Control in System Dynamics E-Book

Alain Oustaloup

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: ISTE Consignment
• Sprache: Englisch

Seitenzahl: 372

Veröffentlichungsjahr: 2024

Beschreibung

In order to ensure a pedagogical presentation of the fundamentals, this book, which is based on 45 years of experience, endeavors to identify the main principles of the control scheme and its dynamics.

The control loop is extensively developed because of the reference it constitutes in control. By establishing the control loop as equivalent to any other control or, more precisely, by making an elementary control loop appear, it becomes possible to reveal a part of this loop in other controls and, thanks to its regulator, qualitatively evaluate the control strategy. A comparative analysis then shows that the complexification of the control scheme does not necessarily go hand in hand with a better control strategy and therefore a better performance.

Since robustness results from innovations in terms of performance desensitization, the CRONE control is presented, in view of the genuine domains of uncertainty taken into account and the small number of parameters to be optimized. The educational nature of this book is enhanced by numerous solved problems that develop examples of synthesis of different controls.

Control in System Dynamics is intended for a very wide audience, including engineers, researchers, teachers and students hoping to expand their knowledge of control and dynamics in automatic control.

Leseprobe

Series EditorJean-Paul Bourrières

Control in System Dynamics

Comparative Analysis of Feedback Strategies

Alain Oustaloup

First published 2024 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2024The rights of Alain Oustaloup to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.

Library of Congress Control Number: 2024943874

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-022-5

PrefaceComplemented with Some Background to the Closed Loop

The subject matter discussed in this book concerns automatic control, or more specifically, control in system dynamics. It is an educational work, both theoretical and applicative in nature, featuring a number of solved problems.

The personalization of the presented content results from a well-known past (which nourishes the present), a desire for innovation and simplifcation, and a long-standing deep thinking based on experience. It is true to say that thinking on this subject has always been conducted based on acquired experience in electronics and automation, through teaching, research and technological-transfer activities within the framework of partnerships with the industrial sector. Driven by a concern for balance, these activities can also be credited for favorably supporting my career, first as an assistant in electronics and automation at the University of Bordeaux from 1978 to 1990, and then as an automation professor at the engineering schools of ENSEIRB (École Nationale Supérieure d’Électronique et de Radioélectricité de Bordeaux) and the Polytechnic Institute of Bordeaux, from 1991 to 2018.

Having had the privilege of presenting the foundations of the control loop at the prestigious Les Houches summer school in physics (France), I had the opportunity very early on to become aware of the favorable perception of our discipline by physicists, who indeed concluded my presentation by interpreting automation as a veritable science, even if my talk remained limited to the subject of the control loop itself. But in the context of the control loop, is automation not considered consistent with automation building based on the famous closed loop, better known by electronics engineers for its feedback (or negative reaction), and the open loop, which simultaneously determines stability and performance in terms of frequency?

The history of the closed loop answers this question, providing testimony through founding works, their context and their development, associated with the main dates. In fact, several elements are drawn in large part from one synthesis document produced by Jean-Claude Trigeassou, based on a contribution by Bennett, a science historian [BEN 79, BEN 93].

In its most widely accepted version, the closed loop can take different forms, notably that of the control loop (with controller), as studied in tracking and regulation in the automation control domain. The concept of the closed loop appears to date back to the 18th century, with the introduction of the mechanical device known as the Watt regulator, by Boulton and Watt in 1788. This device was regularly improved upon, notably by Siemens, and analyzed by physicists interested in its operation: Young in 1807, Airy in 1840 and 1851, and Maxwell in 1868 in his renowned article, “On Governors”, the first ever article on the subject of automation.

As the transcontinental telephone between the East and West Coasts of the United States required reamplification of the signal by repeaters (relay amplifiers for telephone lines), “Bell Laboratories” worked to replace the first (magnetic) amplifiers with low-frequency electronic amplifiers, made possible by the invention of the triode. Black was recruited to Bell in 1925 to improve on electronic tube repeaters, whose main shortcoming was their narrow bandwidth, associated with significant distortion. His idea of introducing a negative reaction to widen the bandwidth and reduce distortion was patented in 1927 and published in 1934 in the Bell System Technical Journal. Even if the concept of a closed loop appears to date back more than a century prior, it is Black specifically who can be credited for introducing the closed loop into the electronics field, through the shaping of feedback, as it is still used today.

But the feedback that could be at the origin of an oscillation, or indeed an instability, led Nyquist, also at Bell, to develop his famous criterion established based on Cauchy's theory and published in 1932 under the title, “Regeneration Theory”. In the case of a stable control loop in open loop, where the Nyquist criterion required the critical point (–1) to not be surrounded for closed-loop stability, a significant decrease in gain accompanied by an (illusory) low phase rotation was then sought.

In 1934, the problem was put to Bode, who demonstrated, for minimal phase-shifting systems (or systems with minimum phase), that a gain curve corresponded to a single phase curve, with this link between gain and phase already presenting a naturally admitted constraint in terms of synthesis and therefore performance: complementary to the Bode diagrams dating back to 1938, this contribution, associated with that of Nyquist, was published in 1940, and went on to be included in a book, published in 1945. Furthermore, the first proposition for the systematic adjustment of the P, I and D actions of the PID regulator should be attributed to Ziegler and Nichols, who published their widely known method in an article back in 1942.

Finally, the control loop was also used for temperature and pressure regulation, as well as for servomechanisms (notably for speed and position), according to the principles set out by Hazen in 1934, very similar to those of Black, although not inspired by them. Research into servomechanisms, which was developed significantly to support the armament effort in the Second World War, was given particular impetus in areas that were a priority for armament, including the domain of control and guidance of anti-aircraft defense guns. In 1940, Brown specified the performances of servo-controls, most specifically in transient state, through the damping coefficient and natural frequency. One of Brown’s researchers, Hall (thesis published in 1943), constructed a chart, known as the Hall chart, plotted in the complex plane (also known as the Nyquist plane), to change from the frequency response in open loop to the frequency response in closed loop. To change directly to closed loop based on the Bode diagrams in open loop, Nichols had the idea of transposing the Hall chart into a new plane, known as the Nichols plane, whose gain in decibels is applied on the ordinate, and whose phase in degrees is applied on the abscissa [JAM 47]. An even older origin of the Nichols plane and chart can be found in the Black plane and chart, dating back to 1934, in the sense that the Nichols plane results from the permutation of the Black gain and phase axes, accompanied by the expression of the gain in decibels (in compliance with the Bode representation). Yet these works, which contributed to the precision (in position and speed) of servomechanisms, were not enough to address the more general problem of pursuit of target, which also required the anticipation of the movements of the target and a prediction of its future behavior. So Wiener developed his famous filter, known as the Wiener filter (1942), used to guide DCA artillery based on radar information, thus replacing manual guiding. Section 1.10.4 provides additional details of a more technical nature.

Having gone through these major points in its history, let us now present the control loop in the context of the present book, specifying that by studying all aspects of the control loop, we are able to present, in a clear, structured manner, all of the functions, properties and performances that can characterize all other controls.

Moreover, the unification exercise considered in Chapter 4, which consists of developing a control loop equivalent to any other control among a set of controls, makes it possible to demonstrate an absence of specific contribution in terms of performances, even with the synthesis method differing, depending on the control.

Certainly expressing a change with respect to the control loop, this methodological contribution, which, in principle, allows us to assume a bonus in terms of performance, and which, subsequently, retains this in the best case scenario, leads us at the very least to think of the principle, according to which it is advisable to change the form (here, the methodology) to preserve the substance (here, performance). It is true that evolution is de rigueur, so if a change is inevitable when it is not necessary, is it not preferable to opt for a change of form that is fundamentally the least detrimental?

With the control loop thus all the more reinforced, as indeed confirmed by its usage in all robustness approaches, it is widely developed in the first two chapters of this book, which accord it due credit. This didactic choice should help the reader to obtain a measure of the major interest of the command loop, in the sense of its own interest and the interest it represents for other controls through the often-implicit place that it occupies, and which is highlighted by establishing the equivalent control loop. This same loop, through the nature of its regulator, makes it possible to objectively appreciate the richness of control strategy, with the control loop thus appearing as an absolute reference. This means that the control loop, which is the proven substrate of automation, must be very widely known to all, including specialists of other controls, at the risk of rediscovering lukewarm water, or even making cold water from it, by achieving less success than the control loop.

The innovative contribution in terms of performance undoubtedly lies in the so-called robust controls, including the CRONE control (the French abbreviation for the Robust Control of Non-Integer Order), which combines three significant advantages in terms of robustness: the choice of dynamic performance to be robustified; taking genuine uncertainty domains into account (unlike the 𝐻∞ approach); and minimal parameterization of the transmittance to be optimized (unlike the QFT approach). Based on the structure of the control loop, it owes its specific nature to a regulator enriched by a non-integer, real or complex order. This order ensures the robustness of the control's stability degree with respect to the (parametric) plant uncertainties.

Represented in the Nichols plane by a straight-line segment whose description transmittance is based on complex, non-integer integration, the generalized template of the third-generation CRONE control is used in the covering technique to build iso-overshoot and isodamping contours. It transpires that the iso-overshoot contours are the Nichols amplitude contours, which are thus validated as iso-overshoot contours. The symmetry, in relation to the abscissa axis of the Nichols plane, of the isodamping contours (of Oustaloup), is a remarkable property that reinforces the relevance of this working plan in terms of analysis and synthesis.

Such contours, called performance contours, significantly enrich the well-known arsenal of tools and methods for frequency synthesis of the regulator (whichever it is) of the control loop. These performance contours are indeed tools of great interest, in the sense that they make it possible to respond, in a deterministic manner, to the performance specifications related to the overshoot and damping of control loop dynamics: surely the presentation of this new bonus in favor of the control loop, offering it an indisputable argument in its favor, is the best way to conclude this preface or, given the context, if we can make room for a little humor: to close the loop?

February 2022

Book Structure and Content

Along with the three appendices covering active noise control, non-integer control and performance contours, respectively, this book consists of five chapters and 13 solved problems, the content of which is discussed here by way of an introduction.

Chapter 1 starts by demonstrating that the plant output disturbance is none other than the effect on its output of the set of so-called “original” disturbances. Given this disturbance, it then develops tracking-regulation duality in the operational and frequency domains, applying the most dual approach possible in order to best structure the presentation of the control loop. After a very thorough and wide-ranging definition of transmittance in open loop, which responds to all of the linear systems forming part of engineering science, the open loop is used to physically present the stable, unstable or barely oscillating nature of the control loop, as well as the left-hand criterion, which governs this character for a stable open loop with minimum phase shifting. Transmittances in closed loop, which are the tracking and regulation transmittances, as well as the resulting tracking and regulation functions, which introduce reference tracking and disturbance rejection, respectively, constitute the substrate of tracking-regulation duality. The structural nature imposed by such a duality is then used to study, in turn, the input sensitivity (of the plant), which results in replacing the academic PID with the non-academic PID, the behavior and performances of the control loop, and its dynamics, before ending with the charts in tracking and regulation, whose duality speaks for itself.

Chapter 2 begins by defining stability, and presenting the fundamental stability condition, which can be verified using algebraic or graphical criteria. Next, the frequency stability margins, that is, the gain and phase margins, are defined. The phase margin is examined in more detail via the determination of an optimal phase margin, in the sense of the minimization of a quadratic criterion based on the steepness time and the first overshoot of the step response in tracking or in regulation. The stability degree, an essential concept in system dynamics, perfectly exemplified by the control loop owing to the coupling between open loop and closed loop, is presented in the time domain, then in the frequency domain, where particular use is made of this coupling by means of the amplitude contours presented in Chapter 1. This approach demonstrates, inter alia, that while the circle centered at (–1) in the complex plane offers (in the mathematical sense) a true measurement of the distance to the critical point, it is nevertheless not significant for a time dynamic performance able to quantify the stability degree in the time domain. When compared with pole-placement control, the frequency approach stands out as favorable in that it satisfies both a reduced first overshoot specification and a damping ratio specification. After defining the precision based on the difference between the sought and actual values of the control loop output, precision is presented in accordance with tracking-regulation duality and with the steady states corresponding to step and ramp tests. Once the stability degree–precision dilemma has been defined, it is highlighted in the Nichols plane, then solved by an open-loop gain, which provides a compromise between stability degree and precision. Next, this compromise is improved and the dilemma is brought into question by a non-integer order. A pragmatic definition of dynamics is provided, after which a “determination of dynamics” section identifies the frequency dynamic performances significant to time dynamic performances, which thus determine the dynamics. The controller synthesis thus forms part of a straightforward, systematic methodological approach that offers the user effecient methods, in user-guide form.

Chapter 3 results from an educational exercise aiming to extract the idea of different linearizing approaches using strategies that stand out because of their simplicity. Forming part of engineering sciences, these strategies, which enrich basic automation and can be applied to the majority of plants, are conducive to favoring the application of linearizing approaches in the most diverse sectors, notably in industry. More precisely, the presentation of each approach is structured by a principle reduced to its simplest expression, and by the application of this principle to one or more nonlinear plants, one of which is used for all approaches; this nonlinear plant, common to all monovariable applications, is a motor shaft that rotates a small mass secured to the shaft by means of a straight arm. In linearization by immersion, extended to a multivariable plant, the principle consists of linearizing the plant with respect to new inputs. In linearization by high gain, the principle is based on a high-gain control used for its disturbance rejection. Although the basic principle of linearization by high gain expresses the idea of a total disturbance rejection, linearization by disturbance rejection expresses the idea of a partial rejection, which is more realistic in terms of the plant input sensitivity. Indeed, its principle is based on an elementary control loop, whose gain in open loop remains compatible with solid plant input immunity. In fact, through its ability to reject disturbance, which in this case represents the plant’s nonlinearity, the control loop has a linearizing character that has come to be frequently used in industry to overcome nonlinearities. While the linearization of the plant around a nominal trajectory constitutes a well-known, relatively widely used linearizing approach, the use made of it here stands out with the introduction of uncertainty domains, with the nonlinearity indeed represented frequentially in the form of uncertainty domains. In fact, this strategy, which displaces uncertainty domains such that they present as little as possible overlap of the areas presenting a low degree of stability, is able to specify the way in which the nonlinearities are taken into account in third-generation CRONE control (see Chapter 5). Finally, the chapter concludes with a rigorous contribution from Brigitte d’Andréa-Novel, which compensates for the extreme simplicity of the strategies proposed.

Chapter 4 presents different control schemes, highlighting their point in common, notably by showing that they are implicitly based on an elementary control loop (the regulator of which is then analyzed). Given that the study domain, frequency or time, differs depending on the nature of the control, it is difficult to conduct a comparative study of the main control principles. More generally, this difficulty is exacerbated by the difference in the study framework, which incorporates both the study domain (frequency or time) and the structure of the control scheme. As such, this chapter aims for a common study framework, seeking to develop a unified approach to control. As developed in the first two chapters, the in-depth study of the control loop, notably in the frequency domain, has seen the development of a tried-and-tested control methodology over time, capable of meeting several temporal dynamic performance specifications, via frequency dynamic performances, significant for these performances. Moreover, such a methodology ought to be interpreted as a reference among the main control methods proposed in this chapter, and indeed this book. It is true that the idea aims to reduce each control scheme to an equivalent control loop, so that they come under the same study framework (that of the control loop), and thus conduct an objective comparative study of the different control strategies: indeed, the nature of the regulator of each equivalent control loop allows an objective appreciation of the richness of each control strategy. One initial conclusion is that the sophistication of the control scheme is not always synonymous with the enriching of control strategy.

While a proportional, high-gain regulator has the benefit of rapidly showing the interest of the high-gain control through its insensitivity (or robustness), ensured by a transfer imposed by that of the reaction chain, this case is nevertheless not realistic since high gain comes at the cost of a large input sensitivity, and therefore low input immunity. In order to make judicious use of a high-gain principle such as this, it is then appropriate to aim, not for an insensitization of the control, but simply for a desensitization, by choosing an acceptable gain. As for the change to the control loop, rewriting the error signal of the high-gain control in an appropriate form makes it possible to transform its functional diagram in accordance with a transfer in cascade with an elementary control loop.

By assuring a unit transfer in tracking, the feedforward control enables differentiation between the dynamics in tracking and regulation, as indeed a reference prefilter does. It is true that appropriately rewriting the plant input enables the functional diagram of the control to be transformed in accordance with a reference prefilter in cascade with an elementary control loop.

In the absence of a disturbance, and in the case where the plant is perfectly modeled, the internal-model control is conducted without feedback and the controller synthesis is then remarkably simple. Appropriately rewriting the plant input enables the internal model scheme to be restructured in the form of an elementary control loop, the regulator of which is expressed. In the case of a “robustification”, which consists of ensuring a reference-output transfer independent of the plant for the control, the (equivalent) elementary control loop thus obtained has the advantage of recognizing the infinite character underlying the control, not only through the regulator gain, but also through the input sensitivity in tracking and regulation.

As a preamble to quadratic and predictive controls, which are based on an optimization with respect to the input, u(t), of the plant, a property that is essential for this optimization is established, a property according to which the output y(t), and its derivatives depend on the past of the input u(t), and not its instantaneous value (such a property resulting from the convolution).

The quadratic-criterion control is presented based on a second-order study plant, with input u(t) or u(k), depending on whether the model is continuous or discrete. The synthesis of the regulator is performed through the minimization of a quadratic criterion, continuous then discrete, which introduces a control weighting factor, δ. A continuous, direct approach based on continuous control and a discrete then continuous approach based on discrete control make it possible to result in the same elementary control loop. One result that is remarkable to say the least, is that the action of the regulator of this loop is reduced to a proportional action, 1/δ, which is none other than the inverse of the control weighting factor.