Practical PID Handbook 9782759826094

Citation preview

Current Natural Sciences

Daniel LEQUESNE

Practical PID Handbook

Printed in France

EDP Sciences – ISBN(print): 978-2-7598-2608-7 – ISBN(ebook): 978-2-7598-2609-4 DOI: 10.1051/978-2-7598-2608-7 All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for private use and not intended for collective use” and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise, “any representation or reproduction – whether in full or in part – without the consent of the author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code. Ó Science Press, EDP Sciences, 2022

Preamble

Proportional Integral Derivative (PID) control is a technique that is well known and widely used in the industrial world. Many authors agree that more than 90% of control loops are of the PID type. It is an old technique that has continued to evolve over more than 70 years: from pneumatic controllers (nozzle-pallet system) through electronic controllers (operational amplifiers) to programmable digital controllers (microprocessors): programmable logic controller (PLC), distributed control systems (DCS). During this development, many adjustment methods were developed: the first dates back to 1935 (Callender, Hartree and Porter), and especially to 1942 (Ziegler and Nichols), and the very way of approaching adjustment has been structured over the years. One might therefore think that PID tuning is well mastered: it is not the case. It is estimated that only 20% of loops work well (Yu, 2006). In many cases the controllers operate manually, or the derivative action is not used or a manufacturer default setting is used. There are several reasons: there are difficulties due to the equipment (imperfection of sensors and actuators such as valves, for example). But we must also add a too weak dissemination of knowledge in the industrial environment linked in particular to the dispersion of the many articles, books, and publications on the subject, to their spread over decades, and to the gap that still remains between theory and practice. This book aims to put theory at the service of practice by providing a minimum base of knowledge to use PID control in the field, with similar approaches, notations, and criteria for comparison and evaluation of performance obtained, in order to have a better global view of the issue. In this book, the processes and controllers are described by transfer functions with Laplace notation. The book is divided into 2 parts and can be addressed to several audiences. DOI: 10.1051/978-2-7598-2608-7.c901 Ó Science Press, EDP Sciences, 2022

Preamble

IV

The first part concerns the main properties of a PID controller with its different aspects (continuous, digital) as well as the basic concepts for the use of PID control. The first chapter recalls the characteristics of the PID controller. These are the usual notions concerning the continuous PID, also called “analogue”, with the basic actions: gain, integral derivative. Its different structures (parallel, mixed, series, with actions on deviation or measurement) and also structure conversions are addressed. This is a must-see chapter. Chapter 2 concerns the digital PID resulting from the discretization of the continuous PID. It is a sampled controller that uses the z-transformation: the concepts of gain, integral, derivative disappear or at least are no longer available: the adjustment is obtained by calculating the coefficients of the sampled transfer function of the controller that can be written in two ways (PID and RST). You will find the correspondence tables to go from continuous to digital. This chapter is necessary for the implementation of controllers and the use of the digital adjustment methods (chapter 11). It can be omitted if only the classic setting is considered. Chapter 3 discusses the implementation of controllers in terms of pseudo-code realization algorithms, close to implementation, with the 2 forms, PID and RST. It is typically aimed at IT developers by offering simple algorithms as the bases of the computer development for PID controllers. The issues of limitations and integral saturation (anti windup) are also discussed. Chapter 4 may appear irrelevant as it concerns the process models used in the setting. This is important, however, because in spite of model-free methods, most of the settings methods refer, near or far, to a model for the process. Here, we discuss only simple, often graphic methods to identify the process to be controlled, without using identification software. This chapter is intended only for the practitioner wishing to identify his process in a simple way. It can be omitted in a first reading. Chapter 5 outlines the main elements defining a setting, as well as different criteria used to characterize them. It is essential, because it contains concepts, terms and symbols commonly used in the literature and throughout the rest of the book. Part 2 focuses on adjustment methods for manual tuning. They are categorized as follows: o o o o o o

Chapter Chapter Chapter Chapter Chapter Chapter

6: Ziegler–Nichols and Associated Methods 7: Cancellation Methods 8: Optimization Methods 9: Pole Placement Methods 10: Frequency Methods 11: Digital Settings Methods

All these methods are exposed from the principles to the final use in the industrial area. They are always illustrated with numerous examples to show their specificity and performances obtained. The last chapter deals with the practical reality of the setting with the consideration of constraints: limitations, non-linearities, sampling period, influence of the model and the parameters of identification of the process.

Preamble

V

Questions of choice (structures, methods) are also addressed. This is an important chapter that will allow any automation engineer, specialist or not, to better understand the performances that can be expected from PID control. Finally, the usual ingredients can be found in the appendix to save time in making use of a method: classic curves concerning the systems of the 1st and 2nd orders, conversion of models, conversion from continuous to digital models, and particular points. This book is above all a practical guide. There are indeed many examples and summary tables allowing immediate use of knowledge, with as less calculations as possible to facilitate the practice of PID control. The book aims to be useful to a wide spectrum of readers interested in PID control ranging from practising technicians and engineers in the industrial area to graduate and undergraduate students.

Contents Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III

CHAPTER 1 Main 1.1 1.2 1.3

Features . . . . . . . . . . . . . . . . . . . . . . Context . . . . . . . . . . . . . . . . . . . . . . Features . . . . . . . . . . . . . . . . . . . . . . PID Actions . . . . . . . . . . . . . . . . . . . 1.3.1 Proportional Action . . . . . . . . 1.3.2 Integral Action . . . . . . . . . . . . 1.3.3 Derivative Action . . . . . . . . . . Different Types of PID . . . . . . . . . . . Equivalence of PID . . . . . . . . . . . . . . 1.5.1 Mixed ↔ Parallel Conversion . 1.5.2 Mixed ↔ Series Conversions . . 1.5.3 Parallel ↔ Series Conversions 1.5.4 Summary Tables . . . . . . . . . . 1.5.5 Examples . . . . . . . . . . . . . . . . PID: Frequency Response . . . . . . . . . 1.6.1 Series PID . . . . . . . . . . . . . . . 1.6.2 Mixed PID . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

1 1 2 5 5 5 7 9 17 19 19 23 25 29 30 30 32

Digital PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Continuous to Digital Transposition . . . . . . 2.1.1 z-Transformation . . . . . . . . . . . . . . . 2.1.2 Backward or BW Approximation . . . 2.1.3 Forward or FW Approximation . . . . 2.1.4 z-Transformation and Approximation 2.2 Basic Actions . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Integral . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Derivative . . . . . . . . . . . . . . . . . . . . 2.2.3 Transposition Tables . . . . . . . . . . . . 2.3 Digital PID . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 PID Form . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

37 37 38 38 38 39 39 39 40 41 42 42

1.4 1.5

1.6

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

CHAPTER 2

Contents

VIII

2.3.2 2.3.3

RST Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correspondence Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 44

CHAPTER 3 Realization Algorithms . . . . . . . . . . 3.1 Sample Processing . . . . . . . . 3.2 Different Algorithms . . . . . . . 3.2.1 PID Forms . . . . . . . . . 3.2.2 RST Form . . . . . . . . . 3.3 Ancillary Features . . . . . . . . . 3.3.1 Manual Mode . . . . . . . 3.3.2 Automatic Mode . . . . 3.3.3 Limitations . . . . . . . . . 3.3.4 Direct/Reverse Choice 3.3.5 Inputs/Outputs . . . . . 3.4 Summary . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

55 56 58 58 63 68 68 68 69 75 76 77

Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Different Process Models . . . . . . . . . . . . . . . . . . . . . 4.1.1 Stable Systems . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Integrating Systems . . . . . . . . . . . . . . . . . . . 4.1.3 Usual Models . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Identification Methods . . . . . . . . . . . . . . . . . 4.2 Broïda Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Open Loop Identification . . . . . . . . . . . . . . . 4.2.2 Closed Loop Identification (Dindeleux, 1989) 4.3 2nd Order Model with Delay . . . . . . . . . . . . . . . . . . 4.3.1 Open Loop Identification . . . . . . . . . . . . . . . 4.3.2 Closed Loop Identification . . . . . . . . . . . . . . 4.4 Strejc Model (Strejc, 1960) . . . . . . . . . . . . . . . . . . . 4.4.1 Open Loop Identification . . . . . . . . . . . . . . . 4.4.2 Closed Loop Identification . . . . . . . . . . . . . . 4.5 Reverse Response Strejc Model . . . . . . . . . . . . . . . . 4.6 Integrating Model with Delay . . . . . . . . . . . . . . . . . 4.6.1 Open Loop Identification . . . . . . . . . . . . . . . 4.6.2 Closed Loop Identification . . . . . . . . . . . . . . 4.7 Integrator and 1st Order Model . . . . . . . . . . . . . . . 4.7.1 Open Loop Identification . . . . . . . . . . . . . . . 4.7.2 Closed Loop Identification . . . . . . . . . . . . . . 4.8 Integrating Model and Order n . . . . . . . . . . . . . . . . 4.8.1 Open Loop Identification . . . . . . . . . . . . . . . 4.8.2 Closed Loop Identification . . . . . . . . . . . . . . 4.9 Integrating Reverse Response Model . . . . . . . . . . . . 4.9.1 Open Loop Identification . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 81 84 85 86 88 88 89 90 90 91 93 93 97 100 103 103 104 104 104 107 110 110 113 115 115

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

CHAPTER 4

Contents

4.9.2 4.9.3

IX

Closed Loop Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Case of Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

CHAPTER 5 Evaluation of Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Process Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Adjustment Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Closed Loop Setting Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Step Response Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 1st or 2nd Order Systems . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Order n Systems: Naslin Criterion (Naslin, 1968, 1962) . 5.3.5 Optimization Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Frequency Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

123 123 123 124 125 126 128 130 131 131 134 135

CHAPTER 6 Ziegler–Nichols and Associated Methods . . . . . . . . . . . . . 6.1 Ziegler and Nichols Closed Loop Method . . . . . . . . 6.2 Ziegler and Nichols Open Loop Method . . . . . . . . 6.3 Cohen and Coon Method (Cohen and Coon, 1953) 6.4 Takahashi Method (Takahashi et al., 1971) . . . . . . 6.5 KT Method of Aström and Hägglund (1995) . . . . . 6.5.1 Ultimate Gain Method . . . . . . . . . . . . . . . . 6.5.2 Open Loop Method . . . . . . . . . . . . . . . . . . 6.5.3 Adjustment Curves . . . . . . . . . . . . . . . . . . . 6.6 Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Stable Process . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Integrating Process . . . . . . . . . . . . . . . . . . . 6.6.3 Process with Delay . . . . . . . . . . . . . . . . . . . 6.7 Overview Summary . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

137 137 139 142 143 144 144 145 147 148 148 153 155 157

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

159 159 159 160 161 161 170 173 173 180

CHAPTER 7 Cancellation Methods . . . . . . . . . . . . . . . . . 7.1 Dindeleux Method (Dindeleux, 1989) 7.1.1 Stable Process . . . . . . . . . . . . 7.1.2 Integrating Process . . . . . . . . . 7.1.3 Ultimate Gain Method . . . . . . 7.1.4 Performances . . . . . . . . . . . . . 7.2 Haalman Method (Haalman, 1965) . . 7.3 Cancellation for 2nd Order . . . . . . . . 7.3.1 Stable Process . . . . . . . . . . . . 7.3.2 Integrating Process . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Contents

X

7.4

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

188 188 189 190 195

Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Integral Criterion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Study of Rovira, Murrill and Smith (Rovira et al., 1967) 8.1.2 Study of Miller, Lopez and Smith (Miller et al., 1967) . . . 8.1.3 Study of Kaya and Scheib (Kaya and Scheib, 1988) . . . . 8.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Chien, Hrones and Reswick Method (Chien et al., 1952) . . . . . . 8.3 Samal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Choice of a Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

197 198 199 199 200 202 203 207 218

7.5

Direct Synthesis Method and IMC Method . . . 7.4.1 Direct Synthesis Method . . . . . . . . . . . 7.4.2 IMC Method (Internal Model Control) . 7.4.3 PID Controller . . . . . . . . . . . . . . . . . . . Synthesis Summary . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

CHAPTER 8

CHAPTER 9 Pole Placement Methods . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Naslin Method (Naslin, 1968, 1962) . . . . . . . . . . . 9.1.1 Original Method . . . . . . . . . . . . . . . . . . . . . 9.1.2 Industrial Method (Chaussard et al., 1967) . 9.1.3 Method for Integrating Process . . . . . . . . . 9.1.4 Variant for Integrating Process . . . . . . . . . . 9.1.5 Method for Integrating Strejc Model . . . . . 9.2 Classical Method (Corriou, 2018; Flaus, 1994) . . . . 9.2.1 Setting Criteria . . . . . . . . . . . . . . . . . . . . . 9.2.2 Performances . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Case of a Derivative on the Measurement . . 9.2.4 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Dominant Poles Method . . . . . . . . . . . . . . . . . . . . 9.3.1 Principle (Aström and Hägglund, 1995) . . . 9.3.2 Application . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Choice of a Method . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

219 219 219 222 240 244 249 252 254 256 259 261 261 261 262 263 265

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

267 268 272 274 276 280 281

CHAPTER 10 Frequency Methods . . . . . . . . . . . . . . . . . . . . 10.1 1st Method: Phase Margin . . . . . . . . . 10.2 2nd Method: Resonance Factor . . . . . . 10.2.1 Examples of Application . . . . . 10.2.2 Case of Integrating Systems . . 10.3 Kessler’s Method (Kessler, 1958, 1955) 10.3.1 2nd Order Process . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

Contents

10.4

10.5

XI

10.3.2 3rd Order Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KLV Method (Kessler, Landau and Voda) (Voda and Landau, 1995) 10.4.1 Basic Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Auto-Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 KLV 1P Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 KLV 2P Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use and Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 287 287 292 292 294 296 298 300 302

CHAPTER 11 Digital 11.1 11.2 11.3

11.4

11.5

11.6

Settings Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice of the Sampling Period . . . . . . . . . . . . . . . . . . . . . . Zero Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cancellation Method (Buhler, 1986) . . . . . . . . . . . . . . . . . . 11.3.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Case of Uncompensated Zeros . . . . . . . . . . . . . . . . 11.3.4 Choice of HD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.5 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.7 Examples of Responses . . . . . . . . . . . . . . . . . . . . . . Pole Placement Method (Borne et al., 1993; Landau, 1988) . 11.4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Zero Cancellation of B . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Pole Placement with RST . . . . . . . . . . . . . . . . . . . . 11.4.4 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.5 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.6 Summary Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracking and Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.5 Summary Tables . . . . . . . . . . . . . . . . . . . . . . . . . . Choice of a Dynamic d1, d2 . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Choice of Tracking Dynamics . . . . . . . . . . . . . . . . . 11.6.2 Choice of a Regulation Dynamics . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

303 303 304 305 305 306 306 306 307 309 309 313 313 314 314 315 315 316 316 323 323 326 326 327 331 331 335 338

CHAPTER 12 Adjustments and Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 12.1 Sampling Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 12.1.1 PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Contents

XII

12.2 12.3

12.4 12.5

12.6

12.1.2 PID Controller . . . . . . . . . . . . . . . Filter Factor N (Visioli, 2006) . . . . . . . . . . Model Compliance . . . . . . . . . . . . . . . . . . 12.3.1 Broïda Model . . . . . . . . . . . . . . . . 12.3.2 Strejc Model . . . . . . . . . . . . . . . . . 12.3.3 Static Gain . . . . . . . . . . . . . . . . . . Control Constraints . . . . . . . . . . . . . . . . . . Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Choice of Controller Type . . . . . . 12.5.2 Choice of an Adjustment Method . Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

351 356 357 357 358 359 360 364 364 365 371

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

373

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

401

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

405

Chapter 1 Main Features 1.1

Context

We place ourselves in the context of a single variable process (SISO: single input single output, figure 1.1): – a command input Uc, often called manipulated variable (MV); – a measurement output M, often called process variable (PV). There may be a disturbance input X. The process is stable or unstable, and characterized by static and dynamic performances (static gain, response time, dead time, etc.).

FIG. 1.1 – Open loop. The addition of a controller (figure 1.2) aims to improve overall performance (stabilization, linearity,…) and more particularly: – obtain a measure which follows the setpoint (Sp) “at best”: this is the behaviour in tracking. Set point is often called Setpoint Variable (SV); – reduce the influence of the disturbance X: this is the behaviour in regulation.

FIG. 1.2 – Closed loop. DOI: 10.1051/978-2-7598-2608-7.c001 © Science Press, EDP Sciences, 2022

Practical PID Handbook

2

1.2

Features

Remember that a PID controller is a controller whose automatic processing is carried out starting from the 3 basic actions: proportional, integral, and derivative. Figure 1.3 summarizes the minimum features of a PID controller whose main properties can be recalled.

FIG. 1.3 – Basic synoptic diagram of a PID controller. Automatic/Manual In manual, it is the operator who controls “on sight” the output (or command), relying on the displays (not shown in figure 1.3) of the measurement and the command. You can also have a command that follows an external signal (tracking) which is not to be confused with a behaviour in tracking. In automatic the command output (MV) is calculated by the PID from the measurement and the setpoint which can be internal or external, analog or digital. Switching to manual generally consists of blocking the output at its last value, the operator being able to regain control by manual control. Switching to automatic can be done in several ways: – activating PID actions from the setpoint (internal or external) without special precautions: this amounts to applying a step (Sp − M) at the input of the PID, Sp and M being the values of the setpoint and the measurement at the time of switching. The consequences can be unfortunate if the difference (Sp – M) is too large or if the actions are poorly adjusted; – activating PID actions from the setpoint with smooth passage: the PID is initialized to make a smooth transition; – activating PID actions from the internal setpoint which is made equal to the measurement when switching to automatic: deviation (Sp − M) is always zero; nothing happens and the loop is checked.

Main Features

3

Direct/Reverse Actions An action is said to be direct if an increase in the measurement must cause an increase in the command (MV). It is said to be reverse if it should cause a decrease in the command. This direct/reverse choice is fundamental and must be made from the start with the setting of the controller; an error here leads to an unstable system getting carried away: the command goes to an extreme value with all the risks that this entails. It is the knowledge of the process that makes this choice. Output Offset When the PID only uses a proportional action, there is always a difference between measurement and setpoint: it can be reduced by this manual auxiliary control, also called bias or out bias input or band centering. This action directly on the automatic control is called feed-forward input when it can be controlled by an external input, in order to correct the effects of a disturbance; the controller can then include static and dynamic processing on this input. Limitations The output limitations may modify the output calculated by the PID: – High and low limits of the control: definition of the high and low stops (in particular avoids completely closing a valve, for example). – Limits of variations of the control (gradient): to avoid variations that are too brutal for the actuator. – Dead band (or hysteresis): the command is kept constant as long as the PID fluctuates in a band (in % of full scale). Type of Inputs Most controllers support all standard inputs: – – – –

Standard thermocouples. Resistance thermometers (PT 100: 100 Ω at 0°). Standard electrical inputs 4–20 mA, 10–50 mA, 0/10 V, 1/5 V etc. Pneumatic inputs: 0.2–1 bar, 3–15 PSI for pneumatic controllers. The configuration of the inputs is done during the initial configuration.

Type of Outputs The main usual outputs are as follows: – Analog output: the output in % of full scale is adapted to the standard of the actuator: electric or pneumatic control with the range of usual scales. It is the most frequent case.

Practical PID Handbook

4

– On/off output: this is an all or nothing modulated output: the percentage S% of the full scale is converted into a duty cycle (proportion of a modulation time or TC cycle time which must be chosen beforehand): S% ¼ ðT1 =TC Þ% This results in the closing of a contact or the presence of a control voltage, during a fraction of the cycle (figure 1.4a). The interest is to generate a progressive command with an on/off actuator; this is the case for electric heating controls in particular (typical example of convectors where the cycle time is around ten to twenty seconds). – PWM (pulse wave modulation ): it is the same principle, but this designation is reserved for systems whose cycle time is much less than the process response time: this is typically the case of position controls for small electric motors powered by a rectangular AC voltage and a frequency of a few kilohertz (figure 1.4b); the motor only sees the average value, which in particular makes it possible to control a zero speed (symmetrical signals). – Split range: the output is distributed over 2 channels that control 2 complementary actuators: typically, a hot water valve and a cold water valve to make lukewarm water.

t

(a)

(b)

FIG. 1.4 – (a) On/off output. (b) Pulse wave modulation output.

In addition to these basic functionalities, controllers often have ancillary functions not shown in figure 1.3 in particular: Processing on the setpoint, such as ramp generators, trajectories, etc. Specific treatments on the measurement (for a corrected flow for example). Filtering (measurement, output). Alarms. Auto adjustment. Communication functions (RS 485, Modbus, Ethernet communication, etc.).

Main Features

1.3

5

PID Actions

PID controller implements 3 basic actions:

1.3.1

Proportional Action

It is expressed by the gain G or the proportional band Bp (in %):

S = GE =

100 E Bp

E

Symbolization:

P

S

The proportional band gives the variation required at the input (in %) to obtain a variation ΔS of 100%. The proportional action is instantaneous, but often insufficient to cancel difference E between setpoint and measurement. For too large gains, we can oscillate the loop.

1.3.2 S¼

1 Ti

Integral Action Z

t

EðtÞ dt

E

Symbolization:

I

S

0

1 E(s). Ti s The integral action is characterized by Ti (in s or min) or Ki = 1/Ti (in repetitions/s or min). Ti, integration time, can be defined on a step response, by the time after which the variation of the output ΔS is equal to the applied step E0 (figure 1.5). In Laplace transformation: S(s) =

S S0+E0 ΔS = E0 S0 0

t Ti

FIG. 1.5 – Step integration. The integral action is all the more rapid or strong as Ti is small. Its role is to cancel the difference between measurement and setpoint in steady state; this is why it is always applied to deviation E. It is a slow action; too fast, it has a destabilizing effect.

Practical PID Handbook

6

Note that the removal of the integral must result in Ti = ∞. In practice we choose a very high value; but sometimes in some software, it is necessary to configure Ti = 0 to remove the integral. The need for an integral action can be examined from figure 1.6 which gives the static aspect of the loop: K represents the static gain of the process.

FIG. 1.6 – Static proportional loop.

Proportional control alone leads to the following transfer relationship: M¼

KG K Sp þ X 1 þ KG 1 þ KG

In tracking, the static gain of the loop is less than 1 and the permanent relative difference is: M  Sp 1 ¼ Sp 1 þ KG The difference is reduced by increasing the gain of the controller, but the difference remains: an integral action will be required in the controller. If the process is integrative (K infinite), deviation M − Sp is canceled: it is therefore useless to add an integral action in the controller: it is the process which plays this role. In case of disturbance X (regulation), the static variation of the measurement is: DM ¼

K DX 1 þ KG

Even if the process is integrative, it remains a difference: DM ¼

DX G

In this case, it will be necessary to use the integral action of the controller. Consequently, the integral action is almost always used, precisely because of the disturbances: it can only be deleted if the process is integrative and without disturbances.

Main Features

1.3.3

7

Derivative Action

S ¼ Td

d EðtÞ dt

Symbolization:

E

D

S

In Laplace transformation: S(s) = Td s E(s). The derivative action is characterized by time Td (in s): the output is proportional to the slope of input E (figure 1.7a where the input is a ramp). Td represents the time after which the variation of the input is equal to the variation of the output. The derivative action has an anticipation effect which can be observed from the function (1 + Derivative) to which a ramp is applied (figure 1.7b): the output copies the input with a jump to the origin: “everything happens as if” the exit was anticipated from Td.

E

S

D

S

E0

E ΔE(Td)

S = ΔE(Td) 0

t

Td (a)

(b)

FIG. 1.7 – (a) Derivative of a ramp input E. (b) 1 + Derivative of a ramp input E.

The purpose of this action is to stabilize the loop. It has no effect on the static regime. Its disadvantage is the amplification of variations of E (noise). In practice, the derivative is filtered by a low-pass filter: S ðs Þ ¼

Td s E ðs Þ 1 þ TNd s

N: gain of the derivative action, or filtering factor, visible on the step response (step E0) in figure 1.8a. In practice N = 1 to 10, sometimes programmable.

Practical PID Handbook

8

(a)

(b)

FIG. 1.8 – (a) Filtered derivative of a step input E0. (b) 1 + Filtered derivative of a ramp

input E.

Figure 1.8b shows the response to a ramp of the function (1 + D 0 ), where D0 represents a filtered derivative. We find the same anticipation time Td. Filtering smooths out sudden variations and limits amplification of noise. The function 1 + D0 is also called phase advance and is then written: 1 þ Td N Nþ 1 s S ðs Þ ¼ E 1 þ TNd s Another way to see the role of the derivative action is illustrated in figure 1.9: the measurement comprises a delay symbolized by a simple time constant τ linked to the sensor, and the derivative is on the measurement. Derivative

FIG. 1.9 – Derivative action on measurement. By adjusting T d N Nþ 1 ¼ s, the numerator of the phase advance compensates for the delay and the set is equivalent to a corrected measurement whose time constant s0 is (N + 1) times smaller: the measurement seen by the controller is less delayed, which has the effect of increasing the stability of the loop. The effect is the same for stability when the derivative is on deviation (Sp − M), but adds an overshoot on a setpoint step.

Main Features

1.4

9

Different Types of PID

We distinguish the parallel, mixed and series forms. The parallel form is also called non interacting form: the 3 actions are independent. The series form is a full interacting form. The mixed form is sometimes called standard form: integral and derivative actions are independent but multiplied by gain G. Usually derivative is on the deviation or on the measure. The derivative on the deviation has the drawback of amplifying a setpoint change, which can result in a significant overshoot. The gain has the same disadvantage; this is why we can find controllers with gain (or derivative and gain) on the measurement. The different possible structures are summarized in table 1.1 with the previous conventions for actions P, I and D. These are obviously so-called “basic” structures, because there are many ways to combine the 3 actions (O’Dwyer, 2006). The synoptic diagrams and the algorithms are given for a controller with reverse action (deviation E = Sp − M). If the derivative is filtered, D = Td s is to be replaced in the Laplace notation by: D0 ¼

Td s 1 þ TNd s

Table 1.1 gives the different algorithms in 2 forms: Absolute form: U = FS(s) Sp − FA(s)M Differential form: U = FE(s) E − FD(s)M With : U ¼ command Sp ¼ setpoint M ¼ measure E ¼ deviation ðSp  M Þ The calculation of FS, FA, FE, FD shows that these functions are always of the form: F ð s Þ ¼ Ax þ

Bx Cx s Dx s Td þ with s ¼ þ 1 þ ss 1 þ ss s N

The term Dx only exists for the series controller and comes from the product G I(s) D0 (s). In the case of a pure derivative: τ = 0 and GI ðs ÞDðs Þ ¼ G TTdi . This constant term is to be grouped in Ax: F ðs Þ ¼ Ax þ Bsx þ C x s. The coefficients Bx and Cx are therefore identical for pure or filtered derivative. The coefficient Ax is also the same except in the case of a series controller. All the results are given in the 2 tables: – Table 1.2: differential form. – Table 1.3: absolute form. The differential form is generally simpler (more zero coefficients) but requires calculating the difference E = Sp − M.

Practical PID Handbook

10

TAB. 1.1 – Different forms of PID.

Main Features

11

TAB. 1.2 – PID algorithm differential form. PID algorithm



U ¼ AE þ

BE s

CEs 1 þ ss

þ

þ

DE 1 þ ss



Differential Form



E  AM þ

Pure derivative: τ = 0 Type

Parallel

BM s

þ

CMs 1 þ ss

þ

DM 1 þ ss

M

E

A

G

0

G

B

1 Ti

0

G Ti

C D

Td 0

0 0

M E Actions on deviation E   0 G 1 þ TTdi 0

GTd 0

Td N

Series Pure D

E

M

Filtered derivative: τ =

Mixed

Index



0 0

G Ti

GTd 0

Filtered D M

E

M

0

G

0

0

G Ti

0

0 0

GTd G TTdi

0 0

G TTdi 0

G

0 0

Derivative on measurement A B C D

G 1 Ti

0 0

0 0 Td 0

G G Ti

0 0

0 0 GTd 0

G G Ti

0 0

Gain on measurement A

0

B

1 Ti

C D

Td 0

G

0

0

G Ti

0 0

A

0

G

B

1 Ti

0

C D

0 0

Td 0

GTd 0

G

0

0

G Ti

0 0

GTd 0

GTd 0  G 1þ

0 0

0 GTd 0

G Ti

0 0

Td Ti



0 0 0

Derivative and gain on measurement  0 G 0 G 1þ G Ti

G Ti

0 GTd 0

Td Ti



0 0

GTd G TTdi

0

G

G Ti

0

GTd 0

0 G TTdi

0

G

G Ti

0

0 0

GTd G TTdi

Practical PID Handbook

12

TAB. 1.3 – PID algorithm absolute form. 

PID algorithm

U ¼ AS þ

BS s

þ

CSs 1 þ ss

þ

DS 1 þ ss





Sp  AM þ

Pure derivative: τ = 0 Type

Parallel

Absolute form BM s

þ

CMs 1 þ ss

þ

DM 1 þ ss

S

Mixed M

S

A

G

G

G

B

1 Ti

1 Ti

G Ti

C D

Td 0

Td 0

GTd 0

M S Actions on deviation E   G G 1 þ TTdi G Ti

GTd 0

G Ti

GTd 0

A

G

G

G

G

G

B

1 Ti

1 Ti

G Ti

G Ti

G Ti

0 0

Td 0

0 0

GTd 0

0 0

Gain on measurement A

0

G

0

G

0

B

1 Ti

1 Ti

G Ti

G Ti

G Ti

C D

Td 0

B

1 Ti

1 Ti

C D

0 0

 G 1þ

Td 0

G Ti

0 0

G Ti

GTd 0

G Ti

0 0

S Td Ti



G Ti

GTd 0  G 1þ

Td Ti



G Ti

GTd 0  G 1þ

Td Ti



G Ti

Derivative and gain on measurement  0 G 0 G 1þ

G

GTd 0

M

GTd 0

0

GTd 0

Filtered D

Td 0

A

Td N

Series

Derivative on measurement

C D

M

Filtered derivative: τ = Pure D

Index



GTd 0

G Ti

GTd 0

Td Ti



M

G

G

G Ti

G Ti

GTd G TTdi

GTd G TTdi

G

G

G Ti

G Ti

0 0

GTd G TTdi

0

G

G Ti

G Ti

GTd 0

GTd G TTdi

0

G

G Ti

G Ti

0 0

GTd G TTdi

Choice of a PID The drawbacks linked to the gain or the derivative on the deviation can be highlighted by the presence of zeros in the transfer function in closed loop and in tracking (setpoint input), while they do not exist in regulation (disturbance).

Main Features

13

FIG. 1.10 – Closed loop. The summary table 1.4a gives the different closed-loop transfer functions corresponding to the diagram in figure 1.10, where the process is represented by a transfer function in the form of a rational fraction: F ðs Þ ¼

KBðsÞ AðsÞ

The most interesting structure is that with gain and derivative on the measurement since it introduces no zero. We know that these zeros (terms additional to the numerator) are often the cause of an overshoot in tracking because the numerator distorts the behavior due to the denominator. The derivative on measurement gives a zero, the other structures give two. In all cases the process zeros (B) are retained. In the case of a filtered derivative, D becomes D0 : D0 ¼

Td s 1 þ TNd s

Table 1.4b is obtained. Filtering shows an additional zero 1 + Td/Ns: – in regulation: in all cases; – in tracking: in the case of derivative or (derivative and gain) on measurement. The structure with derivative and gain on the measurement therefore remains the best for obtaining a compromise setting between tracking and regulation. To remedy the overshoot linked to these zeros, some authors (Aström and Hägglund, 1995) propose to keep actions on the deviation but to weight the setpoint by a coefficient b < 1, for the proportional term. In the example of a mixed PID, the algorithm would be modified as follows: U = P(bSp − M) + I(Sp − M) + D(Sp − M) which can also be written:   U ¼ P ð1 þ I þ DÞ Sp  M  P ð1  bÞSp The standard command is decreased by a constant to limit the response. The transfer function in tracking is written (pure derivative N∞): M ð1 þ bTi s þ Ti Td s 2 ÞB ðs Þ ðs Þ ¼ Sp ð1 þ Ti s þ Ti Td s 2 ÞB ðs Þ þ s Aðs Þ b < 1 decreases the term 1 + Ti s of the numerator which is often a source of overshoot. However, there remains the influence of the term TiTd s2; this term disappears if the derivative is on the measurement.

14

Practical PID Handbook

TAB. 1.4a – Pure derivative closed loop transfer functions.

Main Features

TAB. 1.4b – Filtered derivative closed loop transfer functions.

15

Practical PID Handbook

16

This solution makes it possible to reduce overshoot in tracking, but requires a controller with this weighting, which is not always the case. As for the choice between parallel, mixed or series, we can summarize as follows: The parallel controller is little used in industry, but appears more in computer science or in certain academic works. Its independent actions can be an advantage in certain analytical calculations, but the adjustment is less obvious. It is all the less used as few adjustment methods refer to it. The mixed controller is the most widespread. The serial controller is better suited to frequency analysis and better accounts for the role of each action. It sometimes allows specific simpler settings, but it is less universal than the other 2 (see equivalence of PID). The overshoot in tracking, during a setpoint step is due to the 1st command of the controller which depends only on the structure of the PID. This first command ΔU0 can be easily calculated by doing s → ∞ in the transfer function of the controller, knowing that the measurement is constant before this command. Table 1.5 is obtained which gives the ratio ΔU0/ΔSp for the 1st command as a function of G and N. The case of pure derivative is obtained by doing N → ∞. TAB. 1.5 – First command variation during a unit setpoint step. Controller PID/E D/M P/M P&D/M

Parallel G+N G N 0

Mixed G(1 + N) G GN 0

Series G(1+N) G GN 0

In many cases, the maximum command Umax of the setpoint step response is equal to or close to U0: this table can be useful in choosing a structure, in particular to limit the problems of saturation of the command. It will be noted that the derivative on the measurement can reduce this command independently of N. The gain and the derivative on the measurement gives a 1st zero command, which means that the command starts from initial U0 (no jump): in this case nothing can be said about Umax. This table therefore shows an additional advantage of actions on measurement. Special Case of the Series Controller The series controller has the factor 1 + D which is 1 + Td s. When the derivative is filtered, we sometimes find factors of the type: 1 þ Td s 1 þ Td s or 1 þ Ta s 1 þ Tnd s We can always reduce to the factor 1+

Tds s T 1 þ Ndss s

of the standard series controller.

The transition formulas are given in table 1.6.

Main Features

17

TAB. 1.6 – Passage formulas for non-standard series controllers. 1 + filtered derivative 1 þ Td s 1 þ Ta s 1 þ Td s 1 þ Tnd s

Not standard T d ¼ N Ns þs 1 T ds T a ¼ TNdss T d ¼ N Ns þs 1 T ds

Standard controller T ds ¼ T d  T a

n ¼ Ns þ 1

N s ¼ TT da  1 T ds ¼ n1 n Td Ns ¼ n  1

Note The different PID structures have been represented for a reverse action. For a direct action, just add a specific processing function at the output of the controller (see the carrying out aspect in chapter 3). We have limited ourselves to the study of PID; in operation these algorithms can be largely modified by the non-linearities of the additional functionalities: – – – – –

anti saturation of the integral; upper and lower limits of the output; offset of the output; manual/automatic switching; sampling period (for a discrete PID).

1.5

Equivalence of PID

This involves determining the parameters P, I, D of a controller from those of another type. In practice, the derivative is often filtered and introduces a 4th parameter (the filtering factor N): D0 ¼

D 1 þ D=N

Recall the writing conventions: P ≡ G I ≡ 1/(Ti s) D ≡ Td s. We use the indices p, m, s for parallel, mixed, series. The method consists in writing the algorithms of the different types of controllers in the absolute form which separates setpoint and measurement: U ¼ FS ðs ÞSp  FA ðs ÞM We can then develop functions FS and FA and order: we obtain 4 terms: proportional, integral, derivative, and a filtering term (D/N). Table 1.3 shows that function FA is the same, whatever the actions (D/E, D/M, P/M, P&D/M) for a given type. FS functions are differentiated according to actions. We thus obtain table 1.7. The transition from one type to another is done by identification term by term. The identification for FA functions leads to 3 systems of equations which are written:

Practical PID Handbook

18

8     Ip Dp Im Dm Is Ds > > P þ ¼ P 1 þ 1 þ þ I D ¼ P p m s s s > Np Nm Ns > > < I p ¼ Pm I m ¼ P s I s       P > Dp 1 þ Npp ¼ Pm Dm 1 þ N1m ¼ Ps Ds 1 þ N1s > > > > : Dp ¼ Dm ¼ Ds Np Nm Ns This system is common to the 4 types of action. The identification for FS functions gives, according to the 4 types of action, additional relationships which can be simplified and reduced to the following conditions: D=E : None D=M : Pp ¼ Pm ¼ Ps P=M : Dp ¼ Pm Dm ¼ Ps Ds P&D=M : None These relationships show that for the D/E and P&D/M actions, no additional constraint affects the setpoint; which means that the equivalence relationships of controllers, when verified, will give equivalent controllers in both tracking and regulation. For D/M and P/M actions, the above constraints result in a specific processing of the setpoint (proportional action for D/M, and derivative action for P/M), which is not necessarily compatible with the corresponding actions obtained by identification of FA functions: a complete study shows that this incompatibility exists only for equivalence with the series controller, in which case the equivalence relations cannot remain valid, without additional restrictions, in tracking and in regulation. However, these additional constraints on the setpoint do not occur with established setpoint: the equivalence relationships remain valid in regulation only. TAB. 1.7 – Different terms of the algorithm U = FS(s) Sp − FA(s) M.

Parallel

Terms

P

FA and FS (D/E) FS (D/M) FS (P/M)

P p þ Np p p Pp

FS (P&D/M) FA and FS (D/E) Mixed

FS (D/M) FS (P/M) FS (P&D/M)

Series

FA and FS (D/E) FS (D/M) FS (P/M) FS (P&D/M)

I I D

I p Dp Np



–

P m 1 þ I mNDmm Pm



P m I m Dm Nm

– h  P s 1 þ I s Ds 1 þ Ps P s I s Ds Ns

–

1 Ns

i



D Pp Np

Ip Ip Ip

Dp 1 þ – Dp

Ip Pm Im

–  P m Dm 1 þ

Pm Im Pm Im

– P m Dm

Pm Im

–  P s Ds 1 þ – PsDs

PsIs PsIs PsIs PsIs

–

D/N



Dp Np

–

Dp Np

1 Nm



– Dm Nm

– Dm Nm

1 Ns



–

Ds Ns

– Ds Ns

–

Main Features

19

The following gives the results of the 3 conversions, the method consisting in solving the system common to the 4 actions to which is added the system specific to the D/M and P/M actions.

1.5.1

Mixed ↔ Parallel Conversion

The equivalence systems are written for the parallel ↔ mixed conversion: 9 8   I D > > > > Pp þ pNpp ¼ Pm 1 þ ImNDmm > > > > > >  =  < I p ¼ Pm I m     þ D=M : Pp ¼ Pm P P=M : Dp ¼ Pm Dm > D 1 þ Npp ¼ Pm Dm 1 þ N1m > > > > > p > > > > D D p ; : ¼ m Np

Nm

The resolution shows that all equations are compatible for the 4 actions and leads to equivalence relations: 8 Pp ¼ Pm > > < I p ¼ Pm I m D ¼ Pm D m > > : p N p ¼ Pm N m The equivalence is therefore valid for the 4 actions both in tracking and in regulation. The relationships are simple and allow passage in both directions: we obtain conversion formulas translated in terms of G, Ti, Td, and N, in table 1.8. They remain valid for a pure derivative by making N tend towards infinity. Note that if a pure derivative is approximated by Np = 10, for a parallel controller, the transition to the mixed controller results in Nm = Np/Gp generally less than 10 (G > 1). TAB. 1.8 – Parallel ↔ mixed conversion. Parallel → Mixed Mixed → Parallel Gp ¼ Gm Gm ¼ Gp Tim ¼ Gp Tip Tip ¼ Tim =Gm Tdm ¼ Tdp =Gp Tdp ¼ Gm Tdm Nm ¼ Np =Gp (filtered derivative) Np ¼ Gm Nm

1.5.2

Mixed ↔ Series Conversions

Equivalence systems are written for mixed ↔ series conversions:   9 8  > Pm 1 þ ImNDmm ¼ Ps 1 þ IsNDs s þ Is Ds > > > > > > >  = 

> Pm Dm 1 þ N1m ¼ Ps Ds 1 þ N1s > > > > > > ; : Dm Ds Nm ¼ Ns

Practical PID Handbook

20

From the system of 4 equations, common to the 4 actions, we derive the equivalence formulas: 8 < Pm ¼ P s ð 1 þ I s D s Þ Im ¼ Is = ð 1 þ Is D s Þ : Dm ¼ Nm ¼ Ns Is Ds Ds Ns Ns ð1 þ Is Ds Þ Compatibility with the additional relation for the D/M and P/M actions brings restrictions which will be examined for each of the conversions (series → mixed and mixed → series). In the case of the pure derivative (N → ∞), the additional relationship is the same for the P/M controller (PmDm = PsDs): the mixed ↔ serial equivalence is then ensured both in tracking and in regulation. Only the derivative on measurement brings restrictions in tracking.

1.5.2.1

Series → Mixed Conversion

Common equivalence formulas lend themselves naturally to this conversion. The conservation of the sign of the derivative imposes the condition IsDs < Ns or Tds/Tis < Ns. The calculation of G, Ti, Td, N gives the conversion formulas: 8 Gm ¼ Gs ð1 þ ts Þ with ts ¼ Is Ds ¼ Tds =Tis > > < Tim ¼ Tis ð1 þ ts Þ ¼ Tis þ Tds s Tdm ¼ Tds NsNð1s t > þ ts Þ > : Ns ts Nm ¼ 1 þ t s As an indication, figure 1.11 illustrates the role of the factor ts in the conversion of the derivative, for several values of Ns as a parameter. D/M Case The condition Pm = Ps gives a gain Gm = Gs. Compatibility with the previous results (Gm = Gs (1 + ts)) is only possible if ts = 0, i.e., for a PI (or PD) controller, in which case the actions are identical. An approximated equivalence in tracking is obtained for ts T N 1 ds s > : ¼  1 þ tm 1 þ Nm Tdm Nm To avoid calculations, the mixed → series conversion formulas are translated as curves in figure 1.13; in particular, they show the limits of equivalence.

1.5.3

Parallel ↔ Series Conversions

The equivalence is obtained using the 2 conversions (parallel ↔ mixed) and (mixed ↔ series) in cascade.

1.5.3.1

Series → Parallel Conversion

The equivalence condition remains T ds =T dm \N s . The formulas are simple: see summary table 1.10. It is also possible to use the curves of figure 1.11 to visualize the conversion of the derivative, by multiplying the reading of the ordinate y by Gp, calculated beforehand by the formula Gp = Gs (1 + ts).

1.5.3.2

Parallel → Series Conversion

The condition becomes: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "

#

Tdp Np Np Np Tdp Np 2 tp  \Gp Np 1 þ 2 2 þ 1 or also: \Gp f Tip Gp Gp Gp Tip Gp

24

Practical PID Handbook

FIG. 1.13 – Mixed → series conversion.

Main Features

25

⇒ The limit curve f(Nm) can be used by replacing Nm with Np/Gp. The ordinate Y is T then G 2 Tdp (see mixed → series conversion, figure 1.12). p

ip

The limit value of

2 T dp G p (pure derivative). is T ip 4

We write also: tp ¼

Tdp and Dp ¼ ðGp Np  tp Þ2  4tp Np2 Tip

The results are given in the summary table 1.10. We can also use the curves of the mixed → series conversion (figure 1.13), provided that tm is replaced by tp0 , Nm by Np0 : 8 < 0 t tp ¼ Gp2 p 0 : N p ¼ Np Gp The ordinate Y read: Is the same for the ratio Gs/Gp (lower curves). Must be multiplied by Gp to find the Tis/Tip ratio (lower curves). Must be divided by Gp to find Tds/Tdp and Ns/Np ratios (upper curves). It is also possible to convert first parallel to mixed (table 1.8) then apply the mixed to series conversion (table 1.9).

1.5.4

Summary Tables

All the results are summarized in tables 1.9 and 1.10 for the mixed ↔ series and parallel ↔ series conversions. It appears that series conversions to mixed or parallel are practically always possible because subordinate to the only condition Tds/Tis < Ns, very often realized because Td often Peq = Gs. This relation means that obtaining the tracking equivalence would give a term Peq (whose value here is equal to Gs), but that we actually obtain a term Pc here greater than Peq: the proportional action on the setpoint will be greater (by a factor of 1 + ts); the response will be faster: possible overshoot of the setpoint step response. Conversely, the mixed to series conversion gives a relationship where the term Pc is less than Peq (here equal to Gm): the tracking response will be slower. We can quantify according to the value of g(Nm) calculated or read in figure 1.13. P/M Controller Equivalence here would impose the additional condition PmDm = PsDs: conversion modifies the derivative term Dc (derivative on setpoint) that can be expressed as a function of Deq, Deq being the value that this term should have for equivalence to be achieved in tracking. Example of the series to mixed conversion where the table gives the relationship:

ts \Deq Dc ¼ Deq 1  Ns This relation means that obtaining the tracking equivalence would give a derivative term Deq, but that we actually obtain a term Dc here less than Deq: the derivative action on the setpoint will be weaker (by a factor 1 − ts/Ns): the response will be slower. Conversely, the mixed → series conversion gives a relationship where the term Dc is greater than Deq: the tracking response will be faster (possible overshoot of the setpoint step response). We can quantify if we calculate the expression 1 + Ntmm g (Nm, tm). The tables also give approximated formulas (Approx) which have no other purpose than to save time: The values of the Td/Ti ratios allow us to know if the converted controller is close to the original controller: in practice if Td/Ti ≤ 0.1 we can neglect the corrective factors in the formulas and admit identical actions for the mixed ↔ series conversions or in the gains ratio for conversions with the parallel controller. We can then refine if necessary.

Main Features

29

The approximate formulas also allow orders of magnitude to be determined more quickly. The case of the pure derivative (N → ∞) leads to simpler formulas. In practice, we can consider that we are in this case for N ≥ 10. Only the derivative on the measure does not ensure equivalence in tracking. More generally, even if the tracking equivalence is not achieved, it will still be possible to adjust the loop with the controller resulting from the conversion, even if it means adopting the converted settings, but the equivalence condition must be realized. Otherwise the behaviors obtained will be very different, especially in conversions to a series controller.

1.5.5

Examples

As an indication, figures 1.14 and 1.15 show the effect of a conversion on controllers with derivative on the measurement: the condition of equivalence is achieved: same response in regulation (not shown) but difference in tracking.

FIG. 1.14 – Example of mixed ↔ series conversions on 2nd order process.

Practical PID Handbook

30

FIG. 1.15 – Example of series to mixed conversion for integrator process.

1.6

PID: Frequency Response

We limit the study to PID series and mixed, with filtered derivative on deviation E. We only consider the Bode representation: amplitude (in dB) and phase shift (in degrees) as a function of the frequency ω (logarithmic scale).

1.6.1

Series PID

The transfer function is written: U 1 þ jTi x jTd x ðjxÞ ¼ G 1þ E jTi x 1 þ j TNd x

!

!   1 þ jTi x 1 þ j 1 þ N1 Td x ¼G jTi x 1 þ j TNd x

The characteristic frequencies are: 1 xi ¼ Ti

N xd ¼ ðN þ 1ÞTd

pffiffiffiffiffiffiffiffiffiffiffi x0 ¼ xi xd ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ðN þ 1ÞTi Td

Filtering the derivative introduces a break at N/Td. We can consider 3 cases according to the relative importance of frequencies ωi and ωd.

1.6.1.1

1st Case ωi < ωd (figure 1.16)

It is the most frequent case; oblique asymptotes intersect at ω0. This function goes through a minimum Gmin for a frequency that is all closer to ω0 as ωi and ωd are distant from each other. The limit of Gmin is obtained for N large (pure derivative): Gmin [ G ð1 þ Td =Ti Þ

Main Features

31

FIG. 1.16 to 1.18 – PID series with filtered derivative.

1.6.1.2

2nd Case ωi = ωd (figure 1.17)

This is a special case: x0 ¼ xi ¼ T1i ¼ xd ¼ ðN þN1ÞT d . sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 \2G Gmin ¼ 2G 1  ð N þ 1Þ 2 The phase rotation is faster than before.

1.6.1.3

3rd Case ωi > ωd (figure 1.18)

Same paces as section 1.6.1.1 by reversing Ti and Td.

Practical PID Handbook

32 The gain of the “plateau” is: Gplateau ¼ G

xi N þ 1 Td ¼G N Ti xd

Everything happens as if the gain were multiplied by N Nþ 1 TTdi . However, several shapes of curves can be obtained according to the relative importance of the frequency ωi and N/Td (cut-off frequency of the filter of the derivative). The case where Ti ≤ Td/N is not shown because it corresponds to a PI controller (when Ti = Td/N) or PI with phase delay (when Ti < Td/N) Strictly speaking, the condition for PID i.e., passing through a minimum is: Ti [

Td 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 1 1 2 ðN þ 1Þ

The frequency corresponding to the minimum is greater than ω0.

1.6.2

Mixed PID

U 1 jTd x ðjxÞ ¼ G 1 þ þ E jTi x 1 þ j TNd x

! G

 1 þ Ti þ

  jx  Ti Td 1 þ N1 x2   jTi x 1 þ j TNd x Td N



The numerator may have 2 real roots (in s) corresponding to 2 time constants T1 and T2. The condition Td/Ti ≤ 1/4 for a pure derivative becomes: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 Td  f ðN Þ with f ðN Þ  N 1 þ 2N  2 N ðN þ 1Þ \ 4 Ti We can notice that we find the condition of equivalence to pass the mixed PID to the series. For N J 1, we can use the approximated formula: f ðN Þ 

N 2ð2N þ 1Þ

For the usual values of N (N ≈ 1 to 10), the limit value is between 0.17 and 0.25. The frequencies, ωd, ωi, ω0 are the same as those of the PID series: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 1 N N xi ¼ xd ¼ x0 ¼ xi xd ¼ Ti ðN þ 1ÞTd ðN þ 1ÞTi Td ω0 corresponds to the intersection point of asymptotes whose ordinate is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N þ 1 Td G N Ti

Main Features

33

FIG. 1.19 – Mixed PID with filtered derivative: limit curves. The condition for PID behavior (passing through a minimum) is written: " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# Td 2ðN þ 1Þ \g ðN Þ with g ðN Þ  N 2 1 þ N Ti The graph in figure 1.19 gives the 2 limit curves f(N) and g(N) (for N on the ordinate). As before, we can examine 3 cases:

1.6.2.1

1st Case Td/Ti < f(N) (figure 1.20)

T1 and T2 are given by the following relations where we set t = Td/Ti. 2 ffi3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 Ti t t þ 1þ T1 ¼ 41 þ 4 t 5 N N 2 2 ffi3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ti 4 t t 2 1þ 1þ  T2 ¼ 4 t 5 N N 2 The gain of the plateau is: G

T1 . Ti

Practical PID Handbook

34

FIG. 1.20 to 1.22 – Mixed PID with filtered derivative.

1.6.2.2

2nd Case Td/Ti = f(N) (figure 1.21)

This is the case of 2 time constants identical to the numerator:   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Td ¼ Ti N þ 1  N ðN þ 1Þ ¼ T1 ¼ T2 ¼ N þ 1 þ N ð N þ 1Þ x0 N

Main Features

35

The intersection point for oblique asymptotes has the following ordinate: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii G a ¼ G N þ 1  N ðN þ 1 Þ 4N þ 3 for N J 1. 8N þ 4 is given by:

Approximated value: G a  G

The minimum gain Gmin

h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Gmin 2 ¼ 4 N ð6N þ 5Þ þ 2ð3N þ 1Þ N ðN þ 1Þ G In practice, we can obtain Gmin from Ga (the Gmin/Ga ratio is close to 2): Gmin/Ga = 1.9 to 2 for the usual values of N (1 to 10).

1.6.2.3

3rd Case Td/Ti > f(N) (figure 1.22)

PID (passage through a minimum) supposes Td/Ti < g(N) so that this case corresponds in practice to the double condition: f(N) < Td/Ti < g(N). The minimum Gmin takes place for a frequency ωm > ω0: N þ1 Td x2m ¼ x20 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t with t ¼ Ti N ðN þ 2Þ þ 2t  N

Gmin

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ½A  N ðN þ 1Þ2 þ AðN þ t Þ2 V ¼ N ðN þ 2Þ þ 2t with ¼G N 2 ðN þ 1ÞtV A ¼ NV  t

In practice we can confuse Gmin(ωm) and G(ω0) for the usual values of N (J 1) and for Td < Ti which gives a curve above the descending oblique asymptote. N þt G ðx0 Þ ¼ G pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ½ t þ N ð N þ 1Þ  For high values of Td (J2Ti) the curve is below the oblique asymptote.

Chapter 2 Digital PID Recall that a digital controller is a controller which processes information sampled at period Te. In its use, there are 2 types of controllers: – Continuous controller: The sampling is fast enough for the variations of different quantities to appear continuous, in particular on the command. For the user, the sampling is transparent. The synthesis and settings are defined in continuous terms (transfer functions in s), and in the case of the PID, we reason with the usual parameters G, Ti, Td, N: the PID digital therefore has digital only its implementation. – Discrete controller: Sampling is slower and results in particular in a command in “staircase steps”. The system is no longer continuous: we use the theory of sampled systems with z-transfer functions. The digital PID results initially from the discretization of the continuous PID The discrete digital PID could therefore be defined as a digital controller of degree 2 with an integral action. Note that it is possible to use the continuous controller with slow sampling: the command is in staircase mode, but we reason with the usual notions of the continuous (G, Ti, Td, N). However, this 3rd way is practically not used because most of the adjustment methods assume a continuous variation of the control command.

2.1

Continuous to Digital Transposition

In this paragraph, it is a question of making the correspondence between the continuous PID functions (Laplace transfer functions in s denoted F(s)), and their sampled equivalent: transfer functions in z−1 denoted F(z) (called sampled transfer function: s.t.f.).

DOI: 10.1051/978-2-7598-2608-7.c002 © Science Press, EDP Sciences, 2022

Practical PID Handbook

38

Recall that z is the forward operator and z
1 the backward operator of 1 sample: z ¼ eT e s or z
1 ¼ e
T e s where Te designates the sampling period, at which measurements and actions are carried out. The correspondence supposes that the continuous functions are provided with a zero-order hold block, which means that the command is kept constant between the samples. It can be obtained in several ways:

2.1.1

z-Transformation  
 F ðs Þ F ðz Þ ¼ 1
z
1 Z s

Z

h

F ðs Þ s

i

is the z-transform of FðsÞ s and is obtained by the usual correspondence tables.

This is the classic method which sometimes gives calculations that are a bit long, but which is rigorous: the envelope of the sampled function is the same as the continuous function.

2.1.2

Backward or BW Approximation

It results from the approximation of the derivative at point B (instant k) by the slope of the segment AB (figure 2.1): dM M k
M k
1 M
z
1 M ¼ either in symbolic notation: sM ¼ . dt Te Te
1 1
z . Hence the equivalence by making s ¼ Te

FIG. 2.1 – Sampling approximations.

2.1.3

Forward or FW Approximation

By approximating the derivative to the slope of BC, we obtain another equivalence 1
z
1 by doing: s ¼
1 . z Te

Digital PID

39

Note that alone, this equivalence is not always physically achievable (anticipation); it must then be associated with other elements.

2.1.4

z-Transformation and Approximation

Sometimes z-transformation is combined with an approximation. This is the case in particular for the filtered derivative (Najim and Muratet, 1983): Td s DðsÞ ¼ The equivalence is obtained by replacing the filter by its Td s 1þ N z-transformation, and the derivative by an approximation: 9 1 ð1
aÞz
1 > > ! = Td ð1
aÞð1
z
1 Þ 1
az
1 1 þ TNd s ) D ðz Þ ¼
1 > > 1
az
1 Te > ; : Td s !FW Td 1
z > Te z
1 8 > >
> = 1
z
1 with K ¼ Te i Ki > Ti > ;
1 1
z

Figures 2.2a and 2.2b gives the step response, for the 2 transformations.

Practical PID Handbook

40

–

–

(a)

(b)

FIG. 2.2 – (a) Integral: z-transformation. (b) Integral: BW approximation.

2.2.2

Derivative

The pure derivative can only be an approximation. The most used and the simplest is the BW approximation because physically feasible (unlike the FW approximation when it is alone):
 Td Td s ! Kd 1
z
1 with Kd ¼ Te The filtered derivative can have several forms depending on the equivalences used for the transformation from continuous to digital. However, the filtered derivative is of the form: F ðz Þ ¼

Kd ð1
z
1 Þ 1 þ a1 z
1

Continuous ↔ discrete transposition is only possible if −1 < a1 5).

4.9.2

Closed Loop Identification

Closed loop identification is only possible for the order n = 1. However, it will be necessary to ensure the validity of the model by a prior step response. The method is that of ultimate gain (Gc, Tc): figure 4.41.

FIG. 4.41 – Closed loop identification of a reverse response integrator. The integration time Ti is assumed to be already known (measurement on the open loop step response). cÞ The loop gain is: G b ¼ jTGxc cðð1jhx 1 þ jT xc Þ. i The resolution of the equation Gb(ωc) = −1 leads to the following relationships: G c T 2c G c T 2c ffi 0:0253 2 4p T i Ti Ti h¼ Gc

T¼

4.9.3

Case of Delay

The methods discussed for the identification of reverse response integrator systems do not identify a pure delay. This means that if there is a delay, it is integrated into the model: this is not a drawback since very often when we have a model with delay, the first correction made is to delete it.

122

Practical PID Handbook

In the case n = 1 this delay can be visible, and therefore measurable, on the step response since the start of the response is oblique; still it is necessary that the negative overshoot is sufficiently pronounced. In cases n > 1, it is very difficult to discern the start of the response. In both cases, if for specific reasons we want to obtain a model with an explicit delay L (known or chosen elsewhere), we can still use the graphic identification in open loop provided that the origin of the times is set to t = L in the plot of the step response.

Chapter 5 Evaluation of Performances The evaluation of the performances of a loop supposes that one can compare the performances of the controlled process, without and with controller. It is therefore necessary to define a minimum of quantities which characterize the systems, whether they are stable or unstable. Furthermore, the adjustment methods generally use a model for the process (see, previous chapter) and adjustment criteria, depending on the objectives sought.

5.1

Process Performances

Regardless of the representation models, systems are often characterized by 2 types of response: the step response and the frequency response. These 2 aspects are complementary. The frequency response reveals the hidden aspect of the behaviour (in terms of stability in particular) while the time response is closer to the reality of the operation. It is obviously useful to be able to link the two responses.

5.1.1

Step Response

With constant command, for stable systems, the measurement eventually stabilizes: variation ΔMf for a control variation ΔUc. The speed of the system is usually defined by the response time at 5% on a control step ΔUc: time after which the response remains within a band of ±5% of the final variation (figure 5.1a). For aperiodic processes, the response time is measured at 95% (figure 5.1b). These data are very important for the adjustment of a loop since it conditions the adjustment of the integral and derivative actions. In practice, we apply a rising step then a descending step and we average the 2 results.

DOI: 10.1051/978-2-7598-2608-7.c005 © Science Press, EDP Sciences, 2022

Practical PID Handbook

124

FIG. 5.1 – (a) Response time definition. (b) Case of an aperiodic process. In the case of unstable processes: with constant control, the measurement does not stabilize; the static gain is infinite and the response time is obviously infinite. However, the speed of the system can be quantified by a pseudo response time, defined for a control step, by the time after which the variation of the response is equal to the step applied (figure 5.2).

FIG. 5.2 – Pseudo response time of an integrating system.

5.1.2

Frequency Response

It is the frequency response of the system subjected to an established sinusoidal command (forced state): figure 5.3.

FIG. 5.3 – Frequency response.

Evaluation of Performances

125

Reading the ratio of amplitudes and the phase shift between measurement and control makes it possible to characterize the transfer function: jjFðjxÞjj ¼

S E

Arg FðjxÞ ¼ u These data lead to the well-known curves of Bode, Black, and Nyquist. Note that it is not always possible to oscillate the process and that this method is quite long. In practice, rather, indirect methods are used which make it possible to deduce the frequency response.

5.2

Adjustment Objectives

The addition of a controller (figure 5.4) aims to improve overall performance (stabilization, linearity,…) and more particularly: – Obtain a measurement which copies the setpoint “at best”: it is the behaviour in tracking. – Reduce the influence of the disturbance: this is behaviour in regulation.

FIG. 5.4 – Closed loop. In tracking, performance is usually defined by the response time (in closed loop), therefore from a setpoint step and possibly an overshoot in % of the final variation (figure 5.1a) (or of the setpoint variation since the controller is supposed to ensure equality in static). Sometimes we use a more global notion which is the acceleration factor, defined by the ratio of response times in open loop and closed loop: Fa ¼ trol =trcl In regulation, the performances are defined by amplitude Pmax of the variation of the disturbed measurement (in % of the disturbance X) and by the recovery time trec (or response time): it is defined by the time after which the measurement variation remains less than ±5% of the disturbance X. We can see immediately that these performances are difficult to measure because the disturbance X is rarely accessible; however, the recovery time is used to set objectives for adjusting the loop (figure 5.5).

Practical PID Handbook

126

FIG. 5.5 – Response to a disturbance step in closed loop. In the absence of information on the disturbance, we often use, in practice, the same model for regulation, which amounts to assuming that the disturbance is located at the level of the control according to figure 5.6a. This gives the possibility of generating an artificial disturbance at the input of the process to observe the regulation response. We can then measure the recovery time.

FIG. 5.6a – Representation model.

FIG. 5.6b – Representation model for stable systems.

In the case of stable systems, we sometimes separate the static gains for the command (K1) and for the disturbance (K2), when this is measurable, according to figure 5.6b. This way can also be suitable for unstable systems: K1 and K2 are then static gains of particular actuators of the process.

5.3

Closed Loop Setting Criteria

Tuning criteria are a way of defining a tuning quality that meets the requirements of the operation. The types of process are multiple, often very different: can we compare a distillation column, a speed variator or a hydroelectric dam?

Evaluation of Performances

127

The variety of problems is such that there is no single method or criterion for adjustment. However, the criteria are generally linked to notions of stability whose definition we recall. A system being defined by its transfer function H(s), it generally has poles and zeros: H(s) of the form:
 ðs  z 1 Þðs  z 2 Þ    z i ¼ zeros real or K complex ðs  p1 Þðs  p2 Þ    pi ¼ poles A system is said to be stable if all the poles of its transfer function have a negative real part. If this real part is zero, we are at the limit of stability. Examples (figure 5.7): 8 9 < 1 stable pole p ¼  1 = 1 1 s integrator system ; Ti s ð1 þ ssÞ : 1 unstable pole p2 ¼ 0

1 1 þ T 2 s2

2 poles 

j T

just oscillating system

FIG. 5.7 – Stability area.

In practice, we avoid being at the limit and we define a more restricted area where the poles must be located: angle Ψ ffi 30° which corresponds to a 2nd order system with a damping factor ζ = 0.5.

Practical PID Handbook

128

5.3.1

Stability Criteria

FIG. 5.8 – Closed loop. Term B represents the possible processing on the measurement (derivative and/or gain). The transfer relationship is written as (figure 5.8): M¼

RF F Sp þ X 1 þ RFB 1 þ RFB

Stability is expressed by the loop gain: GB(jω) = R(jω)F(jω)B(jω). We trace the loop gain as a function of the frequency in the different planes: Black, Bode, Nyquist; stability is assessed by the position of the curve relative to the critical point −1 which corresponds to the sustained oscillation pulse ωosc. It is all the better as the loop gain “moves away” from point −1. In practice, the degree of stability can be quantified by the following parameters, illustrated by figure 5.9a–c according to (Guyenot and Hans, 1989): – Gain margin: attenuation of the loop gain for ωosc: Mg ¼

1 jGB jxosc

The usual values vary from 2 to 5 approximately and we typically aim for a margin Mg ≈ 4 (12 dB). – Phase margin: remaining phase to obtain −180° when the module of the loop gain is worth 1 (0 dB): Mu ¼ p þ Arg½GB ðjxÞ0 dB The values range from 30° to 60° and we generally aim for a phase margin of 45°–50° – Module margin: minimum distance between loop gain and point –1: M D ¼ j1 þ G B jmin Figure 5.9c shows that this criterion is the one that best defines the overall stability with respect to the critical point −1. It does not favor either tracking or regulation, and constitutes a compromise. This module margin is directly linked to the sensitivity function S(jω):    1  1  Ms ¼ maxS ðjxÞ  ¼ 1 þ GB  MD

Evaluation of Performances

FIG. 5.9 (a–c) – Practical stability. Source: Guyenot and Hans (1989).

129

Practical PID Handbook

130

The values usually sought for MΔ range from approximately 0.5 to 0.75 (i.e., Ms = 1.3–2). It may be useful to get an idea of the module margin MΔ from the other criteria; from figure 5.9c we obtain limits:   MD \ 2 sinðu=2Þ and MD \1  1=Mg

5.3.2

Step Response Criteria

The step response is widely used because it is simple to implement. The criteria usually used consist in defining the form of the response.

5.3.2.1

Overshoot and Response Time

In tracking, a setpoint step is applied and the response is usually defined by the overshoot and the response time. We prefer one or the other according to the needs: the reduction in response time is often “paid” by an increase in overshoot; a frequent adjustment consists in aiming for an overshoot of less than 5%, in order to have a response time at 95%: figure 5.10a.

FIG. 5.10 (a, b) – Response time and overshoot. In regulation, a disturbance level X is applied to the control input. As in tracking, the reduction in recovery time is also at the expense of the amplitude Pmax. If the response is oscillatory, there is also an interest in obtaining an (negative) overshoot of less than 5% to have a recovery time at 5% (figure 5.10b).

5.3.2.2

Consecutive Overshoots

Some methods use the ratio D1/D2 (for example D1/D2 = 4) of two consecutive overshoots in tracking (figure 5.11a) or in regulation (figure 5.11b). Sometimes the ratios D1/D−1 are used.

Evaluation of Performances

131

FIG. 5.11 (a, b) – Consecutive overshoots. These reports quantify the degree of damping of the response.

5.3.3

1st or 2nd Order Systems

Some adjustment methods allow the response to conform to a 1st or 2nd order model. The 1st order model is particularly simple: we choose the time constant to give the desired speed. The second order is a finer model: we can choose the shape by means of the damping factor ζ (see appendix).

5.3.4

Order n Systems: Naslin Criterion (Naslin, 1968, 1962)

Suppose a system defined by its transfer function: S b0 þ b1 s þ b2 s 2 ðs Þ ¼ E a0 þ a1 s þ a2 s2 þ    þ an sn Numerator: maximum degree 2. Denominator: degree n from 3 to 8. According to Naslin, such a system behaves roughly like a 2nd order if the coefficients ai are such that: ai ¼

a 2i ¼ a where a is a given value a i1 a i þ 1

The condition αi ≥ α (whatever i) constitutes the damping criterion. α is the equivalent of the damping factor ζ (2nd order). Its value is around 2ζ. The practical application is the writing of the transfer function as a function of α, neglecting the influence of the numerator: S ðs Þ ¼ E 1 þ Ts þ

1 T2 2 a s

þ

T3 3 a3 s

þ  þ

Tn n am s

with m ¼

nðn  1Þ 2

Figure 5.12 gives the step responses, according to Naslin, for the same α.

Practical PID Handbook

132

FIG. 5.12 – Naslin’s step responses. Naslin even proposes an approximate relationship between this coefficient α and the overshoot D% of the step response (figure 5.13). He also gives an average rise time, knowing that the rise time is defined as the overshoot, according to figure 5.14: tm 

2:2 ¼ 2:2 T x0

The previous results assume a constant numerator for F(s). In closed loop, transmittances often include a numerator of the 1st order (PI control) or of the 2nd order (PID control) (see chapter 1): F ðs Þ ¼

b0 þ b1 s þ b2 s 2 a0 þ a1 s þ a2 s2 þ    þ an sn

FIG. 5.13 – Average overshoot according to Naslin.

Evaluation of Performances

133

FIG. 5.14 – Definition of the rise time.

For an α chosen as above to obtain a desired overshoot, the presence of the numerator produces a overshoot of the step response greater than expected: the effective damping αe is lower. The rise time is also lower than expected by the formula tm = 2.2/ω0: ω0 must be replaced by a corrected value ω0c. Naslin gives formulas to calculate αe and ω0c: 9 8 1 x00 > > > ða  1:5Þ > = < ae ¼ 1:5 þ b0 4 x0 with x00 ¼ 1st order : 1 1 1 > > b1 > > ; : ¼  x0c x0 2x00 9 8 b0 1 x02 > 02 0 > > > > ða  1:5Þ = < x0 ¼ b 3 x2 16f 0 2 with 2nd order : 2 > > > 1 1 1 b > > > > > : 4f2 ¼ 1 ¼  0 ; : x0c x0 x0 b0 b2 8 > > > < ae ¼ 1:5 þ

Conversely if an ae e is too low, it is possible to use these formulas “upside down”, in order to determine a corrected value ac (>initial α) which gives an ae equal to the initial α. Formulas are written: 1st order : ac ¼ 1:5 þ 4

x0 ða  1:5Þ x00

2nd order : ac ¼ 1:5 þ 16f3

x20 ða  1:5Þ x02 0

Practical PID Handbook

134

5.3.5

Optimization Criteria

Some adjustment methods aim to obtain optimized responses on criteria, the main ones of which are as follows: IE ¼

R1

eðt Þdt R0 1 2 ISE ¼ 0 e ðt Þdt R1 ISTSE ¼ 0 t 2 e2 ðt Þdt R1 IAE ¼ 0 jeðt Þjdt R1 ITAE ¼ 0 t jeðt Þjdt

Integral of Error Integral of Squared Error Integral of Squared Time Error Multiplied by Squared Error Integral of Absolute Value of Error Integral of Time Error Multiplied by Absolute Value of Error

The criteria in ε2(t) or |ε(t)| are better than the first (for which an oscillating response gives a null criterion). The ISTSE or ITAE still gives better results because they take into account the time factor. The analytical application of these criteria is not simple. It is impossible for criteria IAE and ITAE, because of the absolute value. This is why the implementation and calculation of these criteria rather use simulation and optimization by digital computer. Figure 5.15 is an example of illustration of the criterion IAE (Flaus, 1994).

FIG. 5.15 – Criterion IAE in tracking (left) and regulating (right). These criteria are interesting when optimization is a purpose, such as minimizing energy consumption. On the other hand, we lose control of the overshoot or the response time. Another interest stems from the link between the IE criterion and the PID controller; for actions on the deviation e(t) the PID (parallel) can be written: Z t deðtÞ U ðtÞ ¼ kp eðt Þ þ ki eðt Þdt þ kd dt 0 If a disturbance X is applied to the input of the process (assumed to be stabilized), the controller will correct by changing its command U from the initial value U (0) to the final stabilized value U ð1Þ to cancel the disturbance according to the figure 5.16: U ð1Þ  U ð0Þ ¼ X

Evaluation of Performances

135

FIG. 5.16 – Regulation response to a disturbance step. The static error is zero eð1Þ ¼ 0 and the derivative also since the measurement is stabilized. The regulator equation is then written: Z 1 U ð1Þ  U ð0Þ ¼ ki eðt Þdt ¼ ki IE ¼ X 0

This remains true regardless of the structure of the regulator. X We can deduce: jIEj ¼ . ki The criterion will be minimized by maximizing ki. For a mixed PID ki = G/Ti: the criterion leads to decrease in Ti and increase in G. A complementary way consists in seeing that the closed loop transfer function, in regulation, can take the form (see chapter 1, tables 1.4a and 1.4b): M 1 N ðs Þ ðs Þ ¼ X k i D ðs Þ This relation shows that the magnitude of the response will be minimized by maximizing ki.

5.3.6

Frequency Criteria

The stability criteria already mentioned are among them. It is necessary to add the frequency overshoot or resonance factor used in frequency analysis. It is expressed in dB relative to the plateau of the closed loop response. Its interest is the link with the overshoot of the step response: in the case of the 2nd order it results in the curve of figure 5.17. Let us also cite the particular criterion of the symmetrical optimum used by the Kessler method (see chapter 10).

136

Practical PID Handbook

FIG. 5.17 – Overshoots correspondence for a 2nd order system.

Chapter 6 Ziegler–Nichols and Associated Methods This chapter groups together several adjustment methods based on those of Ziegler and Nichols.

6.1

Ziegler and Nichols Closed Loop Method

This method, called ultimate sensitivity or also critical gain or ultimate gain, is identical for stable or unstable processes. It consists in increasing the gain of the controller (in automatic, without integral or derivative), until a sustained oscillation of period Tc corresponding to the critical gain Gc is obtained, as indicated in chapter 4 (figure 6.1).

FIG. 6.1 – Obtaining ultimate gain. Ziegler and Nichols give the optimum setting values for actions P, I, D from Gc and Tc. The adjustment criterion is a depreciation per period of 0.25: D1/D2 = 4. The settings were made on a disturbance step X at the input of the process (figure 6.2). DOI: 10.1051/978-2-7598-2608-7.c006 © Science Press, EDP Sciences, 2022

138

Practical PID Handbook

FIG. 6.2 – Adjustment criterion.   The controller is of the mixed type: G 1 þ T1i s þ Td s The waveforms used as optimal in the famous 1942 article (Ziegler and Nichols, 1942) have been reproduced in figure 6.3.

FIG. 6.3 – Ziegler and Nichols settings. The settings recommended by Ziegler and Nichols are given in table 6.1. The values noted * are those of the original article. It is noted that Ti = 4Td: this ratio facilitates the equivalences between mixed, series and parallel forms. Note that these settings have been optimized on the disturbance. In tracking, they often lead to significant overshoots. This is why we sometimes find (Flaus, 1994; Perry and Chilton, 1973) settings which give better damped responses, with “low” or “no” overshoot. It can be seen that the Ti/Td ratios are less than 4: there is no equivalent series PID; only parallel conversion is possible.

Ziegler–Nichols and Associated Methods

139

TAB. 6.1 – Ziegler and Nichols settings. P

Cont.

PI

Action

PID

Parallel

Mixed

Parallel

Mixed

Series 0.3Gc

G Ti

0.5Gc* ∞

0.45Gc*

0.6Gc

0.6Gc* Tc 2*

Td

0.45Gc Tc 1.85G c

0

0

0

0.075Tc Gc

Tc 8*

Tc 1:2

Tc 1:2Gc

Tc 4 Tc 4

TAB. 6.2 – Different settings of Ziegler and Nichols. Form

Parallel

Mixed

Criteria

G

Ti

Td

G

Ti

Td

Ziegler and Nichols

0.6Gc

Tc 1:2Gc Tc 1.5G c Tc 2:5 G c

0.075Tc Gc

0.6Gc

Tc Gc 9 Tc Gc 15

0.33Gc

Tc 2 Tc 2 Tc 2

Tc 8 Tc 3 Tc 3

Low overshoot

0.33Gc

No overshoot

0.2Gc

0.2Gc

We obtain table 6.2 which also contains the settings of Ziegler and Nichols in order to compare.

6.2

Ziegler and Nichols Open Loop Method

We only take into account the start of the process response to a step ΔE (see figure 6.4) by showing: – a pure delay L; – a slope R% = ΔE/Ta (in %/s).

FIG. 6.4 – Open loop response. This amounts to assimilating the process to the model: FðsÞ ¼

eLs . Ta s

Practical PID Handbook

140

The application of the ultimate gain method to this model (critical gain Gc, oscillations Tc) amounts to writing: Fðjxc Þ ¼ 1 with xc ¼

2p Tc

i:e: Gc

ejLxc ¼ 1 jTa xc

The resolution gives Gc = 1.57Ta/L and Tc = 4L. We can therefore make the link with the results of the closed loop method. Ziegler and Nichols also make the experimental link by obtaining Tc = 4 to 4.6L and define an optimal gain (proportional only): G = Ta/L. The relation G = 0.5Gc of the ultimate gain method (proportional only) then allows to link Gc and L. Table 6.3 is therefore obtained by applying to the initial table 6.1 the relations: Gc ¼ 2Ta =L and

Tc ¼ 4L

TAB. 6.3 – Settings with open loop method. Cont.

P

Action G

Ta L

Ti

∞

Td

0

PI

PID

Parallel

Mixed

Parallel

Mixed

Series

0:9 TLa

0:9 TLa 3.3L

1:2 TLa

1:2 TLa 2L

0:6 TLa L

L 2

L

3:7 TLa 2

0

1:67 TLa

0

2

0.6Ta

We often find tables called Ziegler and Nichols for stable processes; we can show the static gain (figure 6.5): K¼

DS T ¼ DE Ta

In table 6.3, it suffices to replace G by KG and Ta by T; T is the equivalent of a time constant in the Broïda model (figure 6.5).

FIG. 6.5 – Equivalent Broïda model for stable systems.

Ziegler–Nichols and Associated Methods

141

NOTE – The Ziegler and Nichols method is particularly simple and applies to stable or unstable processes. – The 2 aspects of the method (open loop and closed loop) are complementary and allow to limit the uncertainties of the drawing of the tangent at the inflection point (stable processes), using in particular the relation Tc = 4L. – We can also use the frequency response F(jω) of the process alone: see figure 6.6 with: fc ¼ frequency at u ¼ 180 Gc ¼

1 jF ðjfc Þj

FIG. 6.6 – Frequency response. The search for oscillations has been improved by (Aström and Hägglund, 1984): it is the relay method (Flaus, 1994). The proportional controller is replaced by a relay of amplitude h (figure 6.7). The oscillations have an amplitude a, and we go back to the critical gain by the relation (see appendix): Gc ¼

4h pa

FIG. 6.7 – Relay method.

Practical PID Handbook

142

6.3

Cohen and Coon Method (Cohen and Coon, 1953)

Cohen and Coon keep the criterion of Ziegler and Nichols: ratio D1/D2 ffi 4 on a disturbance step (figure 6.2). On the other hand, they involve the static aspect of the process – therefore, a method for stable processes only. The process is identified by a step response where the tangent is plotted at the inflection point (figure 6.8). T is assimilated to a time constant for the Broïda model used in the study: KeLs 1 þ Ts  The controller is of the mixed form: U ¼ G 1 þ F ðs Þ ¼

1 Ti s

 þ Td s ðSp  M Þ.

FIG. 6.8 – Stable process step response.

Cohen and Coon give settings formulas involving the ratio μ = L/T. We obtain table 6.4. In some works we sometimes find more “elegant” or even simpler relationships: we obtain them by rounding up/down some values.

TAB. 6.4 – Cohen and Coon settings. REG G

1 K



1:03 l

P

 þ 0:35

1 K

P(1 + D)   1:24 þ 0:16 l

1 K

P(1 + I)   0:9 þ 0:083 l

P(1 + I + D)   1:35 þ 0:25 l

1 K

Ti

∞

∞

þ 0:83l L 92:7 þ 6l

þ 2:5l L 13:5 5:4 þ 3:3l

Td

0

3:41:1l L 12:4 þ 1:6l

0

L 2:7 þ10:5l

Ziegler–Nichols and Associated Methods

6.4

143

Takahashi Method (Takahashi et al., 1971)

The method takes the basic settings of Ziegler and Nichols by adapting them to the following particularities: – the controller is mixed with derivative and gain on the measurement:   1 E  ð1 þ Td s ÞM U ¼G Ti s where E designates the deviation Sp − M (see chapter 1). – the controller is carried out in digital: sampling period Te. The discretization of the PID above leads to the recurrence equation: Uk  Uk1 ¼ GKi Spk  G ½ð1 þ Ki þ Kd ÞMk  ð1 þ 2Kd ÞMk1 þ Kd Mk2  with Ki ¼ TTei and Kd ¼ TTde . The indices k, k − 1, k − 2 represent the current, previous sampling instants, etc. (see chapter 2). The settings refer to the 2 methods ultimate gain and open loop with integrator model (see figure 6.4): F ðs Þ ¼

eLs Ta s

The interest of these adjustments is the taking into account of the sampling period Te, which remains however small compared to the time constants of the process. Table 6.5 gives directly the actions G, Ti and Td.

TAB. 6.5 – Takahashi settings. Cont.

P

PI Closed loop method GcTc   Gc 0:45  0:54 TTec

G

0.5Gc

Ti Td

∞ 0

G

Ta L þ Te

Ti

∞

3.3L + 1.15Te

Td

0

0

0.83Tc − Te 0 Open loop method LTa   0:9Ta Te L þ 0:5Te 1  0:15 L þ 0:5Te

PID   0.6Gc 1  TTec 0.5 (Tc − Te) T2

c 0:125 Tc T e

1:2Ta L þ Te

2L

a Te  ðL0:3T þ 0:5T Þ2 e

e  0:5 LLT þ Te 0:6 TGa

Practical PID Handbook

144

6.5

KT Method of Aström and Hägglund (1995)

Also called Kappa–Tau tuning the method takes up the 2 open loop and closed loop aspects of Ziegler and Nichols. The controller is of the mixed type with derivative on the measurement and weighting of the setpoint (coefficient b, see chapter 1): U ¼ Gðb Sp  M Þ þ

1 ðSp  M Þ  Td s M Ti s

The setting criteria no longer have anything to do with those of Ziegler and Nichols because the procedure for calculating actions is completely different. The tests were conducted on a battery of process models representative of what is encountered in the industry: F1 ðs Þ ¼

es

ð1 þ Ts Þ2 1 F2 ð s Þ ¼ ð1 þ s Þn F3 ð s Þ ¼ F4 ð s Þ ¼

1 ð1 þ s Þð1 þ as Þð1 þ a2 s Þð1 þ a3 s Þ 1  as ð1 þ s Þ3

T = 0.1–10 n = 3, 4, 8 α = 0.2, 0.5, 0.7 α = 0.1, 0.2, 0.5, 1.2

For integrating processes, the previous models Fj(s) have been taken up by adding an integration: 1 H j ðs Þ ¼ F j ðs Þ s These different processes have been simulated by the same models as Ziegler and Nichols and the adjustment criteria are those of the dominant pole method (Aström and Hägglund, 1995); they lead to choosing a margin of module MΔ: see chapter 9.

6.5.1

Ultimate Gain Method

The method leads to the measurements of the critical gain Gc and of the period Tc of the oscillations (see beginning of the chapter).

6.5.1.1

Stable Processes

Obtaining universal curves introduces the ratio g = 1/KGc where K is the static gain of the process (this ratio is noted by the Greek letter kappa in the study). The settings were made for 2 values of the MΔ module margin. In fact, the authors use the sensitivity function Ms = 1/MΔ: Ms ¼ 1:4 and 2 corresponding to MD ¼ 0:71 and 0:5

Ziegler–Nichols and Associated Methods

145

The authors plot the points as a function of g for the normalized actions K/Gc, Ti/Tc, Td/Tc and b. They deduce average curves which have been approximated by the function:
f ðg Þ ¼ a0 exp a1 g þ a2 g 2 The study is made for the PI and PID controllers, and leads to adjustments which give the normalized actions as a function of g, by means of the coefficients a0, a1, a2, of the function f(g): table 6.6. TAB. 6.6 – Aström and Hägglund settings, by the ultimate gain method, for stable process. Stable process:

K, Gc, Tc

1 g ¼ KG c

f(g) = a0 exp(a1g + a2g2)

Ms = 1.4

PI

PID

6.5.1.2

Actions G/Gc Ti/Tc b G/Gc Ti/Tc Td/Tc b

a0 0.053 0.9 1.1 0.33 0.76 0.17 0.58

a1 2.9 −4.4 −0.0061 −0.31 −1.6 −0.46 −1.3

Ms = 2 a2 −2.6 2.7 1.8 −1.0 −0.36 −2.1 3.5

a0 0.13 0.9 0.48 0.72 0.59 0.15 0.25

a1 1.9 −4.4 0.4 −1.6 −1.3 −1.4 0.56

a2 −1.3 2.7 −0.17 1.2 0.38 0.56 −0.12

Integrating Processes

The previous method cannot be applied: the static gain is infinite. – For the PI regulator we can use the previous formulas by making g = 0. – For the PID, information is missing.

6.5.2

Open Loop Method

6.5.2.1

Stable Process

The stable processes have been modeled by the Broïda model F ðs Þ ¼

KeLs 1 þ Ts

In the study, time constant T is identified from time at 63% of the final response, and not from Ta (figure 6.9). In order to obtain universal curves, the process parameters have been normalized by introducing the 2 dimensionless ratios:

Practical PID Handbook

146

FIG. 6.9 – Identification of the Broïda model. a¼K

L T

and s ¼

L LþT

The actions of the controller have also been normalized: aG, Ti/L, Td/L as well as Ti/T and Td/T; b is unchanged. As previously, the normalized actions are plotted as a function of τ, for all of the tests on the different processes. We obtain points which form an average curve approximated, as before, by the function:
f ðsÞ ¼ a0 exp a1 s þ a2 s2 As an indication, we have reproduced (figure 6.10) the original curves giving the normalized actions for the PI regulator. The dotted lines are the values of Ziegler

FIG. 6.10 – Application of the KT method on different stable processes (PI controller).

Ziegler–Nichols and Associated Methods

147

and Nichols. The points represented by a cross correspond to Ms = 2; the circles correspond to Ms = 1.4.

6.5.2.2

Integrating Processes Ls

The authors use the integrating model of Ziegler and Nichols: F ðs Þ ¼ eTa s and use the Broïda model with integrator to obtain additional information (L0 , T 0 ) that they measure on an impulse response (figure 6.11): 0

Rimp ðs Þ ¼

1 eL s Ta s 1 þ T 0 s L0 s

This is also the step response of H(s) = T1a 1eþ T 0 s As for stable systems, the normalization of the parameters introduces the quantities a and τ, with L = L0 + T 0 : a¼

L0 þ T 0 L ¼ Ta Ta

and

s¼

L0 L0 ¼ L0 þ T 0 L

The plot of the points for the whole of the test gives average curves modelled by the previous function f(τ). We obtain tables 6.7 (stable processes) and 6.8 (integrating processes), that give the normalized actions as a function of τ by f(τ) interposed.

FIG. 6.11 – Integrator system impulse response.

6.5.3

Adjustment Curves

To avoid calculations, in particular of the functions f(τ) or f(g), we have retraced the average curves, so as to read directly the values of the normalized actions (these are the authors’ curves without the points); and this is only for PID controllers. The normalized Ti/T and Td/T parameters were not plotted. Figures 6.12–6.14 are obtained.

Practical PID Handbook

148

TAB. 6.7 – Aström and Hägglund settings, open loop, for stable processes. Process:

KeLs 1 þ Ts

a ¼ K TL

s ¼ L þL T

f(τ) = a0 exp(a1τ + a2τ2)

Ms = 1.4 a0 0.29 8.9 0.79 0.81 3.8 5.2 0.46 0.89 0.077 0.4

Actions aG Ti/L Ti/T b aG Ti/L Ti/T Td/L Td/T b

PI

PID

a1 −2.7 −6.6 −1.4 0.73 −8.4 −2.5 2.8 −0.37 5.0 0.18

Ms = 2 a2 3.7 3.0 2.4 1.9 7.3 −1.4 −2.1 −4.1 −4.8 2.8

a0 0.78 8.9 0.79 0.44 8.4 3.2 0.28 0.86 0.076 0.22

a1 −4.1 −6.6 −1.4 0.78 −9.6 −1.5 3.8 −1.9 3.4 0.65

a2 5.7 3.0 2.4 −0.45 9.8 −0.93 −1.6 −0.44 −1.1 0.051

TAB. 6.8 – Aström and Hägglund settings, in open loop, for integrating processes. Process:

eLs Ta s

L0 s

¼ Ta seð1 þ T 0 sÞ

a ¼ TLa

0

s ¼ LL

f(τ) = a0 exp(a1τ + a2τ2)

Ms = 1.4

PI

PID

6.6 6.6.1

Actions aG Ti/L b aG Ti/L Td/L b

a0 0.41 5.7 0.33 5.6 1.1 1.7 0.12

a1 −0.23 1.7 2.5 −8.8 6.7 −6.4 6.9

Ms = 2 a2 0.019 −0.69 −1.9 6.8 −4.4 2.0 −6.6

Performances Stable Process

We chose a third order process: F ðs Þ ¼

1 ð1 þ sÞ3

a0 0.81 3.4 0.78 8.6 1.0 0.38 0.56

a1 −1.1 0.28 −1.9 −7.1 3.3 0.056 −2.2

a2 0.76 −0.0089 1.2 5.4 −2.3 −0.6 1.2

Ziegler–Nichols and Associated Methods

FIG. 6.12 – Closed loop KT method for stable systems.

149

150

Practical PID Handbook

FIG. 6.13 – Open loop KT method for stable systems.

Ziegler–Nichols and Associated Methods

FIG. 6.14 – Open loop KT method for integrating systems.

151

Practical PID Handbook

152

TAB. 6.9 – Setting values for a stable process. 1 2 3 4 5 6 7 8 9 10 11

Method Z & N CL Gc Tc Z & N OL L Ta Z & N Gc Tc low overshoot Z & N Gc Tc no overshoot Cohen & Coon Takahashi Gc Tc Te = 0.1 Te = 0.1 Takahashi L Ta A & H CL Gc Tc Ms = 1.4 A & H OL L T Ms = 1.4 A & H CL Gc Tc Ms = 2.0 A & H OL L T Ms = 2.0

G 4.80 5.50 2.64 1.60 6.44 4.67 4.74 2.50 2.27 4.80 4.35

Ti 1.80 1.61 1.80 1.20 1.85 1.76 1.57 2.25 2.07 1.83 1.68

Td 0.45 0.40 1.20 1.20 0.29 0.46 0.47 0.56 0.51 0.46 0.42

b – – – – – – – 0.52 0.50 0.27 0.26

Controller

Mixed PID/E

Mixed P&D/M

Mixed D/M b/Sp

From this function, you can determine the parameters necessary for the different adjustment methods, for the equivalent Broïda model (see appendix table A.3). We obtain: L = 0.805 Gc = 8 T 63% = 2.453

Ta = 3.695 Tc = 3.63 τ = 0.247

μ = Tu/Ta = 0.218 (See chapter 4, figure 4.20) a = 0.328

We deduce the actions G, Ti, Td, for different methods: table 6.9. The response curves were plotted with a filtering factor N = 10. Figure 6.15 shows the tracking responses. The command has been divided by 10 on the top figure alone, corresponding to cases 1–5: mixed PID controller with actions on the deviation. Recall that the value of the initial jump of the command is G(N + 1)ΔSp for a mixed controller with actions on the deviation. The 2 Ziegler and Nichols methods give significant overshoots, as does the Cohen and Coon method. The bottom figure shows much better damped curves, due to the actions on the measurement. Overall, the critical gain methods give rather less overshoots. Figure 6.16 gives the regulation responses: they are more grouped. There are few differences between the 2 types of methods (ultimate gain or open loop). We can note a difference in setting of the Aström and Hägglund methods as a function of the Ms sensitivity, which makes it possible to favor tracking or regulation.

Ziegler–Nichols and Associated Methods

153

FIG. 6.15 – Stable process tracking settings.

6.6.2

Integrating Process

The process is F ðs Þ ¼ sð1 þ1 sÞ3 and corresponds to the model

eLs . Ta s

The identification elements are as follows: – Ultimate gain method: Gc = 0.89 Tc = 10.9 (figure 4.36). – Method of Ziegler and Nichols in open loop: L = 3 – KT method of Aström and Häglund: a = L = 4.5

  eLs model . Ta s L0 τ = L0 þ T 0 = 0.18. Ta = 1

The different settings are given in table 6.10. The response curves are given in figure 6.17 with a filtering factor N = 10.

Practical PID Handbook

154

FIG. 6.16 – Stable process settings in regulation. TAB. 6.10 – Setting values for integrating process. G 0.53 0.40

Ti 5.44 6.0

Td 1.36 1.5

b – –

Te = 0.1

0.53

5.39

1.37

–

Mixed P&D/M

Ms = 1.4 Ms = 2

0.32 0.67

14.0 7.6

2.6 1.7

0.33 0.39

Mixed D/M b/Sp

Method 1 Z & N CL Gc Tc 2 Z & N OL L Ta 3 Takahashi Gc Tc 4 KT A & H OL 5 KT A & H OL

Controller Mixed PID/E

The 2 methods of Ziegler and Nichols are quite close and give significant overshoots in tracking. Takahashi’s method with actions practically identical to those of Ziegler and Nichols, limits overshoot in tracking thanks to the structure of the controller.

Ziegler–Nichols and Associated Methods

155

FIG. 6.17 – Responses of an integrating system. The settings of Aström and Häglund give good results and allow you to adjust the speed of response by choosing Ms.

6.6.3

Process with Delay

We take again the Strejc model of order 3 by adding a significant delay: F ðs Þ ¼

e5s ð1 þ sÞ3

We limit ourselves here to the ultimate gain methods. The identification gives the following elements: Gc = 1.25 Tc = 15.7s and leads to the setting table 6.11. The responses were simulated with a filtering factor N = 10, and plotted in figure 6.18.

Practical PID Handbook

156

TAB. 6.11 – Setting values for process with delay. Method 1 Z&N

G 0.75

Ti 7.85

Td 1.96

b –

Controller Mixed PID/E

2 Takahashi Te = 0.1

0.745

7.8

1.98

–

Mixed P&D/M

3 KT A & H Ms = 1.4 4 KT A & H Ms = 2

0.17 0.54

2.63 4.18

0.48 1.1

1.93 0.36

Mixed D/M b/Sp

FIG. 6.18 – Process responses with significant delay. Note that the settings for Ziegler and Nichols and Takahashi are the same; the difference, in tracking, remains linked to the structure of the controller. Processes with delay are difficult to control by the methods of Ziegler and Nichols, and all the less since the delay is significant. Usually we consider that an L/T ratio of the equivalent Broïda model is limited to around 0.5 to ensure control. However, the methods of Aström and Hägglund give very good results even with substantial delays, as shown in figure 6.18.

Ziegler–Nichols and Associated Methods

6.7

157

Overview Summary

All characteristics of the methods discussed in this chapter have been summarized in table 6.12. TAB. 6.12 – Summary of the settings of the Ziegler and Nichols methods and associated.

The methods of Ziegler and Nichols are well known for their simplicity: it suffices to identify a single point (the critical point) to deduce a controller setting, and this regardless of the system (stable or not). The open loop methods require an identification which introduces more uncertainty, in particular by drawing the tangent at the point of inflection. However, all of the responses illustrated in this chapter clearly show that the original methods of Ziegler and Nichols, or Cohen and Coon were designed to give good results in regulation and not in tracking.

158

Practical PID Handbook

Takahashi’s methods are a little better suited to tracking thanks to the P and D actions on measurement. The methods of Aström and Hägglund allow a tracking with little overshoot but also good regulation by adjusting Ms and lead to a robust controller. These methods have the advantage of efficiency, because basically they are pole placement methods, while using the simplicity of Ziegler and Nichols (critical gain in particular for stable systems). Another advantage is the ability to control systems with significant delays. Their disadvantage is the need to have a controller with setpoint weighting. NOTE Tracking responses are always better amortized when the actions are on measurement. We can therefore also use the original methods of Ziegler and Nichols (critical gain in particular) with a P&D/M controller to improve the tracking response, while keeping the response in regulation, which also makes it possible to reduce the command during a setpoint step. The settings of Ziegler and Nichols have often been used to be improved and to give more specific results, especially for the ultimate gain method, because of its simplicity and the absence of a model (O’Dwyer, 2006; Poulin and Pomerleau, 1999; Edgar et al., 1997; Peric et al., 1997; Mac Millan, 1994; Boe and Chang, 1988).

Chapter 7 Cancellation Methods Generally cancellation consists in making the zeros of the controller equal to the poles of the process that we want to compensate, so as to simplify the expression of the loop gain and remain, when possible, 1st or 2nd order in closed loop, to use known results.

7.1 7.1.1

Dindeleux Method (Dindeleux, 1989) Stable Process

Process described by the Broïda model (figure 7.1).

FIG. 7.1 – Control of a stable process.

Series type controller. The calculation method is based on 3 points: (1) the integral action compensates for the time constant T: Ti = T: ⇒ the loop gain GB is simplified and is written: GB ¼

G ð1 þ Td s ÞKeLs Ts

DOI: 10.1051/978-2-7598-2608-7.c007 © Science Press, EDP Sciences, 2022

Practical PID Handbook

160

(2) G and Td are determined to obtain a gain margin Mg = 2 9 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > > > KG 1 þ ð T x Þ > >
 d c 1 = < jG ðjx Þj ¼ KG ð1 þ jTd xÞejLx ¼ B c GB ðjxÞ ¼ jT x ) 2 T xc > > xc ¼ critical frequency > > p > ; : uðxc Þ ¼   Lxc þ arctgTd xc ¼ p > 2 PI controller: Td = 0; we deduce: G ¼ p4 K1 TL PID controller: we choice Td xc ¼ 1 ) arctg Td xc ¼ p4 ( pffiffi 1 T G ¼ 83p 2K L We then find: Td ¼ 4L 3p (3) we check the phase margin The resolution gives: for the PI controller: Mφ = 45° for the PID controller: Mφ ffi 60° From the results giving G and Td for the series PID, we can find the formulas for the other structures (see summary table 7.1). Also given are 3 nomograms making it possible to graphically determine the actions P, I, D from K, L, T, in the case of P, PI and PID series controller (nomograms 1–3).

7.1.2

Integrating Process

Integrator + pure delay process (figure 7.2).

FIG. 7.2 – Control of an integrating process.

Series regulator type PD. G and Td are determined to obtain a gain margin Mg = 2. 9 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > > > G 1 þ ð T x Þ > >
 d c 1 = < G ð1 þ jTd xÞejLx ¼ jGB ðjxc Þj ¼ GB ðjxÞ ¼ jT x ) 2 T xc > > xc ¼ critical frequency > > p > ; : uðxc Þ ¼   Lxc þ arctg Td xc ¼ p > 2

Cancellation Methods

161

We find the previous equations by making K = 1. Hence the results: G ¼ p4 TL Mφ = 45° – P controller: Td = 0 3p T 4L – PD controller: Td xc ¼ 1 G ¼ 8pffiffi2 L Td ¼ 3p Mφ ffi 60° – PID series controller: if an integral action is required to cancel the deviation, because of a disturbance, we will choose Ti ≥ 10 Td (for example Ti ≈ 5L) which hardly changes the stability.

7.1.3

Ultimate Gain Method

The integrator with pure delay process can be identified by the ultimate gain method (see chapter 4): oscillation period Tc for a critical gain Gc in closed loop with proportional controller. Identification leads to relationships: L = 0.25Tc Gc = 2π TTc By replacing T and L function of Gc and Tc in the previous relations, we obtain: pffiffiffi 3 2 Tc Gc Td ¼ G¼ 8 3p and we keep the eventual integral action Ti = 10Td ffi Tc. The advantage of this method is the speed of calculation from real tests in closed loop: does not require prior identification of the open loop process. The settings are gathered in summary table 7.1.

7.1.4

Performances

7.1.4.1

Stable Process

We kept the same process as that used with the Ziegler(and Nichols methods: L ¼ 0:805 Ls 1 corresponding to the Broïda model 1eþ Ts with ð1 þ s Þ3 T ¼ 2:44 The controller is of the series type with actions on the deviation. The derivative action is filtered with N = 10. Table 7.1 gives the actions: G = 2.51 Ti = 2.44 Td = 0.322. The responses are given in figure 7.3 where the response of the Ziegler and Nichols setting was added in closed loop (curve n°2). They show a good compromise between tracking and regulation.

7.1.4.2

Integrating Process

We kept the same process as that used with the Ziegler and Nichols methods:
1 eLs L¼3 with corresponding to the integrating model 3 T ¼1 Ts s ð1 þ s Þ

Practical PID Handbook

162

FIG. 7.3 – Stable process responses. The critical gain method gives: Gc = 0.89 Tc = 10.9. The controller is of the mixed type with actions on the deviation. The derivative is filtered with N = 10. By applying the formulas in table 7.1, we obtain the 2 settings corresponding to the 2 methods: open loop G ¼ 0:28 Ti ¼ 15 Td ¼ 1:2 curves 1: closed loop G ¼ 0:47 Ti ¼ 11 Td ¼ 1:09 curves 2: The responses are given in figure 7.4, where the variation of the measurement, in tracking, has been dilated by 2 for readability. The responses in closed loop of Ziegler and Nichols are also added, in order to compare (curves n° 3). Tracking is better amortized thanks to a slower integral action. The regulation here is average due to the difference between process and model, the more pronounced the higher the L/T ratio; which also explains the difference between the 2 open loop and closed loop settings.

7.1.4.3

Summary

The results are collated in summary table 7.1. They are followed by nomograms 1–6. Dindeleux’s settings give much better damped responses than those of Ziegler and Nichols. They are rather suited to tracking, because of the compensation (which only concerns stable systems) and we obtain a better compromise between tracking and regulation. The results are similar for the integrating processes if the L/T ratio is not too high.

Cancellation Methods

163

FIG. 7.4 – Integrating process responses.

TAB. 7.1 – Summary of Dindeleux settings. PID type

G

Ti Stable process:

P P+I P(1 + I) P+I+D P(1 + I + D) P(1 + I)(1 + D)

0:785 T K L 0:785 T K L 0:785 T K L  0:83 T þ 0:4 K L  0:83 T K L þ 0:4   0:83 T K L

P+I P(1 + I) P+I+D P(1 + I + D) P(1 + I)(1 + D)

Nomo 1

∞

0

1.27KL

0

–

T

0

2

1.2KL

0:33 K T

–

T + 0.4L

0:4L 1 þ 0:4 L=T

–

T

0.4L

3

Unstable process P

Td

KeLs 1 þ Ts

eLs Ts

0:785 TL

∞

0

4

0

–

0:785 TL

6:4 LT 5L

0

5

2 6 LT

0.33T

–

5.4L

0.37L

–

5L

0.4L

6

0 0

– –

0 0.056GcTc

– –

0.096Tc 0.1Tc

– –

0:785 TL 0:9 TL

0:9 TL

0:83 TL

2

Unstable process: ultimate gain GcTc P P+I

0.5Gc 0.5Gc

P(1 + I) P+I+D

0.5Gc 0.58Gc

P(1 + I + D) P(1 + I)(1 + D)

0.58Gc 0.53Gc

∞ Tc 2G c

Tc Tc 1:9 G c

1.1Tc Tc

164

Practical PID Handbook

Cancellation Methods

165

166

Practical PID Handbook

Cancellation Methods

167

168

Practical PID Handbook

Cancellation Methods

169

Practical PID Handbook

170

7.2

Haalman Method (Haalman, 1965)

The compensation is similar to that of Dindeleux but extended to 2nd order models with delay: F ðs Þ ¼

KeLs ð1 þ TsÞð1 þ s sÞ

or

eLs Ts ð1 þ ss Þ

The controller is of the series type with actions on the deviation; pure derivative (N ≥ 10): U ¼G

 ð1 þ Ti s Þð1 þ Td sÞ  Sp  M Ti s

For stable processes, the integral action compensates for T and the derivative action compensates for τ: Ti = T, Td = τ. For integrating processes, the controller is of the PD type: Td = τ. Under these conditions the loop gain is written: GB ¼

GKeLs ðK  1 for integrating processÞ Ts

The Haalman setting consists in choosing the gain G to obtain a loop gain depending only on L: GB ¼ a

eLs aT )G¼ KL Ls

The value of a recommended by Haalman is a = 2/3. It results from the minimization of the ISE criterion (Integral of Square Error), for a pure delay type process, controlled by an integrator controller and on a disturbance step (the minLs imum is obtained for Ti = 1.5L) which gives a loop gain: GB ¼ 23 eLs . This loop gain is kept for the other types of controller (PI, PD and PID). The settings are summarized in table 7.2 with the conversions in the other controller forms.

TAB. 7.2 – Haalman settings: a = 2/3. Stable process

Integrating process

Ls

Cont. G Ti Td

Ls

Ke F ðs Þ ¼ ð1 þ Ts Þð1 þ ss Þ

F ðs Þ ¼ Tsðe1 þ ssÞ

PID controller

PD controller

Parallel

Mixed

Series

Parallel

Mixed/Series

þs a TKL aKL Ts a KL

þs a TKL T+τ

T a KL T τ

T a KL ∞ a TLs

T a KL ∞ τ

Ts T þs

Cancellation Methods

171

Setting a = 2/3 gives the following margins: Gain margin: Mg = 2.36 (7.5 dB) Phase margin: Mφ = 50° Module margin: MΔ = 0.52 Example Figure 7.5 illustrates the Haalman setting (with a = 2/3) on a stable process and shows an almost constant overshoot of around 18% for a Broϊda model (PD controller). For the 2nd order model (series PID controller) the overshoots are greater and less constant, in relation to the filtering of the derivative (here N = 10). The response time essentially depends on delay L. A second adjustment with a = 0.4 makes it possible to limit the overshoots, in particular on the Broïda model. Use The simplicity of the loop gain, after cancellation, gives simple expressions for the gain and phase margins. We get indeed: p MG ¼ 2a M/ ¼

p a 2

As an indication, table 7.3 gives some values of margins and overshoot (D%) in tracking on the Broïda model (PD controller). TAB. 7.3 – Performances for a Broïda model. a

MG

M/ 

1 0.785 0.667 0.6 0.524 0.5 0.4

1.57 2 2.36 2.62 3 3.14 3.92

32.7 45 50 55.6 60 61.4 67.1

D% L = 0.1T to 4T 50 29 17.6 11.9 5.9 4.3 0.12

to to to to to to to

52 31 18.5 12.8 6.6 4.8 0.22

The overshoots are to be increased for 2nd order compensation (PID controller), especially since L is small (L/T < 1). This method is interesting because it separates the actions of the controller: integral and derivative cancel the time constants. The gain can then be adjusted separately according to the T/L ratio to obtain the desired loop gain (gain margin, phase margin) or a desired overshoot of the tracking step response. Given the compensation, this method is rather suitable for tracking, especially for integrating systems since there is no integral action to cancel the disturbance deviation. In regulation, the disturbance Pmax increases with L/T in the usual way. It is well suited for delay processes and provides a robust setting allowing a wide range of delay L.

172

Practical PID Handbook

FIG. 7.5 – Haalman settings responses on stable processes.

Cancellation Methods

7.3

173

Cancellation for 2nd Order

This paragraph concerns stable or integrating 2nd order processes, without delay.

7.3.1

Stable Processes

These are described by the transfer function: F ðs Þ ¼ ð1 þ TsKÞð1 þ ssÞ with T  s It is assumed that the disturbance X is unitary at the input of the process (the static gain K being relative only to the command U). The controller is of the series type because it is the most suitable for the cancellation of the poles of F(s). We assume the filtered derivative (factor N). The loop corresponds to the diagram in figure 7.6, where the controller is expressed as a fraction: the numerator compensates for the denominator of F(s): N þ1 Td ¼ s Ti ¼ T and N

FIG. 7.6 – Loop for cancellation. KG . ð1 þ N sþ 1sÞ We can then calculate the transfer functions in tracking and regulation. Figure 7.6 corresponds to actions on the deviation. For a derivative on the measurement, the derivative block is found in the measurement branch but does not change the cancellation of τ. We finally obtain the transfer functions of table 7.4 where we also considered a pure derivative (N → ∞), for information.

The compensation simplifies the loop gain: GB ðs Þ ¼ Ts

TAB. 7.4 – Closed loop transfer functions. Tracking

M Sp

Regulation Ti = T

Cancellation with pure derivative 1 D/E T s 1 þ KG 1 D/M   T 1 þ KG s ð1 þ ss Þ



1þ D/M



T KG

1 s þ KG ðTNsþ 1Þ s 2

1þ 1þ

T KG

sþ

s N þ1 s

Ts 2 KG ðN þ 1Þ s



ð1 þ ss Þ





T KG

1þ

Td = τ

T KG s

s ð1 þ Ts Þð1 þ ss Þ

Ti = T

Cancellation with filtered derivative D/E

1þ

T KG

M X

 s 1þ

Td ¼ s N þ1 s

T Ts 2 KG s þ KG ðN þ 1Þ s



N s N þ1



ð1 þ Ts Þð1 þ ss Þ

Practical PID Handbook

174

There remain the 2 parameters G and N to be determined according to the desired performance, linked to the 2nd order in s2 at the denominator of the transfer functions: it is characterized by its own frequency ω0 and its damping factor ζ which can be linked to G and N by identification: ( 2f ) ( ) T KG ¼ N4 þf21 Ts x0 ¼ KG ) 1 1 ¼ KG ðTNsþ 1Þ x0 ¼ N2 þ x2 fs 0

The adjustment comes down to the choice of damping ζ and speed ω0 of the desired response.

7.3.1.1

Tracking

The preceding relations make it possible to express the parameters G and N according to ζ and the desired response time trd: ( ) KG ¼ N4fþ2 1 Ts T x 0 t r ð fÞ ) KG ¼ N þ1 2ftrd x0 trd ¼ 2fs trd ω0tr(ζ) is the reduced response time and can be read on curve A18 (where ω0tr is called tr/T), depending on ζ chosen (see appendix A.5.1). In practice we choose ζ between 0.7 and 1 so that we can approximate the curve ω0tr(ζ) to a line: y # 6(ζ  0.2). A simpler solution consists in approximating the tr ðfÞ to a line (figure 7.7). ratio x02f

FIG. 7.7 – Approximation of ω0tr/2ζ.

The compensation settings are then given by the relations: ( ) KG ¼ Tx2ft0 trdr ðfÞ ffi ð1:36 þ fÞ tTrd Ti ¼ T N þ 1 ¼ 4f2 KG Ts

Td ¼ N Nþ 1 s

ð7:1Þ

An example of setting is given in figure 7.8 for ζ = 0.7, 0.8, 1, for a process 1/(1 + s)2 and for unit steps (ΔSp and ΔX). The command U in tracking has been divided by 10 for readability.

Cancellation Methods

175

FIG. 7.8 – Settings example for process 1/(1 + s)2. If N is imposed (by construction for example) it remains only to choose ζ to determine the gain G. The response time is then imposed: tr ¼

1 2fs x0 tr ðfÞ x 0 t r ð fÞ ¼ x0 N þ1

We can see that N + 1 is proportional to ζ2: interest in decreasing ζ to decrease the 1st command G(N + 1) during a setpoint step. Hence a choice ζ # 0.7 (ζ2 = 0.5) which gives an overshoot D ≈ 5% (derivative on the deviation). The corresponding reduced response time is ω0tr = 2.93. 4:14s Hence the response time: tr ¼ 2:07T KG ¼ N þ 1. Figure 7.9 gives the response time tr/τ which makes it possible to choose N, which avoids calculating it. We then deduce G and Td directly from N (table 7.4).

FIG. 7.9 – Filtered derivative in tracking: tr/τ as a function of N for ζ = 0.707.

Practical PID Handbook

176

For the derivative on the measurement, the response is slowed down by the phase 1þ

s

s

þ1 delay ð1 þN ss Þ and the overshoot will be > > > ffi ð1:36 þ fÞ =

> s N > ; : N þ 1 ¼ 4f2 G Td ¼ s> T N þ1 We can also directly choose N and deduce the gain G by the relation: G¼

N þ1T 4f2 s

Cancellation Methods

181

TAB. 7.7 – Integrating process: closed loop transfer functions. Integrating process Tracking

1 Ts ð1 þ ss Þ

M Sp

Regulation

M X

PD controller (without integral) Cancellation with pure derivative: Td = τ 1 1 1 D/E T   s 1þ G T G 1þ G s ð1 þ ss Þ 1  D/M  T 1þ G s ð1 þ ss Þ Td =N Nþ 1 s

Cancellation with filtered derivative: D/E D/M

1 1þ 

T Ts 2 G s þ G ðN þ 1Þ s s 1 þ N þ1 s

1þ

T Ts 2 G s þ G ðN þ 1Þ s



1  G 1þ

1þ

s N þ1 s

T Ts 2 G s þ G ðN þ 1Þ s



ð1 þ ss Þ

ð1 þ ss Þ PID controller

Cancellation with pure derivative: 1 þ Ti s D/E 1 þ Ti s þ TGTi s 2 1 þ Ti s D/M ð1 þ Ti s þ TGTi s2 Þð1 þ ssÞ Cancellation with filtered derivative: D/E

D/M

1 þ Ti s T Ti 2 T Ti s 3 G s þ G ðN þ 1 Þ s   ð1 þ Ti s Þ 1 þ N sþ 1 s   1 þ Ti s þ TGTi s 2 þ G TðNTþi s1Þ s 3 ð1 þ ss Þ

1 þ Ti s þ

Td = τ 1 Ti s   G 1 þ Ti s þ TGTi s 2 ð1 þ ss Þ Td =

N N þ1 s

 Ti s 1 þ 1  G 1 þ Ti s þ T Ti s 2 þ G

s N þ1 s



T Ti s 3 G ðN þ 1Þ s



ð1 þ ss Þ

The choice ζ#0.7 (ζ2 = 0.5) will allow us to use the curves of figure 7.9 to choose N according to the desired speed (tr/τ). The overshoot is 5%. The difference lies mainly in the static error E (figure 7.13) in regulation: E X

2

¼ G1 ¼ N4fþ 1 Ts or also: E% = Bp% (proportional band).

FIG. 7.13 – PD controller and integrating process: disturbance step X.

182

Practical PID Handbook

The static error will be reduced by increasing N, therefore the gain G, which will also increase the speed of the tracking. The adjustment formulas are given in table 7.8 which derives from table 7.5 (K = 1) corresponding to ζ2 = 0.5. Note that this time the equivalences are complete since there is no integral (Ti/Td → ∞). An average adjustment is possible by making N = 1: we can use table 7.5 by making K = 1. This default setting is incorporated in table 7.8, with also the case of the pure derivative. TAB. 7.8 – PD controller and integrating process: cancellation settings (ζ2 = 0.5).

For integrating systems the time constant can be greater than the integration time T, and can lead to large commands. If we aim for example at a response time tr ≈ tr0/2, we obtain an initial command G(N + 1) which varies from 4.6 to 9 for τ = 0.2 to 2 (T = 1). A more practical way to choose N is to fix an order of magnitude of the initial command G0 = G(N + 1): rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð N þ 1Þ 2 T s ) N þ 1 ¼ 2 G0 with G0  GðN þ 1Þ G ð N þ 1Þ ¼ s T 2

Cancellation Methods

183

FIG. 7.14 – PD controller: setting example. Figure 7.14 shows an example of an instantaneous gain setting G0 of the order of 3 for τ/T = 0.2, 1, 2. The control variation has been divided by 2 for readability.

7.3.2.2

PID Controller

7.3.2.2.1 Pure Derivative In the case of a pure derivative, table 7.7 shows that the transmittance in tracking (D/E) is of the form:

1 þ xb0 s b ¼ 2f with 4f2 ¼ G TTi 1 þ x2f0 s þ x12 s2 0

Such a system (β = 2ζ) gives an overshoot 20 dB). Step response for G = 7.3: D ffi 9.2% and tm/T ffi 0.62: figure 10.13. 2nd case: figure 10.12b. FðsÞ ¼

1 Tsð1 þ T 1 sÞ3

with T 1 ¼ T =10

Controller: Ti = T Td = T1. The curve plotted for G = 1 shows that the minimum phase shift is φmin ffi −139°: whatever the gain, the phase margin will be less than 41°. On the other hand, it is not possible to tangent the contour M = 1.1 dB, but only the contour (not traced) M ffi 1.54 dB: translation 8 dB ⇒ G = 2.5. If we refer to the second order, there corresponds an overshoot D ffi 20% (instead of 15% for 1.1 dB). We measure on the translated curve: φ(0 dB) = −141° Mφ = 39°; x0 = 2.5 tm/T ffi 3/2.5 = 1.2; G(−180°) = −16 dB Mg = 16 dB. Step response for G = 2.5: D ffi 40% and tm/T ffi 1.1: figure 10.13.

M/Sp D 9.2% tm/T=0.625 F1(s)=1/Ts(1+T1s)(1+T2s)(1+T3s) R(s)=G(1+1/T1s)(1+T2s) G=7.3 T1=1 T2=0.1 T3=0.01 T4=0.001

M/Sp D 39.5% tm/T=1.1 F2(s)=1/Ts(1+T1s)3

R(s)=G(1+1/Ts)(1+T1s)

T =1

T1=0.1

G=2.5

FIG. 10.13 – Step responses integrating processes case 1 and 2.

Practical PID Handbook

280

In this last example, it would be necessary to visibly modify the actions (increase Td in particular). However, this example clearly illustrates the complementarity of concepts such as phase margin and resonance factor, as well as the synthetic nature of the curves in Black’s chart.

10.3

Kessler’s Method (Kessler, 1958, 1955)

It is a frequency method which applies to second or third order processes: F ðs Þ ¼

K ð1 þ T 1 sÞð1 þ T 2 s Þð1 þ T R sÞ

TΣ represents the sum of all the parasitic time constants and delays. The process can also be integrating: F ðs Þ ¼

K T 1 s ð1 þ T 2 s Þð1 þ T R sÞ

Kessler’s method is based on the criterion of symmetrical optimum (Kessler, 1958): the corrector is adjusted in order to obtain for the GB loop gain, a slope –1 (20 dB per decade) around the frequency ω0dB with 1 octave on the right and m octaves on the left (figure 10.14): m = 1 for a 2nd order process. m = 2 for a 3rd order process. GB represents the modulus of the product RF in the Bode plane. The controller is of the series type with pure derivative. The gain will therefore be chosen to obtain ω0dB = 2T1 R . The other actions will then be determined to obtain 1 or 2 octaves to the left of ω0dB. The method is therefore also applicable to integrating processes. We limit here to the representation of Bode: magnitude (in dB) and phase shift (in degrees) according to the frequency ω (rad/s and logarithmic scale).

Sp

R

1 or 2 octaves

F

M

1 octave ω

ω0dB

1/TΣ GB(ω)

FIG. 10.14 – Kessler’s symmetrical optimum.

(log)

Frequency Methods

10.3.1

281

2nd Order Process

The study can be common to aperiodic and integrating processes by taking as transfer function: K F ðs Þ ¼ ða þ T 1 s Þð1 þ T R sÞ with a ¼ 1 aperiodic process a ¼ 0 and K ¼ 1 integrating process The controller is of the PI type: R(s) = G(1 + 1/Ti s). G heightens F(jω) to obtain ω0dB = 2T1 R . G|F(jω0dB)| =1 and |F(jω0dB)| = K T 1 x1 0dB .

T1 ω0dB = 2T1 R → G ¼ K1 2T (K = 1 for integrator). R One then chooses Ti = 4 TΣ to obtain a symmetrical zone of slope –1 (figure 10.15). This is only possible if T1 ≥ 4TΣ, validity condition for a 2nd order process.

−1

1

PI

G

ω0dB Integrator

1

1

1

4

1

ω

(log)

1

2

KG

−1

−1 1

−2 |F(jω)| for K=1

1

−2

−1

ω

1 1

1 4

1

(log)

1

2

|GB(jω)|

FIG. 10.15 – Kessler’s setting: 2nd order process.

−2

Practical PID Handbook

282

For these settings of KG and Ti, we obtain in closed loop: M 1 þ 4T R ¼ T Sp 1 þ 4T R s þ 8 2R s ða þ T 1 s Þð1 þ T sÞ R T1

10.3.1.1

Integrating Process (a = 0)

10.3.1.1.1 Tracking The transfer function is independent of the integration time T1 and is written: M 1 þ 4T R s ¼ Sp ð1 þ 2T R sÞð1 þ 2T R s þ 4T 2R s 2 Þ The step response gives: D = 43, 4% t m = 5.8 TΣ t r = 14.7 TΣ. (tm is the rise time of the 1st overshoot: figure 10.16). We could reduce this overshoot by putting the gain on the measure, which removes the phase advance 1 + 4TΣs in the numerator. We then obtain: D = 8% tr = 11.9 TΣ.

ΔM Sp

ΔM

±5% X

D%

Pmax% of X ±5% of X

t

t 0

tm

tr

0

tm

trec

FIG. 10.16 – Tracking and regulation.

10.3.1.1.2 Regulation We obtain for a unit disturbance X at the input of the process: M 8T R T Rs ¼ X T 1 ð1 þ 2T R sÞð1 þ 2T R s þ 4T 2R s 2 Þ The step response has a unique shape and an amplitude proportional to TΣ/T1. The most unfavorable case corresponds to the borderline case T1 = 4TΣ for which we measure: Pmax ffi 40%

tm ffi 4:1 TR and trec ffi 9TR

Frequency Methods

10.3.1.2

283

Aperiodic Process (a = 1)

The performances are not constant and depend on TΣ/T1.

10.3.1.2.1

Tracking M 1 þ 4T R s ¼ Sp 1 þ 4T R s þ 8 T 2R s ð1 þ T 1 s Þð1 þ T sÞ R T1

The borderline cases correspond to TΣ/T1 = 0 and TΣ/T1 = 0.25. For TΣ/T1 = 0 we find the performance of the integrator. For TΣ /T1 = 0.25 we obtain D = 4.32% tm = 6.3TΣ tr = 4.14 TΣ. We can also be interested in tr as a function of T1. A more complete simulation gives a limit: tr < 1.75 T1.

10.3.1.2.2

Regulation M 8T R TR s ¼ X T 1 1 þ 4T R s þ 8 T 2R s ð1 þ T 1 s Þð1 þ T sÞ R T1

A simulation shows that the most unfavorable case corresponds to TΣ/T1 = 0.25 for which we measure: Pmax ¼ 29:15% tm ¼ 3:76 TR

10.3.2

trec ¼ 11TR ¼ 2:75 T1

3rd Order Process

The process can be aperiodic or integrating (single or double): F ðs Þ ¼

K ða þ T 1 s Þðb þ T 2 s Þð1 þ T R sÞ

Aperiodic system a = b = 1. Simple integrator system K = 1 a = 0 b = 1. Double integrator system K = 1 a = b = 0. The controller is of the series type: Uc = G(1 + 1/Ti s)(1 + Td s)(Sp − M). The optimum setting must ensure a slope −1 over 2 octaves to the left of ω0dB = 1/2TΣ. This leads to the condition: T1, T2 ≥ 8TΣ and the optimum Kessler setting is: Ti = Td = 8TΣ. If a time constant (T2) is < 8TΣ, it is cancelled by Td and we are brought back to the 1st case with a PI controller. The performances are, however, different in regulation. The graphical representation (figure 10.17) allows to calculate KG such that KG = M1•M2•M3 to ensure a loop gain of 0 dB for the frequency 1/2TΣ:

Practical PID Handbook

284 Integrator Simple Double

−1

+1

PID (G=1)

0dB 1

1

− –1

1

ω (log)

1 8

2

Integrator simple double

F(jω) for K=1

−2

−1

1

0dB

–2

1

1 2

1

1

8

−3

1

2

ω (log)

M1dB M2dB

−3

M3dB

KG(dB) for GB(1/2TΣ)=0dB

−1 ω0dB

GB for KG=1

−2

FIG. 10.17 – Kessler’s setting: 3rd order process. M3: slope −1⇒ M3 = 4.
 T2 3 M2: slope −3⇒ M2 = . 8T R M1: slope −1⇒ M1 =

8T 1 T R . T 22

T 1T 2 . 16KT 2R From the settings of G, Ti and Td, we can deduce the closed loop transfer functions: M ð1 þ 8T R sÞ2 ðs Þ ¼ tracking: Sp DðsÞ M 128T R 3 s ðs Þ ¼ regulation: X T 1 T 2 DðsÞ 128T 3R s ða þ T 1 s Þðb þ T 2 s Þð1 þ T R sÞ: with Dðs Þ ¼ ð1 þ 8T R sÞ2 þ T 1T 2 KG = M1•M2•M3 allows to deduce: G ¼

Frequency Methods

10.3.2.1

285

Double Integrator Process (a = b = 0) 2

3

Dðs Þ ¼ 1 þ 16T R s þ 64T 2R s þ 128T 3R s þ 128T 4R s

4

10.3.2.1.1 Tracking The response to a setpoint step is independent of T1 and T2; a performance measurement gives: D ffi 50%

tm ¼ 5:93TR

tr ¼ 17:7TR

We see here the important influence of the numerator (1 + 8TΣ s)2; it will be appropriate to make the derivative and the gain act on the measurement to limit the overshoot.

10.3.2.1.2 Regulation The form of the response is unique for TΣ given. The amplitude is proportional to TΣ/T1 and TΣ/T2. The most unfavourable case corresponds to TΣ/T1 = TΣ/T2= 1/8. A performance measure gives: Pmax ¼ 12:5%

10.3.2.2

tm ¼ 8:74TR

the

borderline

case:

trec ¼ 18:7TR

Integrating Process (a = 0, b = 1) Dðs Þ ¼ ð1 þ 8T R sÞ2 þ

128T 3R 2 s ð1 þ T 2 s Þð1 þ T R sÞ T2

Performances depend on the TΣ/T2 ratio.

10.3.2.2.1

Tracking M ð1 þ 8T R sÞ2 ðs Þ ¼ Sp DðsÞ

A borderline case is obtained for T2 = 8TΣ and we measure: D ffi 26.75% tm = 6.22 TΣ tr = 13.74 TΣ. The other borderline case is obtained for T2 >> TΣ i.e., TΣ/T2 → 0. We find the double integrator: D ffi 50% tm = 5.93TΣ tr = 17.7TΣ. A more complete simulation shows that the overshoot is all the weaker the closer one gets to the limit T2 = 8TΣ.

Practical PID Handbook

286

On the other hand, the response time decreases with TΣ/T2. There is a minimum towards TΣ/T2 = 0.025 linked to the discontinuity of the passage of the curve at 95%. The value of this minimum is 11.1TΣ (figure 10.18) hence ultimately: 11.1TΣ < tr < 17.7TΣ. tr/TΣ 20

10

TΣ /T2 0 0.01

0.05

0.1

0.125

FIG. 10.18 – Response time as a function of TΣ/T2.

10.3.2.2.2

Regulation M T 2R T R s ðs Þ ¼ 128 X T 1 T 2 DðsÞ

The maximum magnitude is proportional to TΣ/T1. The most unfavourable case is therefore TΣ/T1 = 1/8. A complete simulation shows that the performances deteriorate all the more as one tends towards the limit TΣ/T1 = 1/8 for which we measure: Pmax ffi 11% tm = 9.44 TΣ trec = 21.3 TΣ. Pmax depending on TΣ/T1, and TΣ/T1 < 1/8 allows to write: TR < 11%. Pmax < 88 T1

10.3.2.3

Aperiodic Process (a = b = 1)
Dðs Þ ¼ ð1 þ 8T R sÞ2 þ 128T R s

TR þs T1




 TR þ s ð1 þ TR sÞ T2

The borderline cases correspond to the 2 values 0 and 1/8 for TΣ/T1 and TΣ/T2. When the 2 values tend towards zero, we find the double integrator. The other borderline case corresponds to T1 = T2 = 8TΣ.

10.3.2.3.1

Tracking M ð1 þ 8T R sÞ2 ðs Þ ¼ Sp DðsÞ

Frequency Methods

287

A complete simulation shows that the best performances are obtained at the limit T1 = T2 = 8TΣ; we measure: D = 4.32% tm = 6.28 TΣ tr = 4.14 TΣ. We can also be interested in tr/T1, T1 being the largest of the time constants: tr/T1 tends towards zero when T1 tends towards zero with a maximum around TΣ = T1/10; we then measure tr/T1 ffi 0.99. The limit performances are therefore: 4.32% < D < 50% 4.14 TΣ < tr < 17.7 TΣ or tr < 0.99T1.

10.3.2.3.2

Regulation M T 2R T R s ðs Þ ¼ 128 X T 1 T 2 DðsÞ

Performance deteriorates when approaching the limit T1 = T2 = 8TΣ, for which we measure: Pmax ffi 9.3% tm = 10TΣ trec = 22.4TΣ. Compared to T1, we also obtain the worst performance for T1 = T2 = 8TΣ. ⇒ trec = 2.8 T1.

10.3.3

Performances

The step response curves are plotted in figure 10.19 for stable processes, and figure 10.20 for integrating processes. TΣ = 1 is taken as a unit of time. In tracking, the curves are very similar in their shape: overshoot can be significant (up to 30–40%) but with an almost constant rise time: tm ≈ 6TΣ. In regulation, the responses are also very similar. Kessler’s settings therefore give very robust regulation, the speed of which is practically linked only to TΣ. The limitations are those of the conditions on T1, T2: T1, T2 > 4 or 8 TΣ.

10.3.4

Summary

Tables 10.1 and 10.2 summarize the Kessler settings for aperiodic and integrating processes. Let us recall that the performances are given with the usual conventions (according to figure 10.16). The series controller settings have been converted to the other parallel and mixed structures.

Practical PID Handbook

288

Tracking

Regulation

2nd order

T2 cancelled by Td

3rd order

3rd order

FIG. 10.19 – Kessler’s step responses for stable processes.

Frequency Methods

289

Tracking

Integrator 2nd order

Regulation

1

1 (1 + )

T2 cancelled by Td

Integrator 3rd order

1

(1 +

Integrator 3rd order

1

(1 +

1 2

) (1+ )

1 2

) (1+ )

FIG. 10.20 – Kessler’s step responses for integrating processes.

290

Practical PID Handbook

TAB. 10.1 – Kessler’s settings for stable processes.

Frequency Methods

291

TAB. 10.2 – Kessler’s settings for integrators.

Practical PID Handbook

292

10.4

KLV Method (Kessler, Landau and Voda) (Voda and Landau, 1995)

This is a self-calibration method of the PID whose starting point follows from the Kessler method, and which uses 3 basic settings. The disadvantage of the Kessler method lies in the determination of TΣ. The KLV method allows to be freed from it by making an identification in closed loop by the relay experiment. One can thus make the link between TΣ and the frequency of the oscillations giving a phase shift of −135°: TR ¼

a x135

α is a factor which depends on the process, in particular on the T1/TΣ ratio for a 2nd order. The KLV method uses the 2nd order model as well as the Broïda model to deal with the question of pure delay. The controller is of the series type: U = P(1 + I)(1 + D)(Sp − M).

10.4.1 10.4.1.1

Basic Settings Setting 1: Symmetrical Optimum

The model is of the 2nd order aperiodic (a = 1) or integrator (a = 0 and K = 1): F ðs Þ ¼

K ða þ T 1 s Þð1 þ T R sÞ

The symmetric optimum criterion was studied in the Kessler method and led to the following adjustment formulas (see section 10.3.1) 1 T1 G ¼ 2K T R ðK ¼ 1 for integratorÞ Ti ¼ 4 T R : The validity condition is T1 ≥ 4TΣ. For these settings of KG and Ti, we obtain in a closed loop: M 1 þ 4T R s ¼ Sp 1 þ 4T s þ 8 T 2R s ða þ T s Þð1 þ T sÞ R 1 R T1

10.4.1.2

Setting 2: Modulus Optimum (Kessler, 1955)

The model is of the 2nd aperiodic order: F ðs Þ ¼

K ð1 þ T 1 s Þð1 þ T R sÞ

Frequency Methods

293

The optimum modulus criterion consists in obtaining a slope −1 (−20 dB per decade) over the frequency interval (1/T1, 1/TΣ) which implies the cancellation of T1 by Ti (figure 10.21). The gain is then adjusted to obtain a damping ζ # 0.7. The adjustment formulas are written: ( 1 T1 G ¼ 2K TR Ti ¼ T1 For these settings the modulus margin is MΔ > 0.78 and the phase margin Mφ ≈65°. The closed loop transfer function is: M 1 þ TR ¼ Sp 1 þ 2T R s þ 2T 2R s 2 −1

1

PI

Integrator

G 1 1

−2

1

−1

1

2

ω (log)

KG

−1

F(ω) −2 (K=1)

ω0dB

GB(ω)

ω (log)

−2

FIG. 10.21 – Optimum modulus setting.

10.4.1.3

Setting 3: Case of a Delay

The time constant TΣ is replaced by a pure delay L, and the transfer function corresponds to the Broïda model: F ðs Þ ¼

KeLs T1 with \L  T 1 1 þ T 1s 4

The setting keeps the cancellation of T1 by Ti and the loop gain is written: G B ð jxÞ ¼ The phase shift is: φ(ω) = Lx  90°.

KejLx jT 1 x

Practical PID Handbook

294

The phase margin is: Mφ = 180° φ(ω0dB). The adjustment is made with a phase margin Mφ = 60°. We can deduce: p 1 Lx0dB ¼ # 6 2 T1 0dB G ¼ T 1x # 2KL : K

Hence the settings: Ti = T1

10.4.1.4

Summary Table

The results are collated in table 10.3.

TAB. 10.3 – PI settings for 2nd order processes.

Integ.

Aperiodic

Setting

1 F ðs Þ ¼ T 1 sð1 þ T R sÞ

1

T1 G ¼ 2T R

Ti = 4 TΣ

K F ðs Þ ¼ ð1 þ T 1 sÞð1 þ T R sÞ

1

T1 G ¼ 2KT R

Ti = 4 TΣ

K F ðs Þ ¼ ð1 þ T 1 sÞð1 þ T R sÞ

2

T1 G ¼ 2KT R

Ti = T1

3

T1 G ¼ 2KL

Ti = T1

Ls

F ðs Þ ¼ 1Ke þ T 1s

10.4.2

Mixed/E: U ¼ Gð1 þ T1i sÞ E

Process

Auto-Calibration

The method requires being able to identify TΣ or its equivalent L in the case of a delay.

10.4.2.1

2nd Order Model F ðs Þ ¼

K ða þ T 1 s Þð1 þ T R sÞ

Figure 10.22 illustrates the representation of Bode and shows that for the integrator (a = 0, K = 1), we can identify TΣ by the phase shift of −135°. For the 2nd aperiodic order, this phase shift gives a frequency close to 1/TΣ which can be written: x135 ¼ TaR . The objective is to express the adjustment as a function of point F(j x135 ) characterized in modulus and in phase: jF ð jx135 Þj ¼ Arctg

K a with x135 ¼ TR jð1 þ jT 1 x135 Þð1 þ jT R x135 Þj

T1 a þ Arctg a ¼ 135 TR

Frequency Methods

295

F(ω) (dB) KdB 0

1

1

1

ω

(log)

φ°

ω

0

(log)

−45

Aperiodic

−90

Integrator

−135 −180

FIG. 10.22 – 2nd order process Bode diagram. By posing TΣ/T1 = t and by calling F135 the modulus of F ð jx135 Þ, the 2 relations are written: F 135 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ r  K 2 1 þ at 2 ð1 þ a2 Þ

t

TR a ¼ T 1 tg½135  Arctg a

ð10:1Þ

ð10:2Þ

This makes it possible to link α and F135/K. F 135 cosð135  Arctg aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ K 1 þ a2

10.4.2.2

Broïda Model F ðs Þ ¼

KeLs 1 þ T 1s

In the same way, we look for a relation between L and F ð jx135 Þ: we set α = Lω135 and L/T1 = a. Writing the transfer function for ω135 gives: jF ð jx135 Þj ¼ jð1 þ jTK1 x135 Þj with x135 ¼ La : Lω135 + Arctg T1 ω135 = 135°.

ð10:3Þ

Practical PID Handbook

296 We can deduce: F 135 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi 2 K 1þ a a2

a

L a ¼ T1 tgð3p 4  aÞ

ð10:4Þ

We deduce the relation between F135 and α:
 F 135 3p a ¼ cos 4 K

10.4.2.3

ð10:5Þ

Settings

The relations given by equations (10.2)–(10.5) have been plotted in figure 10.23, which shows zones to appear according to the ratio of the time constants TΣ/T1 and L/T1 to apply settings 1 or 2 or 3. To obtain simple formulas, the curves were approximated by lines. The KLV method is available in 2 versions depending on the values of F135/K.

10.4.3 10.4.3.1

KLV 1P Method PI Controller

This method derives directly from the Kessler setting which assumes TΣ/T 1 ≤ 0.25 and which corresponds to F135/K < 0.1 (figure 10.23). In the case of a delay, L is < 0.1 and can be assimilated to a time constant TΣ. Therefore setting 1 with 2nd order model applies. As t = TΣ/T is ≤ 0.25 we can see that α2/t2 is >> 1, so that equation (10.1) can be written: pffiffiffiffiffiffiffiffiffiffiffiffiffi Kt 1 T1 a 1 þ a2  F 135 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ¼ F 135 Kt KT R a ð1 þ a2 Þ You can then replace in setting 1 of table 10.3 with T R ¼ G¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a ð1 þ a2 ÞF 135

Ti ¼

a . x135

4a x135

The authors simplify the relations by choosing for α an average value 1.15 which corresponds to TT R1 ≈ 12. The case of the integrator is also treated by making α = 1. We obtain table 10.4 which has also been completed for the parallel structure.

Frequency Methods

297

TΣ/T1 L/T1

1

L/T1

TΣ/T1

2.3 g135 −0.3

15 g135 −1.25

0.5

0.1 0

g135=F135/K 0

0.1

0.2

0.3

0.4

0.5

0.6

α

2.5

Time cst

2.0

20 g135 − 0.6

Delay

1.5 1.15

1.15 g135 + 0.75

1.0

0.5

g135=F135/K 0

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 10.23 – Settings as a function of F135/K. TAB. 10.4 – KLV 1P method: PI settings. Settings Integrator Aperiodic

Parallel: U = (G + 0:35 F 135 0:285 G¼ F 135 G¼

1 Ti s)

E

11:5F 135 x135 16F 135 Ti ¼ x135

Ti ¼

Mixed: U = G (1 + 0:35 F 135 0:285 G¼ F 135 G¼

1 T i s)

E

4:6 x135 4:6 Ti = x135 Ti =

The controller adjustment is obtained by identifying the single point F(ω135) which can be obtained with the relay experiment with hysteresis, adjusting it for a phase shift of −135°.

Practical PID Handbook

298

The validity is F135/K < 0.1. It can be checked if necessary on condition of knowing the static gain K of the process, or by using the KLV 2P method.

10.4.3.2

PID Controller

The setting of a PID controller supposes having additional information on the process. As the purpose of the PID is to accelerate the response (compared to the PI ), the authors estimate that one can consider that the process would be of the 3rd order (additional time constant T2) and that the time constant TΣ would be more small (β times) than that estimated, which results in the relations: T2 ¼

1 x135

TR ¼

T2 b

The controller is of the series type where the derivative cancels T2. We are then brought back to the previous PI setting with α = 1 and TΣ = 1/βω135. We obtain table 10.5 where the acceleration factor β must be chosen < 2. The table also contains the settings of the mixed PID. TAB. 10.5 – KLV 1P method: PID settings. Series/E 1 )(1 + T d s) E U = (G + Ti s 1 b pffiffiffi 2 2 F 135

Mixed/E 1 þ T d s) E U = G (1 + Ti s bþ4 b pffiffiffi 8 2 F 135

Ti

4 bx135

bþ4 1 b x135

Td

1 x135

4 1 4 þ b x135

Actions G

10.4.4

KLV 2P Method

The purpose of this method is to extend the previous PI settings beyond the limits of Kessler, i.e., for TΣ > 0.25 T1, and to deal with the case of delays. Figure 10.23 enables zones to be cleared according to F135/K in order to choose one of the settings 1 or 2 or 3 (table 10.3). The curves are approximated by lines. We obtain the following different cases: F135/K ≤ 0.1. This is the PI case studied previously: the settings are those in table 10.4. 0.1 ≤ F135/K < 0.15 both models are possible: – 2nd order model: 0.25 < TΣ/T1 < 1 setting 2 applies: 8 T1 < G ¼ 2KT R : T ¼ T ¼ T1 a i 1 T R x135

(T with

R

T1

 1:15g 135  1:25

a  20g 135  0:6

Frequency Methods

299

We deduce the actions: G¼

1 2K ð15g 135  1:25Þ

Ti ¼

20g 135  0:6 ð15g 135  1:25Þx135

– Broïda model: 0.1 < L/T1 < 0.15. L small can be assimilated to a time constant TΣ( 0.25: setting 3 applies. We obtain with the approximation of the lines: 8 T1 > >  L

> : Ti ¼ T1 ¼ L x135 from which we deduce: G¼

1 2K ð2:3g 135  0:3Þ

Ti ¼

1:15g 135 þ 0:75 ð2:3g 135  0:3Þx135

All the results are summarized in tables 10.7 and 10.8 corresponding to the 2 models used. Note The difficulty is to know if the identification of ω135 corresponds to a time constant or to a delay. The method does not allow to know. The authors recommend to use a priori table 10.7 assuming a process with delay, especially as the range of the values of g135 is greater (0–0.6). Table 10.6 should only be used if you are sure that there is no delay. The ambiguity only arises for 0:1\g 135  0:15. This method requires knowing 2 points of the frequency response: F(ω135) and F(0) = K. The relay experiment allows these 2 identifications in a single closed loop test (Voda and Landau, 1995).

Practical PID Handbook

300

TAB. 10.6 – KLV 2P method: PI settings for process without delay. g 135  F 135 =K

Parallel: U = (G +

1 )E Ti s

0:285 F 135 16F 135 Ti ¼ x135 1 G¼ K ð30g 135  2:5Þ

0:1\g 135  0:15

Ti ¼

1 )E Ti s

0:285 F 135 4:6 Ti ¼ x135 1 G¼ K ð30g 135  2:5Þ G¼

G¼

g 135  0:1

Mixed: U = G (1 +

K ð40g 135  1:2Þ x135

Ti ¼

10g 135  0:3 ð7:5g 135  0:625Þx135

TAB. 10.7 – KLV 2P method: PI settings for process with delay. g 135  F 135 =K

1 )E Ti s

0:285 F 135 16F 135 Ti ¼ x135 0:35 G¼ F 135 11:3F 135 Ti ¼ x135 1 G¼ Kð4:6g 135  0:6Þ Kð2:3g 135 þ 1:5Þ Ti ¼ x135 G¼

g 135  0:1

0:1\g 135  0:25

0:25\g 135  0:6

10.4.5

Parallel: U = (G +

Mixed: U = G (1 +

1 )E Ti s

0:285 F 135 4:6 Ti ¼ x135 0:35 G¼ F 135 4 Ti ¼ x135 1 G¼ Kð4:6g 135  0:6Þ g 135 þ 0:65 Ti ¼ ð2g 135  0:26Þx135 G¼

Performances

Figure 10.24 gives an example of response for 2 processes of 3rd order: F 1 ðs Þ ¼

1 ð1 þ s Þ

3

and

F 2 ðs Þ ¼

1 ð1 þ sÞð1 þ 0:1sÞð1 þ 0:01s Þ

The adjustments have been made according to table 10.7 and are recorded in table 10.8. The controller is of the mixed type. We can compare these responses to those of Ziegler and Nichols for the same process (see chapter 6): the responses are better damped and give a good compromise between tracking and regulation.

Frequency Methods

301

FIG. 10.24 – Example of KLV2P setting on order 3 processes.

TAB. 10.8 – KLV2P settings on 3rd order processes. F(s)

1 ð1 þ s Þ3

1 ð1 þ sÞð1 þ 0:1sÞð1 þ 0:01sÞ

ω135(rad/s) F135 G Ti

0.938 0.4 0.81 2.08

9.52 0.078 3.65 0.48

Practical PID Handbook

302

10.5

Use and Choice

The 2 methods exposed at the beginning of the chapter (phase margin and resonance in Black’s plane) are very simple. Their implementation is closely linked to the tools used: it is necessary to have simulation software fitted with cursors allowing amplitude, phase and frequency readings to be carried out fairly quickly. Their interest is the possibility of personalizing the frequency response by choosing Ti and Td. This is also the difficulty if the process is not well known, because in some cases we only have an experimental response on paper; if the system is stable, we can possibly plot the tangents of slope –1, –2, –3,…, to get an idea of the time constants T1, T2, T3,…, (figure 10.25). The plot is all the more reliable when the time constants are well distributed, which is not necessarily known a priori. 1

1

1

1

2

3

ω (log)

FIG. 10.25 – Frequency reading. The Kessler method leads to robust adjustments and ensures a good compromise between tracking and regulation. It is suitable for a wide range of processes, stable or integrative. It is typically indicated in the case of electromechanical systems where the time constants are very different (electric motors in particular). The KLV methods have their interest in autocalibration for on-line adjustment, since they only require a single identification point (at 135°) for the KLV1P method. The results are quite close to the Kessler method. The KLV2P method requires additional knowledge of static gain. It is necessary to have software allowing the relay test. Compared to the Ziegler and Nichols methods, the KLV method gives more robust settings and a good compromise between tracking and regulation. The interest of frequency methods lies in the overall control of the stability of the loop, when the temporal responses are not a major concern. The frequency methods are studied in numerous works. Some references may be found in the bibliography: (Borne et al., 1993; de Larminat, 1993; Maret, 1993; Poulin and Pomerleau, 1997; Gille et al., 1989, 1987; Guyenot and Hans, 1989; Milsant, 1986; Naslin, 1968, 1962).

Chapter 11 Digital Settings Methods This chapter deals with regulation with a digital PID controller defined as an RS or RST degree 2 controller, with integral action (see chapter 2). The controller is sampled at the sampling period Te; it has a z.o.h.b. (zero-order hold block) which maintains constant command between 2 samples. This results in a control of the actuator in staircase steps of duration Te. The PID digital being of degree 2, it follows that the transfer function in z of the process to be controlled is also limited to degree 2: 1st or 2nd order systems. In practice, the continuous process is identified (see chapter 4) and its sampled model is obtained by applying the continuous/digital correspondence tables (see appendix). However, matching requires choosing the sampling period Te, which generally depends on the desired closed loop performance.

11.1

Choice of the Sampling Period

The choice depends on the bandwidth of the system to be sampled: cutoff frequency fc. The rule adopted in digital control (Landau, 1988) is as follows: fe = 6–25fc. fe is the sampling frequency. fc is the cut-off frequency of the closed loop system, therefore the desired cut-off frequency. We can translate this formula in simple cases where the closed loop system is assimilated to a 1st or 2nd order: this is what we do in practice when we choose the desired closed loop transfer function. – 1st order: time constant T ⇒ fc = 1/2πT ⇒ T/4 < Te < T. This corresponds to a step response which lasts between 2 and 10 samples approximately. – 2nd order: one can refer to the own frequency ω0 and to the damping factor ζ (see appendix):

DOI: 10.1051/978-2-7598-2608-7.c011 © Science Press, EDP Sciences, 2022

Practical PID Handbook

304

Example f ¼ 0:80:9 ) xc ffi 0:8 ) 0:3  x0 Te  1:3. Convenient order of magnitude: ω0Te = 1. You can also choose the sampling period from the response time and the number of samples desired: this is the case for a finite-time response.
Example:

desired t r ¼ 6s response in 3 samples

 ) Te ¼ 2s

t rd An even simpler rule is to choose T e ¼ 510 where trd is the desired response time.

Note: All of these rules make it possible to set orders of magnitude for the sampling period. In the particular case of processes with delay, it is often useful to choose the sampling period so that the delay is an integer multiple of Te. This avoids the appearance of a zero in the s.t.f. (sampled transfer function) of the process and removes the problem of hidden oscillations that can appear if we compensate for this zero. This is notably the case with Broïda’s model: the digital PID can only be suitable if delay L is less than or equal to Te; it will therefore be advantageous to choose Te = L while checking the above rules.

11.2

Zero Cancellation

The discretization of continuous systems results in a s.t.f. with poles and zeros; it is written for the 2nd order: F ðz Þ ¼

B 1 z 1 þ B 2 z 2 z 1 ðB 1 þ B 2 z 1 Þ B1z þ B2 ¼ ¼ 1 þ A1 z 1 þ A2 z 2 ð1  a 1 z 1 Þð1  a 2 z 1 Þ ðz  a 1 Þðz  a 2 Þ

This function contains a zero (−B2/B1) and 2 poles a1 and a2 (resulting from the factoring of the denominator). In the case of a loop, the calculation of the controller sometimes calls for compensation: it simplifies all or part of the poles and zeros of the process. Remember that we can only compensate for poles and zeros of modulus