - l j V + 4 K c 2 K m 2 T m 2 2T i T m 2

^

An analytical solution of Equation (5.12) (to determine cop) is not possible. An approximate analytical solution may be obtained if the following approximation for the arctan function is made: -1

tan

?t

x«—x,

4

I

I

x
.,

,

_i

and tan

7T

7X

x~

' '

2

4x

I I - .

, r- i

x>l

(5.14)

M

^

This is quite an accurate approximation, as is shown in Figures la and lb (page 465). Considering Equation (5.12), four possibilities present themselves if the approximation in Equation (5.14) is to be used. These possibilities, together with the formula for cop that may be determined analytically for each of these cases, are n± (i)

c o ^ > 1, copTm > 1: cop =

In2 >

-4m„

T

T

- ^

'-

(5.15)

n±ln2-~-(0.25n7m+Tm) (ii)

co T j > l , c o p T m < l : co =

L _,

! , 2(0.257iT m +T m )

(5. 1( (5.16)

Chapter 5: Performance and Robustness Issues...

X

T„

(iii)

copT; < 1, copTm > 1: (op = \\4T_mnT

(iv)

co p T,
465

(5-17)

j — — -

(5.18)

The gain and phase margin of the compensated system, for each of the tuning rules, as a function of x m /T m , may be calculated by applying Equations (5.9), (5.11), (5.13) and the relevant approximation for co from Equations (5.15) to (5.18). Figure la: Arctan x and its approximation (Equation 5.14)

0

2

4

6

8

10

Figure lb: % error in taking approximation (Equation 5.14) to arctan(x)

40

T

T

20-

-20

10

Handbook of PI and PID Controller Tuning Rules

466

5.2.2 PID tuning formulae The classical PID controller structure is considered, whose transfer function is given as 1 Y 1 + sT, (5.19) G c (s) = K, 1 + T;s 1 + sceT, and the process model is given by Equation (5.2). Substituting Equations (5.2) and (5.19) into Equations (5.4) and (5.5) gives <j>m = 0.5rc + tan -1 cogTi + tan"1 cogTd - tan -1 cogTm - tan -1 cogocTd - cogxm (5.20) and _ CQpTj m

l(l + wp2Tm2Xl + w p V T d 2 )

(5.21)

(l + cop2Ti2)(l + cop2Td2)

" KcKm y

From Equation (5.6), (i)g is given by the solution of K m K c ^/l + (a g 2 T i 2 > /l + (ag2Td2

'

4

co„TJl + a)„ 2 T„ 2 ^l + a 2 w„ 2 T, 2

_f

(5.22)

"

The equation for a)g may be determined to be a) g 6 +a 1 o) g 4 +a 2 co g 2 +a 3 = 0 with

aj =

T i 2 (T m 2 +a 2 T d 2 )-K c 2 K m 2 T i 2 T d

T^VT,2 ^ - K . X

2

(5.23)

~ , 2™

2

Kc2Km2

^ ^ / )

2^2T

Ti\'a'Td

2

3

2T,

2 „ 2

T

TiX'a'Td

2

Following the procedure outlined by Ho et al. (1996), the analytical solution of Equation (5.23) is given as 0),

=J^/R+VQT7R2"+^R-VQ3+R2

. . ^ 3a, - a . " , „ 9a,a 7 - 2 7 a , - 2 a , with Q = —2- x— and R = — — —. 9 54 From Equation (5.7), co is given by the solution of

-—

(5-24)

Chapter 5: Performance and Robustness Issues...

467

0.5n + tan - 1 copTd + tan" 1 copT; - tan" 1 copTm - tan" 1 copaTd - co p t m = 0 (5.25) As with Equation (5.12), an analytical solution of this equation is not possible. An approximate analytical solution may be obtained if the approximation detailed in Equation (5.14) is made. Looking at Equation (5.25), twelve possibilities present themselves if the approximation in Equation (5.14) is to be used; these possibilities are (1) copT; > 1, ffiT > 1

2 ) c o p T i > l ,
(a) copTd < 1, © p aT d < 1

a) © p T d < 1, co p aT d < 1

(b)co p T d > 1 , co p aT d < 1

b) copTd > 1, co p aT d < 1

( c ) o p T d > 1. ^paTd > 1

c) copTd > 1, co p aT d > 1

(3) cOpT; < 1, copTm > 1

^cOpT^l,

co p T m
(a) © p T d < 1, co p aT d < 1

a) copTd < 1, co p aT d < 1

(b) copTd > 1, © p aT d < 1

b

(c) © p T d > 1, co p aT d > 1

c) copTd > 1, co p aT d > 1

) a>pTd > 1.
The formulae for cop that may be determined for each of these cases are as follows: (1) copT, > 1, copTm > 1 (a) ©pTd < 1, copaTd < 1: 7r±j7t2-7i(4Tm-7iTd(l-a)) C O

P='

(5.26) 4Tm-7iTd(l-a)

(b) copTd > 1, © aT d < 1: 2TC ± ®p=-

( 1

4TI 2 - 7i(4xm + 7iT d a) — + -

4x m +TcT d a

1

-

1^ -

(5.27)

468

Handbook of PI and PID Controller Tuning Rules

(c) copTd > 1, coPp"aT >1 : ^ dd n ±ln

-47ix„

J_ J V ^d

top=-

(2)co p T i >l,

Tj

1

1_

aTd

Tm j

(5.28)

4x„

cpo mT < l :

(a)copTd <1, copp"aT - J dd < 1 : 2n± Un2 + ^-(nTd-nTm-naTd

-4xJ (5.29)

co p =-

7tT m +4x m +7caT d -7cT d

(b)co p T d >l, cop^^d p aT d
J9n-n

con

VTd+Tiy

nT

(KTm+7iaTd+4xm)

(5.30)

m + 4xm + ™Td

(c) copTd > 1, copaTd > 1: 2n±J4n

1

+n V

aT

co„

d

T

1

1

d

Tiy

("T m +4x m )

7tT +4x

(5.31)

(3) copTj < 1, coT > 1: (a) copTd < 1, copaTd < 1: cop =.

'T m (4x m +7r[aT d -T d -T i ])

(5.32)

(b)co p T d >l, co p aT d
%="

(TcT i -7taT d -4x m )

-71

(5.33) rcTj-4xm-7taTd 7t

(c)co p T d >l, c o p a T d > l : o D = ^ T d

7t

71

T

- aTd 7iT. - 4x '^

l

m

(5.34)

Chapter 5: Performance

(4) a) p Tj
and Robustness

Issues...

469

Q)pTm
(a) copTd < 1, copaTd < 1: cop =

2 71

7i(aT d -T d )+^(T m -T i )+4x n

(5.35)

(b) co Td > 1, oo aTd < 1: 2n±Un2 C O

P=-

- A ( „ T m -nT, +naTd + 4 x J

X

(5.36)

7tT„ - TTT, + 4 x „

7iaT,

(c) co Td > 1, oo aT d > 1 n±Jn

-n

vaTd

T

(7iTi-7iTm-4xm)

ay

< » P = -

(5.37)

The gain and phase margin of the compensated system, for each of the tuning rules, as a function of t m /T m may now be calculated by applying Equations (5.20), (5.21), (5.24) and the relevant approximation for oop from Equations (5.26) to (5.37). As Equation (5.19) shows, the procedure applies to tuning rules defined for the classical PID controller structure only; however, Ho et al. (1996) suggest that the method may be applied to tuning rules defined for the ideal PID controller structure, by using equations that approximately convert these tuning rules into their equivalent in the classical controller structure. 5.3 The Analytical Determination of Maximum Sensitivity The maximum sensitivity is the reciprocal of the shortest distance from the Nyquist curve to the (-1,0) point on the Rl-Im axis. It is defined as follows: 1 M (5.38) max = M a x all ra l + Gm(jco)Gc(jco) For a FOLPD process model controlled by a PI controller, |Gm(jco)Gc(jco)| =

Vl + oo'Tj2 K c K r

(5.39)

470

Handbook of PI and PID Controller Tuning Rules

and arg[Gm (j©)Gc (jco)] = -0.5TT - tan"1 coTm + tan"1 oaT; - coxm For a FOLPD process model controlled by a PID controller, V(l + co2Tm2 )(l + a V T d 2 )

(5.40)

»T>

and arg[Gm(jco)Gc(jco)] = - 0.5TC - tan"1 coTm - tan"1 coaTd + tan -1 coT; + tan"1 coTd - coxm (5.42) The maximum sensitivity may be calculated over an appropriate range of frequencies corresponding to phase lags of 100° to 260°. 5.4 Simulation Results Space considerations dictate that only representative simulation results may be provided; an extensive set of simulation results covering 103 PI controller tuning rules and 125 PID controller tuning rules (for the ideal and classical controller structures) are available (O'Dwyer, 2000). The MATLAB package has been used in the simulations. Figures 2 to 7 show how gain margin, phase margin and maximum sensitivity vary as the ratio of time delay to time constant varies, if some PI tuning rules are used (Figures 2 to 4) and corresponding PID tuning rules for the classical controller structure (with a = 0.1) are used (Figures 5 to 7). In these results, Z-N refers to the process reaction curve method of Ziegler and Nichols (1942); W-W refers to the process reaction curve method of Witt and Waggoner (1990); IAE reg, ISE reg and ITAE reg refer to the tuning rules for regulator applications that minimise the integral of absolute error criterion, the integral of squared error criterion and the integral of time multiplied by absolute error criterion, respectively, as defined by Murrill (1967) for PI tuning rules and Kaya and Scheib (1988) for PID tuning rules based on the classical controller structure. Figures 8 to 15 show gain and phase margin comparisons between corresponding PI and PID controller tuning rules. It is clear that the gain margin is generally less when the PID rather than the PI tuning rules are considered, over the ratios of time delay to

Chapter 5: Performance and Robustness Issues...

471

time constant taken; the difference between the phase margins is less clear cut. This suggests that these PID tuning rules should provide a greater degree of performance than the corresponding PI tuning rules, but may be less robust. Comparing the individual tuning rules, it is striking that the ISE based tuning rules have generally the smallest gain margin and have also a small phase margin, suggesting that this is a less robust tuning strategy. The results in Figures 4 and 7 confirm these comments. Figure 2: Gain margin

Figure 3: Phase margin

•Z-U | f 7= JAp ijegi : +4ltAJEreg

'

'

'

---*"

1

1

1

Figure 4: Max. sensitivity

-

/ : t t f V ; i •...

\t;V>° fM--t°--i—1—r--;---:—

... 4+,

;>'! i \ i i

*

o -°oo< 00°

0.2

0.4

0.6

OB

I

1.2

1.4

1.6

IB

0.2

2

'

A//

- ! — ! — 1 — ! — t"-- \ ---;---

0.4

0.6

OB

I

1.4

12

1.6

IB

2

Ratio of t m to T„

Ratio of t m to T„

04

0.6

06

1

12

14

1.6

IB

2

Ratio of t m to T„

Figure 6: Phase margin

Figure 5: Gain margin

02

Figure 7: Max. sensitivity

1- 4 w-w M M : 4 4 iAri reg ; ;

;

+ = lTAEreg! i i +++++t

•^-lis^tegr-r^f-r--

60.. - : . . : . . j / t

;

',i : i >*!*' i j^<\ 40

/ ,0

30°P4°°J

"oofeooiooooo*"?000:

0.2

0.4

0.6

06

1

:

-i-\-f-

: : \

WA

!

'•

'•

14

1.6

1.8

*

J

: : i

1.2

1.4

1.6

Ratio of xm to Tm

18

2

0

0.2

04

0.6

0.8

1

1.2

Ratio of Tm to Tm

02

0.4

06

OB

1

1.2

1.4

1.6

1.8

Ratio of x_ to T„

2

472

Handbook of PI and PID Controller Tuning Rules

No general conclusion can be reached as to the best tuning rule (as expected); it is interesting, though, that a full panorama of simulation results (O'Dwyer, 2000) show that many tuning rules may be applied at ratios of time delay to time constant greater than that normally recommended. One example may be seen in Figures 5 to 7, where the gain margin, phase margin and maximum sensitivity (associated with the use of the PID tuning rule for obtaining minimum IAE in the regulator mode) tends to level out when the ratio of time delay to time constant is greater than 1; normally, the tuning rule is used when the ratio is less than 1 (Murrill, 1967). On the other hand, it is clear from Figures 8 and 9 that there is a significant degradation of performance when using the PID tuning rule of Witt and Waggoner (1990) and the PI tuning rule of Ziegler and Nichols (1942) for large ratios of time delay to time constant, which is compatible with application experience. The decision between the use of a PI and PID controller to compensate the process, depends on the ratio of time delay to time constant in the FOLPD model, together with the desired trade-off between performance and robustness, as expected. It turns out, however, that the analytical method explored allows the calculation of a far wider range of gain and phase margins for PI controllers; it is also true that stability tends to be assured when a PI controller is used (O'Dwyer, 2000). Thus, a cautious design approach is to use a PI controller, particularly at larger ratios of time delay to time constant. Finally, the volume of tuning rules and data generated means that the use of an expert system to recommend a tuning rule based on user defined requirements is indicated; work is ongoing on such an implementation (Feeney and O'Dwyer, 2002).

Chapter 5: Performance and Robustness Issues...

Figure 8: Gain margin comparison

473

Figure 9: Phase margin comparison 120

oO

100

5000 )000( >ooo6ooo(

- - . , - - -vU -

: o : o: o : /

o •/ o / o 7

0

0 .0...

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

vu Ratio of Tv mm to T^ *m

1

1

1

1

0.8

•

:

; 1/

:

;

; . . . „

:

:

iQo°

\: \

: :

0o

j,o" :

: N»O° : : °OOOO°^>~-~L_ ;

0

0.2

0.4

0.6

0.8

:

:

1.8

2

e<

> ° °

Ar :

/? i

. i x^

:

:

' / ! !

: ; ...j

1

1.6

) 0 o6^

\—- — -t~ - : — - : - — :-0Q0-:—\ : : : ; 0 0° : v

1.4

Figure 11: Phase margin comparison

: /

:

1.2

1

: - ± IAE re|g PID i : o =j IAB reg PI: i

. _ _ „ . ; . - - - - : — - - !

1

Ratio of twmm to *•" Tm m

Figure 10: Gain margin comparison I

0.6

1.2

1.4

R a t i o o f !v_ , t o m *"

T_ m

1.6

1.8

2

0

960^ o° ;

|

;

0.2

0.6

0.8

0.4

1

1.2

1.4

1.6

Ratio of i*mm to Tmm

1.8

2

Handbook of PI and PID Controller Tuning Rules

474

Figure 12: Gain margin comparison 2.5,

!

1

1

.

1

1

i

1

1

Figure 13: Phase margin comparison 1

I00r

1SE reg JPIDJ 6 = ISEregPI

in00<

o?° 0° i

—-i—~i ,/

: o

—4—•A~t~ : o : 'o ....

i 1/

0? i o-°4"4--

- — -:- —

o 0 oO° 0 :

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

Ratio of x" mm to Tmm *•"

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ratio of t mm to T*mm * v

Figure 14: Gain margin comparison

Figure 15: Phase margin comparison

c, /

L-.=i.iT^LEregJ m

/ io0 o^oofo°

| 0=1117^E reg PI !

0(

pj*

>" i

1

_n°< 5U

i_oO°(

1 i

„<J0

,0° j

%fafif-

0

0.2

0.4

0.6

•A ! i !

0.6

1

1.2

1.4

Ratio of tfcmm vto w T„ m

1.6

1.8

2

' 6 00
;

0

1

0.2

0.4

0.6

0.8

1.2

1.4

Ratio of xmm to T„m ™ v

x

1.6

1.8

2

Chapter 5: Performance and Robustness Issues...

475

5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for all Values of Delay Normally, the gain and phase margins of the compensated systems tend to increase as the time delay increases, supporting the common view that PI and PID controllers are less suitable for the control of dominant time delay processes. However, for the PI control of a FOLPD process model, O'Dwyer (2000) shows that the tuning rules proposed by Chien et al. (1952), Haalman (1965) and Pemberton (1972a), among others, facilitate the achievement of a constant gain and phase margin as the time delay of the process model varies. All of these tuning rules have the following structure: K c = a T m / K m t m , T i = T m . Following this observation, original approaches to the design of tuning rules for PI, PD and PID controllers are proposed, for a wide variety of process models, which allow constant gain and phase margins for the compensated system. This work organised as follows. In Section 5.5.1, PI controller tuning rules are specified for processes modelled in FOLPD form and IPD form. In Section 5.5.2, PID controller tuning rules are described for processes modelled in FOLPD form, SOSPD form and SOSPD form with a negative zero. Section 5.5.3 deals with the design of PD controller tuning rules for the control of processes modelled in FOLIPD form. 5.5.1 PI controller design 5.5.1.1 Processes modelled in FOLPD form For such processes and controllers, Equations (5.1) and (5.2) apply i.e., K e"STm G m (s) = l + sTm with

G c (s) = K

1+

1

From Section 5.2.1, m =71-0.571 +tan - 1 WgTj -tan" 1 cogTm -u) g T m with GO given by the solution of

(Equation 5.9)

476

Handbook of PI and PID Controller Tuning Rules

K m K c J l + ©„2X2 2m

Q)gTiA/l + cog2T,m

=1

(Equation 5.10)

2

and conT:

Am = — ^ - ,

l + co 2 T

2

V V

(Equation 5.11)

with cop given by the solution of -0.5u + tan_1 c0pT; - t a n - 1 co p T m -co p T m =-n If Kc and T; are designed as follows: aT Kc=—^-

(Equation 5.12)

(5.43)

and T ; =T m Then Equation (5.12) becomes -0.5n-co p x m =-7t i.e.

cop = 7i/2xm

Substituting Equation (5.46) into Equation (5.11) gives A m =7rT m /2K m K c T m Substituting Equation (5.44) into Equation (5.10) gives i.e.

(5.44) (5.45) (5.46) (5.47)

KmKc/cogTm=l

(5.48)

cog=KmKc/Tm

(5.49)

Substituting Equations (5.44) and (5.49) into Equation (5.9) gives (|> m =0.57t-K m K c T m /T m (5.50) Finally, substituting Equation (5.43) into Equation (5.47) gives A m =7r/2a (5.51) and substituting Equation (5.43) into Equation (5.50) gives (t>m =0.57t-a (5.52) Some typical tuning rules are shown in Table 186.

Chapter 5: Performance and Robustness Issues...

411

Table 186: Typical PI controller tuning rules - FOLPD process model a 7l/3

Kc 1.047Tm/KmTm

7t/4

0.785Tm/Kmxm

TC/6

T Tm T

0.524Tm/KmTm

T

m

Am 1.5

7t/6

2.0

7t/4

3.0

7t/3

*m

These rules are also provided in Table 3, Chapter 3. It may also be demonstrated that, if Kc and Tj are designed as follows: K„=

m

,— u u m , / T 2 +4n 2 T

x

(5.53)

and T^Tm (5.54) where K u and Tu are the ultimate gain and ultimate period, respectively, then the constant gain and phase margins provided in Equations (5.51) and (5.52) are obtained. 5.5.1.2 Processes modelled in IPD form A similar analysis to that of Section 5.5.1.1 may be done for the design of PI controllers for processes modelled in IPD form. The process is modelled as follows: G m (s) = K m e - s ^ / s (5.55) Corresponding to Equations (5.4) and (5.9), the phase margin is 71-0.571 +tan ' cOgTj-0.57t-cogxm (5.56) *, with cog given by the solution of K

Kcjl + V ?

=1

(5.57)

cog T, If Kc and T; are designed as follows: a K

=•

Kmxm

and

(5.58)

478

Handbook of PI and PID Controller Tuning Rules

% = bxr

(5.59)

then it may be shown that 0.5 co„ =

a

a

2x m

±— 2 b—x T „ -V;

fa b + 4 2

2

(5.60)

and, substituting Equations (5.59) and (5.60) into Equation (5.56), m=tan-1 0 . 5 a V +0.5abVa 2 b 2 +4

-i0.5

0.5a2+0.5-Va2b2+4 (5.61)

Corresponding to Equations (5.5) and (5.11), the gain margin, ®p

A=-

T

i

(5.62)

K m K c > /l + a) p 2 T i 2 with co given by the solution of -0.57i + tan

(5.63)

cOpT;-0.57t-copTm

i.e. co is given by the solution of tan' COpT; = copxm

(5.64)

An analytical solution to this equation is not possible, though if the -i

approximation tan

i

ft

a^T, « 0.571 P >

i

(when coT: > 1) is used, 4(flpTj

| P .|

>

simple calculations show that the following analytical solution for co may be obtained: 0.5

%="

0.57i + ,|0.257r —

(5.65)

The inequality copTj > 1 may be shown to be equivalent to b > 1.273. Substituting Equations (5.58), (5.59) and (5.65) into Equation (5.62), calculations show that:

Chapter 5: Performance and Robustness Issues...

479

_b_ 4a A

m

(5.66)

=•

1+Some typical tuning rules are shown in Table 187. Table 187: Typical PI controller tuning rules - IPD process model Kc 0.558/Kmxm

T;

Am 1.5

46.2°

2.0

45.5°

*m

0.484/Kmxm

l-4xm 1.55tra

0.458/KmTm

3.35im

3.0

59.9°

0-357/Kmtm

4-3xm

4.0

60.0°

0.305/Kmxm

12.15tm

5.0

75.0°

It may also be shown that the maximum sensitivity is a constant if Kc and Tj are specified according to Equations (5.58) and (5.59). The maximum sensitivity may be calculated to be: M,

l-2acos(co r x m )-

-0.5

a2 2

2

(5.67)

with corxm being a constant obtained numerically from the solution of the equation (a /cor xm ) + cosa>rxm =sin(co r x m )/co r x m . It may also be shown that, if K c and Tj are designed as follows: Kc=aKu

(5.68)

and Ti=bT u (5.69) then the following constant gain and phase margins are determined:

Handbook of PI and PID Controller Tuning Rules

480

2b na A

K

2

i

\n

n

14

4b (5.70)

= 1+-

2

2

7t -2 + JV 4- — 4b

an 27i2a2b2 +27tabV47i2a2b2 +4 and <\>m = tan" - J0.1257i2a2

T).5

+-^-V47r

16b

2

a2b2+4

(5.71)

5.5.2 PID controller design

5.5.2.1 Processes modelled in FOLPD form - classical controller 1 The classical PID controller is given by f 1 Y l + sTd ^ G c (s) = K( 1 + T s v i y 1 + saT,d J

(Equation 5.19)

and the process model is given by Kme-ST™

G m (s) = -

(Equation 5.2)

l + sT„

From Section 5.2.2, m = 0.57r + tan -1 co Tj + tan"1 co Td - tan"' co T - tan"1 co aT d - cog xm (Equation 5.20) with co given by the solution of 2 2 K m K cA /l + cog2T,.2A/l + co T '%<) 1 dH

"^

2

2

„ 22rp 2 cogTiA/l + co g T m ^l + a 22co g T,d

•=1

(Equation 5.22)

Also A,

^ p Tj K

cK m \

l(l + cop2Tm2J(l + co p 2 a 2 T d 2 ) (l + cop2Ti2)(l + cop2Td2)

(Equation 5.21)

Chapter 5: Performance and Robustness Issues...

481

with cop given by tan_1 (OpTj 0.571 + tan"1 copTd ++tan cOpT;-tan"' -tan" 1 co coppTTmm -tan" -tan"11 co cop paTd -co p x m =0 (Equation 5.25) If K c , Tj and Td are designed as follows: Kc =

aTm

(5.72)

Kmxm

T,- = aT m

(5.73)

and Td = T ^m then, Equation (5.25) becomes

(5.74)

0.5TI - ® p ^ m = 0

(5.75)

cop = V2x ra

(5.76)

i.e. and Equation (5.21) becomes A m =<xrcT m /2K m K c T m Equation (5.22) becomes KmKc/ogTm=l

(5.77) (5.78)

i.e. »g=KmKc/Tm

(5.79)

Finally, substituting Equations (5.73), (5.74) and (5.79) into Equation (5.20) gives (t, m =0.57i-K m K c x m /aT m (5.80) Substituting Equation (5.72), (5.73), (5.74) and (5.76) into Equation (5.21) gives A m =7ra/2a (5.81) and substituting Equation (5.72) into Equation (5.80) gives <|>m= 0.5TI-a/a (5.82) This design reduces to the PI controller design when a = 1.

482

Handbook of PI and PID Controller Tuning Rules

5.5.2.2 Processes modelled in SOSPD form - series controller For such processes and controllers, K e"STm G m (s) = 2 (l + sT ml )(l + sT m2 )

(5.83)

with G c (s) = K c

(l + sT d ) T

»

i

s

(5.84)

,

Therefore, Gm(s)Gi(s)=iSi£H±Sl±^>

T |S (l + sTm,Xl + sT m2 ) If Tj and Td are designed as follows: Ti=T m l and Td=Tm2 then, from Equation (5.85) Gm(s)Gc(s)=Km^e

(5.85,

(5.86) (5.87) (5.88)

which is equal to G m (s)G c (s) in Section 5.5.1.1 when T i = T m . Therefore, designing aT Kc= 2(5.89) K

mTm

will allow A m = n/2a and §m = 0.571-a , as before. 5.5.2.3 Processes modelled in SOSPD form with a negative zero classical controller 1 For such processes, K m e- ! T -(l + sTm3) G m (s) = -^S 1 SLL (l + sT ml )(l + sT m2 ) with G c (s) given by Equation (5.19). Therefore,

(5.90)

Chapter 5: Performance and Robustness Issues...

0„(s)G c (s)= K A e ""'(' + S T ^ 1 + ; T 'X 1 + ^

483

(5.9!)

If T, ,T d and a are designed as follows: T
(5.92) (5.93)

« = T m3 /T m2

(5.94)

Ti=T m l =

and therefo ire, designing Kc =

(5.95)

aT

™ Km"1™

will allow A m =n/2a. and <j)m = 0.57t-a, as in Sections 5.5.1.1 and 5.5.2.2. 5.5.3 PD controller design In this case, the process is modelled in FOLIPD form, i.e. Kme-ST~ s(l + sT m )

(5.96)

with G c (s) = K c (l + T d s)

(5.97)

Therefore, G m ( S )G c (s)=

K m K c e" STmV(l + sT6jd ) ' S 1 +

(

sT

(5.98)

m)

Therefore, designing K c -—

aTm K

(5.99)

mTm

and will allow A m and 5.5.2.3.

(5.100) Td=Tm =7i/2aand <\>m = 0.57i-a , as in Sections 5.5.1.1, 5.5.2.2

484

Handbook of PI and PID Controller Tuning Rules

5.6 Conclusions This chapter has considered the performance and robustness of a PI and PID controlled FOLPD process, with the parameters of the controllers determined by a variety of tuning rules. The original contributions of this work are as follows: (a) an expansion of the analytical approach of Ho et al. (1995b; 1996) to determine the approximate gain and phase margin analytically under all operating conditions (b) the analytical determination of the approximate maximum sensitivity of the compensated system and (c) the application of the algorithms using a wide variety of PI and PID tuning rules. The implementation of autotuning algorithms in commercial controllers means that choice of a suitable tuning rule is an important issue; the techniques discussed allow an analytical evaluation to be performed of candidate tuning rules. Finally, the chapter discusses an original approach to design tuning rules for both PI and PID controllers, for a variety of delayed process models, with the objective of achieving constant gain and phase margins for all values of delay. In one of the cases discussed (PI control of an IPD process model), an analytical approximation is used in the development; this approximation may also be used to determine further tuning rules for other process models with delay, and for other PID controller structures.

Appendix 1

Glossary of Symbols Used in the Book

a f l ,a f 2 ,b f ] = Parameters of a filter in series with some PID controllers a,,a 2 ,a 3 ,b[, b 2 ,b 3 = Parameters of a third order process model a 1 ,a 2 ,a 3 ,a 4 ,a 5 ,b 1 ,b2,b3,b 4 ,b 5 = Parameters of a fifth order process model A, =y,(oo), A 2 =y 2 (co), A 3 =y 3 (co), A 4 =y 4 (oo), A 5 =y 5 (oo) Am = Gain margin A p = Peak output amplitude of limit cycle determined from relay autotuning b, c = Weighting factors in some PID controller structures D R = Desired closed loop damping ratio d(t) = Disturbance variable (time domain) du/dt = Time derivative of the manipulated variable (time domain) e(t) = Desired variable, r(t), minus controlled variable, y(t) (time domain) E(s) = Desired variable, R(s), minus controlled variable, Y(s) FOLPD model = First Order Lag Plus time Delay model FOLIPD model = First Order Lag plus Integral Plus time Delay model Fp,F;,Fd = Weights on the desired variable for one PID controller structure Gc (s) = PID controller transfer function G CL (s) = Closed loop transfer function GCL(jco) = Desired closed loop frequency response G (s) = Ideal PID controller transfer function

485

486

Handbook of PI and PID Controller Tuning Rules

G cs (s) = Series PID controller transfer function G m (s) = Process model transfer function G p (s) = Process transfer function Gp(jco) = Process transfer function at frequency co Gp(jco) = Magnitude of Gp(jco) G (jco,!,) = Magnitude of Gp(joo), at frequency co, corresponding to phase lag of § ZG p (jco)= Phase of Gp(jco) + h = Relay amplitude (relay autotuning) IE = Integral of Error = (e(t)dt IAE = Integral of Absolute Error = J~|e(t)|dt IMC = Internal Model Controller IPD model = Integral Plus time Delay model I 2 PD model = Integral Squared Plus time Delay model ISE = Integral of Squared Error = f e2 (t)dt ISTES = Integral of Squared Time multiplied by Error, all to be Squared = |Ve(t)]2dt ISTSE = Integral of Squared Time multiplied by Squared Error =

ft2e2(t)dt ITSE = Integral of Time multiplied by Squared Error = \~ te 2 (t)dt ITAE = Integral of Time multiplied by Absolute Error = T t|e(t)|dt Kc = Proportional gain of the controller Kcp = Proportional gain of the ideal PID controller Kcs = Proportional gain of the series PID controller

Appendix J: Glossary of Symbols Used in the Book

487

K f = Feedback gain used in non-interacting PID controller 10 2 m

KH =

'-p 2 "''ml

2 t m Km xm-

K

TTii^ml^m

xmTm

1.884KcKuxm

|

9co„

2Kmxm (im2 1324

2K m T m m

(Hwang, 1995)

|

49T m 2 324

|

7Tmxm 162

1.775KC2KU2 81a>/

4.71KuKc(xm+1.6Tj 81co„

m

(Hwang, 1995) V

K

H2

" 2K x

T

2

^mxmTm,

1.884KUKCT„

2

9co„

+-

-a

2K

mTm

with a

4

T

:

i

. m

'

, H

T

"l

,

"^m

^m

ml

T

m

* ml

1

. '^mlS

T m

1

324

81

9

m

, ^ ^ m ' m l

Sm

1

a

81

9

and 0.471K„K„

10T m J

|

4T m ,T m (8^ m T m+ 9T ml )

1.775K u 2 K c 2 r m 2

81

81co„2

a = • a„

81

(Hwang, 1995) Kj = Feedback gain term used in non-interacting PID controller 8, 11 K, = Integral-only controller gain (Pinnella et ah, 1986) K L = Gain of a load disturbance process model K m = Gain of the process model K 0 = Weighting parameter used in one PID controller structure K

= Gain of the process

K u = Ultimate proportional gain K^ = Proportional gain when G c (jco)G p (jco) has a phase lag of <>| ku

= Ultimate proportional gain estimate determined from relay autotuning

Handbook of PI and PID Controller Tuning Rules

488

Kj = Feedback gain term used in one PI controller structure (labeled controller with proportional term acting on the output 2) and the PID non-interacting controller 12 structure K25% = Proportional gain required to achieve a quarter decay ratio Kx% = Proportional gain required to achieve a decay ratio of O.Olx K)35o = Controller gain when Gc(jco)Gp(jco)has a phase lag of 135° K 15Q0

= Controller gain when Gc(jco)Gp(jco) has a phase lag of 150°

M s = Closed loop sensitivity Mmax = Maximum value of closed loop sensitivity, M s n = Order of a process model with a repeated pole N = Parameter that determines the amount of filtering on the derivative term on some PID controller structures OS = Closed loop response overshoot PI controller = Proportional Integral controller PID controller = Proportional Integral Derivative controller r = Desired variable (time domain) 0.7071M max 2 -M maxA /l-0.5M max 2 f

' ~

A*

2

i

M

maX - 1 (Chen et al, 1999a) R(s) = Desired variable (Laplace domain) s = Laplace variable SOSPD model = Second Order System Plus time Delay model SOSIPD model = Second Order System plus Integral Plus time Delay model Tar = Average residence time = time taken for the open loop process step response to reach 63% of its final value Td = Derivative time of the controller Tdi = Derivative feedback term used in non-interacting PID controller 8 Tdp = Derivative time of the ideal PID controller Tds = Derivative time of the series PID controller TCL = Desired closed loop system time constant

Appendix 1: Glossary of Symbols Used in the Book

489

TCL2 = Desired parameter of second order closed loop system response TCL3 = Desired parameter of third order closed loop system response, with a repeated pole Tf = Time constant of the filter in series with some PI or PID controllers T; = Integral time of the controller Tip = Integral time of the ideal PID controller Tis = Integral time of the series PID controller TL = Time constant of a load disturbance FOLPD process model Tm = Time constant of the process model TmijT^.T^ = Time constants of second or third order process models, as appropriate. T m i.T mi ,T mi ,T j = Time constants of a general process model Tp = Time constant of the process Tr = Parameter of the filter on the set-point in some PID controller structures TR = Closed loop rise time Ts = Closed loop settling time Tu = Ultimate period A

TU = Ultimate period estimate determined from relay autotuning T25% = Period of the quarter decay ratio waveform, when the closed loop system is under proportional control T x% = Period of the waveform with a decay ratio of O.Olx, when the closed loop system is under proportional control TOLPD model = Third Order Lag Plus time Delay model u(t) = Manipulated variable (time domain), u^ = Final value of the manipulated variable (time domain) U(s) = Manipulated variable (Laplace domain) V = Closed loop response overshoot (as a fraction of the controlled variable final value) x,, x 2 , x 3 , x 4 , x5 = Coefficient values.

490

Handbook of PI and PID Controller Tuning Rules

y = Controlled variable (time domain) y^ = Final value of the controlled variable (time domain) Y(s) = Controlled variable (Laplace domain)

y,(t)= J^K m - ^ j d t , y 2 (t)= J(A, -y,(x))lx, t

t

y 3 (t)= J(A 2 -y 2 (x))iT, y 4 (0= J[A3-y3(T)]iT, y 5 (t)=J[A 4 -y 4 (x)}iT a , P, x = Weighting factors in some PI or PID controller structures 2

£=6Tml

2

+4^ m T m ,x m +K H K m x m

(Rwang; 1995)

2Tml xmcoH =

2T m co u+ K H1 K m x m co u -1.884K u K c

^

0.471KcKu
6T =

2

"^

+4T

2

m1^m+KuKmTm 2x

C2

mTml

( H w a n g ;

1995)

ffl

u

, 6Tm,2(Du +4T ml ^ m t m (0 u + KH2Kmxm2cou -1.884K u K c x m (0.471K u K c x m 2 +2x m T ml 2 co u )co H2

(Hwang, 1995) X = Parameter that determines robustness of compensated system. £, = Damping factor of the compensated system ^m = Damping factor of an underdamped process model ^ e s = Desired damping factor of an underdamped process model £,m j , £m ; = Damping factors of a general underdamped process model K = l/K m K u <) = Phase lag

Appendix 1: Glossary of Symbols Used in the Book

491

<j)c - Phase of the plant at the crossover frequency of the compensated system, with a 'conservative' PI controller (Pecharroman and Pagola, 2000). cpm = Phase margin §a = Phase lag at an angular frequency of CO xL = Time delay of a load disturbance process model Tm = Time delay of the process model co = Angular frequency cobw = -3dB closed loop system bandwidth (Shi and Lee, 2002) co_6dB= Frequency where closed loop system magnitude is -6dB (Shi and Lee, 2004) coc = Maximum cut-off frequency cod = Bandwidth of stochastic disturbance signal (Van der Grinten, 1963) 2

®CL =

"/TCL

cog = Specified gain crossover frequency (Shi and Lee, 2002) 1+ K

coH = |

T

"Km

2 i, 2 Tml ^mm m l xm m^ L

(Hwang, 1995) KHKmxI]

.|

1+ KHlKm

xmTm

|

K H1 K m T m 2

0.942K c K u t m

(Hwang, 1995)

3co„

C0„T

=

[T

2 } 2^mxmTml 3

coM

l + K H2 K m 2 | K H2 K m x m 6

0.942K c K u x r 3cou

(Hwang, 1995) = Frequency where the sensitivity function is maximized (Rotach, 1994)

492

Handbook of PI and PID Controller Tuning Rules

con = Undamped natural frequency of the compensated system coD =

AA rh +0 Sir A (Am — l) , p '- (Hang e/a/. 1993a, 1993b)

lA m 2 -l> m cor = Resonant frequency (Rotach, 1994) cou = Ultimate frequency co^ = Angular frequency at a phase lag of (j) cog()0 = Angular frequency at a phase lag of 90° co'135'o = Angular frequency at a phase lag of 135° co o = Angular frequency at a phase lag of 150°

Appendix 2

Some Further Details on Process Modelling

Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables in Chapters 3 and 4. Some outline details of these modelling strategies are provided, together with the references that describe the modelling method in detail. The references are given in the bibliography. For all models, the label "Model: Method 1" indicates that the model method has not been defined or that the model parameters are assumed known. Of the 1134 tuning rules specified (442 PI controller tuning rules, 692 PID controller tuning rules), it is startling to relate that only 386 (or 34%) of tuning rules have been specified based on a defined process modelling method.

A2.1 FOLPD Model G m ( s ) = K m C

A2.1.1 Parameters estimated from the open loop process step or impulse response Method 2:

Parameters estimated using a tangent and point method (Ziegler and Nichols, 1942). Method 3: K m ,x m determined from the tangent and point method of Ziegler and Nichols (1942); Tm determined at 60% of the

493

494

Handbook of PI and PID Controller Tuning Rules

total process variable change (Fertik and Sharpe, 1979). Method4: K m ,x m assumed known; Tm estimated using a tangent method (Wolfe, 1951). Method 5: Parameters estimated using a second tangent and point method (Murrill, 1967). Method 6: Parameters estimated using a third tangent and point method (Davydov et al, 1995). Method 7: xm and Tm estimated using the two-point method; Km estimated from the open loop step response (ABB, 2001). Method 8: xm and Tm estimated from the process step response: T m = 1 . 4 ( t 6 7 % - t 3 3 % ) , x m = t 6 7 % - l . l T m ; Km assumed known (Chen and Yang, 2000). ' Method 9: xm and Tm estimated from the process step response: Tm = 1.245(t70O/o - t 3 3 % ) , xm = 1.498t33% -0.498t 70% ; Km assumed known (Viteckova et al., 2000b). 2 Method 10: Km estimated from the process step response; x m = time for which the process variable does not change; Tm determined at 63% of the total process variable change (Gerry, 1999). Method 11: Km estimated from the process step response; xm = time for which the process variable has to change by 5% of its total value; Tm determined at 63% of the total process variable change (Kristiansson, 2003). Method 12: Parameters estimated using a least squares method in the time domain (Cheng and Hung, 1985). Method 13: Parameters estimated from linear regression equations in the time domain (Bi et al., 1999). Method 14: Parameters estimated using the method of moments (Astrom and Hagglund, 1995). Method 15: Parameters estimated from the process step response and its first time derivative (Tsang and Rad, 1995). Method 16: Parameters estimated from the process step response using numerical integration procedures (Nishikawa et al., 1984). ' t67o/„, t33% are the times taken by the process variable to change by 67% and 33%, respectively, of its total value. 2 170o/o is the time taken by the process variable to change by 70% of its total value.

Appendix 2: Some Further Details on Process Modelling

495

Method 17: Parameters estimated from the process impulse response (Peng and Wu, 2000). Method 18: Parameters estimated using a "process setter" block (Saito et al, 1990). Method 19: Parameters estimated from a number of process step response data values (Kraus, 1986). Method 20: Parameters estimated in the sampled data domain using two process step tests (Pinnella et ah, 1986). Method 21: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters K m , Tm and x m . Subsequently, xm is estimated from the process step response; then, a parameter labelled x a m =x m -i m (Schaedel, 1997). Method 22: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters Km , Tm and x m . Subsequently, xm is estimated from the process step response (Henry and Schaedel, 2005). A2.1.2 Parameters estimated from the closed loop step response Method 23: Km estimated from the open loop step response. T90% and xm estimated from the closed loop step response under proportional control (Astrom and Hagglund, 1988). Method 24: Parameters estimated using a method based on the closed loop transient response to a step input under proportional control (Sain and Ozgen, 1992). Method 25: Parameters estimated using a second method based on the closed loop transient response to a step input under proportional control (Hwang, 1993). Method 26: Parameters estimated using a third method based on the closed loop transient response to a step input under proportional control (Chen, 1989; Taiwo, 1993).

496

Handbook of PI and PID Controller Tuning Rules

Method 27: Parameters estimated from a step response autotuning experiment - Honeywell UDC 6000 controller (Astrom and Hagglund, 1995). Method 28: Parameters estimated from the closed loop step response when process is in series with a PID controller (Morilla et al, 2000). Method 29: Parameters estimated by modelling the closed loop response as a second order system (Ettaleb and Roche, 2000). A2.1.3 Parameters estimated from frequency domain closed loop information Method 30: Parameters estimated from two points, determined on process frequency response, using a relay and a relay in series with a delay (Tan et al., 1996). Method 31: Tm and xm estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; Km assumed known (Hang and Cao, 1996). Method 32: Tm and xm estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; Km estimated from the process step response (Hang et al, 1993b). Method 33: Tm and xm estimated from the ultimate gain and period, determined using the ultimate cycle method of Ziegler and Nichols (1942); Km estimated from the process step response (Hang et al., 1993b). Method 34: Tm and xm estimated from relay autotuning method (Lee and Sung, 1993); Km estimated from the closed loop process step response under proportional control (Chun et al, 1999). Method 35: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995).

Appendix 2: Some Further Details on Process Modelling

497

Method 36: Ku and Tu estimated from relay autotuning method; xm estimated from the open loop process step response, using a tangent and point method (Wojsznis et al, 1999). Method 37: Parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003). Method 38: Parameters estimated from measurements performed on the manipulated and controlled variables when a relay with hysteresis is introduced in place of the controller (Zhang et al., 1996). Method 39: Non-gain parameters estimated using a relay, with hysteresis, in series with the process in closed loop; Km is estimated with the aid of a small step signal added to the reference (Potocnik et al., 2001). Method 40: Parameters estimated using a relay in series with the process in closed loop (Peric et al, 1997). Method 41: Parameters estimated using an iterative method, based on data from a relay experiment (Leva and Colombo, 2004). Method 42: Km estimated from relay autotuning method; x m , Tm estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al., 2005). A2.1.4 Other methods Method 43: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991). Method 44: Parameter estimates determined graphically from a known higher order process (McMillan, 1984). Method 45: Parameter estimates determined from a known higher order process. Method 46: The parameters of a second order model plus time delay are estimated using a system identification approach in discrete time; the parameters of a FOLPD model are subsequently determined using standard equations (Ou and Chen, 1995).

498

Handbook of PI and PID Controller Tuning Rules

Method 47: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968). Table 188 summaries the most common FOLPD process modelling methods. Table 188: The most common FOLPD process modelling methods Model Method Method 1 Method 2 Method 5 Method 31

PI 112 18 12 6

PID 147 36 26 6

Total 259 54 38 12

It is interesting that in 57% of tuning rules based on the FOLPD process model (252 out of 443), the modelling method has not been specified or the model parameters are known a priori.

A2.2 FOLPD Model with a Negative Zero Gm (s) = - - - -^ + s T mT 2 ^ l + s ml Method 2: Non-delay parameters are estimated in the discrete time domain using a least squares approach; time delay assumed known (Chang et al, 1997). A2.3 Non-Model Specific Modelling methods have not been specified in the tables in most cases, as typically the tuning rules are based on Ku and T u . Modelling methods have been specified whenever relevant data have been estimated using a relay method. Method 2: Ku and Tu estimated from an experiment using a relay in series with the process in closed loop (Jones et al., 1997). Method 3: Ku and Tu estimated with the assistance of a relay (Lloyd, 1994).

Appendix 2: Some Further Details on Process Modelling

499

Method 4: co90„ and Ap estimated using a relay in series with an integrator (Tang et al., 2002).

A2.4 IPD Model Gm (s) =

Km&

s

A2.4.1 Parameters estimated from the open loop process step response Method 2:

xm assumed known; Km estimated from the slope at start of the process step response (Ziegler and Nichols, 1942). Method 3: K m ,x m estimated from the process step response (Hay, 1998). Method 4: K m ,x m estimated from the process step response using a tangent and point method (Bunzemeier, 1998). A2.4.2 Parameters estimated from the closed loop step response Method 5:

Parameters estimated from the servo or regulator closed loop transient response, under PI control (Rotach, 1995). Method 6: Parameters estimated from the servo closed loop transient response under proportional control (Srividya and Chidambaram, 1997). Method 7: Parameters estimated from the closed loop response, under the control of an on-off controller (Zou et al., 1997). Method 8: Parameters estimated from the closed loop response, under the control of an on-off controller (Zou and Brigham, 1998). A2.4.3 Parameters estimated from frequency domain closed loop information Method 9:

Parameters estimated from Ku and Tu values, determined from an experiment using a relay in series with the process in closed loop (Tyreus and Luyben, 1992). Method 10: Parameters estimated from values determined from an

500

Handbook of PI and PID Controller Tuning Rules

experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999). The most common IPD modeling method is the one where the model parameters are known a-priori or the modeling method is unspecified (i.e. Model: Method 1); 46 PI controller tuning rules, and 47 PID controller tuning rules have been specified using the method. Altogether 76% of tuning rules based on the IPD process model (93 out of 122) have been specified using the method.

A2.5 FOLIPD Model G m (s) Method 2:

Method 3: Method 4:

Method 5: Method 6:

Method 7:

Method 8:

K m e" 3(l + sT m )

Parameters estimated from the open loop process step response and its first and second time derivatives (Tsang andRad, 1995). Parameters estimated using the method of moments (Astrom andHagglund, 1995). Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999). Parameters estimated from the open loop response of the process to a pulse signal (Tachibana, 1984). Parameters estimated from the waveform obtained by introducing a single symmetrical relay in series with the process in closed loop (Majhi and Mahanta, 2001). Km estimated from the response of the process to a square wave pulse; other process parameters estimated from the wavefrom obtained by introducing a relay in series with the process in closed loop (Wang et ah, 2001a). Parameters estimated using a relay in series with the process in closed loop (Peric et ah, 1997).

The most common FOLIPD modeling method is the one where the model parameters are known a-priori or the modeling method is

Appendix 2: Some Further Details on Process Modelling

501

unspecified (i.e. Model: Method 1); 25 PI controller tuning rules, and 49 PID controller tuning rules have been specified using the method. Altogether 85% of tuning rules based on the FOLIPD process model (74 out of 87) have been specified using the method.

A2.6 SOSPD Model G m (s) =

K e~ 2 2

T m l s + 2^ m T m l s + l

or K e"STm (l + Tmls)(l + Tm2s)

A2.6.1 Parameters estimated from the open loop process step response Method 2:

K m , Tml and xm are determined from the tangent and point method of Ziegler and Nichols (1942) (Shinskey, 1988, page 151); Tm2 assumed known. Method 3: FOLPD model parameters estimated using a tangent and point method (Ziegler and Nichols, 1942); corresponding SOSPD parameters subsequently deduced (Auslander et al., 1975). Method 4: Tml = T m2 . x m , Tml estimated from process step response: T m l =0.794(t 7 0 % -t 3 3 % ), T m =1.937t 33% -0.937t 70% . Km assumed known (Viteckova et ah, 2000b). Method 5: Tml = T m2 . K m , xm and Tml estimated from the process step response; xm = time for which the process variable does not change; Tml determined at 73% of the total process variable change (Pomerleau and Poulin, 2004). Method 6: Parameters estimated from the underdamped or overdamped transient response in open loop to a step input (Jahanmiri and Fallahi, 1997).

502

Handbook of PI and PID Controller Tuning Rules

A2.6.2 Parameters estimated from the closed loop step response Method 7:

Parameters estimated from a step response autotuning experiment - Honeywell UDC 6000 controller (Astrom and Hagglund, 1995).

A2.6.3 Parameters estimated from frequency domain closed loop information Method 8:

Method 9:

Method 10:

Method 11:

Method 12:

Method 13:

Method 14:

Tm and xm estimated In this method, Tml = Tm2 = T, from K u , Tu determined using a relay autotuning method; Km estimated from the process step response (Hang et al., 1993a). Parameters estimated using a two-stage identification procedure involving (a) placing a relay and (b) placing a proportional controller, in series with the process in closed loop (Sung et al, 1996). Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995). Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999). Parameters estimated from data obtained when the process phase lag is -90° and -180°, respectively (Wang et al., 1999). Model parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003). Km estimated from a relay autotuning method; x m , Tml and 2;m estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al, 2005).

Appendix 2: Some Further Details on Process Modelling

503

A2.6.4 Other methods Method 15: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991). Method 16: Parameter estimates back-calculated from a second discrete time identification method (Wang and Clements, 1995). Method 17: Parameter estimates back-calculated from a third discrete time identification method (Lopez et al, 1969). Method 18: Model parameters estimated assuming higher order process parameters are known. Method 19: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968). The most common SOSPD modeling method is the one where the model parameters are known a-priori or the modeling method is unspecified (i.e. Model: Method 1); 15 PI controller tuning rules, and 93 PID controller tuning rules have been specified using the method. Altogether 68% of tuning rules based on the SOSPD process model (108 out of 160) have been specified using the method.

A2.7 SOSIPD model - Repeated Pole G_(s) = —^ -r s(l + sT m ) 2 Method 2:

Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999).

504

Handbook of PI and PID Controller Tuning Rules

A2.8 SOSPD Model with a Positive Zero G_(s) - — •"-" s T m 3 ^ (l + sTmlXl + sT m2 )

Method 2:

Parameters estimated from a closed loop step response test using a least squares approach (Wang et al., 2001a). Method 3: Tml = T m2 . K m , x m , Tml and Tm3 estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004).

A2.9 SOSPD model with a Negative Zero Gm (s) -

Km

'* + s T m 3 ^ (l + sTmlXl + sT m2 )

Method 2:

Parameters estimated from a closed loop step response test using a least squares approach (Wang et al, 2001a). Method 3: Tml = T m2 . K m , x m , Tml and Tm3 estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004).

A2.10 Unstable FOLPD Model Gm (s) = K™e T m s-1 Method 2: Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980). Method 3: Parameters estimated using a relay feedback approach (Majhi and Atherton, 2000).

Appendix 2: Some Further Details on Process Modelling

Method 4:

505

Parameters estimated using a biased relay feedback test (Huang and Chen, 1999).

K e~STm A2.ll Unstable SOSPD Model G m (s) = -, -^ r or (TmlS-ljO+sTmJ K e- S T „ (T m l s-l)(T m 2 s-l) Method 2:

Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980). Method 3: Parameters estimated by minimising the difference in the frequency responses between the high order process and the model, up to the ultimate frequency. The gain and delay are estimated from analytical equations, with the other parameters estimated using a least squares method (Kwak et al, 2000). Method 4: Parameters estimated using a biased relay feedback test (Huang and Chen, 1999).

A2.12 General Model with a Repeated Pole Gm(s) Method 2:

K e""" 0 + sT m ) n

A FOLPD model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters Km , Tm and xm . Then, n = 10-r L + 1 . Subsequently, xm is estimated from the open loop step response, and Tar is estimated using "known methods" (Schaedel (1997)).

This page is intentionally left blank

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ABB (1996). Operating Guide for Commander 300/310, Section 7. ABB (2001). Instruction manual for 53SL6000 [Online]. Document: PN24991.pdf. Available at www.abb.com [accessed 1 September 2004]. Abbas, A. (1997). A new set of controller tuning relations, ISA Transactions, 36, pp. 183-187. Aikman, A.R. (1950). The frequency response approach to automatic control problems, Transactions of the Society of Instrument Technology, pp. 2-16. Alvarez-Ramirez, J., Morales, A. and Cervantes, I. (1998). Robust proportional-integral control, Industrial Engineering Chemistry Research, 37, pp. 4740^747. Andersson, M. (2000). A MATLAB tool for rapid process identification and PID design, M.Sc. thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden. Ang, K.H., Chong, G. and Li, Y. (2005). PID control system analysis, design and technology, IEEE Transactions on Control Systems Technology, 13(4), pp. 559-576. Argelaguet, R., Pons, M., Quevedo, J. and Aguilar, J. (2000). A new tuning of PID controllers based on LQR optimization, Preprints of Proceedings of PID '00: IF AC Workshop on Digital Control, pp. 303-308. Arvanitis, K.G., Akritidis, C.B., Pasgianos, G.D. and Sigrimis, N.A. (2000). Controller tuning for second order dead-time fertigation mixing process, Proceedings of EurAgEng Conference on Agricultural Engineering, Paper No. OO-AE-011. Arvanitis, K.G., Syrkos, G., Stellas, I.Z. and Sigrimis, N.A. (2003a). Controller tuning for integrating processes with time delay. Part I: IPDT processes and the pseudoderivative feedback control configuration, Proceedings of ll'h Mediterranean Conference on Control and Automation, Paper No. T7-040. Arvanitis, K.G., Syrkos, G., Stellas, I.Z. and Sigrimis, N.A. (2003b). Controller tuning for integrating processes with time delay. Part III: The case of first order plus integral plus dead-time processes, Proceedings of 11' Mediterranean Conference on Control and Automation, Paper No. T7-042. Astrom, K.J. (1982). Ziegler-Nichols auto-tuner, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, Report TFRT-3167.

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Handbook of PI and PID Controller Tuning Rules

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Venkatashankar, V. and Chidambaram, M. (1994). Design of P and PI controllers for unstable first order plus time delay systems, International Journal of Control, 60, pp. 137-144. Visioli, A. (2001). Tuning of PID controllers with fuzzy logic, IEE Proceedings Control Theory and Applications, 148(1), pp. 180-184. Viteckova, M. (1999). Sefizeni cislicovych I analogovych regulatorfl pro regulovane soustavy s dopravnim zpozdSnim (Digital and analog controller tuning for processes with time delay), Automatizace, 42(2), pp. 106-111 (in Czech). Viteckova, M. and Vitecek, A. (2002). Analytical controller tuning method for proportional non-oscillatory plants with time delay, Proceedings of the International Carpathian Control Conference (Malenovice, Czech Republic), pp. 297-302. Viteckova, M. and Vitecek, A. (2003). Analytical digital and analog controller tuning method for proportional non-oscillatory plants with time delay, Proceedings of the 2nd IFAC Conference on Control Systems Design (Bratislava, Slovak Republic), pp. 5964. Viteckova, M., Vitecek, A. and Smutny, L. (2000a). Controller tuning for controlled plants with time delay, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 283-288. Viteckova, V., Vitecek, A. and Smutny, L. (2000b). Simple PI and PID controllers tuning for monotone self-regulating plants, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 289-294. Voda, A. and Landau, I.D. (1995). The autocalibration of PI controllers based on two frequency measurements, International Journal of Adaptive Control and Signal Processing, 9, pp. 395-421. Vrancic, D. (1996). Design of anti-windup and bumpless transfer protection. Part II: PID controller tuning by multiple integration method, PhD thesis (University of Ljubljana, J. Stefan Institute, Ljubljana, Slovenia). Vrancic, D., Kocijan, J. and Strmcnik, S. (2004a). Simplified disturbance rejection tuning method for PID controllers, Proceedings of the 5' Asian Control Conference (Melbourne, Australia). Vrancic, D., Kristiansson, B. and Strmcnik, S. (2004b). Reduced MO tuning method for PID controllers, Proceedings of the 5th Asian Control Conference (Melbourne, Australia). Vrancic, D. and Lumbar, S. (2004). Improving PID controller disturbance rejection by means of magnitude optimum, Report DP-8955 (J. Stefan Institute, Ljubljana, Slovenia). Vrancic, D., Peng, Y., Strmcnik, S. and Hanus, R. (1996). A new tuning method for PI controllers based on a process step response, Proceedings of CESA '96 IMACS MultiConference Symposium on Control, Optimisation and Supervision (Lille, France), 2, pp. 790-794. Vrancic, D., Peng, Y., Strmcnik, S. and Juricic, D. (1999). A multiple integration tuning method for filtered PID controller, Preprints of Proceedings of IFAC I4'h World

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Wang, Y.-G. and Shao, H.-H. (1999). PID autotuner based on gain- and phase-margin specification, Industrial Engineering Chemistry Research, 38, pp. 3007-3012. Wang, Y.-G. and Shao, H.-H. (2000a). Optimal tuning for PI controller, Automatica, 36, pp. 147-152. Wang, Y.-G. and Shao, H.-H. (2000b). PID auto-tuner based on sensitivity specification, Transactions of the Institute of Chemical Engineers, 78(A), pp. 312-316. Wang, Y.-G., Shao, H.-H. and Wang, J. (2000a). PI tuning for processes with large dead time, Proceedings of the American Control Conference (Chicago, U.S.A.), pp. 42744278. Wills, D.M. (1962a). Tuning maps for three-mode controllers, Control Engineering, April, pp. 104-108. Wills, D.M. (1962b). A guide to controller tuning, Control Engineering, August, pp. 9 3 95. Wilton, S.R. (1999). Controller tuning, ISA Transactions, 38, pp. 157-170. Witt, S.D. and Waggoner, R.C. (1990). Tuning parameters for non-PID three-mode controllers, Hydrocarbon Processing, June, pp. 74-78. Wojsznis, W.K. and Blevins, T.L. (1995). System and method for automatically tuning a process controller, U.S. Patent Number 5,453,925. Wojsznis, W.K., Blevins, T.L. and Thiele, D. (1999). Neural network assisted control loop tuner, Proceedings of the IEEE International Conference on Control Applications (HI, U.S.A.), 1, pp. 427-431. Wolfe, W.A. (1951). Controller settings for optimum control, Transactions of the ASME, May, pp. 413-418. Xu, J. and Shao, H. (2003a). Advanced PID tuning for integrating processes with a new robustness specification, Proceedings of the American Control Conference (Denver, CO., U.S.A.), pp. 3961-3966. Xu, J. and Shao, H. (2003b). Advanced PID tuning for unstable processes based on a new robustness specification, Proceedings of the American Control Conference (Denver, CO., U.S.A.), 368-372. Xu, J. and Shao, H. (2003c). A novel method of PID tuning for integrating processes, Proceedings of the 42nd IEEE Conference on Decision and Control (Maui, Hawaii, U.S.A.), pp. 139-142. Xu, J.-H., Sun, R. and Shao, H.-H. (2004). PI controller tuning for large dead-time processes, Kongzhi Yu Juece/Control and Decision, 19(1), pp. 99-101 (in Chinese). Yang, J. C.-Y. and Clarke, D.W. (1996). Control using self-validating sensors, Transactions of the Institute of Measurement and Control, 18, pp. 15-23. Yi, C. and De Moor, B.L.R. (1994). Robustness analysis and control system design for a hydraulic servo system, IEEE Transactions on Control Systems Technology, 2(3), pp. 183-197. Young, A.J. (1955). An introduction to process control system design (Longman, Green and Company, London).

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Yu, C.-C. (1999). Autotuning of PID controllers (Advances in Industrial Control Series, Springer-Verlag London Ltd.). Yu, S. W. (1988). Optimal PI tuning for load disturbances, Journal of the Chinese Institute of Chemical Engineers, 19(6), pp. 349-357. Zhang, D. (1994). Process dynamics and controller selection of DCS, Proceedings of the ISA Advances in Instrumentation and Control Conference (Anaheim, U.S.A), 49(2), pp. 231-240. Zhang, W., Xi, Y., Yang, G. and Xu, X. (2002). Design PID controllers for desired timedomain or frequency-domain response, ISA Transactions, 41, pp. 511-520. Zhang, W. and Xu, X. (2002). H,, PID controller design for runaway processes with time delay, ISA Transactions, 41, pp. 317-322. Zhang, W., Xu, X. and Sun, Y. (1999). Quantitative performance design for integrating processes with time delay, Automatica, 35, pp. 719-723. Zhang, W.D., Sun, Y.X. and Xu, X.M. (1996). Modified PID controller based on HM theory, Proceedings of the IEEE International Conference on Industrial Technology, pp. 9-12. Zhong, Q.-C. and Li, H.-X. (2002). 2-degree-of-freedom proportional-integral-derivativetype controller incorporating the Smith principle for processes with dead time, Industrial Engineering Chemistry Research, 41, pp. 2448-2454. Zhuang, M. and Atherton, D.P. (1993). Automatic tuning of optimum PID controllers, IEE Proceedings, Part D, 140, pp. 216-224. Ziegler, J.G. and Nichols, N.B. (1942). Optimum settings for automatic controllers, Transactions oftheASME, November, pp. 759-768. Zou, H., Hedstrom, K.P., Warrior, J. and Hays, C.L. (1997). Field based process control system with auto-tuning, U.S. Patent Number 5,691,896 . Zou, H. and Brigham, S.E. (1998). Process control system with asymptotic auto-tuning, U.S. Patent Number 5,818,714.

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Index

ABB (1996), 237 ABB (2001), 15, 29,42, 71, 224,225 Abbas (1997), 47, 169 Aikman(1950), 72 Alfa-Laval, 12, 73 Allen Bradley, 7, 15 Alvarez-Ramirez et al. (1998), 81 Andersson (2000), 58, 81 Argelaguet et al. (2000), 229 Arvanitis et al. (2000), 327, 332, 341, 371 Arvanitis et al. (2003a), 85, 86, 274, 275 Arvanitis et al. (2003b), 94-100, 298302 Astrom (1982), 73,236 Astrom and Hagglund (1984), 236 Astrom and Hagglund (1988), 155, 236 Astrom and Hagglund (1995), 45, 64, 70, 73, 75, 76, 93, 185, 236, 245, 259, 284, 349 Astrom and Hagglund (2000), 150 Astrom and Hagglund (2004), 150, 167-168, 223, 260, 272 Atkinson and Davey (1968), 235 Auslandere/a/. (1975), 309 Bailey, 7, 9, 12, 13 Bailey Fisher and Porter, 13 Bain and Martin (1983), 38, 43 Bateson (2002), 73, 238, 258 Bekkeretal. (1991), 45 Belanger and Luyben (1997), 88, 267 Bequette (2003), 151, 181, 450

Bialkowski (1996), 58, 81 Bi et al. (1999), 47 Bi et al. (2000), 47, 327 Blickley (1990), 235 Boe and Chang (1988), 60 Bohl and McAvoy (1976a), 372, 373 Bohl and McAvoy (1976b), 311, 330, 352,353 Borresen and Grindal (1990), 29, 155 Brambilla et al. (1990), 58, 118, 178, 328 Branica et al. (2002), 176 Bryant et al. (1973), 42, 114-115, 151 Buckley (1964), 151 Bunzemeier(1998), 77, 266 Calcev and Gorez (1995), 71, 236 Callender et al. (1935/6), 26, 149, 154, 449 Camachoefa/. (1997), 169 Carr (1986), 235 Chandrashekar et al. (2002), 133, 136, 408,409,417,418 Changed al. (1997), 69 Chao et al. (1989), 106, 107, 111, 342, 344-349 Chau (2002), 237 Chen and Seborg (2002), 48, 81, 172, 261,280,327-328 Chen and Yang (2000), 56 Chen etal. (1997), 59 Chen et al. (1999a), 51, 80, 261, 280

536

Handbook of PI and PID Controller Tuning Rules

Chen etal. (1999b), 51,324 Chen et al. (2001), 193 Cheng and Hung (1985), 157, 164 Cheng and Yu (2000), 80 Chesmond(1982), 73, 238 Chidambaram (1994), 79 Chidambaram (1995a), 175-176 Chidambaram (1995b), 131,404 Chidambaram (1997), 131, 140, 405 Chidambaram (2000a), 63, 226 Chidambaram (2000b), 84, 271 Chidambaram (2000c), 132, 135, 416 Chidambaram and Kalyan (1996), 433 Chidambaram and Sree (2003), 79, 261,271 Chidambaram et al. (2005), 49 Chien (1988), 58, 81, 190, 206, 211, 264, 267, 269, 285, 291, 294, 386, 387, 389, 393, 394, 395 Chien and Fruehauf (1990), 81, 206, 211, 264, 267, 269, 285, 291, 294, 386, 387, 389, 393, 394, 395 Chien era/. (1952), 27, 155 Chien et al. (1999), 65, 86, 229, 275 Chiue/a/. (1973), 42, 326 Chun etal. (1999), 59 Cluett and Wang (1997), 54-55, 78, 170-171,260 Cohen and Coon (1953), 28, 155 Concept, 9 Connell(1996), 156 ControlSoft Inc. (2005), 156 Coon (1956), 76, 89 Coon (1964), 76, 89 Corripio (1990), 236, 253 Cox etal. (1994), 73 Cox etal. (1997), 55 Davydov et al. (1995), 46, 188 De Paor (1993), 236 De Paor and O'Malley (1989), 131,404 Devanathan(1991), 37, 161 Direct synthesis, 24-25, 42-58, 61, 62, 63-64, 65, 67, 68, 71-72, 75, 78-81, 84, 86, 90-91, 93, 100, 114-118,

123-124, 125, 126-127, 129, 131133, 135-136, 138-139, 141, 142, 143-144, 146, 147, 148, 151, 152, 153, 168-178, 181, 183-184, 185, 186, 188-189, 193, 203-205, 208, 210, 212, 219-220, 225, 226, 227229, 230, 242, 245, 246-251, 260261, 263, 264, 265, 268, 271, 275, 277, 279-280, 284, 285, 291, 292, 293, 296, 302, 303, 304, 306, 307, 308, 321-328, 332, 338, 341, 349350, 351, 353, 354, 367, 370, 371, 372, 378, 379, 380, 385, 387, 388, 390, 391, 392, 394, 396, 397, 399, 400, 404-407, 408-409, 410-413, 415, 416^119, 427, 433, 439^141, 442, 445^146, 447^148, 452, 453, 454, 455^456, 457^459, 460-461 Frequency domain criteria, 49-58, 64, 79-81, 124, 152, 174-178, 181, 183-184, 193, 208, 268, 292, 321324, 332, 399, 400, 433, 453 Time domain criteria, 42-49, 61, 62, 63-64, 65, 67, 68, 78-79, 123, 141, 142, 143-144, 147, 148, 168-173, 227-229, 230, 265, 306, 325-328, 332, 372, 380, 385, 415, 433, 445446, 447-448 Dmonetal. (1997), 240 ECOSSE Team (1996a), 37 ECOSSE Team (1996b), 73, 238 Edgar et al. (1997), 31, 71, 197, 224, 253, 257 Ettaleb and Roche (2000), 47 expert system, 472 Farrington (1950), 235 Fanuc, 7 Fertik(1975), 150 Fertik and Sharpe (1979), 29 Fischer and Porter, 9, 12, 17 Fisher-Rosemount, 17 Ford (1953), 259 Foxboro, 9, 11, 12, 17, 209

Index Frank and Lenz (1969), 30, 33, 34, 35 Friman and Waller (1997), 52, 71, 178 Fruehauf etal. (1993), 46, 78, 169 Gaikward and Chidambaram (2000), 132,406 Gain margin, analytical calculation, 463-469 Gain margin, figures, 471, 473-474 Gain and phase margins, constant, 475483 PI controller, FOLPD model, 475^77 PI controller, IPD model, 477-480 PD controller, FOLIPD model, 483 PID controller, FOLPD model, classical controller 1, 480-481 PID controller, SOSPD model, series controller, 482 PID controller, SOSPD model with a negative zero, classical controller 1, 482-483 Gallier and Otto (1968), 38, 108, 162, 315 Garcia and Castelo (2000), 239 Genesis, 15 Geng and Geary (1993), 60, 179 Gerry (1988), 149, 159 Gerry (1999), 58, 178 Gerry (2003), 37 Gerry and Hansen (1987), 149 Gong et al. (1996), 190 Goodwin etal. (2001), 189 Gorecki et al. (1989), 45, 49, 168, 174 Gorez (2003), 48, 173,185 Gorez and Klan (2000), 326,454 Haalman (1965), 33, 77, 149, 279, 352 Haeri (2002), 225 Haeri (2005), 49 Hagglund and Astrom (1991), 73 Hagglund and Astrom (2002), 56, 64, 84, 136 Hang and Astrom (1988a), 220-221 Hang and Astrom (1988b), 236 Hang and Cao (1996), 221

537 Hanged/. (1991), 45, 221 Hang et al. (1993a), 50, 321, 332 Hang et al. (1993b), 50, 70, 211, 255 Hange/a/. (1994), 338 Hansen (1998), 370 Hansen (2000), 151, 271, 379 Harriott (1964), 238 Harris and Tyreus (1987), 206, 234 Harrold (1999), 195,253 Hartmann and Braun, 9 Hartreeera/. (1937), 451 Hassan (1993), 106,312 Hay (1998), 29, 73, 76-77, 156, 238, 259 Hazebroek and Van der Waerden (1950), 27, 32-33, 77 Heck (1969), 28-29 Henry and Schaedel (2005), 52-53 Hiroi and Terauchi (1986), 217, 222 Ho and Xu (1998), 132,433 Ho era/. (1994), 321 Ho etal. (1995a), 321-322 Ho etal. (1997), 322 Wo etal. (1998), 158, 163 Wo etal. (1999), 51, 158, 163 no etal. (2001), 234 Honeywell, 7, 11, 15, 16, 18, 45, 349 Horned/. (1996), 181, 184 Hougen (1979), 49, 79, 90, 115, 126, 208,268,292,351 Hougen (1988), 116,268,351 Huang and Chao (1982), 195, 197, 198, 199,201,203 Huang and Chen (1997), 415, 433 Huang and Chen (1999), 415, 433 Huang and Jeng (2002), 38, 150 Huang and Jeng (2003), 167, 320-321 Huang and Lin (1995), 420, 434^138 Huang et al. (1996), 32, 38, 103-104, 108-110,355-362 Huang e^a/. (1998), 58, 329 Huang et al. (2000), 212, 219, 354, 367,390,391,396,397 Huang et al. (2002), 182

538

Handbook of PI and PID Controller Tuning Rules

Huang et al. (2005), 58, 60, 205, 207, 332,335 Huba and Zakova (2003), 79, 91 Hwang (1995), 36, 39, 107, 112-113, 159-160, 165-166, 314-315, 316319 Hwang and Chang (1987), 70 Hwang and Fang (1995), 37, 41, 160, 166, 167

Kristiansson and Lennartson (1998b), 248 Kristiansson and Lennartson (2000), 72, 249-250 Kristiansson and Lennartson (2002), 72,250-251,264,285 Kristiansson et al. (2000), 243 Kuhn(1995),46, 188 Kwake/a/. (2000), 439-141

Intellution, 7 Isaksson and Graebe (1999), 59

Landau and Voda (1992), 330-331 Lee and Edgar (2002), 66, 87, 101, 140, 182, 231, 232, 276, 278, 303, 305, 333, 422,425,426 Lee and Shi (2002), 62, 206 Lee et al. (1998), 59, 178, 328, 329 Lee et al. (2000), 407, 413, 428, 431, 443, 444 Leeds and Northup, 7 Lennartson and Kristiansson (1997), 248 Leonard (1994), 260 Leva (1993), 73, 246 Leva (2001), 53, 58, 178, 192 Leva (2005), 220 Leva and Colombo (2000), 191 Leva and Colombo (2004), 220 Leva et al. (1994), 57, 322-323 Leva etal. (2003), 57, 118, 192, 324 Li etal. (1994), 177 Li et al. (2004), 385 Lime/a/. (1985), 61,332 Liptak (1970), 238 Liptak(2001), 30, 156 Liu et al. (2003), 286, 287-288, 374375,376-377 Lloyd (1994), 239 Lopez et al. (1969), 106,310 Loron(1997), 153 Luoetal. (1996), 237 Luyben (1996), 267 Luyben(1998), 132 Luyben (2000), 124, 181 Luyben and Luyben (1997), 237

Jahanmiri and Fallahi (1997), 336 Jhunjhunwala and Chidambaram (2001), 135,404,413 Jonese/a/. (1997), 73,239 Juang and Wang (1995), 169 Kamimura et al. (1994), 46, 151 Karaboga and Kalinli (1996), 237 Kasahara e? a/. (1997), 219 Kasahara et al. (1999), 190-191 Kasaharae/o/. (2001), 229 Kaya (2003), 277 Kaya and Atherton (1999), 277 Kaya and Scheib (1988), 196, 197, 198, 199,201,215-216,233 Keane et al. (2000), 320 Keviczky and Csaki (1973), 43, 111, 150, 151 Khan and Lehman (1996), 39-40, 45, 47 Khodabakhshian and Golbon (2004), 124 Kim et al. (2004), 219-220 Kinney (1983), 253 Klein etal. (1992), 29 Kookos etal. (1999), 79 Kraus (1986), 209 Kristiansson (2003), 57, 72, 119, 183184,241,244,330,334,337 Kristiansson and Lennartson (1998a), 248

Index MacLellan(1950), 72, 238 Maffezzoni and Rocco (1997), 190 Majhi and Atherton (2000), 130, 423424 Majhi and Mahanta (2001), 304 Majhi and Litz (2003), 353 Manned/. (2001), 44, 172 Mantz and Tacconi (1989), 252 Marchetti and Scali (2000), 329 Marlin (1995), 31,38,157, 162 Matausek and Kvascev (2003), 57 Matsuba et al. (1998), 60, 179 Maximum sensitivity, analytical calculation, 469^170 Maximum sensitivity, figures, 471 Maximum sensitivity, constant, 479480 McAnany(1993), 45 McAvoy and Johnson (1967), 105, 235, 343 McMillan (1984), 91, 146, 179, 281, 429 McMillan (1994), 29, 60, 70, 73, 236, 238 Mizutani and Hiroi (1991), 14, 15 Modicon, 9 Modcomp, 15 Mollenkamp et al. (1973), 42, 326 Moore, 13, 17 Morari and Zafiriou (1989), 190 Morilla et al. (2000), 171 Moras (1999), 27, 156 Murrill (1967), 28, 30, 33, 34, 155, 157, 158 NI Labview (2001), 74, 240 Normey-Rico et al. (2000), 186, 263 Normey-Rico et al. (2001), 265 O'Connor and Denn (1972), 161 O'Dwyer (2001a), 51, 80-81, 204, 280, 353, 387 O'Dwyer (2001b), 209-210, 254 Ogawa(1995), 59, 81 Ogawa and Katayama (2001), 230

539 Omron, 13 Oppelt(1951),27 Other tuning rules, 72-74, 82, 151, 237-240, 258 Oubrahim and Leonard (1998), 289, 340 Ou and Chen (1995), 199, 203 Ozawa et al. (2003), 229 Panda et al. (2004), 119, 329, 332, 334, 350 Paraskevopoulos et al. (2004), 137-139 Parked/. (1998), 421 Parr (1989), 70, 73, 155, 235, 238 Pecharroman (2000), 84, 92, 121, 122, 365-366 Pecharroman and Pagola (2000), 121, 270, 295, 365, 381-382 Pemberton (1972a), 30, 325 Pemberton (1972b), 325, 326 Peng and Wu (2000), 157 Penner(1988), 82 Performance index minimisation, 24, 30-42, 63, 65, 77-78, 84, 85, 89-90, 92, 94-100, 102-113, 120-121, 122, 128, 130, 137-138, 145, 149-150, 157-168, 187, 195-203, 213-214, 215-216, 218, 223, 224-225, 229, 233, 241, 243-244, 253, 256, 257, 259-260, 266-267, 270-271, 272, 273, 274-275, 279, 290-291, 295296, 297, 298-302, 310-321, 342349, 352, 355-362, 363-367, 368369, 381-382, 383-384, 398, 401402, 403^104, 414, 420, 423^124, 432,434-438 Regulator tuning, 30-37, 77, 85, 8990, 94-97, 102-107, 137, 145, 149-150, 157-161, 195-199, 215216, 224, 233, 253, 257, 259, 266267, 273, 274, 279, 290-291, 297, 298-299, 310-315, 342-346, 352, 355-358, 363-365, 368-369, 403, 414,420,432,434^35 Minimum error, 37, 149, 159

540

Handbook of PI and PID Controller Tuning Rules

Minimum IAE, 30-32, 77, 89, 102105, 149-150, 157, 195-197, 215, 224, 233, 253, 257, 266267, 273, 290, 297, 310, 342343, 355-358, 368-369, 414, 420, 434^435 Minimum IE, 37, 161, 363-365 Minimum ISE, 32-34, 77, 85, 94, 96, 105, 137, 149, 158, 197,215, 233, 259, 274, 279, 298, 299, 343-344, 352, 403 Minimum ISTES, 36, 159 Minimum ISTSE, 36, 158-159, 259, 403 Minimum ITAE, 34-35, 77, 90, 106, 145, 158, 198-199, 216, 233, 291, 310-313, 344-345, 352,432 Minimum ITSE, 35, 107, 199, 259, 345-346, 403 Minimum IAE or ISE, 37 Minimum performance index other, 85, 95, 96-97, 161, 274, 298-299 Modified minimum IAE, 157 Nearly minimum IAE and ITAE, 37, 160 Nearly minimum IAE, ISE and ITAE, 36, 107, 159-160, 314315 Servo tuning, 38^11, 85, 97-100, 108-113, 130, 137, 150, 162-166, 187, 199-203, 213-214, 216, 229, 233, 260, 274-275, 300-302, 315319, 346-349, 359-362, 383-384, 403^104, 420, 423^24, 436^38 Approximately minimum ITAE, 40 Minimum IAE, 38-39, 108-110, 150, 162, 187, 199-200, 216, 233, 315, 346, 359-362, 383, 420, 436^138 Minimum ISE, 39-^10, 85, 97, 99, 111, 137, 150, 162-163, 187, 201, 213, 216, 229, 233, 260, 274,300,301,347,384,403

Minimum ISTES, 40, 165, 214 Minimum ISTSE, 40, 164-165, 213-214,260,404 Minimum ITAE, 40, 111, 163-164, 187, 201-203, 216, 233, 315316,347-348,384 Minimum ITSE, 111, 130, 203, 260, 348-349, 403, 423^124 Minimum performance index other, 85, 98, 99-100, 274-275, 300-302 Modified minimum ITAE, 164 Nearly minimum IAE and ITAE, 41, 166 Nearly minimum IAE, ISE and ITAE, 112-113, 165-166, 316319 Small IAE, 39 Other tuning, 41^12, 63, 65, 77-78, 84, 92, 120-121, 122, 128, 138, 150, 167-168, 218, 223, 224-225, 241, 243-244, 256, 260, 270-271, 272, 295-296, 319-321, 365-367, 381-382,398,401-402,404 Peric et al. (1997), 60, 91, 179, 281 Pessen (1953), 254 Pessen (1954), 254 Pessen (1994), 70, 157,210 Petitt and Carr (1987), 235 Phase margin, analytical calculation, 463-469 Phase margin, figures, 471, 473^174 PI controller structures Controller with a double integral term, 6-7, 22, 88 Controller with proportional term acting on the output 1, 6, 22, 65, 85-86,94-100,137-139 Controller with proportional term acting on the output 2, 6, 22, 66, 87, 101, 140 Controller with set-point weighting, 6, 22, 63-64, 75, 84, 92-93, 120121, 122, 128-129, 135-136, 143144, 148

541 Ideal controller, 6, 22, 26-60, 67, 7074, 76-82, 89-91, 102-119, 123124, 126-127, 130-134, 141, 145146, 147, 149-151, 152, 153 Ideal controller in series with a first order lag, 6, 22, 61, 68, 69, 83, 125, 142 Ideal controller in series with a second order filter, 6, 22, 62 PID controller structures Alternative controller 1,18, 24, 307 Alternative controller 2, 18, 24, 308 Alternative controller 3, 18, 24 Alternative controller 4, 18, 24, 372373 Blending controller, 11, 23, 193 Classical controller 1, 11, 23, 194207, 253, 266-267, 290-291, 342350,387,394,414,432,451 Classical controller 2, 12, 23, 208, 268,292,351 Classical controller 4, 12, 23, 211, 255, 269, 294, 389, 395 Controller with filtered derivative, 89, 23, 187-192, 246-251, 264, 285, 338,386,393,400,457-459 Controller with filtered derivative and dynamics on the controlled variable, 10,23,265 Controller with filtered derivative in series with a second order filter, 9, 23, 339 Controller with filtered derivative with setpoint weighting 1, 9, 23, 286 Controller with filtered derivative with setpoint weighting 2, 9-10, 23,374-375 Controller with filtered derivative with setpoint weighting 3, 10, 23, 287-288 Controller with filtered derivative with setpoint weighting 4, 10, 23, 376-377

Ideal controller, 5, 7, 19, 23, 24, 154180, 235-240, 259-261, 279-281, 309-331, 383-384, 392, 398, 403407, 427-429, 442^143, 449, 452, 454, 455-456 Ideal controller in series with a first order filter, 8, 23, 336 Ideal controller in series with a first order lag, 8, 23, 181-182, 241-242, 262, 282-283, 332-335, 385, 399, 408^109, 430, 445-446, 450, 453 Ideal controller in series with a second order filter, 8, 23, 183-184, 243-244, 337 Ideal controller with first order filter and dynamics on the controlled variable, 8, 23 Ideal controller with first order filter and setpoint weighting 1, 8, 23, 186 Ideal controller with first order filter and setpoint weighting 2, 8, 23, 263 Ideal controller with setpoint weighting 1,10, 23, 252, 289, 340, 410-413,431,444 Ideal controller with setpoint weighting 2, 10, 23, 341 Ideal controller with weighted proportional term, 8, 23, 185, 245, 284 Industrial controller, 17-18, 24, 233234, 306, 380 Non-interacting controller 1, 12-13, 23,212,354-362,390,396 Non-interacting controller 2a, 13, 23, 213-214 Non-interacting controller 2b, 13, 23, 215-216 Non-interacting controller 3, 15, 23, 420, 434^138 Non-interacting controller 4, 15, 23, 224-225, 257, 273, 297, 368-369 Non-interacting controller 5, 16, 23, 226, 370

542

Handbook of PI and PID Controller Tuning Rules

Non-interacting controller 6 (I-PD controller), 16, 23, 227-229, 274275, 298-302, 371 Non-interacting controller 7, 16, 23, 230,460^161 Non-interacting controller 8, 16, 23, 276,303,421-422,439^141 Non-interacting controller 9, 16, 23, 258 Non-interacting controller 10, 16, 24, 277, 423^124 Non-interacting controller 11, 16-17, 24,231,304 Non-interacting controller 12, 17, 24, 232, 278, 305, 425^126 Non-interacting controller 13, 17 Non-interacting controller based on the two degree of freedom structure 1, 13, 23, 217-221, 256, 270-271, 295-296, 363-367, 379, 381-382, 391, 397, 401^02, 416—419, 447448 Non-interacting controller based on the two degree of freedom structure 2, 13-14, 23, 222 Non-interacting controller based on the two degree of freedom structure 3, 14, 23, 223, 272 PD-I-PD (only PD is DOF) incomplete 2DOF algorithm, 14 PI-PID (only PI is DOF) incomplete 2DOF algorithm, 14 Series controller (classical controller 3), 12, 19, 23, 209-210, 254, 293, 352-353,378,388,415,433 Super (or complete) 2DOF PID algorithm, 15 Pinnella e« a/. (1986), 44 Polonyi (1989), 398 Pomerleau and Poulin (2004), 118, 123, 125 Potocnik etal. (2001), 53-54 Poulin and Pomerleau (1996), 77, 90, 145,291,432 Poulin and Pomerleau (1997), 291, 432

Poulin and Pomerleau (1999), 78, 90 Poulin et al. (1996), 117, 146, 291, 350, 387, 394 Pramod and Chidambaram (2000), 406 Prashanti and Chidambaram (2000), 410,411,412-413 Process modelling Delay model, 21, 149-151, 449^51 Fifth order system plus delay model, 21,455-461 FOLIPD model, 20, 22, 23, 24, 89101,279-308,500-501 FOLPD model, 20, 22, 23, 24, 26-66, 154-234,493^98 FOLPD model with a negative zero, 20, 69,498 FOLPD model with a positive zero, 20, 67-68 General model with a repeated pole, 21, 152,452-453,505 General model with integrator, 22, 153 General stable non-oscillating model with a time delay, 22, 454 IPD model, 20, 22, 23, 24, 76-88, 259-278, 499-500 I 2 PD model, 20, 374-380 Non-model specific, 20, 22, 23, 24, 70-75, 235-258, 498-499 SOSIPD model, 20, 122, 381-382, 503 SOSPD model, 20, 22, 23, 24, 102121, 309-373, 501-503 SOSPD model with a negative zero, 21,125,392-397,504 SOSPD model with a positive zero, 21, 123-124,383-391,504 Third order system plus time delay model, 21, 126-129, 398^102 Unstable FOLPD model, 21, 130140, 403^126, 504-505 Unstable FOLPD model with a positive zero, 21, 141-144

Index Unstable SOSPD model (one unstable pole), 21, 145-146, 427441,505 Unstable SOSPD model with a positive zero, 21, 147-148, 445448 Unstable SOSPD model (two unstable poles), 21, 442^144, 505 Process reaction curve methods, 24, 26-30, 76-77, 89, 149, 154-156, 194-195, 209-210, 211, 217, 222, 224,259,266,309,449,451 Pulkkinene/a/. (1993), 59 Reswick(1956),28 Rice and Cooper (2002), 262, 267 Rivera etal. (1986), 58, 178 Rivera and Jun (2000), 61, 83, 280, 283, 329, 333 Robbins (2002), 71,237 Robust, 25, 58-59, 61, 62, 66, 69, 8 1 82, 83, 86, 87, 101, 118-119, 133134, 140, 151, 178, 181-182, 184, 190-192, 206, 211, 220, 231, 232, 234, 261, 262, 264, 267, 269, 276, 278, 280, 282-283, 285, 286, 287288, 291, 294, 303, 305, 328-330, 332-334, 336, 337, 338, 339, 374375, 376-377, 386, 387, 389, 393, 394, 395, 407, 413, 421-422, 425426,428,430,431,444,450 Rotach (1994), 238-239 Rotach(1995), 78, 261 Rotstein and Lewin (1991), 133-134, 428 Rovira et al. (1969), 38,40,162, 163 Rutherford (1950), 72,237 Sadeghi and Tych (2003), 162, 163, 164 Sain and Ozgen (1992), 156 Saitoh a/. (1990), 169 Sakaie/a/. (1989), 29 S ATT Instruments, 17

543 Schaedel (1997), 52, 118, 152, 181, 189,204,350,399,400,453 Schneider (1988), 45 Seki et al. (2000), 236 Setiawan et al. (2000), 65 Siemens, 9 Shen (1999), 363 Shen (2000), 364, 365 Shen (2002), 218, 256 Shi and Lee (2002), 206 Shi and Lee (2004), 339 Shigemasa et al. (1987), 227-228 S h i n e s / . (1997), 240 Shinskey (1988), 31, 32, 77, 150, 196, 197, 224, 266, 273, 310, 342, 414 Shinskey (1994), 32, 77, 89, 102, 150, 197, 225, 266, 273, 290, 297, 343, 369 Shinskey (1996), 31, 32, 77, 102, 105, 150, 197, 224, 267, 273, 343, 368, 369 Shinskey (2000), 29, 195 Shinskey (2001), 29 Sklaroff(1992),45, 349 Skoczowski and Tarasiejski (1996), 452 Skogestad (2001), 77, 78, 150 Skogestad (2003), 48, 78, 79, 150, 151, 293,353,378 Skogestad (2004a), 82 Skogestad (2004b), 79, 306, 327, 380 Smith (1998), 55 Smith (2002), 40, 58, 59, 81,82 Smith (2003), 164, 237, 330 Smith and Corripio (1997), 30, 38, 43, 197, 200, 203 Smith et al. (1975), 43, 326, 349 Somani et al. (1992), 50, 116-117, 174 Square D, 15 Sree and Chidambaram (2003a), 67, 68 Sree and Chidambaram (2003b), 142, 143, 144 Sree and Chidambaram (2003c), 141, 144

544

Handbook of PI and PID Controller Tuning Rules

Sree and Chidambaram (2004), 147, 148, 445^46, 447-448 Sree and Chidambaram (2005a), 419 Sree and Chidambaram (2005b), 261 Sree et al. (2004), 49, 133, 173, 406407 Srividya and Chidambaram (1997), 79 Srinivas and Chidambaram (2001), 64, 219,417 St. Clair (1997), 29, 195, 253 Streeter et al. (2003), 245 Sung era/. (1996), 313, 316 Suyama(1992),43, 169,326 Suyama (1993), 227-228 Syrcos and Kookos (2005), 161 Tachibana (1984), 279 Taguchi and Araki (2000), 63, 84, 92, 120-121, 122, 128, 218, 271, 296, 366-367, 382, 401-402 Tan etal. (1996), 58, 117, 177 Tan et al. (1998a), 282-283 Tan et al. (1998b), 83, 182, 282 Tan et al. (1999a), 209, 237, 254 Tan era/. (1999b), 407 Tan era/. (1999c), 134 Tan era/. (2001), 237, 254 Tang et al. (2002), 74, 240 Tavakoli and Fleming (2003), 39 Tavakoli and Tavakoli (2003), 187 Taylor, 17 Thomasson (1997), 58, 59, 81 Tinham (1989), 235 Toshiba, 11, 16 Tsang and Rad (1995), 203, 307 Tsang et al. (1993), 62, 203, 210, 307, 308 Turnbull, 12 Tyreus and Luyben (1992), 78 Ultimate cycle, 25, 60, 70-71, 88, 91, 124, 146, 179-180, 207, 210, 220221, 235-237, 252, 253, 254, 255, 257, 267, 281, 289, 330-331, 335, 340, 353, 373, 429

Valentine and Chidambaram (1997a), 47, 171 Valentine and Chidambaram (1997b), 405 Valentine and Chidambaram (1998), 131 VanDoren (1998), 224, 257 Van der Grinten (1963), 151, 168, 279, 325 Venkatashankar and Chidambaram (1994), 131 Visioli (2001), 259, 260, 403, 404 Viteckova (1999), 43, 79, 280, 326327 Viteckova and Vitecek (2002), 44, 173 Viteckova and Vitecek (2003), 44, 173 Viteckova et al. (2000a), 43, 79, 280, 326-327 Viteckova et al. (2000b), 43, 327 Voda and Landau (1995), 43, 51-52 Vrancic (1996), 71, 75, 127, 129, 247, 457 Vrancic and Lumbar (2004), 459, 460461 Vrancic era/. (1996), 71, 127 Vrancic et al. (1999), 247, 455^56, 458 Vrancic et al. (2004a), 127, 459, 460461 Vrancic et al. (2004b), 127 Wade (1994), 73, 238 Wang and Clements (1995), 326 Wang and Cai (2001), 296 Wang and Cai (2002), 296, 418 Wang and Cluett (1997), 78, 260 Wang and Cluett (2000), 54-55, 170171 Wang and Jin (2004), 427, 442 Wang and Shao (1999), 323 Wang and Shao (2000a), 56 Wang and Shao (2000b), 324 Wang era/. (1995a), 162, 164, 383, 384 Wang era/. (1995b), 71,242 Wang era/. (1999), 323

Index Wang et al. (2000a), 55 Wang et al. (2000b), 384, 392 Wang et al. (2001a), 384, 392 Wang et al. (2001b), 303 Wang et al. (2002), 204-205 Wills (1962a), 315, 325, 330 Wills (1962b), 310, 315 Wilton (1999), 41-42, 319-320 Witt and Waggoner (1990), 194-195, 196, 198,200,202 Wojsznis and Blevins (1995), 179 Wojsznis et al. (1999), 179-180, 237 Wolfe (1951), 28, 76, 149 Xu and Shao (2003a), 296 Xu and Shao (2003b), 419 Xu and Shao (2003c), 296 Xu et al (2004), 58 Yang and Clarke (1996), 321, 353 Yi and De Moor (1994), 246

545 Yokogawa, 8, 11, 13 Young (1955), 72 Yu (1988), 30-31, 33-34, 35 Yu (1999), 237 Zhang (1994), 46, 388, 394 Zhang and Xu (2002), 139, 430 Zhang era/. (1996), 206 Zhang etal. (1999), 81,282 Zhang et al. (2002), 182, 192, 206, 333 Zhong and Li (2002), 59, 81, 338 Zhuang and Atherton (1993), 33, 36, 39,40, 158, 159, 163, 164-165, 176177,213-214 Ziegler and Nichols (1942), 27, 70, 76, 154, 235 Zou and Brigham (1998), 261 Zou etal. (1997), 178,261

. .ic vast majority ol automatic controllers used to compensate industrial processes arc ol I'l or I'll) type. This hook comprehensively compiles, using a unified notation, tuning rules for these controllers proposed H a n d b o o k of

over the last sewn decades i 1935-2005). The tuning rides are carefully categorized and application

information about each rule is given. The book ''•"•"sses controller architecture and process modeling ;. as well as the performance and robustness of

Controller Timing Rules

loops compensated with l'l or I'll) controllers. This

2nd Edition

unique publication brings together in an easv-to-u.se format material previously published in a large number ol papers and hooks. This wholly revised second edition extends the presentation ol l'l and I'll) controller tuning rules,

lor single variable processes with time delays, to include additional rules compiled since the first edition was published in 2008. Key Features • Addresses the needs of a niche market where no comparable book is available A c o m p r e h e n s i v e compilation of PI and FID controller tuning rules Makes t h e t u n i n g rules easily accessible to researchers and practitioners through unified notation Highlights the marked increase in the number of tuning rules compiled, from 600 in the first edition to 1,134 in this second edition

P424 he ISBN 1 - 8 6 0 9 4 <>?'/ 1

Imperial College Press www.icpress.co.uk