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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
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 3.33T -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
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
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 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
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)
TTT( 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 —2L1 T 6 dB gain margin dominant delay process
Tm
.(71)
_
1.3 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
Tmm
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
-^ 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 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; t m
2 a + xm
a not specified
2a + i m Km(a + tm)2
2b
71
Tea
4b
Am = 71
1+-
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/Tmc
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 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)
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„
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
^2-0. A ,rt+-r^i.| A , 3 - 2 zK. x„vA^, ,A ,)' ~ ™ - 3 — ' " 2 rv 3
Km+-
K 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-"- , (TCL2S + 0 2
0.025 + 1.75-
A. obtained graphically - sample values below
X
T ra ^
'
Tm,i2. - T ml 2 + (Tm3xm - T ml 2 )(T0.25t 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 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>—— = 0.6K, 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 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,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 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
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)
0.10.5.
- + 0.64 •7.82j
T
,(358) _ ,
TT
n
0.29 < — T
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
• 0.2T,,, 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 _ 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
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°
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
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. + 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- 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"
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
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" 2 , $m > 60°
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
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)
•
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