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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Explore the foundational and advanced subjects associated with proportional-integral-derivative controllers from leading authors in the field In PID Passivity-Based Control of Nonlinear Systems with Applications, expert researchers and authors Drs. Romeo Ortega, Jose Guadalupe Romero, Pablo Borja, and Alejandro Donaire deliver a comprehensive and detailed discussion of the most crucial and relevant concepts in the analysis and design of proportional-integral-derivative controllers using passivity techniques. The accomplished authors present a formal treatment of the recent research in the area and offer readers practical applications of the developed methods to physical systems, including electrical, mechanical, electromechanical, power electronics, and process control. The book offers the material with minimal mathematical background, making it relevant to a wide audience. Familiarity with the theoretical tools reported in the control systems literature is not necessary to understand the concepts contained within. You'll learn about a wide range of concepts, including disturbance rejection via PID control, PID control of mechanical systems, and Lyapunov stability of PID controllers. Readers will also benefit from the inclusion of: * A thorough introduction to a class of physical systems described in the port-Hamiltonian form and a presentation of the systematic procedures to design PID-PBC for them * An exploration of the applications to electrical, electromechanical, and process control systems of Lyapunov stability of PID controllers * Practical discussions of the regulation and tracking of bilinear systems via PID control and their application to power electronics and thermal process control * A concise treatment of the characterization of passive outputs, incremental models, and Port Hamiltonian and Euler-Lagrange systems Perfect for senior undergraduate and graduate students studying control systems, PID Passivity-Based Control will also earn a place in the libraries of engineers who practice in this area and seek a one-stop and fully updated reference on the subject.
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Table of Contents
Cover
Series Page
Title Page
Copyright
Dedication
Author Biographies
Preface
Bibliography
Acknowledgments
Acronyms
Notation
1 Introduction
2 Motivation and Basic Construction of PID Passivity‐Based Control
2.1 ‐Stability and Output Regulation to Zero
2.2 Well‐Posedness Conditions
2.3 PID‐PBC and the Dissipation Obstacle
2.4 PI‐PBC with and Control by Interconnection
Bibliography
Notes
3 Use of Passivity for Analysis and Tuning of PIDs: Two Practical Examples
3.1 Tuning of the PI Gains for Control of Induction Motors
3.2 PI‐PBC of a Fuel Cell System
Bibliography
Notes
4 PID‐PBC for Nonzero Regulated Output Reference
4.1 PI‐PBC for Global Tracking
4.2 Conditions for Shifted Passivity of General Nonlinear Systems
4.3 Conditions for Shifted Passivity of Port‐Hamiltonian Systems
4.4 PI‐PBC of Power Converters
4.5 PI‐PBC of HVDC Power Systems
4.6 PI‐PBC of Wind Energy Systems
4.7 Shifted Passivity of PI‐Controlled Permanent Magnet Synchronous Motors
Bibliography
Notes
5 Parameterization of All Passive Outputs for Port‐Hamiltonian Systems
5.1 Parameterization of All Passive Outputs
5.2 Some Particular Cases
5.3 Two Additional Remarks
5.4 Examples
Bibliography
Note
6 Lyapunov Stabilization of Port‐Hamiltonian Systems
6.1 Generation of Lyapunov Functions
6.2 Explicit Solution of the PDE
6.3 Derivative Action on Relative Degree Zero Outputs
6.4 Examples
Bibliography
Notes
7 Underactuated Mechanical Systems
7.1 Historical Review and Chapter Contents
7.2 Shaping the Energy with a PID
7.3 PID‐PBC of Port‐Hamiltonian Systems
7.4 PID‐PBC of Euler‐Lagrange Systems
7.5 Extensions
7.6 Examples
7.7 PID‐PBC of Constrained Euler–Lagrange Systems
Bibliography
Notes
8 Disturbance Rejection in Port‐Hamiltonian Systems
8.1 Some Remarks on Notation and Assignable Equilibria
8.2 Integral Action on the Passive Output
8.3 Solution Using Coordinate Changes
8.4 Solution Using Nonseparable Energy Functions
8.5 Robust Integral Action for Fully Actuated Mechanical Systems
8.6 Robust Integral Action for Underactuated Mechanical Systems
8.7 A New Robust Integral Action for Underactuated Mechanical Systems
8.8 Examples
Bibliography
Notes
Appendix A Passivity and Stability Theory for State‐Space Systems
A.1 Characterization of Passive Systems
A.2 Passivity Theorem
A.3 Lyapunov Stability of Passive Systems
Bibliography
Note
Appendix B Two Stability Results and Assignable Equilibria
B.1 Two Stability Results
B.2 Assignable Equilibria
Bibliography
Appendix C Some Differential Geometric Results
C.1 Invariant Manifolds
C.2 Gradient Vector Fields
C.3 A Technical Lemma
Bibliography
Appendix D Port–Hamiltonian Systems
D.1 Definition of Port‐Hamiltonian Systems and Passivity Property
D.2 Physical Examples
D.3 Euler–Lagrange Models
D.4 Port‐Hamiltonian Representation of GAS Systems
Bibliography
Index
End User License Agreement
List of Tables
Chapter 7
Table 7.1 System parameters.
Table 7.2 Initial conditions.
Table 7.3 Gains sets.
List of Illustrations
Chapter 2
Figure 2.1 Block diagram representation of the closed‐loop system of Proposi...
Chapter 3
Figure 3.1 Feedback decomposition of the closed‐loop system.
Figure 3.2 Compressor map: the curves are parameterized by
, and grow when ...
Figure 3.3 Convergence of the three states to the equilibrium point.
Figure 3.4 Simulation response of the closed‐loop system with the PI control...
Chapter 4
Figure 4.1 Schematic of the quadratic boost converter.
Figure 4.2 Schematic diagram of the equivalent circuit of a VSR in
frame....
Figure 4.3 Wind energy system.
Figure 4.4 Function
.
Chapter 5
Figure 5.1 Two‐tanks system.
Chapter 7
Figure 7.1 Time histories of the (a) position of the cart
and (b) angle of...
Figure 7.2 Time histories of the (a) velocity of the cart
and (b) angular ...
Figure 7.3 Time histories of the (a) input force
and (b) the nonlinear gai...
Figure 7.4 Time histories of the (a) position of the cart
and (b) angle of...
Figure 7.5 Time histories of the (a) velocity of the cart
and (b) angular ...
Figure 7.6 Time histories of the (a) input force
and (b) the nonlinear gai...
Figure 7.7 Time histories of the (a) position of the cart
and (b) angle of...
Figure 7.8 Time histories of the (a) velocity of the cart
and (b) angular ...
Figure 7.9 Time histories of (a) the input force
and (b) the nonlinear gai...
Figure 7.10 Captures of a video animation of the cart‐pendulum on an incline...
Figure 7.11 Single ultraflexible link with base excitation.
Figure 7.12 Simulation results for
.
Figure 7.13 Simulation results for
.
Figure 7.14 Simulation results for
.
Figure 7.15 Inverted flexible pendulum.
Figure 7.16 Comparison of simulation and experimental results for
.
Chapter 8
Figure 8.1 Graph of the state space showing two sheets of the invariant foli...
Figure 8.2 (a, b) Angle of the arm
and position of the hand
, and (c, d) ...
Figure 8.3 (a, b) States of the controller, (c, d) control torque on the arm...
Figure 8.4 (a, b) Angle of the arm
and position of the hand
, and (c, d) ...
Figure 8.5 (a, b) States of the controller, (c, d) control torque on the arm...
Figure 8.6 Time histories of the Acrobot angles
and
with the IDA‐PBC plu...
Figure 8.7 Time histories of the Acrobot angular velocities
and
with the...
Figure 8.8 Time history of the matched disturbance
, and the controller sta...
Figure 8.9 Idealized scheme of the Disk‐on‐Disk system (a) and real setup (b...
Figure 8.10 Time history of coordinate
with the IDA‐PBC controller plus th...
Figure 8.11 Time history of the coordinate
with the IDA‐PBC controller plu...
Figure 8.12 Time history of the matched disturbance
, and the controller st...
Figure 8.13 Time history of the control input and the disturbance with the s...
Appendix A
Figure A.1 Standard feedback configuration.
Guide
Cover
Series Page
Title Page
Copyright
Dedication
Author Biographies
Preface
Acknowledgments
Acronyms
Notation
Table of Contents
Begin Reading
Appendix A Passivity and Stability Theory for State‐Space Systems
Appendix B Two Stability Results and Assignable Equilibria
Appendix C Some Differential Geometric Results
Appendix D Port–Hamiltonian Systems
Index
WILEY END USER LICENSE AGREEMENT
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IEEE Press
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Editor in Chief
Jón Atli Benediktsson
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Diomidis Spinellis
David Alan Grier
Andreas Molisch
Sarah Spurgeon
Elya B. Joffe
Saeid Nahavandi
Ahmet Murat Tekalp
PID Passivity‐Based Control of Nonlinear Systems with Applications
Romeo OrtegaInstituto Tecnológico Autónomo de México
José Guadalupe RomeroInstituto Tecnológico Autónomo de México
Pablo BorjaUniversity of Groningen
Alejandro DonaireThe University of Newcastle
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Library of Congress Cataloging‐in‐Publication Data
Names: Ortega, Romeo, 1954‐ author. | Romero, J. G., author. | Borja, Pablo, author. | Donaire, Alejandro, author.Title: PID passivity‐based control of nonlinear systems with applications / Romeo Ortega, José Guadalupe Romero, Pablo Borja, Alejandro Donaire.Description: Hoboken, New Jersey : Wiley‐IEEE Press, [2021] | Includes index.Identifiers: LCCN 2021024088 (print) | LCCN 2021024089 (ebook) | ISBN 9781119694168 (cloth) | ISBN 9781119694175 (adobe pdf) | ISBN 9781119694182 (epub)Subjects: LCSH: PID controllers. | Nonlinear systems.Classification: LCC TJ223.P55 O66 2021 (print) | LCC TJ223.P55 (ebook) | DDC 629.8/36–dc23LC record available at https://lccn.loc.gov/2021024088LC ebook record available at https://lccn.loc.gov/2021024089
Cover Design: WileyCover Image: © marty8801/iStock/Getty Images
To the memory of my mother – Romeo.
To the memory of my mother; and to Salua, Patricia, Liliana and Vianca with all my love – José Guadalupe.
To Celia with all my love. To my parents, Diana and Mario – Pablo.
To Delfina, Phoebe and Cristina – Alejandro.
Author Biographies
Romeo Ortega was born in Mexico. He obtained his BSc in Electrical and Mechanical Engineering from the National University of Mexico, Master of Engineering from Polytechnical Institute of Leningrad, USSR, and the Docteur D‘Etat from the Polytechnical Institute of Grenoble, France in 1974, 1978 and 1984 respectively.
He then joined the National University of Mexico, where he worked until 1989. He was a Visiting Professor at the University of Illinois in 1987–1988 and at McGill University in 1991–1992, and a Fellow of the Japan Society for Promotion of Science in 1990–1991. He was a member of the French National Research Council (CNRS) from June 1992 to July 2020, where he was a “Directeur de Recherche” in the Laboratoire de Signaux et Systemes (CentraleSupelec) in Gif‐sur‐Yvette, France. Currently, he is a full time Professor at ITAM in Mexico. His research interests are in the fields of nonlinear and adaptive control, with special emphasis on applications.
He has published five books and more than 350 scientific papers in international journals, with an h‐index of 84. He has supervised more than 35 PhD thesis. He is a Fellow Member of the IEEE since 1999 (Life 2020) and an IFAC Fellow since 2016. He has served as chairman in several IFAC and IEEE committees and participated in various editorial boards of international journals. He is currently Editor in Chief of International Journal of Adaptive Control and Signal Processing and Senior Editor of Asian Journal of Control.
José Guadalupe Romero was born in Tlaxcala, Mexico in 1983. He received the BS degree in electronic engineering from the University of Zacatecas, Zacatecas, Mexico, in 2006 and the MSc degree in robotics and advanced manufacturing from the Centre for Research and Advanced Studies, National Polytechnic Institute (CINVESTAV), Mexico, in 2009. He obtained the PhD degree in Control Theory from the University of Paris‐Sud XI, France in 2013.
He was a postdoctoral fellow at Schneider electric and EECI in Paris France and at the Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM) in 2014 and 2015, respectively. Currently, he is an associate professor and researcher at the Digital Systems Department at the Instituto Tecnológico Autónomo de México (ITAM) and since 2019 he is the Director of undergraduate mechatronics engineering program.
His research interests are focused on nonlinear and adaptive control, stability analysis and the state estimation problem, with application to mechanical systems, aerial vehicles, mobile robots and multi‐agent systems.
Pablo Borja was born in Mexico City, Mexico. He obtained the B.Eng. in Electrical and Electronic Engineering and the M.Eng. from the National Autonomous University of Mexico in 2011 and 2014, respectively, and the PhD in Control Systems from the University Paris Saclay, France, in 2017.
From 2017 to 2018, he was a postdoctoral researcher member of the Engineering and Technology Institute Groningen (ENTEG) at the University of Groningen (RUG). Since 2018 he is a fellow of the Faculty of Science and Engineering and ENTEG member at the RUG. His research interests encompass the control and analysis of nonlinear systems, passivity‐based control, control of physical systems, passivity and its role in control theory, and model reduction.
Alejandro Donaire received the Electronic Engineering and PhD degrees in 2003 and 2009, respectively, from the National University of Rosario, Argentina. His work was supported by the Argentine National Council of Scientific and Technical Research, CONICET. In 2009, he joined the Centre for Complex Dynamic Systems and Control at The University of Newcastle, Australia, and in 2011, he received the Postdoctoral Research Fellowship of the University of Newcastle, Australia. From 2015 to March 2017 he was with the PRISMA Lab at the University of Naples Federico II, and from 2017 to 2019 with the Institute for Future Environments, School of Electrical Engineering and Computer Science, Queensland University of Technology, Australia. In 2019, he joined the School of Engineering, The University of Newcastle, Australia, where he conducts his academic activities. His research interests include nonlinear and energy‐based control theory with application to electrical drives, multi‐agent systems, robotics, smart micro‐grids networks, marine and aerospace mechatronics, and power systems.
Preface
It is widely recognized thatproportional‐integral‐derivative (PID) control offers the simplest and yet most efficient solution to many real‐world control problems. It is said to be a universal controller in the sense that the integral action takes care of the past, the proportional one of the present, and the derivative term has a predictive effect. Since the invention of PID control in 1910, the popularity of PID control has grown tremendously (Ang et al., 2005; Åstrom and Hägglund, 1995, 2006; Samad, 2017).
It is interesting to quote a 2018 report of Karl Åstrom (Åstrom, 2018) where he points out the following:
In spite of the predictions that other control techniques, e.g.
model predictive control
(MPC), will make, the PID obsolete, more than 90% of industrial controllers are still implemented based around PID algorithms.
In a report of Bill Bialkowski of the Canadian consulting company Entech, it is indicated that out of 3000–5000 control loops in the paper mill industry, 97% use
proportional‐derivative
(PI) and the remaining 3% are MPC, adaptive, etc.
In the same report it is indicated that, out of the 50% of the PIDs that
do not work
well, 30% are due to
bad tuning
.
Indeed, it is very well known that PID controllers yield, in general, a satisfactory performance provided they are well tuned. The need to fulfill this requirement has been the major driving force of the research on PID control, with the vast majority of the reports related to the development of various PID tuning techniques, which are customarily based on a linear approximation of the plant around a fixed operating point (or a given trajectory). When the range of operation of the system is large, the linear approximation is invalidated and the procedure to tune the gains of PID regulators is a challenging task. Although gain scheduling, auto tuning, and adaptation provide some help to overcome this problem, they suffer from well‐documented drawbacks that include being time‐consuming and fragility of the design. The interested reader is referred to Ang et al. (2005) for a recent, detailed account of the various trends and topics pertaining to PID tuning.
The present book is devoted to the study of PID passivity‐based control (PBC), which provides a solution to the tuning problem of PID control of nonlinear systems. The underlying principle for the operation of PID‐PBC is, as the name indicates, the property of passivity, which is a fundamental property of dynamical systems. One of the foundational results of control theory is the passivity theorem (Desoer and Vidyasagar, 2009; Khalil, 2002; van der Schaft, 2016), which states that the feedback interconnection of two passive systems ensures convergence of the output to zero and stability (in the sense) of the closed‐loop. On the other hand, it is well‐known (van der Schaft, 2016) that PID controllers are (output strictly) passive systems – for all positive PID gains. Therefore, wrapping the PID around a passive output yields a stable system for all PID gains. Clearly, this situation simplifies the gain‐tuning task, since the designer is left with the only task of selecting, among all positive gains, those that ensure the best transient performance.
PID‐PBCs have been successfully applied to a wide class of physical systems, see e.g. Aranovskiy et al. (2016), Castaños et al. (2009), Cisneros et al. (2015, 2016), De Persis and Monshizadeh (2017), Hernández‐Gómez et al. (2010), Meza et al. (2012), Romero et al. (2018), Sanders and Verghese (1992), and Talj et al. (2010). However, their application has mainly been restricted to academic circles. It is the authors' belief that PID‐PBCs have an enormous potential in engineering practice and should be promoted among practitioners. The main objective of the book is then to give prospective designers of PID‐PBCs the tools to successfully use this technique in their practical applications. Toward this end, we provide a basic introduction to the theoretical foundations of the topic, keeping the mathematical level at the strict minimum necessary to cover the material in a rigorous way, but at the same time to make it accessible to an audience more interested in its practical application. To fulfill this objective, we have skipped technically involved theoretical proofs – referring the interested reader to their adequate source – and we have included a large number of practical examples.
We are aware that aiming at penetrating current engineering practice is a very challenging task. It is our strong belief that combining the unquestionable dominance of PIDs in applications with the fundamental property of passivity, which in the case of physical systems captures the universal feature of energy conservation, yields an unbeatable argument to justify its application.
Bibliography
K. H. Ang, G. Chong, and Y. Li. PID control system analysis, design and technology.
IEEE Transactions on Control Systems Technology
, 13(4): 559–576, 2005.
S. Aranovskiy, R. Ortega, and R. Cisneros. A robust PI passivity‐based control of nonlinear systems and its application to temperature regulation.
International Journal of Robust and Nonlinear Control
, 26(10): 2216–2231, 2016.
K. J. Åstrom. Advances in PID control. In
XXXIX Jornadas de Automatica
, Badajoz, Spain, 2018.
K. J. Åstrom and T. Hägglund.
PID Controllers: Theory, Design, and Tuning
. 2nd edition. Instrument Society of America, 1995.
K. J. Åstrom and T. Hägglund.
Advanced PID control
. ISA‐The Instrumentation, Systems, and Automation Society, Research Triangle Park, NC 27709, 2006.
F. Castaños, B. Jayawardhana, R. Ortega, and E. García‐Canseco. Proportional plus integral control for set point regulation of a class of nonlinear RLC circuits.
Circuits, Systems and Signal Processing
, 28(4): 609–623, 2009.
R. Cisneros, M. Pirro, G. Bergna‐Díaz, R. Ortega, G. Ippoliti, and M. Molinas. Global tracking passivity‐based PI control of bilinear systems and its application to the boost and modular multilevel converters.
Control Engineering Practice
, 43(10): 109–119, 2015.
R. Cisneros, R. Gao, R. Ortega, and I. Husain. PI passivity‐based control for maximum power extraction of a wind energy system with guaranteed stability properties.
International Journal of Emerging Electric Power Systems
, 17(5): 567–573, 2016.
C. De Persis and N. Monshizadeh. Bregman storage functions for microgrid control.
IEEE Transactions on Automatic Control
, 63(1): 53–68, 2017.
C. A. Desoer and M. Vidyasagar.
Feedback Systems: Input‐Output Properties
. Academic Press, New York, 2009.
M. Hernández-Gómez, R. Ortega, F. Lamnabhi‐Lagarrigue, and G. Escobar. Adaptive PI stabilization of switched power converters.
IEEE Transactions on Control Systems Technology
, 18(3): 688–698, 2010.
H. Khalil.
Nonlinear Systems
. Prentice‐Hall, Upper Saddle River, NJ, 2002.
C. Meza, D. Biel, D. Jeltsema, and J. M. A. Scherpen. Lyapunov‐based control scheme for single‐phase grid‐connected PV central inverters.
IEEE Transactions on Control Systems Technology
, 20(2): 520–529, 2012.
J. G. Romero, A. Donaire, R. Ortega, and P. Borja. Global stabilisation of underactuated mechanical systems via PID passivity‐based control.
Automatica
, 96(10): 178–185, 2018.
T. Samad. A survey on industry impact and challenges thereof.
IEEE Control Systems Magazine
, 37(1): 17–18, 2017.
S. R. Sanders and G. C. Verghese. Lyapunov‐based control for switched power converters.
IEEE Transactions on Power Electronics
, 7(1): 17–24, 1992.
R. Talj, D. Hissel, R. Ortega, M. Becherif, and M. Hilairet. Experimental validation of a PEM fuel cell reduced order model and a moto‐compressor higher order sliding mode control.
IEEE Transactions on Industrial Electronics
, 57(6): 1906–1913, 2010.
A. J. van der Schaft.
‐Gain and Passivity Techniques in Nonlinear Control
. Springer‐Verlag, Berlin, 3rd edition, 2016.
Acknowledgments
This book is the result of extensive research collaborations during the last 10 years. Some of the results of these collaborations have been reported in the papers (Bergna‐Díaz et al., 2019; Borja et al., 2016, 2020; Castaños et al., 2009; Chang et al., 2000; Cisneros et al., 2013, 2015, 2016, 2020; Donaire and Junco, 2009; Donaire et al., 2016a, 2017; Escobar et al., 1999; Ferguson et al., 2017a; Ferguson et al., 2017b, 2018, 2020; Gandhi et al., 2016; Hernández‐Gómez et al., 2012; Jaafar et al., 2013; Jayawardhana et al., 2007; Jung et al., 2015; Monshizadeh et al., 2019; Ortega et al., 2020; Pérez et al., 2004; Talj et al., 2009, 2010, 2011; Wu et al., 2020; Zhang et al., 2015, 2018; Zonetti and Ortega, 2015; Zonetti et al., 2015). We are grateful to our co‐authors, S. Aranovskiy, A. Astolfi, A. Allawieh, D. Bazylev, M. Becherif, A. Benchaib, G. Bergna‐Díaz, A. Bobtsov, F. Castaños, G. Chang, R. Cisneros, M. Crespo, G. Duan, D. Efimov, G. Escobar, G. Espinosa‐Pérez, J. Espinoza, J. Ferguson, P. Gandhi, R. Gao, E. García‐Canseco, E. Godoy, M. Hernández‐Gómez, M. Hilairet, D. Hissel, I. Husain, A. Jaafar, B. Jayawardhana, D. Jeltsema, S. Junco, F. Kazi, F. Lamnabhi‐Lagarrigue, Z. Liu, F. Mancilla‐David, R. Mehra, E. Mendes, R. H. Middleton, N. Monshizadeh, P. Monshizadeh, M. Pérez, M. Pirro, A. Pyrkin, S. Sánchez, S. Satpute, J. Scherpen, B. Siciliano, M. Singh, H. Su, R. Talj, E. Tedeschi, A. van der Schaft, D. Wu, M. Zhang, D. Zonetti, for several stimulating discussions and for their hospitality while visiting their institutions.
Some of the topics of this book have been taught by the first author at the EECI Graduate School on Control in Istanbul, Turkey, in 2016, in the Winter Course of the Mexican Association of Automatic Control in 2016, in the Summer School of the Institute of Control Problems of the Academy of Sciences in Moscow, Russia, in 2017 and in the University of Chile, Santiago, Chile, in 2018. A workshop on this topic was organized in Zhejiang University, Hangzhou, China, in 2017.
A large part of this work would not have been possible without the financial support of several institutions. The first author would like to thank ITMO University in Saint Petersburg, Russia, for having sponsored part of this work and the Instituto Tecnológico Autónomo de México (ITAM) for opening its doors for the continuation of his scientific career in Mexico. The second author wishes to thank the Ecole Doctorale‐Sciences et Technologies de l'Information des Télécommunications et des Systèmes (ED‐STITS) for having funded his doctoral studies and the ITAM for supporting his research activities. The third author wants to thank the National Council of Science and Technology (CONACyT), the Mexican Secretary of Public Education (SEP), and the University of Groningen for all the support received during his academic career. The fourth author wants to thank the University of Newcastle for supporting his research and academic activities.
Mexico/Groningen/Newcastle
Romeo Ortega
José Guadalupe Romero
Pablo Borja
Alejandro Donaire
Acronyms
AC
alternate current
AMM
assumed modes method
CbI
control by interconnection
CL
controlled Lagrangians
DAC
digital‐to‐analog converter
DC
direct current
DOF
degree(s)‐of‐freedom
EL
Euler–Lagrange
FOC
field‐oriented control
GAS
globally asymptotically stable
GES
global exponential stability
HVDC
high‐voltage direct current
IA
integral action
IDA
interconnection and damping assignment
IISS
integral input‐to‐state stability
ISS
input‐to‐state stability
LMI
linear matrix inequality
LTI
linear time‐invariant
MDICS
matched disturbance integral controlled system
PBC
passivity‐based control
PDE
partial differential equation
PEM
proton exchange membrane
pH
port‐Hamiltonian
PD
proportional‐derivative
PI
proportional‐integral
PID
proportional‐integral‐derivative
PMSG
permanent magnet synchronous generator
PMSM
permanent magnet synchronous motor
PWM
pulse‐width modulation
SPR
strictly positive real
VSR
voltage source rectifiers
VTOL
vertical take‐off and landing
Notation
Given a vector , the symbol denotes its Euclidean norm, i.e. . We denote the th element of as . The th element of the canonical basis of is represented by . To ease the readability, column vectors are also expressed as .
Consider the matrix , then denotes the th column of , the th row of , and the th element of . Moreover, denotes the transpose of . Given a square matrix , . To simplify the notation, we express diagonal matrices as , where are the diagonal elements of the matrix.
The symbol denotes the identity matrix. The symbol refers to the th eigenvalue of . In particular, , denote the largest and the smallest eigenvalue of , respectively. A matrix is said to be positive semidefinite if and for all , and is said to be positive definite if the inequality is strict, i.e. for all . is negative (semi)definite if is positive (semi)definite. For a positive definite matrix and a vector , we denote the weighted Euclidean norm as . The notation used for constant matrices is directly extended to the nonconstant case.
Unless something different is stated, all the functions treated in this book are assumed to be smooth. Moreover, the symbol is reserved to express time, where we assume . Then, given a function that depends on time, the symbol denotes the differentiation with respect to time of , i.e. where . The and norms of signals are denoted and , respectively.
Given a function and a vector , we define the differential operator and . For a function , we define the th element of its
1Introduction
Motivated by current practice, in this book, we explore the possibility of applying the industry‐standard proportional‐integral‐derivative (PID) controllers to regulate the behavior of nonlinear systems. As is well known, PID controllers are universal, in the sense that they incorporate knowledge of the system's past, present, and future, and they are overwhelmingly dominant in engineering practice. PIDs are highly successful when the main control objective is to drive a given output signal to a constant value. PIDs, however, have two main drawbacks, first, the task of tuning the gains is far from obvious when the system's operating region is large; second, in some practical applications, the control objective cannot be captured by the behavior of output signals.
In this book we show that, for a wide class of systems, these two difficulties can be overcome by exploiting the property of passivity, which in the case of physical systems captures the universal feature of energy conservation. To achieve this end, we propose a new class of controllers called PID passivity‐based controls (PBCs), whose main construction principle is to wrap the PID around a passive output of the plant. Since PIDs define (output strictly) passive systems for all positive gains, and the feedback interconnection of passive systems is stable, the proposed architecture yields a highly robust design that preserves stability for all tuning gains – considerably simplifying the task of commissioning the controller. To enable potential designers to use PID‐PBCs, we present in the book a comprehensive coverage of this topic.
Since passivity for physical systems is simply a reformulation of energy balancing, it is possible in many practical examples to easily identify some passive outputs. However, in many examples, either these outputs are not the ones we would like to regulate, and/or their desired value is not equal to zero. To address the first problem, we explore in the book the possibility of adding an integral action to nonpassive outputs preserving some stability properties. For the second problem, we propose to wrap the PID around the error of the output signal, and then we investigate whether the system is passive with respect to this error signal – a property called shifted passivity.
Another scenario of practical interest is when the control objective is to drive the full system state to a desired constant value. A classical example is mechanical systems, whose passive outputs are the actuated velocities, but in many applications – e.g. robotics – the objective is to drive all positions to some desired constant values. To formulate mathematically this objective, we aim at achieving Lyapunov stability of the desired equilibrium, a task that entails the need to construct a Lyapunov function, i.e., a nonincreasing function of the state with a minimum at the desired equilibrium. The approach we adopt in the book to solve this new task is to identify passive outputs whose integral
