Group Coordination and Cooperative Control (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari 336 K.Y. Pettersen J.T. Gravdahl H. ...

Author: Kristin Y. Pettersen | Jan Tommy Gravdahl | Henk Nijmeijer

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Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

336

K.Y. Pettersen J.T. Gravdahl H. Nijmeijer (Eds.)

Group Coordination and Cooperative Control With 92 Figures

Series Advisory Board

F. Allg¨ower · P. Fleming · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · A. Rantzer · J.N. Tsitsiklis

Editors Professor Kristin Y. Pettersen Professor Jan Tommy Gravdahl Norwegian University of Science and Technology (NTNU) Faculty of Information Technology, Mathematics and Electrical Engineering Department of Engineering Cybernetics O.S. Bragstads plass 2D 7491 Trondheim Norway

Professor Dr. Henk Nijmeijer Eindhoven University of Technology Department of Mechanical Engineering Dynamics & Control Group P.O. Box 513 5600 MB Eindhoven The Netherlands [email protected]

[email protected] [email protected]

ISSN 0170-8643 ISBN-10 3-540-33468-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33468-2 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006923820 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by editors. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany (www.ptp-berlin.com) Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 89/3141/Yu - 5 4 3 2 1 0

Preface

Group coordination and cooperative control are topics that are currently receiving a lot of interest in a variety of research communities, including biology, robotics, communications and sensor networks, artiﬁcial intelligence, automatic control etc. Coordinating the motion of a group of relatively simple and inexpensive agents can cover a larger operational area and achieve complex tasks that exceeds the abilities of one single agent. Group coordination and cooperative control are enabling technologies for applications such as teams of cooperative robots performing demining operations, aerospace formation ﬂying for imaging and survey operations, ﬂeets of AUVs doing oceanographic surveys and environmental surveillance, and ships doing coordinated towing operations. Inspired by the progress in the ﬁeld, a workshop on Group Coordination and Cooperative Control was organised in Tromsø, Norway, 2006. The objective of this workshop was to focus on the control theoretic challenges that group coordination and cooperation raise. By bringing together a small number of researchers in control systems, providing an intimate ambience with presentations of recent results and discussions of theoretical challenges, industrial needs, unresolved problems and future research directions, the aim was to lay the ground for future research and cooperation on the topics of Group Coordination and Cooperative control. This volume contains the contributions of the workshop. The contributions cover a wide range of subjects within the area of group coordination and cooperative control. We hope this book forms a valuable and up-to-date text on the newer trends in group coordination and formation control, and that it may serve as a source of inspiration for the interested reader.

VI

Preface

Acknowledgements Financial support for the workshop was provided by the Strategic University Programme Computational Methods in Motion Control (CM-in-MC), Centre for Ships and Ocean Structures (CESOS), Norsk Hydro Fond, and Department of Engineering Cybernetics, NTNU. We are also most grateful to all the reviewers who assisted us in a timely manner.

Trondheim and Eindhoven, February 2006

Kristin Y. Pettersen Jan Tommy Gravdahl Henk Nijmeijer

Contents

1 Formation Control of Marine Surface Vessels Using the Null-Space-Based Behavioral Control Filippo Arrichiello, Stefano Chiaverini, and Thor I. Fossen . . . . . . . . . . . . . . . .

1

2 Passivity-Based Agreement Protocols: Continuous-Time and Sampled-Data Designs Emrah Bıyık and Murat Arcak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3 Cross-Track Formation Control of Underactuated Autonomous Underwater Vehicles Even Børhaug, Alexey Pavlov, and Kristin Y. Pettersen . . . . . . . . . . . . . . . . . . .

35

4 Kinematic Aspects of Guided Formation Control in 2D Morten Breivik, Maxim V. Subbotin, and Thor I. Fossen . . . . . . . . . . . . . . . . . .

55

5 ISS-Based Robust Leader/Follower Trailing Control Xingping Chen and Andrea Serrani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

6 Coordinated Path Following Control of Multiple Vehicles Subject to Bidirectional Communication Constraints Reza Ghabcheloo, Antonio Pascoal, Carlos Silvestre, and Isaac Kaminer . . . . .

93

7 Robust Formation Control of Marine Craft Using Lagrange Multipliers Ivar-Andr´e Flakstad Ihle, J´erˆ ome Jouﬀroy, and Thor I. Fossen . . . . . . . . . . . . . 113 8 Output Feedback Control of Relative Translation in a Leader-Follower Spacecraft Formation Raymond Kristiansen, Antonio Lor´ıa, Antoine Chaillet, and Per Johan Nicklasson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9 Coordinated Attitude Control of Satellites in Formation Thomas R. Krogstad, and Jan Tommy Gravdahl . . . . . . . . . . . . . . . . . . . . . . . . . . 153

VIII

Contents

10 A Virtual Vehicle Approach to Underway Replenishment Erik Kyrkjebø, Elena Panteley, Antoine Chaillet, and Kristin Y. Pettersen . . . 171 11 A Study of Huijgens’ Synchronization: Experimental Results Ward T. Oud, Henk Nijmeijer, and Alexander Yu. Pogromsky . . . . . . . . . . . . . . 191 12 A Study of Controlled Synchronization of Huijgens’ Pendula Alexander Yu. Pogromsky, Vladimir N. Belykh, and Henk Nijmeijer . . . . . . . . . 205 13 Group Coordination and Cooperative Control of Steered Particles in the Plane Rodolphe Sepulchre, Derek A. Paley, and Naomi Ehrich Leonard . . . . . . . . . . . . 217 14 Coordinating Control for a Fleet of Underactuated Ships Anton Shiriaev, Anders Robertsson, Leonid Freidovich, and Rolf Johansson . . 233 15 Decentralized Adaptation in Interconnected Uncertain Systems with Nonlinear Parametrization Ivan Tyukin and Cees van Leeuwen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 16 Controlled Synchronisation of Continuous PWA Systems Nathan van de Wouw, Alexey Pavlov, and Henk Nijmeijer . . . . . . . . . . . . . . . . . 271 17 Design of Convergent Switched Systems Roel A. van den Berg, Alexander Yu. Pogromsky, Gennady A. Leonov, and Jacobus E. Rooda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

List of Contributors

Murat Arcak Department of Electrical, Computer, and Systems Engineering. Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA. [email protected]

Even Børhaug Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway. [email protected]

Filippo Arrichiello Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale. Universit` a degli Studi di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italy. [email protected]

Antoine Chaillet CNRS-LSS, Sup´elec, 3 rue Joliot Curie, 91192 Gif s/Yvette, France. [email protected]

Vladimir N. Belykh Volga State Transport Academy, Department of Mathematics, N. Novgorod, Russia. [email protected]

Xingping Chen Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Ave, 205 Dreese Lab, Columbus, OH 43221, USA. [email protected]

Emrah Bıyık Department of Electrical, Computer, and Systems Engineering. Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA. [email protected]

Stefano Chiaverini Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale. Universit` a degli Studi di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italy. [email protected]

Morten Breivik Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

Thor I. Fossen Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

X

List of Contributors

Leonid Freidovich Department of Applied Physics and Electronics, University of Ume˚ a, SE-901 87 Ume˚ a, Sweden. [email protected]

Thomas R. Krogstad Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

Reza Ghabcheloo Institute for Systems and Robotics/Instituto Superior T´ecnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. [email protected]

Erik Kyrkjebø Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

Jan Tommy Gravdahl Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected] Ivar-Andr´ e Flakstad Ihle Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected] Rolf Johansson Department of Automatic Control, LTH, Lund University, PO Box 118, SE-221 00 Lund, Sweden. [email protected] ome Jouﬀroy J´ erˆ Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected] Isaac Kaminer Department of Mechanical and Astronautical Engineering, Naval Postgraduate School, Monterey, CA 93943, USA. [email protected] Raymond Kristiansen Department of Computer Science, Electrical Engineering and Space Technology, Narvik University College, Lodve Langesgt. 2, N-8505, Norway. [email protected]

Naomi Ehrich Leonard Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA. [email protected] Gennady A. Leonov St. Petersburg State University, Mathematics and Mechanics Faculty, Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia. [email protected] Antonio Lor´ıa CNRS-LSS, Sup´elec, 3 rue Joliot Curie, 91192 Gif s/Yvette, France. [email protected] Per Johan Nicklasson Department of Computer Science, Electrical Engineering and Space Technology, Narvik University College, Lodve Langesgt. 2, NO-8505, Norway. [email protected] Henk Nijmeijer Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]

List of Contributors

Ward T. Oud Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Derek A. Paley Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA. [email protected] Elena Panteley C.N.R.S, UMR 8506, Laboratoire de Signaux et Syst`emes, 91192 Gif s/Yvette, France. [email protected] Antonio Pascoal Institute for Systems and Robotics/Instituto Superior T´ecnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. [email protected] Alexey Pavlov Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway. [email protected] Kristin Y. Pettersen Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway. [email protected] Alexander Yu. Pogromsky Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]

XI

Anders Robertsson Department of Automatic Control, LTH, Lund University, PO Box 118, SE-221 00 Lund, Sweden. [email protected] Jacobus E. Rooda Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Rodolphe Sepulchre Electrical Engineering and Computer Science, Universit´e de Li`ege, Institut Monteﬁore B28, B-4000 Li`ege, Belgium. [email protected] Andrea Serrani Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Ave, 205 Dreese Lab, Columbus, OH 43221, USA. [email protected] Anton Shiriaev Department of Applied Physics and Electronics, University of Ume˚ a, SE-901 87 Ume˚ a, Sweden. [email protected] Carlos Silvestre Institute for Systems and Robotics/Instituto Superior T´ecnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. [email protected] Maxim V. Subbotin Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA. [email protected] Ivan Tyukin RIKEN Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan. [email protected]

XII

List of Contributors

Nathan van de Wouw Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Roel A. van den Berg Eindhoven University of Technology, Department of Mechanical Engineering,

P.O.Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Cees van Leeuwen RIKEN Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan. [email protected]

Preface

Group coordination and cooperative control are topics that are currently receiving a lot of interest in a variety of research communities, including biology, robotics, communications and sensor networks, artiﬁcial intelligence, automatic control etc. Coordinating the motion of a group of relatively simple and inexpensive agents can cover a larger operational area and achieve complex tasks that exceeds the abilities of one single agent. Group coordination and cooperative control are enabling technologies for applications such as teams of cooperative robots performing demining operations, aerospace formation ﬂying for imaging and survey operations, ﬂeets of AUVs doing oceanographic surveys and environmental surveillance, and ships doing coordinated towing operations. Inspired by the progress in the ﬁeld, a workshop on Group Coordination and Cooperative Control was organised in Tromsø, Norway, 2006. The objective of this workshop was to focus on the control theoretic challenges that group coordination and cooperation raise. By bringing together a small number of researchers in control systems, providing an intimate ambience with presentations of recent results and discussions of theoretical challenges, industrial needs, unresolved problems and future research directions, the aim was to lay the ground for future research and cooperation on the topics of Group Coordination and Cooperative control. This volume contains the contributions of the workshop. The contributions cover a wide range of subjects within the area of group coordination and cooperative control. We hope this book forms a valuable and up-to-date text on the newer trends in group coordination and formation control, and that it may serve as a source of inspiration for the interested reader.

VI

Preface

Acknowledgements Financial support for the workshop was provided by the Strategic University Programme Computational Methods in Motion Control (CM-in-MC), Centre for Ships and Ocean Structures (CESOS), Norsk Hydro Fond, and Department of Engineering Cybernetics, NTNU. We are also most grateful to all the reviewers who assisted us in a timely manner.

Trondheim and Eindhoven, February 2006

Kristin Y. Pettersen Jan Tommy Gravdahl Henk Nijmeijer

Contents

1 Formation Control of Marine Surface Vessels Using the Null-Space-Based Behavioral Control Filippo Arrichiello, Stefano Chiaverini, and Thor I. Fossen . . . . . . . . . . . . . . . .

1

2 Passivity-Based Agreement Protocols: Continuous-Time and Sampled-Data Designs Emrah Bıyık and Murat Arcak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3 Cross-Track Formation Control of Underactuated Autonomous Underwater Vehicles Even Børhaug, Alexey Pavlov, and Kristin Y. Pettersen . . . . . . . . . . . . . . . . . . .

35

4 Kinematic Aspects of Guided Formation Control in 2D Morten Breivik, Maxim V. Subbotin, and Thor I. Fossen . . . . . . . . . . . . . . . . . .

55

5 ISS-Based Robust Leader/Follower Trailing Control Xingping Chen and Andrea Serrani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

6 Coordinated Path Following Control of Multiple Vehicles Subject to Bidirectional Communication Constraints Reza Ghabcheloo, Antonio Pascoal, Carlos Silvestre, and Isaac Kaminer . . . . .

93

7 Robust Formation Control of Marine Craft Using Lagrange Multipliers Ivar-Andr´e Flakstad Ihle, J´erˆ ome Jouﬀroy, and Thor I. Fossen . . . . . . . . . . . . . 113 8 Output Feedback Control of Relative Translation in a Leader-Follower Spacecraft Formation Raymond Kristiansen, Antonio Lor´ıa, Antoine Chaillet, and Per Johan Nicklasson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9 Coordinated Attitude Control of Satellites in Formation Thomas R. Krogstad, and Jan Tommy Gravdahl . . . . . . . . . . . . . . . . . . . . . . . . . . 153

VIII

Contents

10 A Virtual Vehicle Approach to Underway Replenishment Erik Kyrkjebø, Elena Panteley, Antoine Chaillet, and Kristin Y. Pettersen . . . 171 11 A Study of Huijgens’ Synchronization: Experimental Results Ward T. Oud, Henk Nijmeijer, and Alexander Yu. Pogromsky . . . . . . . . . . . . . . 191 12 A Study of Controlled Synchronization of Huijgens’ Pendula Alexander Yu. Pogromsky, Vladimir N. Belykh, and Henk Nijmeijer . . . . . . . . . 205 13 Group Coordination and Cooperative Control of Steered Particles in the Plane Rodolphe Sepulchre, Derek A. Paley, and Naomi Ehrich Leonard . . . . . . . . . . . . 217 14 Coordinating Control for a Fleet of Underactuated Ships Anton Shiriaev, Anders Robertsson, Leonid Freidovich, and Rolf Johansson . . 233 15 Decentralized Adaptation in Interconnected Uncertain Systems with Nonlinear Parametrization Ivan Tyukin and Cees van Leeuwen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 16 Controlled Synchronisation of Continuous PWA Systems Nathan van de Wouw, Alexey Pavlov, and Henk Nijmeijer . . . . . . . . . . . . . . . . . 271 17 Design of Convergent Switched Systems Roel A. van den Berg, Alexander Yu. Pogromsky, Gennady A. Leonov, and Jacobus E. Rooda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

List of Contributors

Murat Arcak Department of Electrical, Computer, and Systems Engineering. Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA. [email protected]

Even Børhaug Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway. [email protected]

Filippo Arrichiello Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale. Universit` a degli Studi di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italy. [email protected]

Antoine Chaillet CNRS-LSS, Sup´elec, 3 rue Joliot Curie, 91192 Gif s/Yvette, France. [email protected]

Vladimir N. Belykh Volga State Transport Academy, Department of Mathematics, N. Novgorod, Russia. [email protected]

Xingping Chen Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Ave, 205 Dreese Lab, Columbus, OH 43221, USA. [email protected]

Emrah Bıyık Department of Electrical, Computer, and Systems Engineering. Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA. [email protected]

Stefano Chiaverini Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale. Universit` a degli Studi di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italy. [email protected]

Morten Breivik Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

Thor I. Fossen Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

X

List of Contributors

Leonid Freidovich Department of Applied Physics and Electronics, University of Ume˚ a, SE-901 87 Ume˚ a, Sweden. [email protected]

Thomas R. Krogstad Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

Reza Ghabcheloo Institute for Systems and Robotics/Instituto Superior T´ecnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. [email protected]

Erik Kyrkjebø Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]

Jan Tommy Gravdahl Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected] Ivar-Andr´ e Flakstad Ihle Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected] Rolf Johansson Department of Automatic Control, LTH, Lund University, PO Box 118, SE-221 00 Lund, Sweden. [email protected] ome Jouﬀroy J´ erˆ Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected] Isaac Kaminer Department of Mechanical and Astronautical Engineering, Naval Postgraduate School, Monterey, CA 93943, USA. [email protected] Raymond Kristiansen Department of Computer Science, Electrical Engineering and Space Technology, Narvik University College, Lodve Langesgt. 2, N-8505, Norway. [email protected]

Naomi Ehrich Leonard Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA. [email protected] Gennady A. Leonov St. Petersburg State University, Mathematics and Mechanics Faculty, Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia. [email protected] Antonio Lor´ıa CNRS-LSS, Sup´elec, 3 rue Joliot Curie, 91192 Gif s/Yvette, France. [email protected] Per Johan Nicklasson Department of Computer Science, Electrical Engineering and Space Technology, Narvik University College, Lodve Langesgt. 2, NO-8505, Norway. [email protected] Henk Nijmeijer Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]

List of Contributors

Ward T. Oud Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Derek A. Paley Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA. [email protected] Elena Panteley C.N.R.S, UMR 8506, Laboratoire de Signaux et Syst`emes, 91192 Gif s/Yvette, France. [email protected] Antonio Pascoal Institute for Systems and Robotics/Instituto Superior T´ecnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. [email protected] Alexey Pavlov Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway. [email protected] Kristin Y. Pettersen Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway. [email protected] Alexander Yu. Pogromsky Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]

XI

Anders Robertsson Department of Automatic Control, LTH, Lund University, PO Box 118, SE-221 00 Lund, Sweden. [email protected] Jacobus E. Rooda Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Rodolphe Sepulchre Electrical Engineering and Computer Science, Universit´e de Li`ege, Institut Monteﬁore B28, B-4000 Li`ege, Belgium. [email protected] Andrea Serrani Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Ave, 205 Dreese Lab, Columbus, OH 43221, USA. [email protected] Anton Shiriaev Department of Applied Physics and Electronics, University of Ume˚ a, SE-901 87 Ume˚ a, Sweden. [email protected] Carlos Silvestre Institute for Systems and Robotics/Instituto Superior T´ecnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. [email protected] Maxim V. Subbotin Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA. [email protected] Ivan Tyukin RIKEN Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan. [email protected]

XII

List of Contributors

Nathan van de Wouw Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Roel A. van den Berg Eindhoven University of Technology, Department of Mechanical Engineering,

P.O.Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] Cees van Leeuwen RIKEN Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan. [email protected]

1 Formation Control of Marine Surface Vessels Using the Null-Space-Based Behavioral Control F. Arrichiello1 , S. Chiaverini1, and T.I. Fossen2 1

2

Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale. Universit` a degli Studi di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italy. {f.arrichiello,chiaverini}@unicas.it Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway. [email protected]

Summary. In this paper the application of the Null-Space-Based behavioral control (NSB) to a ﬂeet of marine surface vessels is presented. From a marine applications point of view, the NSB can be considered as a guidance system that dynamically selects the motion reference commands for each vessel of the ﬂeet. These motion commands are aimed at guiding the ﬂeet in complex environments simultaneously performing multiple tasks, i.e., obstacle avoidance or keeping a formation. In order to apply the guidance system to a ﬂeet of surface vessels through the entire speed envelope (fully-actuated at low velocities, under-actuated at high velocities), the NSB works in combination with a low-level maneuvering control that, taking care of the dynamic models of the vessels, elaborates the motion commands to obtain the generalized forces at the actuators. The guidance system has been simulated while successfully performing complex missions in realistic scenarios.

1.1 Introduction The ﬁeld of cooperation and coordination of multi-robot systems has been object of considerable research eﬀorts in the last years. The basic idea is that multi-robot systems can perform tasks more eﬃciently than a single robot or can accomplish tasks not executable by a single one. Moreover, multi-robot systems have advantages like increasing tolerance to possible vehicle fault, providing ﬂexibility to the task execution or taking advantage of distributed sensing and actuation. In the context of this paper, the term multi-robot system means a generic system of autonomous vehicles like Unmanned Grounded Vehicles (UGVs), Unmanned Aerial Vehicles (UAVs), Unmanned Underwater Vehicles (UUVs) or autonomous surface vessels. Thus, applications of multi-robot systems can be widely diﬀerent such as, e.g., exploration of an unknown environment with a team of mobile robots [10], navigation and formation control with multiple UAVs [29], UUVs [12] and surface vessels [18, 19] and object transportation with multiple UGVs [30].

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 1–19, 2006. © Springer-Verlag Berlin Heidelberg 2006

2

F. Arrichiello, S. Chiaverini, and T.I. Fossen

When controlling a multi-robot system, a common requirement is the motion coordination among the vehicles; that is, the vehicles have to move in the environment keeping a suitable relative conﬁguration. Among other possible methods, in this paper behavior-based approaches to coordinate a multi-robot system are considered. Behavior-based approaches, widely studied for robotic applications [5], are useful to guide a multi-robot system in an unknown or dynamically changing environment. These approaches give the system the autonomy to navigate in complex environments avoiding low-level path planning, using sensors to obtain instantaneous information of the environment and increasing ﬂexibility of the system. Among the behavioral approaches, seminal works are reported in the papers [9] and [4], while the textbook [5] oﬀers a comprehensive state of the art. A behavioral approach designed for exploration of planetary surfaces has been investigated in [17], while in [20] the experimental case of an autonomous navigation in outdoor natural terrain is presented. Lately, behavioral approaches have been applied to the formation control of multi-robot systems as in, e.g. [23], [22] and [6]. The use of techniques inherited from inverse kinematics for industrial manipulators is described in [7, 3]. In [26] an architecture for dynamic changes of the behavior selection strategies is presented. In the behavior-based approach, the mission of the system is usually decomposed in elementary sub-problems, eventually solvable in parallel, whose solutions need to be composed in one single motion command for each robot. The sub-problems are commonly termed behaviors or tasks. In presence of multiple behaviors, each task output is designed so as to achieve its speciﬁc goal but it is generally impossible that a single motion command to the robot can accomplish all the assigned behaviors at the same time. In particular, when a motion command cannot reduce simultaneously the value of all the task functions there is a conﬂict among the tasks that must be solved by a suitable policy. The behavior-based approach proposed in this paper, namely the Null-SpaceBased behavioral control (NSB), diﬀers from the other existing methods in the behavioral coordination method, i.e., in the way the outputs of the single elementary behaviors are assembled to compose a complex behavior. The NSB can be seen as a centralized system aimed at guiding a platoon of autonomous vehicles in different scenarios and to achieve diﬀerent missions. Following the main behavioral approaches, the mission of the platoon is decomposed in elementary tasks, i.e., move the barycenter of the platoon, avoid obstacles, keep a formation. For each task, a function that measures its degree of fulﬁlment (e.g., a cost or a potential function) can be deﬁned; thus, in a static environment, the task is achieved when its output is constant at a value that minimizes the task function. Using a suitable policy to manage multiple tasks, the NSB is aimed at elaborating the instantaneous motion references for each vehicle. How to follow these motion references, instead, is the aim of a low-level maneuvering control system. In this way, the NSB does not take care of the typology of the vehicles and let the maneuvering control system take into consideration the kinematics and the dynamics of the vehicles. This characteristic lets the NSB be applicable to diﬀerent kinds of vehicles like autonomous robots (both holonomic and nonholonomic), underwater robots or surface vessels (fully actuated and underactuated).

1 Formation Control of Marine Surface Vessels Using the NSB v NSB,1 σd , σ˙ d

NSB

v NSB v NSB,n

env η

Maneuvering c.1 .. .

τ1

ηn , νn

η1 , ν1

w1

η1 , ν1

Maneuvering c.n

Vessel 1 .. .

τn

Vessel n

3

η, ν ηn , νn

wn

Fig. 1.1. Sketch of the guidance system control schema

In this paper the application of the NSB to a ﬂeet of marine surface vessels is presented. The proposed case considers vessels that are underactuated at high velocities and fully-actuated at low velocities. That is, the propulsive action of the vessels can span all the generalized directions at low speeds (i.e., both force in the surge and sway direction an torque in the yaw direction), and part of them at high speeds. The proposed guidance system has been widely tested in numerical simulations and a case study of a ﬂeet of vessels performing complex missions in realistic scenarios is presented in order to validate the eﬀectiveness of the approach.

1.2 Guidance System for Marine Surface Vessels The guidance system proposed in this paper has to solve simultaneously diﬀerent problems: it is responsible for ships’ safety, that is, it has to make the ships able to avoid static and dynamic obstacles doing a dynamic path generation (path generation on-the-ﬂy); it has to achieve diﬀerent kinds of missions, e.g., navigation keeping a ﬁxed relative formation or changing the route or the formation considering weather and environmental conditions; it has to maneuver fully/under-actuated ships (ships equipped with 3/less than 3 actuators). To this purpose, the guidance system (see Figure 1.1) is decomposed in two main blocks: the Null-Space-Based behavioral control and the maneuvering control. The NSB takes into consideration the parameters of the mission, the environmental condition and the status of the ﬂeet to elaborate the desired velocities for each vehicle. These velocities represent the reference input for the maneuvering controls that, taking into consideration kinematics and dynamics of the ships, have to deﬁne the generalized forces applied by the actuators. In the following both the blocks will be extensively explained. 1.2.1 Null-Space-Based Behavioral Control for Autonomous Vehicles The Null-Space-Based behavioral approach is a centralized system aimed at guiding a platoon of generic autonomous vehicles in diﬀerent scenarios and to achieve different missions. As explained in [1, 2], the NSB can be considered as a behavioral

4

F. Arrichiello, S. Chiaverini, and T.I. Fossen

approach that, diﬀerently from others, uses a hierarchy based logic to combine multiple conﬂicting tasks. In particular, the NSB is able to fulﬁll or partially fulﬁll each task according to their position in the hierarchy and according to the eventual conﬂicts with the highest priority tasks. The basic concepts of the NSB, with reference to a generic platoon of autonomous vehicles, will be recalled in the following. By deﬁning as σ ∈ IRm the task variable to be controlled and as p ∈ IRn the system conﬁguration, the task function f : IRn → IRm is: σ = f (p)

(1.1)

with the corresponding diﬀerential relationship: σ˙ =

∂f (p) v = J (p)v , ∂p

(1.2)

where J ∈ IRm×n is the conﬁguration-dependent task Jacobian matrix and v ∈ IRn is the system velocity. Notice that n depends on the speciﬁc autonomous system considered and the term system conﬁguration simply refers to the vessel position/orientation (in case of a material point n = 2, in the case of a platoon of z surface vessels n = 3z). An eﬀective way to generate smooth motion references pd (t) for the vehicles, starting from smooth desired values σ d (t) of the task function, is to act at the differential level by inverting the (locally linear) mapping (1.2); in fact, this problem has been widely studied in robotics (see, e.g., [27] for a tutorial). A typical requirement is to pursue minimum-norm velocity, leading to the least-squares solution: v N SB = J † σ˙ d ,

(1.3) −1

. where the pseudoinverse is deﬁned as J † = J T J J T At this point, the vehicle motion controller needs a reference position trajectory besides the velocity reference; this can be obtained by time integration of v d . However, discrete-time integration of the vehicle’s reference velocity would result in a numerical drift of the reconstructed vehicle’s position; the drift can be counteracted by a so-called Closed Loop Inverse Kinematics (CLIK) version of the algorithm, namely, (1.4) v N SB = J † σ˙ d + Λσ , where Λ is a suitable constant positive-deﬁnite matrix of gains and σ is the task error deﬁned as σ = σ d −σ. The Null-Space-Based behavioral control intrinsically requires a diﬀerentiable analytic expression of the tasks deﬁned so that it is possible to compute the required Jacobians. Considering the case of multiple tasks, on the analogy of (1.4) the single task velocity is computed as (1.5) v i = J †i σ˙ i,d + Λi σ i , where the subscript i denotes i-th task quantities. If the subscript i also denotes the degree of priority of the task (e.g. Task 1 being the highest-priority one), in a case of 3 tasks, according to [11] the CLIK solution (1.4) is modiﬁed into

1 Formation Control of Marine Surface Vessels Using the NSB Task #A sensors

Task #B Task #C

vA

supervisor

!

v1

vB

v2

vC

v3

!

5

vd

I − J †1 J 1

I − J †2 J 2

Fig. 1.2. Sketch of the Null-Space-Based behavioral control in a 3-task example. The supervisor is in charge of changing the relative priority among the tasks

v N SB = v 1 + I − J †1 J 1

v 2 + I − J †2 J 2 v 3 .

(1.6)

Remarkably, equation (1.6) has a nice geometrical interpretation. Each task velocity is computed as if it were acting alone; then, before adding its contribution to the overall vehicle velocity, a lower-priority task is projected (by I − J †i−1 J i−1 ) onto the null space of the immediately higher-priority task so as to remove those velocity components that would conﬂict with it. The Null-Space-Based behavioral control always fulﬁls the highest-priority task at nonsingular conﬁgurations. The lower-priority tasks, on the other hand, are fulﬁlled only in a subspace where they do not conﬂict with the ones having higher priority, that is, each task reaches a sub-optimal condition that optimizes the task respecting the constraints imposed by the highest-priority tasks. A functional scheme of this architecture is illustrated in Figure 1.2. The presence of a supervisor might be considered in order to dynamically change the relative task priorities. For instance, in absence of close obstacles, an obstacle-avoidance task could be locally ignored. It must be remarked that, with this approach, a full-dimensional highest-priority task would subsume the lower-priority tasks. In fact, with a full-dimensional fullrank J 1 matrix, its null space would be empty and the whole vector v 2 would be ﬁltered out. In case of non-conﬂicting tasks, all the tasks are totally fulﬁlled. Stability of the NSB for Material Points Under Two Tasks Assuming that the vessels perfectly follow the desired motion references v = v N SB (that is, the vessels behave as material points), the stability analysis of the NSB is reduced to only prove convergence of the task functions to the desired value. Let us consider the case of 2 tasks acting simultaneously. The velocity of the ships are: v = v N SB = J †1 σ˙ 1,d + Λ1 σ 1 + I − J †1 J 1 J †2 σ˙ 2,d + Λ2 σ 2 .

(1.7)

Considering the Lyapunov function V1 = 12 σ 21 and writing the equation of V˙ 1 , then, by (1.2):

6

F. Arrichiello, S. Chiaverini, and T.I. Fossen T ˙ ˙ 1,d − J 1 v) V˙ 1 = σ T 1 σ 1 = σ 1 (σ

˙ 1,d − J 1 J †1 σ˙ 1,d + Λ1 σ 1 − J 1 I − J †1 J 1 J †2 σ˙ 2,d + Λ2 σ 2 V˙ 1 = σ T 1 σ ˙ 1,d − J 1 J †1 σ˙ 1,d + Λ1 σ 1 = σT 1 σ

= −σ T 1 Λ1 σ 1 .

Recalling that Λ1 is positive deﬁnite, then the convergence to 0 of σ 1 is proved; that is, the ﬁrst task is always achieved. Considering the Lyapunov function for the secondary task V2 = 12 σ 22 T ˙ ˙ 2,d − J 2 v) V˙ 2 = σ T 2 σ 2 = σ 2 (σ

˙ 2,d − J 2 J †1 σ˙ 1,d + Λ1 σ 1 − J 2 I − J †1 J 1 J †2 σ˙ 2,d + Λ2 σ 2 = σT 2 σ † ˙ 1,d + Λ1 σ 1 + J 2 J †1 J 1 J †2 σ˙ 2,d + Λ2 σ 2 = σT 2 −Λ2 σ 2 − J 2 J 1 σ

Like widely explained in Section 1.2.1, the fulﬁllment of the secondary task is veriﬁed only when it is not conﬂicting with the primary one, that is, the product J 2 J †1 is null. In this case V˙ 2 = −σ T 2 Λ2 σ 2 and the convergence to 0 of σ 2 is proved. 1.2.2 Maneuvering Control of Marine Surface Vessels A maneuvering control is a on board controller aimed at steering the vessel along a desired path and moving it with a desired velocity [13, 14]. Thus, receiving motion reference commands, the maneuvering control has to dynamically elaborate the generalized forces applied by the actuators. For fully actuated ships, the maneuvering control problem is extensively explained in [28]. For underactuated ships, equipped with less than 3 actuators (only 1-2 actuators are used to control the surge, sway and yaw modes), the maneuvering control is a challenging problem. In works [24, 21, 25] the main control problems regarding underactuated ships are recalled, while in [15] a nonlinear path-following for underactuated marine craft is presented.

Fig. 1.3. Tugboat with two main azimuth thrusters aft and one tunnel thruster in the bow. Courtesy of Roll-Royce Marine

1 Formation Control of Marine Surface Vessels Using the NSB

7

In this paper the case of surface vessels underactuated at high velocities and fully-actuated at low velocities is considered. The only relevant types of vessels to consider are the ones which become underactuated in the sway direction (lateral direction) at high speed. These vessels are typically equipped with a number of tunnel thrusters, designed to assist at low-speed maneuvers, which are inoperable at high speeds due to the relative water speed past their outlets; therefore, at high speeds, the only mean of actuation are the main thrusters. An example of such a vessel is a tugboat from Rolls-Royce Marine (Figure 1.3). For the vessels, the following 3 DOF nonlinear maneuvering model [13], is considered: η˙ = R(ψ) ν

(1.8)

M ν˙ + N (ν)ν = τ + RT(ψ) w,

(1.9)

where η = [n e ψ]T is the position and attitude vector in the North-East-Down T (NED) reference frame; ν = [u v r] is the linear and angular velocity vector in the Body-ﬁxed (BODY) reference frame; R(ψ) ∈ SO(3) is the rotation matrix from NED to BODY; M is the vessel inertia matrix; N (ν) is the sum of the centrifugal and Coriolis matrix and of the hydrodynamic damping matrix; τ is the vessel propulsion force and torque; w is the vector of the environmental forces (wind, currents, etc.) acting on the vessel in the NED reference system. In the proposed guidance system, the maneuvering control is a velocity and heading controller aimed at making each vehicle follow its velocity reference command elaborated by the NSB, that, following the schema of Figure 1.4, can be converted from v N SB (elaborated considering the vessels as material points) to UN SB and χN SB .

ψ

n

χ u β

{B} v

U

e Fig. 1.4. Motion reference model of the surface vessel

Following the approach proposed in [8], a nonlinear, model-based velocity and heading controller is designed referring to an adaptive backstepping technique.

8

F. Arrichiello, S. Chiaverini, and T.I. Fossen

Start by deﬁning the error variables z1 ∈ IR and z 2 ∈ IR3 : z1 = χ − χN SB = hT η + β − χN SB = hT η − ψN SB z 2 = ν − α, where hT = [ 0 0 1 ]T , ψN SB = χN SB − β and α = [ α1 α2 α3 ]T ∈ IR3 is a vector of stabilizing functions to be speciﬁed later. Step 1: Deﬁne the Control Lyapunov Function as: V1 =

1 k1 z12 , 2

(1.10)

where k1 > 0. Diﬀerentiating V1 with respect to time V˙ 1 = k1 z1 z˙1 = k1 z1 hT η˙ − ψ˙ N SB V˙ 1 = k1 z1 hT ν − ψ˙ N SB , since η˙ = Rν and hT Rν = hT ν. By deﬁnition of z 2 , then: V˙ 1 = k1 z1 hT (z 2 + α) − ψ˙ N SB = k1 z1 hT z 2 + k1 z1 α3 − ψ˙ N SB . Choosing α3 = ψ˙ N SB − z1 ,

(1.11)

V˙ 1 = −k1 z12 + k1 z1 hT z 2

(1.12)

then,

Step 2: Deﬁne the Control Lyapunov Function as: V2 =

1 1 1 T −1 ˜ ˜ Γ w k1 z12 + z T M z2 + w 2 2 2 2

˜ ∈ IR3 is parameter error due to environmental disturbances deﬁned as where w ˜ =w ˆ − w with w ˆ being estimate of w, and Γ = Γ T > 0. w ˜ and assuming w ˙ = 0. then: Diﬀerentiating V2 along trajectories of z1 , z 2 and w ˙2 + w ˜ T Γ −1 w, V˙ 2 = −k1 z12 + k1 z1 hT z 2 + z T ˆ˙ 2 Mz ˜˙ = w. ˆ˙ since M = M T and w Since

1 Formation Control of Marine Surface Vessels Using the NSB

9

˙ M z˙ 2 = M (ν˙ − α) ˙ = τ − N (ν)ν + RT(ψ) w − M α then: T ˙ +w ˜ T Γ −1 w. V˙ 2 = −k1 z12 + z T ˆ˙ 2 hk1 z1 + τ − N ν + R w − M α

˜ =w ˆ − w, then: Being ν = z 2 + α and w T T ˆ − Mα ˙ V˙ 2 = −k1 z12 − z T 2 N z 2 + z 2 hk1 z1 + τ − N α + R w

˜ T Γ −1 w ˆ˙ − Γ Rz 2 . +w Assigning

ˆ − hk1 z1 − K 2 z 2 ˙ + N α − RT w τ = Mα

where K 2 > 0, and choosing: then:

(1.13)

w ˆ˙ = Γ Rz 2

(1.14)

V˙ 2 = −k1 z12 − z T 2 (N + K 2 )z 2 .

(1.15)

As stated in [16], since the system is persistently excited, choosing smooth α1 , α˙ 1 , α2 , α˙ 2 ∈ L∞ then the proposed control and disturbance adaption law makes the origin of the ˆ UGAS/ULES. error system [z1 , z 2 , w] The choice of α1 , α2 depends on the actuation system of the vessels, i.e., when the vessel is fully-actuated α1 = UN SB cos βN SB α2 = UN SB sin βN SB ; while when the vessel is under-actuated α1 = UN SB cos βN SB and α2 is deﬁned such as τ2 = 0. ˆ converges The proposed controller guarantees that z1 , z 2 go to zero, and that w to w. Thus, the vessel velocity converges to the desired values {UN SB , χN SB } only when the vessel is fully-actuated; for an underactuated vessel, instead, u converges to UN SB cos βN SB while v is uncontrolled. It is worth noting that, while √ an underu actuated ship executes a straight motion at cruise speed it is U = u2 + v 2 since the sway velocity v is much smaller than u; on the other hand, when v cannot be neglected with respect to u (as, e.g., in turning maneuvers) it is (UN SB cos βN SB )2 +v 2 . Nevertheless, in real applications, ships move usU = ing way-point tracking and turning maneuvers are used for short time periods to adjust changes of course; thus, the convergence properties of the proposed control law are acceptable in practice.

10

F. Arrichiello, S. Chiaverini, and T.I. Fossen

When implementing the control law (1.13) it is important to avoid expressions involving the time derivatives of the states. To this purpose, the time derivative of the stabilizing function α is conveniently computed by diﬀerentiating along the trajectories of the states [13]. Moreover, to obtain high-order derivatives of ψN SB in (1.11), it is worth noticing that ψN SB is deﬁned as ψN SB = χN SB − β, where χN SB is the reference input obtained by the NSB and β is measured using GPS velocities. Thus, the high-order derivatives ψ˙ N SB , ψ¨N SB are computed through a low-pass ﬁlter. More details of the described maneuvering control can be found in [8].

1.3 Tasks According to the behavioral control approach, the mission of the ﬂeet is decomposed in three elementary tasks: move the barycenter of the ﬂeet, keep a formation relative to the barycenter, and avoid collisions with obstacles and among vehicles. In this section, the corresponding task functions are presented. 1.3.1 Barycenter The barycenter of a platoon expresses the mean value of the vehicles positions. In a 2-dimensional case (like for material points) the task function is expressed by: σ b = f b (p1 , . . . , pn ) =

1 n

n

pi . i=1

T

where pi = [ηi,1 ηi,2 ] is the position of the vehicle i. Deriving the previous relation: n

σ˙ b = i=1

∂f b (p) v i = J b (p) v, ∂pi

where the Jacobian matrix J b ∈ IR2×2n is Jb =

1 n

10 10 . ... 01 01

Following equation (1.4), the output of the barycenter task function is v b = J †b σ˙ b,d + Λb σ b ,

(1.16)

where the desired value of the task function represents the desired trajectory of the barycenter.

1 Formation Control of Marine Surface Vessels Using the NSB

11

1.3.2 Rigid Formation The rigid formation task moves the vehicles to a predeﬁned formation respect to the barycenter. The task function is deﬁned as:   p1 − pb   .. σf =  , . pn − pb

where pn are the coordinates of the vehicle n and pb are the coordinates of the barycenter. Writing for simplicity the vector p as [ η1,1 . . . ηn,1 η1,2 . . . ηn,2 ]T , then the Jacobian matrix J f ∈ IR2n×2n is: AO , OA

Jf = where A ∈ IRn×n is:



1 1− n

− n1 . . . − n1

(1.17)



 1   − n 1 − n1 . . . − n1   A=  .. .. . . . .  . . ..  . − n1 − n1 . . . 1 − n1

(1.18)

Because the Jacobian matrix is singular, the pseudoinverse can not be calculated as J T JJ T

−1

but as a matrix that veriﬁes the following properties: J J †J = J †;

J †J J † = J

and with JJ † and J † J symmetric. Since J f is symmetric and idempotent, J †f = J f . The desired value σ f,d of the task function describes the shape of the desired formation; that is, once deﬁned the formation, the elements of σ f,d represent the coordinates of each vehicle in the barycenter reference frame. The output of the formation task function, in the case of ﬁxed desired formation (σ˙ f,d = 0), is: (1.19) v f = J †f Λf σ f 1.3.3 Obstacle Avoidance The obstacle avoidance task function is built individually to each vehicle, i.e., it is not an aggregate task function. In fact, an obstacle in the environment may be close to some vehicle but far from some other; moreover, each vehicle is an obstacle for the others in the team but not for itself. With reference to the generic vehicle in the team, in presence of an obstacle in the advancing direction, the task function has to elaborate a driving velocity, aligned to the vehicle-obstacle direction, that keeps the vehicle at a safe distance d from the obstacle. Therefore, it is:

12

F. Arrichiello, S. Chiaverini, and T.I. Fossen

σo = p − po

σo,d = d

J o = rˆT ,

where po is the obstacle position and ˆ= r

p − po p − po

is the unit vector aligned with the obstacle-to-vehicle direction. According to the above choice,equation (1.5) simpliﬁes to v o = J †o λo σ o = λo (d − p − po ) rˆ,

(1.20)

where λo is a suitable constant positive scalar. It is worth noting that, being ˆrˆ T , N (J o ) = I − r the tasks of lower priority than the obstacle avoidance are only allowed to produce motion components tangent to the circle of radius d and centered in po , so as to not interfere with the enforcement of the safe distance d. While the implementation of the proposed obstacle-avoidance task function is the same for both punctual environmental obstacles and other vehicles, in the case of continuous obstacles it changes a bit. In particular, for convex- or straight-line obstacles, po represents the coordinates of the closest point of the obstacle to the ship at the current time instant. In the frequent case of multiple obstacles acting simultaneously (e.g., both an obstacle in the environment and the other vehicles of the team) a priority among their avoidance should be deﬁned; a reasonable choice is to assign the currently closest obstacle the highest priority. In critical situations the obstacle avoidance function may give a null-velocity output; this causes delay to the mission or loss of vehicles to the formation but increases safety of the approach.

1.4 Simulations In this section results of two simulations in complex realistic scenarios will be presented. In the ﬁrst proposed scenario (see Figure 1.5) a ﬂeet of 7 vessels has to overtake a bottleneck (that can represent the costal proﬁle of a fjord) and a small obstacle, in presence of environmental disturbances, keeping a V-formation (a conﬁguration widely used for Navy and military applications). The mission is decomposed in three elementary tasks: move the barycenter of the ﬂeet, keep a formation relative to the barycenter, and avoid collisions with obstacles and among vehicles. The obstacle-avoidance is the highest priority task because its achievement is of crucial importance to preserve the integrity of the vessels, while the barycenter and the rigid formation are respectively the secondary and tertiary tasks. Following the stability analysis consideration of Section 1.2.1, it is easy to prove that the barycenter and the rigid formation task function are not conﬂicting (J f J †b = 0). The obstacle avoidance task function has to ensure each vessel a safe distance of 60 m from the environmental obstacles and from other vessels. For each vessel, the

1 Formation Control of Marine Surface Vessels Using the NSB

500 n [m]

n [m]

500

0

−500

0

0

0

0

500 e [m]

0

500 e [m]

1000

0

500 e [m]

1000

0

500 e [m]

1000

0

−500

1000

500

0

−500

1000

500

n [m]

500

500 e [m]

500 e [m]

0

−500

1000

0

−500

0

500

n [m]

n [m]

500 e [m]

0

−500

1000

0

500

n [m]

500 e [m]

n [m]

n [m]

500

−500

13

1000

0

−500

Fig. 1.5. Navigation in complex environment. Obstacle Avoidance-Barycenter-Rigid Formation

14

F. Arrichiello, S. Chiaverini, and T.I. Fossen a

500

b

25

! o [m] σ

n [m]

20

0

15 10 5 0

−500

0

500

−5

1000

e [m] c

50

400

600

800

1000

600

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1000

t [s] d

100

! f [m] σ

30

! b [m] σ

200

150

40

20 10 0

50 0 −50 −100

−10 −20

0

0

500

1000

t [s]

1500

2000

−150

0

200

400

t [s]

Fig. 1.6. a) Paths followed by the ships; b) error of the obstacle avoidance task function; c) error of the barycenter task function; d) error of the rigid formation task function

task function is activated only when the distance from environmental obstacles or other vessels becomes lower than 60 m. When the vessel is simultaneously close to (i.e., under 60 m from) multiple obstacles, then the closest obstacle has the highest priority. Since a moving obstacle is assumed to be more dangerous than a ﬁxed one, if the vessel is simultaneously close to an environmental obstacle and another vessel, the avoidance of the other vessel takes higher priority if the distance from it is greater than the distance from the environmental obstacle multiplied by a gain chosen (by trial and error) equal to 0.3 . The desired trajectory of the barycenter is a rectilinear segment that connects the initial position [ 0 0 ]T m to the ﬁnal position [ 800 0 ]T m according to a ﬁfth-order polynomial time law of 700 s duration with null initial and ﬁnal velocity. The vessels have to attain and keep a V-formation around the barycenter and, once reached the ﬁnal conﬁguration, the vessels has to keep the formation and orient themselves in the opposite direction to the current. The gain matrices of the barycenter task function and of the rigid formation task function are respectively a 2-dimensional diagonal matrix with coeﬃcient 0.1 and a 14-dimensional diagonal matrix with coeﬃcient 0.1 . The vessels have the dynamic model described in Section 1.2.2, where the matrices M and N are deﬁned as:

1 Formation Control of Marine Surface Vessels Using the NSB 500

n [m]

n [m]

500

0

−500 −200

0

200

400

e [m]

600

800

0

−500 −200

1000

0

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0

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e [m]

500

n [m]

n [m]

500

0

−500 −200

15

0

200

400

e [m]

600

800

0

−500 −200

1000

e [m]

Fig. 1.7. Navigation in complex environment. Obstacle Avoidance-Barycenter-Rigid Formation

  M =    N =

2

0.0035 Ns m

0

0

0.0046 Ns m

0

−0.0009 Ns m

0.0007 0 0

Ns m

0

2

0 0.0071 Ns m

2



 9 −0.0009 Ns2   ∗ 10

1.0740 Ns2  0  0.1812 Ns  ∗ 108

−0.1079 Ns m 1.8490 Ns

To simulate the under-actuation at the high velocities, the force in sway direction (τ2 ) is saturated by a value depending on the velocity in the surge direction (τ1 , τ3 are saturated by ﬁxed values representing the realistic limits of the actuators). The environmental disturbances are represented by a force (constant in the NED reference frame) of w = [−10000 − 30000 0]N. Figure 1.5 shows that the vessels do overtake the bottleneck avoiding collisions and reach the ﬁnal conﬁguration (both barycenter and formation). In particular, the vessels, starting from the initial V-formation, loose the conﬁguration to avoid the obstacles and, once overtaken, reach again the desired formation. To avoid os-

16

F. Arrichiello, S. Chiaverini, and T.I. Fossen a

500

b

30

! o [m] σ

n [m]

25

0

20 15 10 5

−500

0

500

c

20 15

60

10

40

5 0

−10

−40 500

1000

t [s]

400

1500

2000

600

800

1000

600

800

1000

t [s] d

0 −20

0

200

20

−5

−15

0

80

! f [m] σ

! b [m] σ

0

1000

e [m]

−60

0

200

400

t [s]

Fig. 1.8. a) Paths followed by the ships; b) error of the obstacle avoidance task function; c) error of the barycenter task function; d) error of the rigid formation task function

cillations around the ﬁnal conﬁguration, when the vessels are close to the desired conﬁguration then the desired velocities are imposed equal to 0 and the desired orientation equal to the estimated current opposite direction. In this way, the vessels orient themselves in the opposite direction to the current and keep their positions compensating the current with the only main propellers. It is worth noticing that the proposed approach permits also switching of the formations, thus, in a similar environment, it can be useful changing the formation to overtake the obstacles (i.e., reducing the angle of the V or reaching a line conﬁguration) and come back to the desired conﬁguration once far from the obstacles. Figure 1.6 shows the errors of the task functions during the all mission. The obstacle avoidance is activated only close to the obstacles (in middle of the mission). The barycenter function error starts from a low value (the barycenter of the initial conﬁguration is close to the desired value), increases during the avoidance of the bottleneck obstacle and decreases once overtaken. The rigid formation function error starts from a null value, increases during the obstacle avoidance and converges to zero once overtaken the obstacles. In the second scenario(see Figure 1.7) a ﬂeet of 7 vessels has to navigate through an environment full of punctual obstacles, in presence of environmental disturbances, keeping a V-formation. The mission parameters are the same of the previous example but the great number of obstacles and the presence of current permits to test the

1 Formation Control of Marine Surface Vessels Using the NSB

17

navigation system while performing missions in extreme conditions. Figures 1.7, 1.8 show that the mission is successfully performed also in this scenario. It is worth noticing that the error of the obstacle avoidance task function reaches a peak of 25m, thus each vessel keeps a safe distance from obstacles and other vessels always greater than 30m (the peak is veriﬁed when a vessel is simultaneously close to an obstacle and another vessel). Figure 1.9.a) shows that the estimation of the current ˆ does converge to the real value w = [−10000 − 30000 0]N while Figure 1.9.b) w shows that in the ﬁnal conﬁguration all the ships orient themselves in the opposite direction to the current. Videos of the performed simulations can be found at http : //webuser.unicas.it/arrichiello/video/ a)

4

3

x 10

2

2

θ [rad]

ˆ [N] w

1 0 −1 −2

1 0 −1 −2

−3 −4

b)

3

0

1000

t [s]

2000

3000

−3

0

1000

t [s]

2000

3000

Fig. 1.9. a) Current estimation; b) Orientation of the ships

1.5 Conclusion In this paper the null-space-based behavioral control has been presented to guide a ﬂeet of autonomous marine surface vessels in complex environments. The NSB works in combination with the low-level maneuvering controls of each ship to take into consideration the dynamics of the ﬂeet. The guidance system has been simulated in realistic missions involving the attainment of a formation while moving through obstacles in presence of sea current; the obtained results show the eﬀectiveness of the proposed method.

Acknowledgment This research has been partially supported by a Marie Curie Fellowship of the European Community programme CyberMar Marie Curie Training Site in Trondheim. The authors appreciate the comments from Gianluca Antonelli, Ivar-Andr´e F. Ihle and Fabio Celani.

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References 1. G. Antonelli, F. Arrichiello, and S. Chiaverini. Experimental kinematic comparison of behavioral approaches for mobile robots. In Proceedings of the 16th IFAC World Congress, Prague, CZ, July 2005. 2. G. Antonelli, F. Arrichiello, and S. Chiaverini. The null-space-based behavioral control for mobile robots. In Proceedings of the 2005 IEEE International Symposium on Computational Intelligence in Robotics and Automation, Espoo, Finland, June 2005. 3. G. Antonelli and S. Chiaverini. Kinematic control of a platoon of autonomous vehicles. In Proceedings of the 2003 IEEE International Conference on Robotics and Automation, pages 1464–1469, Taipei, TW, Sept. 2003. 4. R.C. Arkin. Motor schema based mobile robot navigation. The International Journal of Robotics Research, 8(4):92–112, 1989. 5. R.C. Arkin. Behavior-Based Robotics. The MIT Press, Cambridge, MA, 1998. 6. T. Balch and R.C. Arkin. Behavior-based formation control for multirobot teams. IEEE Transactions on Robotics and Automation, 14(6):926–939, 1998. 7. B.E. Bishop. On the use of redundant manipulator techniques for control of platoons of cooperating robotic vehicles. IEEE Transactions on Systems, Man and Cybernetics, 33(5):608–615, Sept. 2003. 8. M. Breivik and T.I. Fossen. A uniﬁed concept for controlling a marine surface vessel through the entire speed envelope. In Proceedings of the 2005 IEEE International Symposium on Mediterrean Conference on Control and Automation, pages 1518– 1523, Limassol, Cyprus, June 2005. 9. R.A. Brooks. A robust layered control system for a mobile robot. IEEE Journal of Robotics and Automation, 2:14–23, 1986. 10. W. Burgard, M. Moors, C. Stachniss, and F.E. Schneider. Coordinated multi-robot exploration. IEEE Journal of Robotics, 21(3):376–386, June 2005. 11. S. Chiaverini. Singularity-robust task-priority redundancy resolution for real-time kinematic control of robot manipulators. IEEE Transactions on Robotics and Automation, 13(3):398–410, 1997. 12. E. Fiorelli, N.E. Leonard, P. Bhatta, D. Paley, R. Bachmayer, and D.M. Fratantoni. Multi-auv control and adaptive sampling in monterey bay. In Proceedings IEEE Autonomous Underwater Vehicles 2004: Workshop on Multiple AUV Operations, pages 134–147, Sebasco, ME, June 2004. 13. T.I. Fossen. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, 2002. 14. T.I. Fossen. A nonlinear uniﬁed state-space model for ship maneuvering and control in a seaway. Journal of Bifurcation and Chaos, 2005. 15. T.I. Fossen, M. Breivik, and R. Skjetne. Line-of-sight path following of underactuated marine craft. In Proceedings of the 2003 IFAC Conference on Maneuvering and Control of Marine Craft, Girona, Spain, September 2003. 16. T.I. Fossen, A. Loria, and A. Teel. A theorem for ugas and ules of (passive) nonautonomous systems: Robust control of mechanical systems and ships. International Journal of Robust and Nonlinear Control, JRNC-11:95–108, 2001. 17. E. Gat, R. Desai, R. Ivlev, J. Loch, and D.P. Miller. Behavior control for robotic exploration of planetary surfaces. IEEE Transactions on Robotics and Automation, 10(4):490–503, 1994. 18. I.-A. F. Ihle, J. Jouﬀroy, and T.I. Fossen. Formation control of marine surface craft using lagrange multipliers. In Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, 2005.

1 Formation Control of Marine Surface Vessels Using the NSB

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19. I.-A. F. Ihle, J. Jouﬀroy, and T.I. Fossen. Robust Formation Control of Marine Craft using Lagrange Multipliers. Submitted to the Workshop on Group Coordination and Cooperative Control, Tromso, Norway, springer-verlag’s lecture notes in control and information systems series edition, May 2006. 20. D. Langer, J.K. Rosenblatt, and M. Hebert. A behavior-based system for oﬀ-road navigation. IEEE Transactions on Robotics and Automation, 10(6):776–783, 1994. 21. E.L. Lefeber, K.Y. Pettersen, and H. Nijmeijer. Tracking control of an underactuated ship. IEEE Transactions on Control Systems Technology, 11(1):52–61, 2003. 22. M.J. Mataric. Behavior-based control: Examples from navigation, learning, and group behavior. J. Experimental and Theoretical Artiﬁcial Intelligence, 9(2-3):323 336, 1997. 23. L.E. Parker. On the design of behavior-based multi-robot teams. Advanced Robotics, 10(6):547–578, 1996. 24. K.Y. Pettersen and T.I. Fossen. Underactuated dynamic positioning of a ship- experimental results. IEEE Transactions on Control Systems Technology, 8(5):856–863, 2000. 25. K.Y. Pettersen, F. Mazenc, and H. Nijmeijer. Global uniform asymptotic stabilization of an underactuated surface vessels: Experimental results. IEEE Transactions on Control System Technology, 12(6):891–903, 2004. 26. M. Scheutz and V. Andronache. Architectural mechanisms for dynamic changes of behavior selection strategies in behavior-based systems. IEEE Transactions on Systems, Man and Cybernetics, 34(6):2377–2395, Dec. 2004. 27. B. Siciliano. Kinematic control of redundant robot manipulators: A tutorial. Journal of Intelligent Robotic Systems, 3:201–212, 1990. 28. R. Skjetne. The Maneuvering Problem. PhD.Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2005. 29. D.M. Stipanovic, G. Inalhan, R. Teo, and C.J. Tomlin. Decentralized overlapping control of a formation of unmanned aerial vehicles. Automatica, 40:1285–1296, 2004. 30. Z. Wang, E. Nakano, and T. Takahashi. Solving function distribution and behavior design problem for cooperative object handling by multiple mobile robots. IEEE Transactions on Systems, Man, and Cybernetics, Part A, 33(5):537–549, 2003.

2 Passivity-Based Agreement Protocols: Continuous-Time and Sampled-Data Designs E. Bıyık and M. Arcak Department of Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute, Troy, NY 12180-3590 E-mails: [email protected], [email protected] Summary. We pursue a group agreement problem where the objective is to achieve synchronization of group variables. We assume a bidirectional information ﬂow between members, and study a class of feedback laws that are implementable with local information available to each member. We ﬁrst review a unifying passivity framework for the group agreement problem in continuous-time. Then, we extend the results to a class of sampleddata systems by exploiting the passivity properties of the underlying continuous-time system, and prove semiglobal asymptotic stability as the sampling period and a feedback gain are reduced.

2.1 Introduction In [1], a unifying passivity framework for several group coordination problems is presented. The group agreement problem, where the interest is in achieving synchronization of group variables (position, heading, phase of oscillations, etc.) is given as a special case. In this chapter, we study the agreement problem in greater detail. We ﬁrst restate the general coordination results in [1] for the agreement problem and give an independent proof of stability. We next extend these passivity-based designs to sampled-data systems, which are not studied in [1]. Sampled-data implementation is of interest in cooperative control, because data exchange between members is carried out over a communication channel. Although data is transmitted in discrete-time, controllers and internal dynamics of the members may be continuous-time, which means that a sampled-data analysis of the feedback system is necessary. Because the exact discrete-time model of the underlying continuous-time plant is seldom available in nonlinear systems, the sampled data studies in [2, 3, 4, 5, 6, 7] determine stability from properties of the continuoustime model. This approach typically yields “semiglobal” asymptotic stability results, which show that the region of attraction enlarges as the sampling period and possibly other design parameters are reduced. In this chapter, we exploit the passivity property of the underlying continuoustime plant and make use of results from [8] on the preservation of dissipativity properties under sampling. Our main result establishes semiglobal asymptotic stability as the sampling period and a feedback gain are reduced. One of the motivations for

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 21–33, 2006. © Springer-Verlag Berlin Heidelberg 2006

22

E. Bıyık and M. Arcak

this study is a path-following problem studied in [9] where the path parameters are to be synchronized for vehicles in a formation. In this application the path parameters may be updated in discrete-time, while the path-following controllers may be implemented locally in continuous-time, thus resulting in a sampled-data feedback system that ﬁts our framework. The subsequent sections are organized as follows: In Section 2.2, we present the communication topology within the group and introduce the basic structure of the feedback laws to be studied. We then develop a passivity-based design for these feedback laws and give a stability result for continuous-time agreement protocols. In Section 2.3, we review the discrete-time agreement protocols in [1] to set the stage for the new sampled-data design. The main result in Section 2.4 proves semiglobal asymptotic stability by combining passivity tools with results from sampled-data systems [7, 8]. Conclusions are given in Section 2.5.

2.2 Continuous Time Passive Protocols for Group Agreement We consider a group of N members, where each member i = 1, · · · , N is represented by a vector xi ∈ Rp that consists of variables to be synchronized with the rest of the group. The communication structure between these members is described by a graph. We say that the ith and jth members are “neighbors” if they have access to the relative information xi − xj , in which case we let the ith and jth vertices of the graph be connected by an edge. To simplify our derivations we assign an orientation to the graph by considering one of the vertices to be the positive end of the edge. The choice of orientation does not change the results because we assume the information ﬂow between neighbors to be bidirectional. Denoting by M the total number of edges, we recall from [10] that the N ×M incidence matrix D of the graph is deﬁned as:   +1 if ith vertex is the positive end of the kth edge (2.1) dik = −1 if ith vertex is the negative end of the kth edge  0 otherwise. Because the sum of its rows is zero, the rank of D is at most N − 1. Indeed, the rank is N − 1 when the graph is connected, that is when a path exists between every two distinct vertex, and less than N − 1 otherwise. The columns of D are linearly independent when no cycles exist in the graph. In the agreement problem the objective is to develop coordination laws that are implementable with local information (the ith member can use the information xi − xj if the jth member is a neighbor) and that guarantee lim |xi (t) − xj (t)| → 0.

t→∞

(2.2)

To achieve this objective [1] proposes a class of feedback laws of the form M

x˙ i = −Hi

dik ψk (zk ) k=1

+ v(t)

i = 1, · · · , N

(2.3)

2 Passivity-Based Agreement Protocols

23

where zk denotes the diﬀerence variable N

zk :=

dlk xl = l=1

xi − xj xj − xi

if i is the positive end if j is the positive end

(2.4)

for the ith and jth members connected by edge k, v(t) ∈ Rp is the reference velocity for the group, ψ : Rp → Rp is a multivariable nonlinearity to be designed, and Hi {ui } denotes the output at time t of a static or dynamic block with input ui . This block may represent the additional dynamics or nonlinearities inherent in the xi -subsystem, or a ﬁlter introduced to improve performance or robustness of the design (2.3). As an example, consider fully-actuated point masses x¨i = fi ,

i = 1, · · · , N

(2.5)

where xi ∈ R2 is the position of each mass and fi ∈ R2 is the force input. With the preliminary feedback ˙ − ui , fi = −Ki (x˙ i − v(t)) + v(t)

Ki = KiT > 0

(2.6)

and the change of variables ξi = −x˙ i + v(t), we bring (2.5) into the form x˙ i = −ξi + v(t) ξ˙i = −Ki ξi + ui ,

(2.7) (2.8)

where the ξi subsystem with input ui and output yi = ξi plays the role of Hi in (2.3). Note that the input of Hi in (2.3) M

ui :=

dik ψk (zk )

(2.9)

k=1

depends only on the neighbors of the ith member (dik = 0), thus, the feedback law (2.3) is implementable with local information. We note from (2.4) that the concatenated vectors x := [xT1 · · · xTN ]T satisfy

z := [z1T · · · uTM ]T

z = (DT ⊗ Ip )x

(2.10) (2.11)

where Ip is the p × p identity matrix and “ ⊗ ” represents the Kronecker product, which means that z is constrained to lie in the range space R(DT ⊗ Ip ). Therefore, if the columns of D are linearly dependent, i.e. if there are cycles in the graph, then zk ’s are mutually dependent. To obtain a block diagram representation for (2.3) we represent the output of Hi by the variable yi := Hi (ui ), (2.12) and denote

24

E. Bıyık and M. Arcak T T ] . ψ := [ψ1T · · · ψM

u := [uT1 · · · uTN ]T

T T ] y := [y1T · · · yN

Then, noting from (2.9) that

(2.13)

u = (D ⊗ Ip )ψ,

(2.14)

and denoting by 1p the p-dimensional column vector with all entries equal to 1, we obtain the right-hand side of (2.3) from the output of the summing junction in Figure 1. !

1p ⊗v(t)

x˙

−

y

D T ⊗ Ip

z˙

ψ1

!

z ..

ψ2

. !

ψ ..

.

D ⊗ Ip

u

ψM

H1 ..

. HN

Fig. 2.1. A block diagram representation for the interconnected system (2.3). The vectors x, y, z, u and ψ are as deﬁned in (2.10) and (2.13), Ip is the p × p identity matrix, 1p is the p-dimensional column vector with all entries equal to 1, and “⊗” represents the Kronecker product.

Because post-multiplication by D ⊗ Ip and pre-multiplication by its transpose DT ⊗ Ip preserve passivity properties, we proceed with a passivity-based design of the ﬁlters Hi , i = 1, · · · , N , and nonlinearities ψk , k = 1, · · · , M . We then establish (2.2) with the help of Passivity Theorem [11, 12, 13] which guarantees the stability for the negative feedback interconnection of two passive systems. We restrict the feedforward nonlinearities ψk (zk ) to be of the form ψk (zk ) = ∇Pk (zk )

(2.15)

where Pk (zk ) : Rp → R≥0 is a positive deﬁnite, radially unbounded and C 2 function with the property ∇Pk (zk ) = 0 ⇔ zk = 0. (2.16) In the feedback path, if Hi is a static block, we restrict it to be of the form yi = hi (ui )

(2.17)

where hi : Rp → Rp is a locally Lipschitz function satisfying uTi hi (ui ) > 0

∀ui = 0.

(2.18)

2 Passivity-Based Agreement Protocols

25

If Hi is a dynamic block of the form ξ˙i = fi (ξi , ui ) ξi ∈ Rni yi = hi (ξi , ui ),

(2.19)

we assume fi (·, ·) and hi (·, ·) are locally Lipschitz functions such that hi (0, 0) = 0 and (2.20) fi (0, ui ) = 0 ⇒ ui = 0. Our main restriction on (2.19) is that it be strictly passive with a C 1 , positive deﬁnite, radially unbounded storage function Si (ξi ) satisfying S˙ i ≤ −Wi (ξi ) + uTi yi

(2.21)

for some positive deﬁnite function Wi (·). Theorem 2.1. Consider the interconnected system (2.1), (2.3) and (2.4) represented with a block diagram in Figure 1. Suppose v(t) is uniformly bounded and piecewise continuous, and ψk , k = 1, · · · , M and Hi , i = 1, · · · N are designed as in (2.15)-(2.16) and (2.17)-(2.21). If the columns of D are linearly independent, then the origin (z, ξ) = 0 is uniformly globally asymptotically stable. When D has linearly dependent columns, uniform global asymptotic stability of the origin (z, ξ) = 0 is achieved if Pk (zk ) is further constrained as: zkT ∇Pk (zk ) > 0

∀zk = 0.

(2.22)

Asymptotic stability of the origin (z, ξ) = 0 means that if a path exists between the ith and jth members in the group, then the trajectories xi (t) and xj (t) in (2.3) synchronize with each other. It further implies that the feedback term in the right hand side of (2.3) converges to zero, which means that all members achieve a common group velocity x˙ i = v(t). If the graph is connected all variables xi are synchronized in the limit as t → ∞. Proof of Theorem 1: For the feedforward and feedback paths in Figure 1 we take the storage functions M

Vf (z) :=

Pk (zk ) and Vb (ξ) := k=1

Si (ξi )

(2.23)

i∈I

respectively, where I denotes the subset of indices i = 1, · · · , N which correspond to dynamic blocks Hi . For the feedforward path we take the time-derivative of the storage function Vf and use the fact that (DT ⊗ Ip )(1p ⊗ v(t)) = 0 to obtain: V˙ f = ψ T z˙ = ψ T (DT ⊗ Ip )x˙ = ψ T (DT ⊗ Ip ){1p ⊗ v(t) − y} = −ψ T (DT ⊗ Ip )y = −((D ⊗ Ip )ψ)T y = −uT y.

(2.24)

26

E. Bıyık and M. Arcak

Likewise, the time-derivative of the storage function Vb for the feedback path satisﬁes: V˙ b =

S˙ i ≤ − i∈I

i∈I

uTi yi .

Wi (ξi ) +

(2.25)

i∈I

To prove asymptotic stability of the origin we use the Lyapunov function V (z, ξ) = Vf (z) + Vb (ξ).

(2.26)

From (2.18), (2.24) and (2.25), the time derivative of this Lyapunov function satisﬁes V˙ ≤ −

uTi yi = −

Wi (ξi ) − i∈I

i∈I

i∈I /

uTi hi (ui ),

Wi (ξi ) −

(2.27)

i∈I /

which is negative semideﬁnite because of (2.18) and implies that the trajectories (z(t), ξ(t)) are bounded on the maximal interval of deﬁnition of (2.3). Because v(t) in (2.3) is also bounded, we conclude that x(t) is deﬁned for all t ≥ 0 and, thus, from (2.27) the origin (z, ξ) = 0 is stable. We now apply the Invariance Principle1 to prove asymptotic stability. To investigate the largest invariant set where V˙ (z, ξ) = 0, we note from (2.20) that if ξi = 0 holds identically then ui = 0. Likewise, the static block satisﬁes (2.18), which means that the right hand side of (2.27) vanishes when ui = 0, i = 1, · · · , N . We thus conclude that u = 0, which means, from (2.14), that ψ(z) lies in the null space N (D ⊗ Ip ). When the columns of D are independent, however, N (D ⊗ Ip ) = 0 and, hence, ψ(z) = 0. It then follows from (2.15) and (2.16) that origin is the largest invariance set where V˙ (z, ξ) = 0, and consequently, asymptotically stable. Uniformity of asymptotic stability follows from the time-invariance of the (z, ξ) dynamics. When D has linearly dependent columns, the null space of N (D⊗Ip ) is nontrivial. To show that ψ(z) = 0 is the only feasible solution, we let z ∈ R(DT ⊗ Ip ) in (2.11) be such that ψ(z) ∈ N (D ⊗ Ip ). Since R(DT ⊗ Ip ) and N (D ⊗ Ip ) are orthogonal to each other, z ∈ R(DT ⊗ Ip ) and ψ(z) ∈ N (D ⊗ Ip ) together imply z T ψ(z) = 0.

(2.28)

We then conclude, from (2.15), (2.16) and (2.22) that, (2.28) holds if and only if z = 0, which establishes the uniform asymptotic stability of the origin (z, ξ) = 0. When p = 1, that is when xi ’s and zk ’s are scalars, condition (2.22) already follows from (2.16) and positive deﬁniteness of Pk . When p ≥ 2, however, (2.22) further restricts the choice of Pk . Indeed, if Pk has nonconvex level sets then the gradient ∇Pk and zk can be more than 90◦ apart, in which case (2.22) fails.

2.3 Discrete-Time Protocols Before presenting the sampled-data design we review the discrete-time agreement protocols in [1], which are of the form 1

The Invariance Principle is indeed applicable because the dynamics of (z, ξ) are autonomous. Although v(t) appears in the block diagram in Figure 1, it is cancelled in the z˙ equation because (DT ⊗ Ip )(1p ⊗ v(t)) = 0.

2 Passivity-Based Agreement Protocols

27

M

xi (n + 1) − xi (n) = −Hi

dik ψk (zk (n))

+ v(n),

(2.29)

k=1

where n = 0, 1, 2, · · · is the time index, and Hi is a discrete-time dynamic block or a static nonlinearity. The block diagram representation of this discrete-time model is as in Figure 1, with integral blocks replaced by summation blocks, and x˙ and z˙ replaced by, respectively, δx = x(n + 1) − x(n)

and δz = z(n + 1) − z(n).

(2.30)

Passivity of the feedforward path cannot be achieved in discrete-time because the phase lag of the summation block exceeds 90◦ . Indeed, with the further restriction ∇2 Pk (zk ) ≤ γI

∀zk ∈ Rp ,

γ>0

(2.31)

it is shown in [1] that the storage function Vf (z) in (2.23) satisﬁes Vf (z(n + 1)) − Vf (z(n)) ≤ −uT y +

γλN T y y, 2

(2.32)

where λN denotes the largest eigenvalue of the graph Laplacian matrix DDT . In particular, the second term on the right hand side of (2.32) quantiﬁes the shortage of passivity. To compensate for this shortage, we design Hi ’s to achieve excess of passivity in the feedback path. If Hi is a static block yi = hi (n, ui ), then we restrict it by uTi yi − µyiT yi ≥ wi (ui )

(2.33)

where wi (·) is a positive deﬁnite function and the constant µ > 0 quantiﬁes the excess of passivity. Likewise, if Hi is a dynamic block of the form ξi (n + 1) = fi (ξi (n), ui (n)) yi = hi (ξi , ui )

ξ ∈ R ni

(2.34)

we assume that (2.20) holds and that there exists a positive deﬁnite and radially unbounded storage function Si (ξi ) satisfying Si (ξi (n + 1)) − Si (ξi (n)) ≤ −Wi (ξ) + uTi yi − µyiT yi

(2.35)

for some positive deﬁnite function Wi (·). We then guarantee stability of the feedback system by choosing γλN . (2.36) µ≥ 2 Theorem 2.2. [1]: Consider members i = 1, 2, ..., N interconnected as described by the graph representation (2.1), and let zk , k = 1, 2, ..., M denote the diﬀerences between the variables xi of neighboring members as in (2.4). Let Pk (zk )’s be C 2 , positive deﬁnite, radially unbounded functions satisfying, (2.16), (2.22) and (2.31). Suppose that Hi ’s are either dynamic blocks (2.34) satisfying (2.35)-(2.36), or static

28

E. Bıyık and M. Arcak

blocks yi = hi (n, ui ) satisfying (2.33)-(2.36), or possibly a combination of such static and dynamic blocks. Then the feedback law (2.29) with ψk (zk ) = ∇Pk (zk ) achieves global asymptotic stability of the origin (z, ξ) = 0. In particular, if a path exists between the ith and jth vertices in the graph representation (2.1), then xi (n) − xj (n) → 0

as

t → ∞.

(2.37)

2.4 Sampled-Data Design We now study a sampled-data design in which xi are updated in discrete-time as in (2.29), and the Hi blocks consist of a combination of continuous- and discretetime dynamics as depicted in Figure 2. The structure of Hi is in part motivated by a formation design studied in [9] where each vehicle is driven by a path-following controller, and the path parameters (xi in our notation) are synchronized in discretetime according to (2.29). The continuous-time vehicle dynamics play the role of the ξi -block in Figure 2, and ηi is a discrete-time correction term to achieve synchronization.

ηi (nT )

yi (nT )

ZOH

ηi

ξ˙i = fi (ξi , ηi ) νi = hi (ξi )

νi

νi (nT ) T

Γi

ui (nT )

Hi Fig. 2.2. A block diagram representation of the sampled-data dynamic block Hi where Γi is a constant feedback gain matrix, ZOH stands for “Zero-Order-Hold” and T is the sampling period of νi .

In this structure, Γi is a constant gain matrix, and the ξi -subsystem ξ˙i = fi (ξi , ηi )

(2.38)

νi = hi (ξi )

(2.39)

is such that ξi ∈ Rni , and fi (·, ·) and hi (·) are locally Lipschitz functions such that hi (0) = 0. We further assume that (2.38)-(2.39) is strictly passive with a C 2 , positive deﬁnite storage function Si (ξi ) such that

2 Passivity-Based Agreement Protocols

S˙ i ≤ −W (ξi ) + νiT ηi

29

(2.40)

for some positive deﬁnite function W (·). As a preparation for our main stability result, the following lemma characterizes to what extent the dissipation property (2.40) is preserved under sampling: Lemma 2.1. Suppose that Hi is a sampled-data dynamic block with the structure given in Figure 2. Assume that the continuous-time ξi -subsystem in (2.38)-(2.39) is passive with a C 2 , positive deﬁnite storage function Si satisfying (2.40). Then, given compact sets Dξi and Dηi there exist positive constants Ti∗ , K1i , K2i such that for all sampling periods T ∈ (0, Ti∗ ) and for all (ξi (nT ), ηi (nT )) ∈ Dξi × Dηi , the storage function Si satisﬁes Si (ξi (nT + T )) − Si (ξi (nT )) ≤ − W (ξi (nT )) + T K1i ξi (nT )T ξi (nT ) T 1 +ui (nT )Tyi (nT )−( −T K2i)yi (nT )Tyi (nT ), σΓi (2.41) where the design parameter σΓi denotes the largest singular value of Γi . Proof of Lemma 1: Since ξi (nT ) and ηi (nT ) lie in the compact sets Dξi and Dηi , respectively, we can ﬁnd a pair of strictly positive numbers (∆ξi , ∆ηi ) such that |ξi (nT )| ≤ ∆ξi , |ηi (nT )| ≤ ∆ηi . Then, from [8, Prop. 3.4] and [8, Cor. 5.3], there exist Ti∗ > 0 and positive constants K1i , K2i such that for all T ∈ (0, Ti∗ ), |ξi (nT )| ≤ ∆ξi and |ηi (nT )| ≤ ∆ηi , we have: Si (ξi (nT + T )) − Si (ξi (nT )) ≤− W (ξi (nT )) + νi (nT )T ηi (nT ) T + T K1i ξi (nT )Tξi (nT )+T K2iηi (nT )T ηi (nT ). (2.42) For the given structure of the feedback path in Figure 2 we note that ηi (nT ) = −yi (nT ) and νi (nT ) = −ui (nT ) + Γi−1 yi (nT ).

(2.43)

To incorporate the feedback path from ui (nT ) to yi (nT ) with the storage function in (2.42), we substitute (2.43) into (2.42) and obtain: Si (ξi (nT + T ))−Si (ξi (nT )) ≤ − W (ξi (nT )) + T K1i ξi (nT )T ξi (nT ) T 1 +ui (nT )Tyi (nT )−( −T K2i)yi (nT )Tyi (nT ), σΓi (2.44) which completes the proof. To study the stability of the interconnected system (2.1), (2.4) and (2.29) we employ the storage functions

30

E. Bıyık and M. Arcak

Vb (ξ) :=

1 T

N

M

Si (ξi ) and Vf (z) := i=1

Pk (zk ),

(2.45)

k=1

for the feedback and feedforward paths, respectively, where Pk ’s are designed as in Section 3. We then prove that the origin (z, ξ) = 0 is semiglobally asymptotically stable in the sampling period T and in the feedback gain σΓ , which means that an arbitrarily large region of attraction for the origin can be achieved by making both T and σΓ suﬃciently small. Theorem 2.3. Consider members i = 1, 2, ..., N interconnected as described by the graph representation (2.1), and let zk , k = 1, 2, ..., M denote the diﬀerences between the variables xi of neighboring members as in (2.4). Let Pk (zk )’s be designed as C 2 , positive deﬁnite, radially unbounded functions satisfying (2.16), (2.22) and (2.31). Suppose that Hi ’s are sampled-data dynamic blocks as in Figure 2, and satisfy (2.40) and the following two assumptions: A1. W (ξi ) in (2.40) is lower bounded by W (ξi ) ≥ C|ξi |2 for some C > 0, A2. ξi -subsystems, i = 1, · · · , N are input-to-state stable (ISS) from ηi to ξi , i.e. there exist class − KL and class − K functions β(·, ·) and α(·), respectively, such that (2.46) |ξi (t)| ≤ β(|ξi (t0 )|, t − t0 ) + α( sup |ηi (τ )|). t0 ≤τ ≤t

¯ such Then given compact sets Dz and Dξ there exist positive constants T and σ that for all sampling periods T < T and σmax (Γ ) < σ ¯ , the feedback law (2.29) with ψk (zk ) = ∇Pk (zk ) achieves asymptotic stability of the origin (z, ξ) = 0 with a region of attraction that includes Dz × Dξ . Proof of Theorem 3: For the interconnected system we take the Lyapunov function V (z, ξ) = Vf (z) + Vb (ξ), and let S =: Sz × Sξ denote a level set of V (z, ξ), such that S ⊃ Dz × Dξ . We note from Figure 2, (2.14) and (2.39) that M

ηi (nT ) = −Γi (ui (nT )+νi (nT )) = −Γi

dik ψk (zk (nT ))+hi (ξi (nT )) .

(2.47)

k=1

Therefore, given a compact set S and a positive number σ ∗ , there exists a compact set Dη such that σmax (Γ ) < σ ∗

and

(z(nT ), ξ(nT )) ∈ S =⇒ η(nT ) ∈ Dη .

(2.48)

We then apply Lemma 1 with (ξ(nT ), η(nT )) ∈ (Sξ × Dη ), and obtain T ∗ , K1 and K2 , such that, from (2.44) and (2.45), for all sampling periods T < T ∗ , N

Vb (ξ(nT + T )) − Vb (ξ(nT )) ≤ −

W (ξi (nT )) − T K1 ξi (nT )T ξi (nT )

i=1

+u(nT )Ty(nT )−

1 −T K2 y(nT )Ty(nT ). σmax (Γ ) (2.49)

2 Passivity-Based Agreement Protocols

31

We then deﬁne

γλN 1 , := T K2 + σ ∗∗ 2 and note that for all T < T ∗ and σmax (Γ ) < σ ∗∗ , (2.49) and A1 yields

(2.50)

Vb (ξ(nT + T )) − Vb (ξ(nT )) ≤−(C − T K1 )ξ(nT )T ξ(nT ) +u(nT )Ty(nT ) (2.51) −κy(nT )T y(nT ), where κ > γλ2N from (2.50). To make the ﬁrst term in the right hand side of (2.51) negative deﬁnite, we deﬁne C , (2.52) T ∗∗ := K1 and then note that for all sampling periods T < T ∗∗ , (C − T K1 )ξ(nT )T ξ(nT ) > 0

∀ξ(nT ) ∈ Sξ − {0}.

(2.53)

Finally, we deﬁne σ ¯ =: min{σ ∗ , σ ∗∗ }

and T := min{T ∗ , T ∗∗ },

(2.54)

and conclude from (2.32), (2.51) and (2.53) that for all σmax (Γ ) < σ ¯ , T < T , and ξ(nT ) ∈ Sξ − {0}, V (z, ξ) satisﬁes: V (z(nT +T ),ξ(nT +T )) − V (z(nT ),ξ(nT )) ≤ − (C − T K1 )ξ(nT )Tξ(nT ) γλN )y(nT )T y(nT ) < 0, − (κ − 2 (2.55) which implies that the trajectories (z(nT ), ξ(nT )) are bounded and stay in S. It further follows from standard arguments that there exists a class − K function α1 such that for all (z(0), ξ(0)) ∈ S, |z(nT ), ξ(nT )| ≤ α1 (|z(0), ξ(0)|).

(2.56)

To study the behavior of the trajectories between the sampling points we ﬁrst note from (2.47) that |η(nT )| ≤ ||Γ || Lh |ξ(nT )| + ||D ⊗ Ip ||Lψ |z(nT )| =: α2 (|ξ(nT ), z(nT )|),

(2.57)

where Lh and Lψ are Lipschitz constants of h(·) and ψ(·) in S, respectively, and α2 is a class − K function by construction. From the ISS property in (2.46), we obtain for all t ∈ [nT, nT + T ] |ξ(t)| ≤ β(|ξ(nT )|, 0) + α(α2 (|ξ(nT ), z(nT )|)) =: α3 (|ξ(nT ), z(nT )|),

(2.58)

where α3 is also a class − K function. Next, because z(t) is constant between the sampling points, we can combine (2.58) and (2.56) and obtain another class − K function α4 such that for all (z(0), ξ(0)) ∈ S,

32

E. Bıyık and M. Arcak

|z(t), ξ(t)| ≤ α4 (|z(0), ξ(0)|),

(2.59)

which proves stability of the origin (z, ξ) = 0. Finally, to conclude asymptotic stability with a region of attraction containing S we note from (2.55) that as n → ∞ , ξ(nT ) → 0 and y(nT ) → 0, hence, u(nT ) → 0 from (2.43) and the property hi (0) = 0. It then follows from similar arguments in the proof of Theorem 1 that u(nT ) → 0 implies ψk (zk ) = ∇Pk (zk ) → 0, k = 1, · · · , M , which proves that z(nT ) → 0. We thus conclude, using (2.58), that (z(t), ξ(t)) → 0 as t → ∞.

2.5 Conclusions We have reviewed a unifying passivity framework for the group agreement problem and extended the results to a class of sampled-data systems. The sampled-data result is of particular importance because, in practice, the communication channel between the members of the group may necessitate a discrete-time implementation in the feedforward path, whereas in the feedback path the local controllers or internal dynamics of each member may be continuous-time. Our result showed that in such sampled-data implementations the size of the region of attraction is determined by the sampling period and the feedback gain.

Acknowledgment This research was supported in part by National Science Foundation under grant no ECS-0238268.

References 1. M. Arcak. Passivity as a design tool for group coordination. Submitted, 2005. 2. F.H. Clarke, Y.S. Ledyaev, E.D. Sontag, and A.I. Subbotin. Asymptotic controllability implies feedback stabilization. IEEE Transactions on Automatic Control, 42(10), 13941407, 1997. 3. B.Castillo, S. Di Gennaro, S.Monaco and D. Normand-Cyrot. On regulation under sampling. IEEE Transactions on Automatic Control, 42(6), 864-868, 1997. 4. A.R. Teel, D.Nesic, and P.V. Kokotovic. A note on input-to-state stability of sampleddata nonlinear systems. In Proceedings of IEEE Conference on Decision and Control, 2473-2478, Tampa, FL, 1998. 5. I.M.Y. Mareels, H.B. Penfold, and R.J. Evans. Controlling nonlinear time-varying systems via euler approximations. Automatica, 28(4), 681-696, 1992. 6. D. Neˇsi´c, A.R. Teel and P.V.Kokotovi´c. Suﬃcient conditions for stabilization of sampleddata nonlinear systems via discrete-time approximations. Systems and Control Letters, 38(5), 259-270, 1999. 7. D. Neˇsi´c and A.R. Teel. A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE Transactions on Automatic Control, 49(7), 1103-1034, 2004.

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8. D.S. Laila, D. Neˇsi´c, A.R. Teel. Open and closed loop dissipation inequalities under sampling and controller emulation. European Journal of Control, 8(2), 109-125, 2002. 9. I-A.F. Ihle, M. Arcak, T.I. Fossen. Passivation designs and robustness for synchronized path following. Submitted, 2005. 10. N. Biggs. Algebraic Graph Theory. Cambridge University Press, second edition, 1993. 11. A. J. van der Schaft. L2 -gain and Passivity Techniques in Nonlinear Control. SpringerVerlag, New York and Berlin, second edition, 2000. 12. R. Sepulchre, M. Jankovi´c, and P. Kokotovi´c. Constructive Nonlinear Control. SpringerVerlag, New York, 1997. 13. H.K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ, third edition, 2002.

3 Cross-Track Formation Control of Underactuated Autonomous Underwater Vehicles E. Børhaug, A. Pavlov, and K.Y. Pettersen Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway [email protected], [email protected], [email protected] Summary. The problem of 3D cross-track control for underactuated 5-degrees-of-freedom (5-DOF) autonomous underwater vehicles (AUV) is considered. The proposed decentralized controllers make the AUVs asymptotically constitute a desired formation that follows a given straight-line path with a given forward speed proﬁle. The proposed controllers consist of two blocks. The ﬁrst block, which is based on a Line of Sight guidance law, makes every AUV asymptotically follow straight line paths corresponding to the desired formation motion. The second block manipulates the forward speed of every AUV in such a way that they asymptotically converge to the desired formation and move with a desired forward speed proﬁle. The results are illustrated with simulations.

3.1 Introduction Formation control of marine vessels is an enabling technology for a number of interesting applications. A ﬂeet of multiple autonomous underwater vehicles (AUVs) moving together in a prescribed pattern can form an eﬃcient data acquisition network for surveying at depths where neither divers nor tethered vehicles can be used, and in environments too risky for manned vehicles. This includes for instance oceanographic surveying at deep sea, operations under ice for exploration of Arctic areas and eﬃcient monitoring sub-sea oil installations. In this paper we study the problem of 3D cross-track formation control for underactuated 5-DOF AUVs that are independently controlled in surge, pitch and yaw. Roughly speaking, this problem can be formulated as follows: given a straight line path, a desired formation pattern, and a desired speed proﬁle, control the AUVs such that asymptotically they constitute the desired formation which then moves along the given path with the desired speed. The relevance of such a formation control problem is justiﬁed by the fact that a desired path for autonomous vehicles is usually given by straight lines interconnecting way-points, see, e.g., [9]. At the same time, the speed proﬁle of an autonomous vehicle is often speciﬁed independently of the desired path. This makes it possible to decouple the mission planning into two stages: a geometric path planning stage and a dynamic speed assignment stage.

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 35–54, 2006. © Springer-Verlag Berlin Heidelberg 2006

36

E. Børhaug, A. Pavlov, and K.Y. Pettersen

The cross-track control problem or, more generally, the path following control problem has been investigated in a number of publications. In [10, 3] this problem is considered for a single underactuated 3-DOF surface vessel and the proposed controllers are validated in experiments. In [11], the straight line cross-track control problem is considered for a single 3-DOF underactuated surface vessel. In that paper a nonlinear control law based on so-called Line of Sight (LOS) guidance is proposed and global κ-exponential stability of the cross-track error to a desired straight line path is proven. In [2] the straight line cross-track control problem for 5-DOF underactuated underwater vehicles is solved using an LOS guidance law and a nonlinear feedback controller rendering global κ-exponential stability of the cross-track error. A general path following problem for 6-DOF underactuated underwater vehicles is considered in [7]. The proposed nonlinear control strategy guarantees asymptotic convergence to a desired reference path. Formation control and cooperative control of various systems have been studied in a large number of recent publications. For a necessarily incomplete list of publications on this subject, see, e.g., [15, 17, 1, 19, 18, 8, 5, 4] and the references therein. Most of the existing works on formation control focus on the high level analysis of achieving a formation. In this case the systems to be controlled are often assumed to have simple dynamics (like fully actuated point masses), which makes controlling every individual system an easy task and allows one to focus on the dynamics of the whole group of controlled systems rather than on the dynamics of individual systems. However, in marine applications the dynamics of individual systems (e.g., ships or AUVs) can be rather complicated to control, especially for underactuated systems. In this case the problem can not be considered only at the level of formation control alone, but must be analyzed both at the levels of individual systems and the whole group. This makes the problem more challenging. For marine vehicles the formation control problem is considered in [21, 12, 13]. In [21] formation control for a ﬂeet of fully actuated surface ships is considered. The proposed maneuvering-based controllers make each vehicle follow a given parameterized path with an assigned speed. The speed assignment, which depends on the states of all vehicles in the formation, guarantees exponential convergence to the desired formation. Similar ideas are used in [12] for the case of fully actuated AUVs. The control scheme proposed in [12] consists of a feedback controller that stabilizes each vehicle to a given path and a coordination controller that coordinates the motion of the vehicles along the paths. Graph theory is used to allow for diﬀerent communication topologies in the formation. The problem of formation control for fully actuated marine surface vessels is also considered in [13]. The proposed exponentially stabilizing formation control laws are derived by imposing holonomic inter-vessel constraints and using tools from analytical mechanics. The main contribution of this paper is the development of a cross-track control scheme for formation control of underactuated AUVs. This control problem can be decomposed into two sub-problems: a) a cross-track control problem and b) a coordination control problem. Given a desired formation pattern and a desired straight line path to be followed by the formation, we can deﬁne parallel desired straight line paths for each individual AUV, see Fig. 3.1. Then for every AUV we design LOS-based cross-track controllers that make the AUVs converge to the correspond-

3 Cross-Track Formation Control of Underactuated AUVs

37

Fig. 3.1. Formation of AUVs.

ing paths. However, without controlling the forward speed of the vehicles in some coordinated manner, the desired formation pattern will not be achieved. To asymptotically achieve the formation pattern, each vehicle must adapt its forward speed in such a way that asymptotically all vehicles constitute the desired formation and move with the desired speed. This is the coordination control problem. The low bandwidth of underwater communication links form a serious constraint for cooperative control. To overcome this problem the proposed control scheme requires communication of only one position variable of each AUV among the vehicles. Moreover, it does not require communication links between all vehicles, thus signiﬁcantly reducing the inter-vehicle communication. Similar ideas for 2D formation control of fully actuated marine vehicles are considered in [21] and [12]. In this work, we consider the case of 3D formation control for underactuated underwater vehicles with full dynamic models. The paper is organized as follows. In Section 3.2 we present the AUV model and control problem statement. In Section 3.3 we recall a result on cascaded systems that will be used throughout the paper. Section 3.4 contains a solution of the crosstrack control problem for one AUV. In Section 3.5 we solve the coordination control problem for AUV formations. Simulation results are presented in Section 3.6 and conclusions are presented in Section 3.7.

3.2 Vehicle Model and Control Objective In this section we present the kinematic and dynamic model describing the motion of the class of AUVs studied in this paper. Moreover, we deﬁne the notation used throughout the paper and state the control problem to be solved. 3.2.1 AUV Model We consider an autonomous underwater vehicle (AUV) described by the 5-DOF model [9] η˙ = J(η)ν M ν˙ + C(ν)ν + D(ν)ν + g(η) = Bτ ,

(3.1) (3.2)

b where η col(pi , Θ) ∈ R5 and ν col(v b , ωib ) ∈ R5 . Here pi = [x y z]T is the inertial position of the AUV in Cartesian coordinates and Θ = [θ ψ]T is the Euler-angle representation of the orientation of the AUV relative to the inertial frame, where θ and ψ are the pitch and yaw angles, respectively. The roll is assumed

38

E. Børhaug, A. Pavlov, and K.Y. Pettersen

to be zero. Vector v b = [u v w]T is the linear velocity of the AUV in the bodyﬁxed coordinate frame, where u, v and w are the surge, sway and heave velocities b respectively, and ωib = [q r]T is the angular velocity of the AUV in the body-ﬁxed coordinate frame, where q and r are the pitch and yaw velocities respectively. The matrix J(η) ∈ R5×5 is the transformation matrix from the body-ﬁxed coordinate frame b to the inertial coordinate frame i. Moreover, M = M T > 0 is the mass and inertia matrix, C(ν) is the Coriolis and centripetal matrix, D(ν) is the damping matrix and g(η) are the restoring forces and moments due to gravity and buoyancy. The vector τ = [τu τq τr ]T is the control input, where τu is the surge control, τq is the pitch control and τr is the yaw control. The matrix B ∈ R5×3 is the actuator matrix. Note that the AUV is underactuated, as only 3 independent controls are available to control 5 degrees of freedom. By elaborating the diﬀerential kinematic equations in (3.1), we obtain ([9]): x˙ = u cos ψ cos θ − v sin ψ + w cos ψ sin θ

(3.3a)

y˙ = u sin ψ cos θ + v cos ψ + w sin θ sin ψ

(3.3b)

z˙ = −u sin θ + w cos θ θ˙ = q 1 r. ψ˙ = cos θ

(3.3c) (3.3d) (3.3e)

Due to the Euler angle singularity, Eq. (3.3e) is not deﬁned for |θ| = π2 . However, the normal operating conditions for AUVs are θ ∈ − π2 , π2 (for π2 < |θ| < π the AUV is upside-down and, moreover, for conventional AUVs the vertical dive corresponding to θ = − π2 is physically impossible). Therefore, our state space for θ will be considered to be θ ∈ − π2 , π2 . In this paper we will use the dynamics model (3.2) in a modiﬁed form: ν˙ = −M −1 (C(ν)ν + D(ν)ν + g(η)) + M −1 Bτ

f (ν, η) + τ¯ ,

(3.4)

where f = [fu , fv , fw , fq , fr ]T and τ¯ M −1 Bτ = [¯ τu , τ¯v , τ¯w , τ¯q , τ¯r ]T . Notice that for a large class of underwater vehicles, the body-ﬁxed coordinate system can be chosen such that τ¯v = τ¯w = 0. This is possible for AUVs having port/starboard symmetry, under the assumption of zero roll. For the corresponding technique applied to surface vessels, see, e.g., [11] and [6]. Moreover, we assume that det [e1 e4 e5 ]M −1 B = 0, such that the mapping (τu , τq , τr ) → (¯ τu , τ¯q , τ¯r ) is invertible. Therefore instead of designing controllers for (τu , τq , τr ), we will design controllers for (¯ τu , τ¯q , τ¯r ). We assume that for |u| ≤ Umax , where Umax > 0 is the maximal surge velocity, the sway and heave velocities v and w satisfy the following assumptions |v| ≤ Cv Umax |r|,

|w| ≤ Cw Umax |q|,

(3.5)

|w| ≤ Umax .

(3.6)

for some Cv > 0, Cw > 0, and |v| ≤ Umax ,

3 Cross-Track Formation Control of Underactuated AUVs

39

These assumptions can be justiﬁed for a slender AUV (with its length much larger than the width/height). In this case, the damping for sway and heave directions (v and w) will be much larger compared to the surge direction (u). Therefore, the forward velocity (satisfying |u| ≤ Umax ) becomes dominant, see (3.6). Assumption (3.5) means that for the case of the angular velocities q and r converging to zero (i.e., the heading of the AUV has almost no change), the speeds v and w are damped out because of the hydrodynamical drag in the sway and heave direction and also converge to zero. 3.2.2 Control Objective In this paper we deal with cross-track control for formations of AUVs. We will design decentralized control laws for n AUVs such that, after transients, the AUVs form a desired formation and move along a desired straight-line path with a given velocity proﬁle, as illustrated in Fig. 3.2. The desired formation is characterized by ud

u1

z

x y

p(t)

rp1 rp2

u2

ud

L

rp3 u3

ud

Fig. 3.2. Formation control of AUVs.

a formation reference point p(t) and a set of vectors rpj , j = 1, . . . , n giving the desired relative positions of the AUVs with respect to the point p(t). The desired path of the formation is given by a straight line L. The desired velocity proﬁle is given by a diﬀerentiable function ud (t). The control objective is to guarantee that asymptotically, i.e., in the limit for t → +∞, a) the AUVs constitute the formation, i.e., r1 (t) − rp1 = . . . = rn (t) − rpn =: p(t), where rj , j = 1, . . . , n, are the position vectors of the AUVs; b) p(t) follows the desired path L with the desired velocity proﬁle ud (t), i.e., p(t) ∈ ˙ L and |p(t)| = ud (t), and the orientation of the AUVs are aligned with the desired straight line paths. By choosing an inertial coordinate system with the x-axis coinciding with the desired straight-line path, i.e., L = {(x, y, z) : x ∈ R, y = 0, z = 0} (see Fig. 3.2), the control objective can be formalized as follows

40

E. Børhaug, A. Pavlov, and K.Y. Pettersen

lim yj (t) − Dyj = 0,

j = 1, . . . , n,

lim zj (t) − Dzj = 0,

j = 1, . . . , n,

t→+∞

t→+∞

lim θj (t) = 0,

j = 1, . . . , n,

lim ψj (t) = 0,

j = 1, . . . , n,

t→+∞ t→+∞

lim x1 (t) − Dx1 = · · · = lim xn (t) − Dxn = Const +

t→+∞

t→+∞

(3.7)

(3.8) t 0

ud (s)ds,

(3.9)

where [xj , yj , zj ]T and [Dxj , Dyj , Dzj ]T are the coordinates of the AUV position vectors rj and relative position vectors rpj , respectively, in the chosen inertial coordinate system. The above stated control problem will be solved in two steps. First, for each AUV we design independent cross-track controllers guaranteeing that the cross-track control goal (3.7) is achieved, the orientation of the AUVs satisfy (3.8), and that the remaining dynamics in the x direction track certain speed reference commands ucj (j corresponds to the jth AUV). At the second stage, we specify control laws for ucj that coordinate the AUVs in the x-direction to asymptotically constitute the desired formation and make the speed of the formation track the desired speed proﬁle ud (t), as speciﬁed in the coordination control goal (3.9).

3 Cross-Track Formation Control of Underactuated AUVs

41

3.3 Preliminaries In this section we recall some results on cascaded systems of the form x˙ 1 = f1 (x1 , t) + g(x1 , x2 , t)x2 x˙ 2 = f2 (x2 , t).

(3.10) (3.11)

Prior to formulating the results we give the following deﬁnition. Deﬁnition 3.1 ([16]). System x˙ = f (x, t) is called exponentially stable in any ball if for any r > 0 there exist k = k(r) > 0 and α = α(r) > 0 such that |x(t)| ≤ k|x(t0 )|e−α(t−t0 ) . Theorem 3.1 ([16]). System x˙ = f (x, t) is exponentially stable in any ball if and only if it is globally uniformly asymptotically stable and locally exponentially stable. The next result directly follows from [20] (Theorem 7 and Lemma 8). It will be used throughout this paper. Theorem 3.2. System (3.10), (3.11) is exponentially stable in any ball if the following conditions are satisﬁed: a) systems x˙ 1 = f1 (x1 , t) and (3.11) are both exponentially stable in any ball; b) there exists a quadratic positively deﬁnite function V : Rn → R satisfying ∂V f1 (x1 , t) ≤ 0, ∂x1

∀ x 1 , t ≥ t0 ;

(3.12)

c) the interconnection term g(x1 , x2 , t) satisﬁes ∀t ≥ t0 |g(x1 , x2 , t)| ≤ ρ1 (|x2 |) + ρ2 (|x2 |)|x1 |,

(3.13)

where ρ1 , ρ2 : R≥0 → R≥0 are some continuous functions.

3.4 Cross-Track Control of One AUV In this section, we develop a cross-track controller for one AUV based on a line of sight (LOS) guidance algorithm and nonlinear controller design. Since we are dealing with only one AUV, the index j referring to the AUV’s number is omitted. 3.4.1 Line of Sight Guidance Line of sight (LOS) guidance is often used in practice for path control of marine vehicles. In this section we propose to use an LOS guidance law to meet the crosstrack control goal (3.7) and the orientation control goal (3.8). First, following (3.7), we deﬁne the cross-track error as ey ez

y − Dy . z − Dz

(3.14)

42

E. Børhaug, A. Pavlov, and K.Y. Pettersen

To study the cross-track error dynamics, we diﬀerentiate (3.14) with respect to time and use (3.1) to obtain: e˙ y e˙ z

=

u sin ψ cos θ + v cos ψ + w sin θ sin ψ . −u sin θ + w cos θ

(3.15)

The right-hand side of system (3.15) contains no control inputs. To regulate the cross-track error to zero, we will control the surge speed u, the pitch angle θ and the yaw angle ψ in such a way that the cross-track error converges to zero. This will be done with the help of LOS guidance. For LOS guidance, we pick a point that lies a distance ∆ > 0 ahead of the vehicle, along the desired path. The angles describing the orientation of the xz- and xy-projection of the line of sight are referred to as the LOS angles. With reference to Fig. 3.3, the LOS angles are given by the following two expressions: ez (t) ∆

θLOS (t) = tan−1

ψLOS (t) = tan−1

,

−ey (t) ∆

.

(3.16)

L ∆ x Dz Dy

θLOS y

ψLOS

ez ey

z Fig. 3.3. Illustration of the LOS angles.

In the next subsections we will propose three controllers. The ﬁrst controller regulates the surge speed u to asymptotically track some commanded speed signal uc (t). The second controller makes the pitch angle θ track θLOS . We will show that this will result in the cross-track error ez and pitch angle θ exponentially converging to zero. The third controller makes the yaw angle ψ asymptotically track ψLOS . This will make the cross-track error ey and the yaw angle ψ exponentially converge to zero. 3.4.2 Surge Control The AUV considered in this work is underactuated. It can be actuated only in surge, pitch and yaw. In order to make the system controllable in other degrees of freedom (sway and heave), we need to ensure that the surge speed u(t) is separated from zero. This will be achieved by a controller that makes u(t) asymptotically track a speed reference command uc (t) that lies strictly within the bounds uc (t) ∈ (Umin , Umax ),

43

3 Cross-Track Formation Control of Underactuated AUVs

t ≥ t0 , for some Umax > Umin > 0. The speed reference command uc satisfying the above mentioned condition will be speciﬁed and used at a later stage to achieve the coordination control goal (3.9). The controller for tracking the commanded signal uc is given by τ¯u := −fu (ν, η) + u˙ c − ku (u − uc ),

(3.17)

where ku > 0 is the controller gain. As follows from (3.4), this yields the linear GES tracking error dynamics u ˜˙ = −ku u˜, where u ˜ := u − uc . Notice that since uc (t) lies strictly within (Umin , Umax ), ∀t ≥ 0, and u(t) → uc (t) exponentially and without overshoot, there exists a t0 ≥ 0 such that u(t) ∈ [Umin , Umax ], ∀t ≥ t0 . Therefore, in the following sections we assume that u(t) ∈ [Umin , Umax ], ∀t ≥ t0 . 3.4.3 Pitch Control In this section, we propose a control law for the pitch control τ¯q that guarantees θ(t) → θLOS (t). We derive the pitch tracking error dynamics by diﬀerentiating θ˜ := θ − θLOS with respect to time and using (3.3d), (3.16): ˙ θ˜ = θ˙ − θ˙LOS = q −

∆ ∆ e˙ z = q − 2 (−u sin θ + w cos θ). e2z + ∆2 ez + ∆2

(3.18)

We choose q as a virtual control input with the desired trajectory for q given by qd =

∆ ˜ (−u sin θ + w cos θ) − kθ θ, e2z + ∆2

(3.19)

where kθ > 0. Inserting q = qd + q˜ into (3.18), where q˜ := q − qd , then gives: ˙ θ˜ = −kθ θ˜ + q˜.

(3.20)

The pitch rate error dynamics is obtained by diﬀerentiating q˜ with respect to time and using (3.4): q˜˙ = q˙ − q˙d = fq (ν, η) + τ¯q − q˙d (3.21) We choose the feedback linearizing control law where kq > 0. This results in

τ¯q = q˙d − fq (ν, η) − kq q˜,

(3.22)

q˜˙ = −kq q˜.

(3.23)

Notice that the closed-loop dynamics (3.20), (3.23) is a linear system with eigenvalues −kθ < 0 and −kq < 0. Therefore, (3.20) and (3.23) is GES. The controller (3.22), with qd given by (3.19), requires the measurement of u˙ and w. ˙ For AUVs equipped with inertial navigation systems, these accelerations are measured. Next, we derive an estimate of |w|, which will be used in the next section. Notice ˜ q|. One can compute ˜ Thus, |q| ≤ |qd |+|q−qd | ≤ |θ˙LOS |+kθ |θ|+|˜ that qd = θ˙LOS −kθ θ. θ˙LOS from (3.16) and obtain the estimate |θ˙LOS | ≤ ∆|e˙ z |/(e2z + ∆2 ) ≤ |e˙ z |/∆. Substitution of this estimate into the obtained estimate of |q| and then into (3.5) gives ˜ + |˜ q |). (3.24) |w| ≤ Cw Umax (|e˙ z |/∆ + kθ |θ|

44

E. Børhaug, A. Pavlov, and K.Y. Pettersen

3.4.4 Analysis of the ez Dynamics In this section we analyze the ez -dynamics of the AUV and show that the controller (3.22) by making θ(t) → θLOS (t) also makes ez (t) converge exponentially to zero. Subsequently this implies that θLOS (t) and therefore θ(t) converge to zero (see (3.16)). The diﬀerential equation for ez given in (3.15) can be written as ˜ e˙ z = −u sin θLOS + w cos θLOS + δ(θ, θLOS , u, w)θ,

(3.25)

˜ Notice that where δ(θ, θLOS , u, w) := (u(sin θLOS − sin θ) + w(cos θ − cos θLOS ))/θ. by the mean value theorem and by assumption (3.6) we have |δ(θ, θLOS , u, w)| ≤ |u| + |w| ≤ 2Umax ,

(3.26)

provided |u| ≤ Umax . Notice that this condition holds because u(t) ∈ [Umin , Umax ], ∀t ≥ t0 , as discussed in Section 3.4.2. In the sequel we will write δ without its arguments. Substituting the expressions of θLOS from (3.16) into (3.25), we obtain e˙ z = −u

ez e2z + ∆2

+w

∆ e2z + ∆2

˜ + δ θ.

(3.27)

System (3.27) can be considered as a nominal system perturbed through the term δ θ˜ by the GES dynamics (3.20), (3.23). The next theorem provides a result on stability of these systems. Theorem 3.3. Consider system (3.27) in cascade with (3.20), (3.23). Let w satisfy assumptions (3.5) and (3.6) and suppose u(t) ∈ [Umin , Umax ] for all t ≥ t0 with Umin > 0 and Umax < ∆/Cw . Then system (3.27), (3.20), (3.23) is exponentially stable in any ball. Proof: Consider the Lyapunov function candidate V := 1/2|ez |2 . Its derivative along solutions of (3.27) satisﬁes u|ez |2 + ∆|ez ||w| −u|ez |2 + ∆ez w ˜ + ez δ θ˜ ≤ − + |ez ||δ||θ| V˙ = e2z + ∆2 e2z + ∆2 ≤−

u|ez |2 e2z + ∆2

˜ + |ez ||w| + |ez |2Umax |θ|.

(3.28)

In the last inequality we have used inequality (3.26) and the upper bound on u. Substituting inequality (3.24), which holds since w satisﬁes assumption (3.5), into (3.28) and using the upper bound on u, we obtain V˙ ≤ −

u|ez |2 ˜ + |˜ ˜ + |ez |Cw Umax (|e˙ z |/∆ + kθ |θ| q |) + |ez |2Umax |θ|. e2z + ∆2

(3.29)

Notice that |V˙ | = |ez ||e˙ z |. Therefore, Umin |ez |2 Cw Umax ˙ ˜ V˙ ≤ − |V | + |ez |(αq |˜ + q | + αθ |θ|), 2 2 ∆ ez + ∆

(3.30)

3 Cross-Track Formation Control of Underactuated AUVs

45

where αq := Cw Umax and αθ := (Cw kθ + 2)Umax . Hence, V˙

1 − signV˙

Cw Umax ∆

≤−

Umin |ez |2 ˜ + |ez |(αq |˜ q | + αθ |θ|). e2z + ∆2

(3.31)

Denote β + := (1 + Cw Umax /∆)−1 and β − := (1 − Cw Umax /∆)−1 . Since Umax < ∆/Cw , we have β + > 0 and β − > 0. Hence Umin |ez |2 ˜ V˙ ≤ −β + + β − |ez |(αq |˜ q | + αθ |θ|). e2z + ∆2

(3.32)

Next we consider the comparison system √ Umin 2V ˜ V˙ = −β + √ + β − 2V (αq |˜ q | + αθ |θ|). 2 2V + ∆

(3.33)

If we show that system (3.33), (3.20), (3.23) is exponentially stable in any ball provided V (t0 ) ≥ 0, then by the comparison lemma [14] we conclude that system (3.27), (3.20), (3.23) is exponentially stable in any ball. Notice that system (3.33), (3.20), (3.23) can be considered as a cascaded connection of the nominal system 2β + Umin V, V˙ = − √ 2V + ∆2

(3.34)

√ q |+ with GES system (3.20), (3.23) through the interconnection term g := β − 2V (αq |˜ ˜ One can easily see that system (3.34) is exponentially stable in any set αθ |θ|). V (t0 ) ∈ [0, R], R ≥ 0 with the quadratic Lyapunov function V 2 . The intercon˜ Therefore by nection term g can be estimated by |g| ≤ (1 + 2V )β − (αq |˜ q | + αθ |θ|). Theorem 3.2 the cascade (3.33) and (3.20), (3.23) is exponentially stable in any ball. By the comparison lemma [14] the system (3.27), (3.20), (3.23) is exponentially stable in any ball.This concludes the proof of the theorem. 3.4.5 Yaw Control In this section, we propose a control law for the yaw control τ¯r that guarantees that ψ → ψLOS exponentially. We derive the yaw tracking error dynamics by diﬀerentiating ψ˜ := ψ − ψLOS with respect to time and using (3.3e): ∆ 1 r+ 2 e˙ y cos θ ey + ∆2 ∆ 1 r+ 2 (u sin ψ cos θ + v cos ψ + w sin θ sin ψ). = cos θ ey + ∆2

˙ ψ˜ = ψ˙ − ψ˙ LOS =

(3.35) (3.36)

We choose r as a virtual control input and choose the desired trajectory for r as rd = − cos θ

∆ (u sin ψ cos θ + v cos ψ + w sin θ sin ψ) − kψ ψ˜ cos θ, e2y + ∆2

(3.37)

46

E. Børhaug, A. Pavlov, and K.Y. Pettersen

where kψ > 0. Inserting r = rd + r˜ into (3.36), where r˜ := r − rd , then gives ˙ ψ˜ = −kψ ψ˜ +

1 r˜. cos θ

(3.38)

We derive the yaw rate error dynamics by diﬀerentiating r˜ with respect to time and using (3.4): r˜˙ = r˙ − r˙d = fr (ν, η) + τ¯r − r˙d . (3.39) We choose the feedback linearizing control law τ¯r = r˙d − fr (ν, η) − kr r˜,

(3.40)

where kr > 0. This results in the GES linear dynamics r˜˙ = −kr r˜.

(3.41)

System (3.38), (3.41) can be viewed as a cascade of two GES linear systems interconnected through the term r˜/ cos θ(t). If θ(t) lies in a compact subset of (− π2 , π2 ) for all t ≥ t0 , then 1/ cos θ(t) is bounded and the interconnection term satisﬁes |˜ r / cos θ(t)| ≤ C r˜ for some constant C > 0. Therefore system (3.38), (3.41) is GES. Just like in the case of pitch control, here we give an estimate of |v|, which will be used in the next section. Notice that r = cos θψ˙ = cos θψ˙ LOS + cos θ(ψ˙ − ψ˙ LOS ). ˜ Therefore, As follows from (3.36) and (3.37), cos θ(ψ˙ − ψ˙ LOS ) = r˜ − kψ cos θψ. ˜ One can compute ψ˙ LOS from (3.16) and obtain the |r| ≤ |ψ˙ LOS | + |˜ r | + kψ |ψ|. estimate |ψ˙ LOS | ≤ ∆|e˙ y |/(e2y + ∆2 ) ≤ |e˙ y |/∆. Substitution of this estimate into the obtained estimate of |r| and then into (3.5) gives ˜ + |˜ r|). |v| ≤ Cv Umax (|e˙ y |/∆ + kψ |ψ|

(3.42)

3.4.6 Analysis of the ey Dynamics In this section we consider the ey -dynamics of the AUV and show that the controller (3.40) by making ψ(t) → ψLOS (t) also makes ey (t) converge exponentially to zero. This, in turn, implies that ψLOS (t) and ψ(t) converge to zero. The diﬀerential equation for ey given in (3.15) can be written as e˙ y = u sin ψLOS + v cos ψLOS + δψ (u, v, ψ, ψLOS )ψ˜ + δy (u, w, ψ, θ)θ

(3.43)

where, δψ (u, v, ψ, ψLOS ) := u(sin ψ − sin ψLOS )/ψ˜ + v(cos ψ − cos ψLOS )/ψ˜ and δy (u, w, ψ, θ) := (u sin ψ(cos θ − 1) + w sin θ sin ψ)/θ. Notice that by the mean value theorem and by assumption (3.6) we have |δy (u, w, ψ, θ)| ≤ |u| + |w| ≤ 2Umax |δψ (u, v, ψ, ψLOS )| ≤ |u| + |v| ≤ 2Umax provided |u| ≤ Umax . The pitch angle θ can be written as θ = θ˜ + θLOS = θ˜ + δz (ez )ez ,

(3.44) (3.45)

3 Cross-Track Formation Control of Underactuated AUVs

47

where δz (ez ) := tan−1 (ez /∆)/ez is a globally bounded function. In the sequel we will write δz , δy and δψ without their arguments. Substituting the expression for θ into (3.43) together with the expression of ψLOS from (3.16) gives e˙ y = −u

ey e2y

+

∆2

+v

∆ e2y

+ ∆2

+ δψ ψ˜ + δy θ˜ + δy δz ez .

(3.46)

System (3.46) can be viewed as a nominal system perturbed through the terms ˜ r˜)-dynamics (3.38), (3.41), which is GES provided ˜ δy θ˜ and δy δz ez by the (ψ, δψ ψ, ˜ q˜)θ(t) lies in a compact subset of (− π2 , π2 ) for all t ≥ t0 , and by the ez and (θ, dynamics (3.27), (3.20), (3.23), which is exponentially stable in any ball by virtue of Theorem 3.3. Notice that the nominal dynamics of system (3.46) is identical to the nominal dynamics of system (3.27) and the estimate (3.42) is identical to the estimate (3.24). Therefore, using the same arguments as in the proof of Theorem 3.3 we can formulate the following result for the ey -dynamics. ˜ r˜)-dynamics Theorem 3.4. Consider the ey -dynamics (3.46) in cascade with the (ψ, ˜ (3.38), (3.41) and with the ez and (θ, q˜)-dynamics (3.27), (3.20), (3.23). Let w and v satisfy assumptions (3.5) and (3.6) and suppose u(t) ∈ [Umin , Umax ] for all t ≥ t0 with Umin > 0 and Umax < min{∆/Cw , ∆/Cv }. Then the overall closed-loop system system (3.46), (3.38), (3.41), (3.27), (3.20), (3.23) is exponentially stable in any ball provided θ(t) lies in a compact subset of (− π2 , π2 ) for all t ≥ t0 .

Under the conditions of this theorem, we obtain that the cross-track errors ey (t) and ez (t) and the orientation angles θ(t) and ψ(t) converge to zero exponentially, i.e., the cross-track control goal (3.7) and the orientation control goal (3.8) are achieved. The remaining dynamics is in the x-direction. These dynamics are analyzed in the next section. 3.4.7 Analysis of the x Dynamics The x-dynamics can be written as x˙ = u + u(1 − cos ψ cos θ) − v sin ψ + w cos ψ sin θ = uc + u˜ + u

(1 − cos θ) sin ψ sin θ (1 − cosψ) ψ + u cos ψ θ−v ψ + w cos ψ θ. ψ θ ψ θ (3.47) −1

tan (−ey /∆) Recall that ψ = ψLOS + ψ˜ = ey + ψ˜ and θ = θLOS + θ˜ = ey Substituting these expressions into (3.47), we obtain

x˙ = uc + h(ez , ey , θ, ψ, u, w, v)χ,

tan−1 (ez /∆) ˜ ez + θ. ez

(3.48)

˜ ψ) ˜ T . The interconnection matrix h can easily be obtained where χ := (˜ u, ez , ey , θ, from the expressions given above. Notice that since the functions sin α/α, (1 − cos α)/α and tan−1 (α)/α are globally bounded, and because of assumption (3.6) |v| and |w| are bounded by Umax , we conclude that h is globally bounded. As follows from the previous sections, the variable χ exponentially converges to zero provided that u(t) and θ(t) satisfy the conditions in Theorem 3.4. Therefore, the speed in the x-direction asymptotically tracks the speed reference command uc (t).

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E. Børhaug, A. Pavlov, and K.Y. Pettersen

3.5 Coordination Control of Multiple AUVs In the previous sections we have designed a cross-track controller that guarantees that every AUV in closed loop with this controller achieves the cross-track control goal (3.7) with the remaining dynamics in the x-direction given by (3.48). In order to achieve the coordination control goal (3.9), for each AUV we will use the freedom we have in choosing the commanded speed signal ucj , j = 1, . . . , n. Since in this section we are dealing with multiple AUVs, we will use the subscript j to denote the AUV’s number. Recall that the cross-track control goal (3.7) is achieved provided that the commanded speed for each AUV lies inside the set (Umin , Umax ), i.e., ucj (t) ∈ (Umin , Umax ) ,

∀ t ≥ 0, j = 1, . . . , n.

(3.49)

In this section we must therefore design control laws for ucj , j = 1, . . . , n, that satisfy these constraints at the same time as they guarantee that all AUVs achieve the coordination control goal (3.9). To satisfy (3.9) the AUVs have to adjust their forward speed to asymptotically converge to the desired formation pattern and move with the desired speed proﬁle ud (t). This means that they may either have to speed up or wait for other AUVs to obtain the desired formation before they collectively reach the desired speed ud (t). Here, we make a natural assumption that the desired speed proﬁle lies within (Umin , Umax ), i.e., there exists a > 0 such that ud (t) ∈ [Umin + a, Umax − a],

∀t ≥ 0.

(3.50)

To solve the coordination problem (3.9), we propose the following control law for ucj : n

γji (xj − xi − dji ) ,

ucj = ud (t) − g

j = 1, . . . , n.

(3.51)

i=1

Here dji Dxj − Dxi correspond to the distances along the x-axis between the jth and ith AUVs in the formation. The linkage parameters γji are nonnegative and satisfy γij = γji , γii = 0. The function g(x) is a continuously diﬀerentiable non-decreasing function with a bounded derivative satisfying g (0) > 0, g(0) = 0 and g(x) ∈ (−a, a), where a is the parameter deﬁned in (3.50). Notice that under these assumptions on g and under the assumption on the desired speed proﬁle ud (t) (3.50), the proposed ucj satisfy the condition (3.49) for all values of its arguments. The function g can be chosen, for example, equal to g(x) 2a/πtan−1 (x). The dynamics of (3.48) in closed loop with (3.51) are given by the equations n

γji (xj − xi − dji )

x˙ j = ud (t) − g

+ hj χj ,

j = 1, . . . , n.

(3.52)

i=1

It can be easily veriﬁed that after the change of coordinates x ¯j t u (s)ds, j = 1, . . . , n, system (3.52) is equivalent to d 0

xj − Dxj −

3 Cross-Track Formation Control of Underactuated AUVs

x ¯˙ j = −g

49

n

γji (¯ xj − x¯i )

+ hj χj ,

j = 1, . . . , n.

(3.53)

i=1

To simplify this system, we will rewrite it in the vector form. To this end, let us ¯ := (¯ introduce the following notations x x1 , . . . , x ¯n )T , the function g(¯ x) and the matrix Γ given by  n    −γ12 · · · −γ1n j=1 γ1j g(¯ x1 ) n  −γ21 −γ2n  j=1 γ2j · · ·     g(¯ x) :=  ...  , Γ :=  . .. .. .. . .   . . . . g(¯ xn ) n −γn2 · · · γ −γn1 j=1 nj Then, system (3.53) can be written in the vector form ¯˙ = −g(Γ x ¯ ) + Hε, x

(3.54)

Where ε := [χT1 , . . . , χTn ]T and the matrix H is a block-diagonal matrix with hj , j = 1, . . . , n, on its diagonal. Since hj , is a globally bounded function of the states of the jth AUV, j = 1, . . . , n, the matrix H is a globally bounded function of the states of all n AUVs. Notice that matrix Γ has the property Γ v1 = 0, where v1 := (1, 1 . . . , 1)T . Therefore, Γ has a zero eigenvalue with v1 being the corresponding eigenvector. For system (3.54) the control goal (3.9) can be stated in the equivalent form ¯ (t) → ηv1 , as t → +∞ x

(3.55)

for some η ∈ R. Now we can formulate the main result of this section. Theorem 3.5. Consider system (3.54) coupled with the cross-track dynamics of every AUV through Hε. Suppose the conditions of Theorem (3.4) hold for every AUV and the zero eigenvalue of matrix Γ has multiplicity one. Then control goal (3.55) is achieved. Proof: Consider the matrix Γ . From the structure of Γ , by Gershgorin’s theorem [22] we obtain that all eigenvalues of Γ lie in the closed right half of the complex plane. Since Γ is symmetric, all its eigenvalues are real and, by the condition of the theorem, only one of them is zero. Therefore, all the other eigenvalues are positive. Since Γ is symmetric, one can choose an orthogonal matrix S, i.e., such that S −1 = S T , satisfying Γ = SΛS T , where Λ=

00 0 I(n−1)

,

with I(n−1) being the (n−1)-dimensional identity matrix. The corresponding matrix S equals S := [v1 , v2 , . . . , vn ], where v1 is the eigenvector corresponding to the zero eigenvalue and v2 , . . . , vn are the orthogonal eigenvectors corresponding to the remaining positive eigenvalues λj , j = 2, . . . , n, and normalized according to |vj |2 = 1/λj . Substituting this factorization of Γ into (3.54), we obtain, after the change of ¯, ˆ := S T x coordinates x

50

E. Børhaug, A. Pavlov, and K.Y. Pettersen

ˆ˙ = −S T g (SΛˆ x x) + S T Hε. T

(3.56)

T

Denote Ξ := [v2 , v3 , . . . , vn ]. Then S = [v1 , Ξ] . Denote the ﬁrst component of ˆ by ζ and the (n − 1)-dimensional vector of the remaining components by ξ, i.e., x ˆ = [ζ, ξ T ]T . By the structure of Λ we have SΛˆ x x = Ξξ. With this new notation we can write system (3.56) in the following form: ζ˙ = −v1T g (Ξξ) + v1T Hε ξ˙ = −Ξ T g (Ξξ) + Ξ T Hε.

(3.57) (3.58)

From these equations we see that the ξ-dynamics are decoupled from ζ. The ξdynamics can be considered as the nominal dynamics ξ˙ = −Ξ T g (Ξξ)

(3.59)

coupled through Ξ T Hε with the cross-track dynamics of the variables eyj , vj , ezj , wj , θj , qj , ψj , rj , u˜j of every AUV. These cross-track dynamics are exponentially stable in any ball provided that uj (t) ∈ [Umin , Umax ] and θj (t) lies in a compact subset of (− π2 , π2 ), for j = 1, . . . , n. We will show that system (3.58) in cascade with the cross-track dynamics of all AUVs is exponentially stable in any ball provided that the above mentioned conditions on uj (t) and θj (t) are satisﬁed. This will be shown using Theorem 3.2. Since the coupling matrix Ξ T H is globally bounded and the cross-track dynamics of all AUVs is exponentially stable in any ball (under the conditions on uj (t) and θj (t) stated above), we only need to show that the nominal system is exponentially stable in any ball with a quadratic Lyapunov function satisfying (3.12). Consider the Lyapunov function V (ξ) = 1/2|ξ|2 . It’s derivative along solutions of (3.59) equals V˙ = −ξ T Ξ T g (Ξξ) . (3.60) Denote ϑ := Ξξ. Then V˙ = −ϑT g(ϑ). By elaborating this expression we obtain V˙ = − ni=1 ϑi g(ϑi ). Notice that by the conditions imposed on the function g we have xg(x) > 0 for all x ∈ R satisfying x = 0. Therefore V˙ = 0 if and only if ϑ = 0. At the same time, since rankΞ = (n − 1), it holds that ϑ = Ξξ = 0 if and only if ξ = 0. Hence, V˙ as a function of ξ is negative deﬁnite. This implies that system (3.59) is GAS. In fact, since system (3.59) is autonomous, it is GUAS. Let us show that system (3.59) is locally exponentially stable (LES). The system matrix A of system (3.59) being linearized at the origin equals A = −Ξ T ∂∂gx¯ (0)Ξ. By the ¯ we obtain ∂∂gx¯ (0) = g (0)In . Since Ξ consists of orthogonal construction of g(x), eigenvectors vi normalized by |vi |2 = 1/λi , where λi > 0, i = 2, . . . , n, we obtain A = −g (0)diag(1/λ2 , . . . , 1/λn ), which is Hurwitz, because g (0) > 0 by the definition of g(x). Therefore, the linearized system (3.59) is GES, which implies that system (3.59) itself is LES. By Theorem 3.1 system (3.59) is exponentially stable in any ball. Applying Theorem 3.2, we conclude that system (3.58) in cascade with the cross-track dynamics of all AUVs is exponentially stable in any ball provided that uj (t) ∈ [Umin , Umax ] and θj (t) lies in a compact subset of (− π2 , π2 ), for j = 1, . . . , n. This, in turn, implies that the right-hand side of (3.57) exponentially tends to zero.

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51

By integrating (3.57), we obtain that ζ(t) → η, where η ∈ R is some constant. Recall ¯ = Sx ˆ = v1 ζ + Ξξ. Since ξ(t) → 0 and ζ(t) → η, we obtain x ¯ (t) → ηv1 . that x Remark 1. Note that for the jth AUV, the overall controller, which consists of the cross-track controller and the coordination controller (3.51), requires only the communication of the x positions from those AUVs that correspond to non-zero entries in the jth row of the interconnection matrix Γ . At the same time, the condition on Γ imposed in Theorem 3.5 allows for many zero entries in Γ , meaning that all-to-all communication is not required. These properties of the proposed controllers are very important in low bandwidth and unreliable underwater communication. Remark 2. The proposed controllers (3.17), (3.22) and (3.40) are based on feedback linearization. In practice, however, exact cancelation is not possible due to inevitable model uncertainties. This can lead to steady-state errors. Partly this problem can be solved by omitting the cancelation of the dissipative damping terms.

3.6 Simulations The proposed formation control scheme has been implemented in Simulink! and simulated using a 6-DOF model of the HUGIN AUV from FFI and Kongsberg Maritime. The control scheme is simulated for the case of three AUVs. The desired formation is chosen to be given by (Dx1 , Dy1 , Dz1 ) = (0, 10, 0), (Dx2 , Dy2 , Dz2 ) = (20, −10, 0), (Dx3 , Dy3 , Dz3 ) = (20, 0, 0), see Section 3.2.2. The desired straight-line path coincides with the x-axis. The desired formation speed is chosen as ud = 2.0 m/s. The initial cross-track errors are chosen as (ey1 (0), ez1 (0)) = (20, 20), (ey2 (0), ez2 (0)) = (−25, −25) and (ey3 (0), ez3 (0)) = (20, 20). The initial surge speed is chosen as uj (0) = 0.5 m/s, j = 1, 2, 3, while the initial sway and heave velocities are chosen as vj (0) = wj (0) = 0, j = 1, 2, 3. All vehicles are given zero initial pitch and yaw angle, i.e., θj (0) = ψj (0) = 0, j = 1, 2, 3. The controller gain ku in (3.17) is chosen as ku = 10, kθ and kq in (3.22) are chosen as kθ = 3 and kq = 50 and the controller gains kψ and kr in (3.40) are chosen as kψ = 3.5 and kr = 50. The linkage parameters γij are set to γ12 = γ13 = γ23 = 4 and the function g(x) is chosen as g(x) = π2 tan−1 (x) ∈ [−1, 1]. The simulation results are shown in Fig. 3.4(a)-3.4(c). Figure 3.4(a) shows the cross-track error norm of each vehicle, Fig. 3.4(b) shows the inertial velocity in the x-direction, i.e., x˙ j , of each vehicle and Fig. 3.4(c) shows the xy-trajectory of each vehicle. The presented simulation results clearly show that the control goals (3.7)(3.9) are achieved. The AUVs asymptotically constitute the desired formation that moves along the desired path with the prescribed velocity proﬁle.

3.7 Conclusions In this paper we have considered the problem of 3D cross-track formation control for underactuated AUVs. The proposed decentralized control laws guarantee convergence of the underwater vehicles to a desired formation moving with a desired

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E. Børhaug, A. Pavlov, and K.Y. Pettersen 40

3

Cross-track error norm AUV1: ||e1 (t)|| Cross-track error norm AUV1: ||e2 (t)|| Cross-track error norm AUV3: ||e3 (t)||

35

2

25

x˙ j (t) [m/s]

||ej (t)|| [m]

30

20 15

1.5 1 0.5

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5 0 0

Forward velocity AUV1: x˙ 1 (t) Forward velocity AUV2: x˙ 2 (t) Forward velocity AUV3: x˙ 3 (t)

2.5

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40

t [s]

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70

−0.5 0

80

10

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t [s]

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(b) Forward velocity x˙ j (t), j = 1, 2, 3.

(a) Cross-track error norms. 40

xy-trajectory AUV1 xy-trajectory AUV2 xy-trajectory AUV3

30 20

y [m]

10 0 −10 −20 −30 −40 0

20

40

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x [m]

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120

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(c) xy-trajectories. Fig. 3.4. Simulation results

formation speed along a desired straight-line path. This control problem has been solved in two steps. The ﬁrst step is the design of a cross-track controller. For each AUV such a controller, which is based on the Line of Sight guidance law, guarantees convergence to the desired path for this AUV in the formation. It has been proved that for any initial condition of an AUV, the convergence to the desired path is exponential (yet depending on the initial conditions). Moreover, this controller guarantees that the forward velocity of the AUV tracks some speed reference command, which is designed at the second step. Control laws for the speed reference command are designed for each AUV. These controllers asymptotically align the AUVs in the direction of the desired path in such a way that they constitute the desired formation and move synchronously with the desired speed proﬁle. The performance of the proposed formation control scheme has been investigated for the case of three AUVs through numerical simulations in Simulink!with a model of the HUGIN AUV. The simulations have demonstrated the validity of the obtained theoretical results.

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53

References 1. R. Bachmayer and N. Leonard. Vehicle networks for gradient descent in a sampled environment. In Proc. 41st IEEE Conference on Decision and Control, pages 112–117, Las Vegas, NV, USA, December 2002. 2. E. Børhaug and K.Y. Pettersen. Cross-track control for underactuated autonomous vehicles. In Proc. 44th IEEE Conference on Decision and Control, pages 602 – 608, Seville, Spain, December 2005. 3. M. Breivik and T.I. Fossen. Path following of straight lines and circles for marine surface vessels. In Proc. 6th IFAC Control Applications in Marine Systems (CAMS), pages 65–70, Ancona, Italy, July 2004. 4. J. Cortes, S. Martinez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Trans. on Robotics and Automation, 20(2):243–255, 2004. 5. J.P. Desai, J.P. Ostrowski, and V. Kumar. Modeling and control of formations of nonholonomic mobile robots. IEEE Trans. on Robotics and Automation, 17(6):905– 908, 2001. 6. K.D. Do and J. Pan. Global tracking control of underactuated ships with oﬀ-diagonal terms. In Proc. 42nd IEEE Conference on Decision and Control, pages 1250–1255, Maui, Hawaii, USA, December 2003. 7. P. Encarna¸ca ˜o and A.M. Pascoal. 3D path following for autonomous underwater vehicle. In Proc. 39th IEEE Conference on Decision and Control, pages 2977–2982, Sydney, Australia, December 2000. 8. J.A. Fax. Optimal and Cooperative Control of Vehicle formations. PhD thesis, California Institute of Technology, 2001. 9. T.I. Fossen. Marine Control Systems. Marine Cybernetics, Norway, 2002. 10. T.I. Fossen, M. Breivik, and R. Skjetne. Line-of-sight path following of underactuated marine craft. In Proc. 6th IFAC Manoeuvring and Control of Marine Craft (MCMC), pages 244–249, Girona, Spain, September 2003. 11. E. Fredriksen and K.Y. Pettersen. Global κ-exponential way-point manoeuvering of ships. In Proc. 43rd IEEE Conference on Decision and Control, pages 5360–5367, Bahamas, December 2004. 12. R. Ghabcheloo, A. Pascoal, C. Silvestre, and D. Carvalho. Coordinated motion control of multiple autonomous underwater vehicles. In Proc. Int. Workshop on Underwater Robotics, pages 41–50, Genoa, Italy, November 2005. 13. I.F Ihle, J. Jouﬀroy, and T.I. Fossen. Formation control of marine surface craft using lagrange multipliers. In Proc. 44th IEEE Conference on Decision and Control, pages 752–758, Sevilla, Spain, December 2005. 14. H.K. Khalil. Nonlinear Systems. Pearson Inc., NJ, USA, 3rd edition, 2000. 15. V. Kumar, N. Leonard, and A.S. Morse, editors. Cooperative Control. Springer-Verlag, 2005. 16. E. Lefeber. Tracking Control of Nonlinear Mechanical Systems. PhD thesis, University of Twente, The Netherlands, 2000. 17. N. Leonard and E. Fiorelli. Virtual leaders, artiﬁcial potentials and coordinated control of groups. In Proc. 40th IEEE Conference on Decision and Control, pages 2968–2973, Orlando, FL, USA, December 2001. 18. P. Ogren, M. Egerstedt, and X. Hu. A control Lyapunov function approach to multiagent coordination. IEEE Trans. on Robotics and Automation, 18(5):847–851, 2002. 19. P. Ogren, E. Fiorelli, and N. Leonard. Formations with a mission: Stable coordination of vehicle group maneuvers. In Proc. 15th Int. Symposium on Mathematical Theory of Networks and Systems [CD-ROM], IN, USA, August 2002.

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20. E. Panteley, E. Lefeber, A. Loria, and H. Nijmeijer. Exponential tracking control of mobile car using a cascaded approach. In Proc. IFAC Workshop on Motion Control, pages 221–226, Grenoble, France, 1998. 21. R. Skjetne, S. Moi, and T.I. Fossen. Nonlinear formation control of marine craft. In Proc. 41st IEEE Conference on Decision and Control, pages 1699–1704, Las Vegas, NV, USA, December 2002. 22. R.S. Varga. Gerschgorin and His Circles. Springer-Verlag, Berlin, 2004.

4 Kinematic Aspects of Guided Formation Control in 2D M. Breivik1 , M.V. Subbotin2 and T.I. Fossen3 1 2 3

Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway; e-mail: [email protected] Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA; e-mail: [email protected] Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway; e-mail: [email protected]

Summary. This paper addresses fundamental kinematic aspects of planar formation control. A so-called guided formation control scheme based on a guided path following approach is developed by combining guidance laws with synchronization algorithms and collision avoidance techniques. The result is collision-free formation assembly and path following, inspired by the helmsman behavior of human pilots. All regularly parametrized paths are rendered feasible, and geometric constraints of formation control problems are illustrated. Finally, a simulation is employed to display transient behavior.

4.1 Introduction Formation control concepts are becoming increasingly important both commercially, scientiﬁcally, and militarily. This is directly related to the increased feasibility of such concepts for mechanical vehicle systems, which is facilitated by the ongoing and rapid development within sensor, communication, and computer technology. Today, relevant applications utilizing formation control can be found everywhere; at sea, on land, in the air, and in space. Oceanwise, oceanographers utilize platoons of autonomous underwater vehicles (AUVs) to eﬃciently gather timely, spatially-distributed data for the construction of sea topography maps relevant in their research. Formation control is also required for underway ship replenishment operations, where fuel and goods need to be transported from one ship to another while still in transit. Landwise, military applications include platoons of scout vehicles, coordinated clearing of mine ﬁelds, and mobile communication and sensor networks. Airwise, unmanned aerial vehicles (UAVs) are currently playing an increasingly important role, especially in military operations, both strategically and tactically. Furthermore, ﬂying sensor networks acting as eyes 1

This work was supported by the Research Council of Norway through the Centre for Ships and Ocean Structures at the Norwegian University of Science and Technology.

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 55–74, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Fig. 4.1. The Royal Air Force aerobatic team The Red Arrows ﬂying in formation with four Euroﬁghter Typhoons. Courtesy of BAE Systems, http://www.baesystems.com/.

in the sky open up new possibilities for information gathering and distribution, for instance in search-and-rescue missions. More prosaic formation ﬂying applications include aerial acrobatics, see e.g. Figure 4.1. Spacewise, a recent trend is the concept of large sensor arrays consisting of multiple satellites. Spacecraft cooperate to act as a single instrument, thus achieving a level of resolution previously unheard of, and practically impossible to obtain by employing any single satellite. Other advantages include reconﬁgurability properties, increased robustness, and reduced costs of implementation and operation. 4.1.1 Previous Work Control system researchers have always been fascinated by the natural world, and sought inspiration from it in their analysis and design whenever applicable. This especially holds true for concepts relating to group coordination and cooperative control. Hence, the aggregate behavior observed in natural phenomena such as ﬂocks of birds, schools of ﬁsh, herds of ungulates, and swarms of bees, has been and still remains the focus of a concerted eﬀort by researchers to unlock some of the secrets of the natural world in order to utilize them for artiﬁcial systems. Speciﬁcally, Reynolds [21] originally proposed a distributed behavioral model for bird-like simulation entities (boids) to be able to naturally attain realistic motion patterns in animation of animal groups. Simple motion primitives were suggested for each group member (i.e. collision avoidance, velocity matching, and neighbor tracking), resulting in complex motion behavior resembling that found in nature. In principle, this kind of emergent behavior is similar to that illustrated for cellular automata, see e.g. Wolfram [30], where the assignment of simple behavioral rules to each cell in an automata results in extensive and unpredicted overall complexity. However, this kind of rule-based paradigm hardly lends itself to rigid mathematical analysis, nor can it easily be applied as a constructive tool for designing cooperative behavior with guaranteed

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performance for mechanical vehicle systems. Still, some relevant analysis results can be found in Jadbabaie et al. [12], and Tanner et al. [27], [28]. In any case, the fact of the matter is that by applying such suggested motion primitives, the group ultimately degenerates into a steady state condition (stationary speed and orientation) if it is not roaming in a complex environment capable of continuously exciting it. This fact has been put to good use with the concept of virtual leaders, see e.g. Leonard and Fiorelli [15], where such leaders are employed to herd and control a group formation acting on the basis of artiﬁcial potential ﬁelds. Relevant work in the vein of behavioral assignments are also reported by Balch and Arkin [2]. Interesting and useful as it may be, achieving cooperative control through enforcing local motion primitives does not generally guarantee a speciﬁc global behavior. However, guaranteeing a speciﬁc global behavior is what formation control is all about. It is a high precision application which entails the assembly and maintenance of a rigid geometric formation structure. Relevant work can be found in Lewis and Tan [16], where the concept of a virtual structure is put forward. In their framework, the formation itself is considered analogous to a real rigid-body geometric structure, where the geometric constraints of the structure are enforced by feedback control of the mechanical vehicle systems involved. Furthermore, the design of the approach is such that if a vehicle fails, the rest of the vehicles will reconﬁgure accordingly in order to maintain the desired virtual structure, thus automatically allowing for the weakest link. A conceptually similar procedure can be found in Egerstedt and Hu [8], where the vehicles involved are required to track a desired virtual structure composed of virtual vehicles which follows a virtual leader tracking a desired reference path. Interesting work are also reported by Ihle et al. [11], who propose utilizing constraint forces originating from constraint functions in order to maintain a formation as a virtual structure. This approach is rooted in analytical mechanics for multi-body dynamics, and facilitates a ﬂexible and robust formation control scheme. The concept of virtual structures is also considered by Ren and Beard [20]. Formation control with application towards marine craft can be found in Encarna¸c˜ao and Pascoal [9], where the authors consider a leader-follower approach in which an autonomous underwater vehicle is tracking the 2D projection of a surface craft onto its nominal path, while the surface craft itself is following its own nominal path at sea. Furthermore, Skjetne et al. [25], [24] consider formation control of multiple maneuvering systems, both in a centralized and decentralized manner. The approach yields a robust formation scheme with dynamic adjustment to the weakest link in the formation. Marine applications are also considered by Lapierre et al. [14], where a formation of two vehicles, each equipped with its own path following controller, is treated. Coordinated vehicle motion is enforced by augmenting a simple path parameter synchronization algorithm to the existing path following controller of one of the vehicles, which consequently constitutes the follower in a leader-follower scheme. Other concepts have also been reported in the literature, for instance by Beard et al. [3], where a supervisory formation control architecture based on so-called coordination states is proposed. Desai et al. [7] utilize graph theory to obtain adaptable formation conﬁgurations within a leader-follower framework, while Olfati-Saber and Murray [18] treats distributed formation control based on graph-theoretical and

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Lyapunov-based methods. Furthermore, a framework for investigating the motion feasibility of formations is reported in Tabuada et al. [26] by using so-called formation graphs, while Gazi [10] combines the use of artiﬁcal potential ﬁelds with slidingmode control to ensure robust formation control for swarms of mechanical vehicle systems. Finally, a state-of-the-art presentation of formation-related concepts can be found in Kumar et al. [13], while Bekey [4] thoroughly treats all relevant aspects of autonomous robot control, including formation control concepts. 4.1.2 Main Contribution The main contribution of this paper is a concept denoted guided formation control. This concept is developed within the framework of path following, under the assumption that one is granted the freedom to design the path to be followed. All regularly parametrized paths are rendered feasible by the approach. The design involves constructing guidance laws to solve the path following problem for single agents, devising synchronization algorithms in order to coordinate the motion of multiple agents, and suggesting collision avoidance techniques to ensure that no collisions occur during the formation assembly phase. The agents constituting the formation are assumed to instantly being able to attain any assigned motion behavior, hence they are essentially kinematic particles. This fact contributes to illustrate the geometric essence of the proposed approach. Hence, in a sense the resulting theory becomes generally valid. For the sake of intuition and page limitation, the paper only considers the planar 2D case, while extensions towards the spatial 3D case will be followed up separately.

4.2 Motion Control: Guided Path Following for Single Agents This section derives the guidance laws required to solve a path following problem for a single agent, represented by a kinematic particle. First, a preliminary discussion elaborates on why a path following concept is preferable over a trajectory tracking concept when given the freedom to design the path to be followed. Subsequently, the path following problem is formally stated, and ultimately it is solved by developing the required guidance laws for a kinematic particle moving on a planar surface. 4.2.1 Tracking Versus Following The most basic form of motion control of mechanical vehicle systems relates to the ability to accurately maneuver along a given path. In this aspect, the concept of trajectory tracking is often employed. It basically boils down to chasing a timevarying reference position, see e.g. Athans and Falb [1]. However, if the reference position just traces out the pattern of a pre-designed geometric curve, the tracking concept seems like a bad idea. This relates to the fact that when given the freedom to design a path, it inherently mixes the space and time assignments into one single assignment; demanding that a vehicle is located at a speciﬁc point in space at a speciﬁc, pre-assigned instant in time. Hence, if for some reason the original

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time-parametrization of the path becomes dynamically infeasible, for instance pertaining to changes in the weather or the propulsion capability of the vehicle, it must be temporarily reparametrized to avoid saturating the actuators and rendering the path-vehicle system unstable. However, this does not seem like an intuitive way to act, and clearly does not correspond to the way in which a human pilot would adapt dynamically to changing conditions. A human does not aim at tracking a conceptual point in front of his vehicle, especially if he realizes that it would be equivalent with risking lives or damaging the vehicle. Enter the concept of path following. It entails the separate construction of the geometric curve and the dynamic assignment, where the task objective is ﬁrst and foremost to follow the path, see e.g. Samson [22]. Thus, if the dynamic assignment cannot be satisﬁed, the vehicle will still be able to follow the path. This clearly represents a more ﬂexible and robust approach than the tracking concept, and corresponds well with how a human driving a car chooses to negotiate a road. His main concern is to stay on the road, while continually adjusting the vehicle speed according to the traﬃc situation and road conditions. Consequently, the path following concept appears as the most favorable on which to found motion control systems for accurate and safe maneuvering of mechanical vehicle systems when given the freedom to frame the path to be followed. 4.2.2 Problem Statement Before stating the (guided) path following problem, some relevant denotation will have to be introduced. Assume that we have designed a desired geometric path in the plane. Since we are considering the pure geometric aspects, an agent is represented by a kinematic particle whose goal it is to converge to and follow the desired geometric path. It is free to roam the plane unrestricted. A path particle, on the other hand, belongs to the desired geometric path, restricted to move along it at all times. As already stated, the path following concept involves the separate construction of the geometric path and the dynamic assignment; emphasizing spatial localization as a primary task objective, while considering the dynamic aspect to be of secondary importance, sacriﬁcable if necessary. Speciﬁcally, this paper treats a path following approach where the dynamic assignment is associated with the agent (its assigned speed), and where the path particle is designed to evolve according to the agent motion. Hence, the path particle can be regarded as an active collaborator which never leaves the agent behind. This entails a closed-loop type of solution to the problem at hand, with the path system adjusting itself to the agent system. Such an approach inherently requires that guidance laws are applied to guide the agent towards the geometric path, stemming from the fact that the path system lacks a self-propelled attractor. Consequently, the concept is termed guided path following. Then, by using the convenient task classiﬁcation scheme of Skjetne [23], the (guided) path following problem can be expressed by the following two task objectives: Geometric Task: Make the position of the agent converge to and follow the desired geometric path. Dynamic Task: Make the speed of the agent converge to and track the desired speed assignment.

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The geometric task implies that the agent must move at non-zero speed (towards the collaborating path particle), while the dynamic task states that the dynamic assignment is associated with the agent (inherently implying that its motion must be guided if the geometric task is to be fulﬁlled). Finally, when both task objectives for some reason cannot be met simultaneously, the geometric one should have precedence over the dynamic one. 4.2.3 Guided Path Following for Single Agents This part develops the guidance laws required to solve the planar 2D case of the guided path following problem for single agents represented by kinematic particles. The theory mainly stems from Breivik and Fossen [5].

Fig. 4.2. The geometric principle of the proposed guided path following scheme in 2D.

Denote the position and velocity vectors of an agent in the INERTIAL frame (I) ˙ y] by p = [x, y] ∈ R2 and p˙ = [x, ˙ ∈ R2 , respectively. The velocity vector has two 1 ˙ 2 (the characteristics; size and orientation. Denote the size by U = |p| ˙ 2 = (p˙ p) speed), and let the orientation be characterized by the angular variable: χ = arctan

y˙ x˙

,

(4.1)

which is denoted the azimuth angle. These are the variables that must be manipulated in order to solve the problem at hand as far as the agent is concerned. Furthermore, since a kinematic particle is without dynamics, it can instantly attain any assigned motion behavior. Hence, rewrite U and χ as Ud and χd , respectively.

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Then consider a geometric path continuously parametrized by a scalar variable ∈ R, and denote the position of its path particle as pp ( ) ∈ R2 . Consequently, the geometric path can be expressed by the set: P = p ∈ R 2 | p = pp ( ) ∀

∈R ,

(4.2)

where P ⊂ R2 . For a given , deﬁne a local reference frame at pp ( ) and name it the PATH frame (P). To arrive at P, we need to positively rotate the INERTIAL frame an angle: yp ( ) χp ( ) = arctan (4.3) xp ( ) about its z-axis, where the notation xp ( ) = can be represented by the rotation matrix: Rp,z (χp ) =

dxp d (

) has been utilized. This rotation

cos χp − sin χp sin χp cos χp

∈ SO(2),

(4.4)

used to state the error vector between p and pp ( ) expressed in P by: (4.5)

ε = Rp (p − pp ( )),

where Rp is short for Rp,z , and ε = [s, e] ∈ R2 consists of the along-track error s and the cross-track error e; see Figure 4.2. The along-track error represents the (longitudinal) distance from pp ( ) to p along the x-axis of the PATH frame, while the cross-track error represents the (lateral) √ distance along the y-axis. Also, recognize the notion of the oﬀ-track error |ε|2 = s2 + e2 . It is clear that the geometric task is solved by driving the oﬀ-track error to zero. Consequently, by diﬀerentiating ε with respect to time, we obtain:

where:

˙ p (p − pp ) + Rp (p˙ − p˙ p ), ε˙ = R

(4.6)

˙ p = R p Sp R

(4.7)

with: Sp = S(χ˙ p ) =

0 −χ˙ p , χ˙ p 0

(4.8)

which is skew-symmetric; Sp = −Sp . We also have that: p˙ = p˙ dv = Rdv vdv ,

(4.9)

where: Rdv = Rdv,z (χd ) =

cos χd − sin χd sin χd cos χd

∈ SO(2)

(4.10)

represents a rotation from I to a frame attached to the agent with its x-axis along the assigned velocity vector. Let this frame be called the DESIRED VELOCITY

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frame (DV). Hence, the vector vdv = [Ud , 0] ∈ R2 represents the agent velocity with respect to I, decomposed in DV. Additionally, we have that: p˙ p = Rp vp ,

(4.11)

where vp = [Up , 0] ∈ R2 represents the path particle velocity with respect to I, decomposed in P. By then expanding (4.6) in light of the recent discussion, we get: ε˙ = (Rp Sp ) (p − pp ) + Rp (Rdv vdv − Rp vp ) = Sp ε + Rp Rdv vdv − vp .

(4.12)

Now deﬁne the positive deﬁnite and radially unbounded Control Lyapunov Function (CLF): 1 1 Vε = ε ε = (s2 + e2 ), (4.13) 2 2 and diﬀerentiate it with respect to time along the trajectories of ε to obtain: V˙ ε = ε ε˙ =ε

Sp ε + Rp Rdv vdv − vp

=ε

Rp Rdv vdv − vp ,

(4.14)

since the skew-symmetry of Sp leads to ε Sp ε = 0. By further expansion, we get: V˙ ε = s(Ud cos(χd − χp ) − Up ) + eUd sin(χd − χp ).

(4.15)

At this point, it seems natural to consider the path particle speed Up as a virtual input for stabilizing s. Thus, by choosing Up as: Up = Ud cos(χd − χp ) + γs,

(4.16)

where γ > 0 becomes a constant gain parameter in the guidance law, we achieve: V˙ ε = −γs2 + eUd sin(χd − χp ),

(4.17)

which shows that the task of the path particle is to ensure that the along-track error s converges to zero, hence continuously tracking the moving agent. What remains is thus to ensure that the cross-track error e also converges to zero. This is the responsibility of the agent itself, and beyond anything that the path particle can achieve. Equation (4.17) shows that (χd − χp ) can be considered a virtual input for stabilizing e. Denote this angular diﬀerence by χr = χd − χp . Intuitively, it should depend on the cross-track error itself, such that χr = χr (e). An attractive choice for χr (e) could then be the physically motivated: χr (e) = arctan −

e e

,

(4.18)

where e becomes a time-varying guidance variable shaping the convergence behavior towards the longitudinal axis of P. Such a variable is often referred to as a

4 Kinematic Aspects of Guided Formation Control in 2D

63

lookahead distance in literature dealing with planar path following along straight lines [19], and the physical interpretation can be derived from Figure 4.2. Note that other sigmoid shaping functions are also possible candidates for χr (e), for instance the tanh function. Consequently, the desired azimuth angle is given by: χd ( , e) = χp ( ) + χr (e),

(4.19)

with χp ( ) as in (4.3) and χr (e) as in (4.18). Since is the actual path parametrization variable that we control for guidance purposes, we need to obtain a relationship between and Up to be able to implement (4.16). By using the kinematic relationship given by (4.11), we get that: ˙ = =

Up xp2 + yp2 Ud cos χr + γs xp2 + yp2

,

(4.20)

which is non-singular for all regular paths. Hence, by utilizing trigonometric relationships from Figure 4.2, the derivative of the CLF ﬁnally becomes: V˙ ε = −γs2 − Ud

e2 +

e2

2 e

,

(4.21)

which is negative deﬁnite under the assumptions that the speed of the agent is positive and lower-bounded, and that the lookahead distance is positive and upperbounded. Consequently, with the recent considerations pertaining to (4.20) and (4.21), we can state the following relevant assumptions: A.1 The path is regularly parametrized; 0 <

x( )p2 + y( )p2 < ∞ ∀

∈ R.

A.2 The agent speed is positive and lower-bounded; Ud (t) ∈ [Ud,min, ∞ ∀t ≥ 0. Note that it is non-negative by deﬁnition. A.3 The lookahead distance is positive and upper-bounded; e ∈ 0, e,max] ∀t ≥ 0. This basically means that χr (e)sign(e) ∈ 0, π2 . Elaborating on these facts, we ﬁnd that the total dynamic system, which consists of the agent system and the path system, can be represented by the states ε and . Hence, a complete stability analysis cannot be carried out merely by considering (4.13) and (4.21) by themselves. Moreover, the involved dynamics are non-autonomous since Ud and e can be time-varying. However, by reformulating the time dependence through the introduction of an extra state l, the augmented system can be made autonomous: l˙ = 1, l0 = t0 ≥ 0,

(4.22)

see e.g. Teel et al. [29]. Hence, the augmented system can be represented by the state vector x = ε , , l ∈ R2 × R × R+ , and with dynamics represented by the nonlinear time-invariant ordinary diﬀerential equation:

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M. Breivik, M.V. Subbotin, and T.I. Fossen

x˙ = f (x).

(4.23)

The time variable for the augmented system is denoted t with initial time t = 0, such that l(t) = t + t0 . The motivation for this reformulation is that it facilitates the utilization of set stability analysis for time-invariant systems when concluding on whether the task objectives in the problem statement have been met or not. Hence, deﬁne the closed, but unbounded set: E = x ∈ R2 × R × R+ | ε = 0 ,

(4.24)

which represents the dynamics of the augmented system when the agent has converged to the path particle, i.e. travels along the desired geometric path. Also, let: |x|E = inf {x − y | y ∈ E} = (ε ε)

1 2

(4.25) (4.26)

represent a function measuring the distance from x to E, i.e. the oﬀ-track error. Making |x|E converge to zero is equivalent to solving the geometric task of the guided path following problem, and the following proposition can now be stated: Proposition 4.1. The error set E is rendered uniformly globally asymptotically and locally exponentially stable (UGAS/ULES) under assumptions A.1-A.3 if χr is equal to (4.18), and is updated by (4.20). Proof. Since the set E is closed, but not bounded, we initially have to make sure that the dynamic system (4.23) is forward complete [29], i.e. that for each x0 the solution x(t, x0 ) is deﬁned on [0, ∞ . This entails that the solution cannot escape to inﬁnity in ﬁnite time. By deﬁnition, l cannot escape in ﬁnite time. Also, (4.13) and (4.21) shows that neither can ε. Consequently, (4.20) shows that cannot escape in ﬁnite time under assumptions A.1 and A.2. The system is therefore forward complete. We also know that ∀x0 ∈ E the solution x(t, x0 ) ∈ E ∀t ≥ 0 because ε0 = 0 ⇒ ε˙ = 0. This renders E forward invariant for (4.23) since the system is already shown to be forward complete. Now, having established that (4.23) is forward complete and that E is forward invariant, and considering the fact that Vε = 12 ε ε = 12 (|x|E )2 , we can derive our stability results by considering the properties of Vε , see e.g. [23]. Hence, we conclude by standard Lyapunov arguments that the error set E is rendered UGAS. U e2 for the error dynamics at ε = 0, Furthermore, V˙ ε = −γs2 − Ude e2 ≤ −γs2 − d,min e,max which proves ULES. By stabilizing the error set E, we have achieved the geometric task. The dynamic task is trivially fulﬁlled by assigning a desired speed Ud satisfying assumption A.2. In total, we have now solved the planar guided path following problem. Note that by choosing the speed of the agent equal to: Ud = κ

e2 +

2, e

(4.27)

where κ > 0 is a constant gain parameter, we obtain: V˙ ε = −γs2 − κe2 , which results in the following proposition:

(4.28)

4 Kinematic Aspects of Guided Formation Control in 2D

65

Proposition 4.2. The error set E is rendered uniformly globally exponentially stable (UGES) under assumptions A.1 and A.3 if χr is equal to (4.18), is updated by (4.20), and Ud satisﬁes (4.27). Proof. The ﬁrst part of the proof is identical to that of Proposition 4.1. Hence, we conclude by standard Lyapunov arguments that the error set E is rendered UGES. Although very powerful, this result is clearly not achievable by physical systems since these exhibit natural limitations on their maximum attainable speed. In this regard, Proposition 4.1 states the best possible stability property a planar physical system like a watercraft can hold.

4.3 Formation Control: Guided Formation Path Following for Multiple Agents This section utilizes the guidance laws from the preceding section in order to ensure that individual agents are able to converge to and follow their assigned formation positions. Subsequently, a coordination scheme which synchronizes all the agents to obtain formation assembly and maintenance is proposed. A key feature of this scheme is that it inherently makes allowance for the weakest link in the formation, even though such a capability is not explicitly called for due to the kinematic nature of the consideration. Finally, a collision avoidance strategy which ensures collisionfree transient formation assembly is suggested. The combined net result is a guided formation control scheme based on a guided formation path following approach. 4.3.1 Problem Statement Analogous to the (guided) path following problem statement, the (guided) formation path following problem can be formally expressed by the following two task objectives: Geometric Task: Make the position of the formation converge to and follow the desired geometric path. Dynamic Task: Make the speed of the formation converge to and track the desired speed assignment. This problem statement is formulated at the highest level of abstraction, thus leaving considerable freedom to the implementation details. It also inherently implies the notion of the formation as a rigid-body-like entity, which under nominal operating conditions is precisely the objective. When all the formation agents have converged to their assigned positions relative to the path, as well as being synchronized to each other, they indeed constitute the rigid-body-like entity denoted a formation in this high-level problem statement. As in the single agent case, the geometric task should have precedence over the dynamic task if both cannot be satisﬁed simultaneously.

66

M. Breivik, M.V. Subbotin, and T.I. Fossen

Fig. 4.3. The geometric principle of the proposed guided formation path following scheme in 2D for an individual formation agent and its support particle.

4.3.2 Guided Formation Path Following for Multiple Agents Consider a formation consisting of n members. Each formation member, or agent, is assigned its own unique position within the formation. This position is deﬁned in and related to the PATH frame, which is natural given that the formation is expected to propagate along a desired geometric path. Every member is expected to converge to and follow its own formation position, which when ultimately synchronized with the rest constitute the rigid-body formation structure that is the goal. To be able to achieve convergence to the formation position, the guidance laws from Section 4.2 are applied. However, it is not the agent itself that will adhere directly to these laws, but an associated support particle. It is then this support particle which ultimately converges to the path. Consequently, the individual agents must satisfy a position constraint deﬁned relative to the support particles such that when these converge to and follow the path, the formation agents converge to and follow their formation positions. This entails that we only have to derive the dynamic relationship between a formation agent and its associated support particle, i.e. the dynamic equations that the formation agent must adhere to given that its support particle employs the guidance laws from the preceding section. Denote the position and velocity vectors of the ith support particle by ps,i = [xs,i , ys,i ] ∈ R2 and p˙ s,i = [x˙ s,i , y˙ s,i ] ∈ R2 , respectively, where i ∈ I = {1, ..., n}. Then denote the position constraint corresponding to the formation position of the ith formation agent by εf,i = [sf,i , ef,i ] ∈ R2 . Consequently, the INERTIAL frame position of this formation agent can be stated by: pf,i = ps,i + Rsv,i εf,i ,

(4.29)

4 Kinematic Aspects of Guided Formation Control in 2D

67

which is illustrated in Figure 4.3. Here Rsv,i = R(χs,i ) is of the same form as (4.10), while χs,i adheres to (4.19). This means that each formation agent must satisfy the following dynamic relationship: ˙ sv,i εf,i + Rsv,i ε˙ f,i p˙ f,i = p˙ s,i + R = p˙ s,i + Rsv,i Ssv,i εf,i + Rsv,i ε˙ f,i ,

(4.30)

in order to converge to and follow its assigned position in the formation. Here Ssv,i = S(χ˙ s,i ) is of the same skew-symmetric form as (4.8). Note that (4.30) allows for a time-varying formation position. However, in practice it suﬃces to consider that ε˙ f,i = 0, implying that formation reconﬁgurations nominally are conducted discretely. Consequently, we typically have that: p˙ f,i = p˙ s,i + Ssv,i Rsv,i εf,i ,

(4.31)

where the relationship Rsv,i Ssv,i = Ssv,i Rsv,i has been applied. It is straightforward to conﬁrm that (4.31) states the inertial relationship between two points in a translating and rotating rigid body (i.e. of the form v + ω × r), which is indeed the relationship between a formation agent and its support particle in a ﬁxed rigid-body formation. 4.3.3 Synchronization of Multiple Guided Agents By deﬁnition, a formation is assembled when all the support particles of the formation have converged to each other and propagate along the path at the same speed. They then constitute the centre of the rigid-body formation structure. Clearly, a synchronization algorithm is required in order to achieve such convergence. The synchronization essentially occurs on the one-dimensional manifold that is the path, and requires coordinating the motion of the path particles associated with the support particles. This naturally translates into synchronizing the involved path parametrization variables i , i ∈ I. Hence, a formation is assembled when: i

=

j,

˙ i = ˙ j , ∀i, j ∈ I.

(4.32)

We will employ a leader-follower type of approach, and denote the path parametrization variable associated with the leader as l . A formation can either have a virtual leader or a leader associated with a speciﬁc formation agent. The dynamic assignment of a virtual leader corresponds to: ˙l=

Ul 2 xp,l

2 + yp,l

,

(4.33)

where Ul (t) ∈ [Ul,min , ∞ ∀t ≥ 0 is the assigned formation speed along the path, and xp,l = xp ( l ). A virtual leader is just an independent path particle. In contrast, the dynamic assignment of a leader selected among the formation agents corresponds to:

68

M. Breivik, M.V. Subbotin, and T.I. Fossen

Up,l

˙l=

2 +y2 xp,l p,l

Ul cos χr,l + γl sl

=

2 +y2 xp,l p,l

,

(4.34)

since the involved support particle does not move identically along the path at all times. By deﬁnition, the leader does not care about the location of its followers; it is the prime mover that drives the whole formation forward. Consequently, the synchronization responsibility is placed entirely on the followers. The dynamic assignment of the ith follower corresponds to: Up,i

˙i= =

2 +y2 xp,i p,i

Ud,i cos χr,i + γi si 2 +y2 xp,i p,i

,

(4.35)

where the speed Ud,i of the associated support particle represents the input that can be eﬀectively assigned in a synchronization context. In this regard, consider the positive deﬁnite and radially unbounded CLF: V˜i = where:

˜i =

1 2 ˜ , 2 i

(4.36)

−

(4.37)

i

l,

and consider the situation when both the leader and the ith follower travels identically along the path, which is the relevant case as far as the pure synchronization aspect is concerned. Then, diﬀerentiate (4.36) with respect to time along the trajectories of ˜ i to obtain: V˙ ˜ i = ˜ i ˜˙ i  = ˜i and choose:

2 +y2 xp,i p,i

−

= Ul 1 − σi

˜ ,i (t)

∈ 0,

Ul 2 +y2 xp,l p,l



 Ud,i

where:

 Ud,i

˜i ˜ i2 +

˜ ,i,max ]

2 ˜ $,i



,

2 +y2 xp,i p,i 2 +y2 xp,l p,l

(4.38)

,

and σi (t) ∈ 0, 1 , ∀t ≥ 0

(4.39)

(4.40)

are synchronization variables shaping the transient coordination behavior, to ﬁnally obtain:

4 Kinematic Aspects of Guided Formation Control in 2D

σi Ul

V˙ ˜ i = −

2 +y2 xp,l p,l

= −µi

69

˜ i2 ˜ i2 +

˜ i2 2 ˜ ,i

˜ i2 +

2 ˜ ,i

,

(4.41)

which is negative deﬁnite since: µi (t) =

σi (t)Ul (t) 2 +y2 xp,l p,l

∈ 0, ∞ ∀t ≥ 0.

(4.42)

Selecting (4.39) means that the ith follower will speed up when located behind the leader, and slow down when located in front of the leader. Also, the assignment is suitably bounded, such that the pursuit will be restrained. The synchronization scheme can be analysed by considering the state vector zi = εs,i , ˜ i ∈ R2 ×R, composed of the oﬀ-track vector of the ith support particle and the path parametrization variable of its collaborating path particle. Thus, the following proposition can be stated: Proposition 4.3. The origin of zi is rendered UGAS/ULES under assumptions A.1-A.3 if (4.19) and (4.20) are employed for the ith support and path particle, respectively, while Ud,i is updated by (4.39), where ˜ ,i and σi are chosen in accordance with (4.40). Proof (Sketch). The proof can be carried out by recognizing the relevant dynamic system as a cascade (see e.g. Lor´ıa [17]), where the εs,i -subsystem perturbs the ˜ i subsystem. By stabilizing zi ∀i ∈ I, the entire formation is structurally stabilized if the formation agents simultaneously adhere to (4.29) and (4.30). Consequently, we have achieved the geometric task in question. The dynamic task is trivially satisﬁed by assigning a desired leader speed Ul satisfying assumption A.2. In total, we have now solved the planar guided formation path following problem. Furthermore, and analogous to Proposition 4.2, note that the origin of zi also can be rendered UGES. In a purely kinematic case it does not really matter whether the formation leader is virtual or a speciﬁc formation agent. However, for a real dynamic case followers can start lagging behind due to e.g. reduced thrust capability, resulting in a partly disintegrated formation unable to maintain its originally assigned structure. Consequently, consider an approach which automatically adjusts to the weakest member of the formation, i.e. a scheme where the current formation leader at any time t is selected as the hindmost formation agent: l (t)

= min i

i (t),

(4.43)

which does not aﬀect the result of Proposition 4.3. However, transientwise such a scheme can result in rapid replacements of the formation leader due to overtaking situations mainly pertaining to a diﬀerence in the cross-track errors of the involved

70

M. Breivik, M.V. Subbotin, and T.I. Fossen

agents. To alleviate such a behavior, the synchronization variable σi is suggested to be updated by: σi =

σ,i

e2i

+

2 σ,i

,

(4.44)

which satisﬁes (4.40) when σ,i (t) ∈ 0, σ,i,max ] ∀t ≥ 0. Hence, the synchronization term does not manifest itself before the cross-track error of the ith follower has reached a certain threshold value. This does not alter the asymptotic behavior of the synchronization law (4.39), but constitutes an intelligent weighing between the two more or less separate tasks of convergence towards the formation position on the one hand, and synchronization towards the formation leader on the other hand. 4.3.4 Collision Avoidance Among Multiple Guided Agents An obvious prerequisite for avoiding collision is that εf,i = εf,j ∀i = j, where i, j ∈ I. Hence, no two agents should be assigned the same position in the formation. Furthermore, we do not consider the general case where the environment contains both static and dynamic obstacles, but purely contemplate avoiding collisions among formation members, which is only relevant during the formation assembly phase. Hence, we know that every agent is an evader, and that there are no pursuers around. We propose a collision avoidance scheme for each formation agent in which the goal of formation assembly is temporarily disabled when a collision situation is detected. Furthermore, both the agent speed Uf,i and orientation angle χf,i is utilized to avoid a collision, thus constituting what can be termed the evasion space. Since the number of formation agents is ﬁnite, we know that the number of collision encounters will be ﬁnite if each agent utilize an evasion algorithm which works in ﬁnite time. This means that the structural formation stabilization result of Proposition 4.3 is retained. Hence, we have a ﬁnite number of formation agents cooperating to avoid collisions during formation assembly, each executing a collision avoidance algorithm which works in ﬁnite time for a ﬁnite amount of times, consequently aﬀecting the transient formation assembly behavior, but not the asymptotic one. However, the details of the deﬁnite collision avoidance scheme are omitted due to space limitations.

4.4 Discussion A fundamental aspect to consider when designing guidance and control systems for mechanical vehicle systems is their actuation capability. Basically, there are two relevant categories; fully actuated and underactuated vehicles. Fully actuated vehicles are independently actuated in all the required degrees-of-freedom (DOFs) simultaneously, while underactuated vehicles are not. In fact, most mechanical vehicle systems are underactuated; cars are intrinsically underactuated, as are aircraft and missiles. Hence, it is desirable to approach the formation control problem such that the proposed concept does not exclude application for underactuated vehicles. In fact, it has been the intention behind this paper to propose something which inherently lends itself to utilization for both fully actuated and underactuated vehicles. By facilitating a guided scheme, which essentially redeﬁnes the output-to-be-controlled from

4 Kinematic Aspects of Guided Formation Control in 2D

71

position and heading (which is only applicable to fully actuated vehicles) to linear velocity and heading (which is equally applicable to fully actuated and underactuated vehicles), this goal can be attained. A relevant example can be found in Breivik and Fossen [6], which considers the design of a single control law capable of maneuvering marine surface vessels at sea irrespective of whether they are fully actuated or not. The scheme is inspired by the real-life behavior of a helmsman, who employs the vessel velocity to maneuver. He does not think in terms of the vessel position (as the output to be controlled), but in his mind feeds any position error back through the velocity assignment; indirectly ensuring position control through direct velocity control. Such reasoning is also found in [2], discussing formation control aspects pertaining to (inherently underactuated) unmanned ground vehicles (UGVs), and suggesting heuristic speed and steering primitives for the assemby and maintenance of a UGV formation. By deﬁnition, a formation control scheme is not practically applicable to a very large number of agents, i.e. to swarms. This relates to the fact that the required communication and computational need becomes unsustainable. Thus, it can be concluded that the proposed guided formation control approach is suitable for a relatively small group of vehicles who need to perform high-precision formation maneuvers in order to satisfy a high-level task objective like e.g. seabed mapping, ship replenishment, or formation ﬂying.

4.5 Simulation To illustrate the dynamic performance of the proposed guided formation control scheme, a simulation is carried out in which three agents assemble and maintain a V-shaped formation while chasing a virtual leader along a sinusoidal path in the plane. Speciﬁcally, the desired path is parametrized as xp ( ) = 10 sin(0.2 ) and yp ( ) = ; the control parameters are chosen as γi = 100, e,i = ˜ ,i = 1 and σi = 0.9, ∀i ∈ {1, 2, 3}; and the speed of the virtual leader is ﬁxed at Ul = 0.25. Figure 4.4 illustrates the transient behavior of the formation agents as they assemble and maintain a V-shaped formation deﬁned by εf,1 = [−1, −1] , εf,2 = [0, 0] and εf,3 = [−1, 1] , while synchronizing with the virtual leader, as is further illustrated in Figure 4.5.

4.6 Conclusions This paper has addressed fundamental kinematic aspects of formation control. A socalled guided formation control scheme based on a guided path following approach was developed by combining guidance laws with synchronization algorithms and collision avoidance techniques. Consequently, collision-free formation assembly and path following was attained, inspired by the helmsman behavior of human pilots controlling mechanical vehicle systems. All regularly parametrized paths were rendered feasible by the approach, which also illustrated fundamental geometric constraints inherent in formation control problems. Furthermore, some relevant aspects of the guided formation control concept were discussed, while ﬁnally a numerical simulation was employed to illustrate the transient behavior of the proposed scheme.

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M. Breivik, M.V. Subbotin, and T.I. Fossen Guided formation control of 3 kinematic particles pursuing a virtual leader

10

X

I

5

0

−5

−10

0

5

10

YI

15

20

25

Fig. 4.4. The transient behavior of 3 agents utilizing guided formation path following to assemble and maintain a V-shaped formation while tracking a virtual leader. Synchronization of the path parametrization variables towards ϖl 30

25

ϖi

20

15

10

ϖ l ϖ1 ϖ 2 ϖ3

5

0

20

40

60

80

100 Time [s]

120

140

160

180

200

Fig. 4.5. The transient behavior of the involved path parametrization variables, which synchronizes with l .

4 Kinematic Aspects of Guided Formation Control in 2D

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References 1. M. Athans and P. Falb. Optimal Control: An Introduction to the Theory and Its Applications. McGraw-Hill Book Company, New York, 1966. 2. T. Balch and R. C. Arkin. Behavior-based formation control for multirobot teams. IEEE Transactions on Robotics and Automation, 14(6):926–939, 1998. 3. R. W. Beard, J. Lawton, and F. Y. Hadaegh. A feedback architecture for formation control. In Proceedings of the ACC’00, Chicago, Illinois, USA, 2000. 4. G. A. Bekey. Autonomous Robots: From Biological Inspiration to Implementation and Control. The MIT Press, Massachusetts, 2005. 5. M. Breivik and T. I. Fossen. Principles of guidance-based path following in 2D and 3D. In Proceedings of the 44th IEEE CDC, Seville, Spain, 2005. 6. M. Breivik and T. I. Fossen. A uniﬁed concept for controlling a marine surface vessel through the entire speed envelope. In Proceedings of the ISIC-MED’05, Limassol, Cyprus, 2005. 7. J. P. Desai, J. P. Ostrowski, and V. Kumar. Modeling and control of formations of nonholonomic mobile robots. IEEE Transactions on Robotics and Automation, 17(6):905– 908, 2001. 8. M. Egerstedt and X. Hu. Formation constrained multi-agent control. IEEE Transactions on Robotics and Automation, 17(6):947–950, 2001. 9. P. Encarna¸ca ˜o and A. Pascoal. Combined trajectory tracking and path following: An application to the coordinated control of autonomous marine craft. In Proceedings of the 40th IEEE CDC, Orlando, Florida, USA, 2001. 10. V. Gazi. Swarm aggregations using artiﬁcial potentials and sliding-mode control. IEEE Transactions on Robotics, 21(6):1208–1214, 2005. 11. I.-A. F. Ihle, J. Jouﬀroy, and T. I. Fossen. Formation control of marine surface craft using lagrange multipliers. In Proceedings of the 44th IEEE CDC, Seville, Spain, 2005. 12. A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6):988–1001, 2002. 13. V. Kumar, N. E. Leonard, and A. S. Morse, editors. Cooperative Control. Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin Heidelberg, 2005. 14. L. Lapierre, D. Soetanto, and A. Pascoal. Coordinated motion control of marine robots. In Proceedings of the 6th IFAC MCMC, Girona, Spain, 2003. 15. N. E. Leonard and E. Fiorelli. Virtual leaders, artiﬁcial potentials and coordinated control of groups. In Proceedings of the 40th IEEE CDC, Orlando, Florida, USA, 2001. 16. M. A. Lewis and K.-H. Tan. High precision formation control of mobile robots using virtual structures. Autonomous Robots, 4:387–403, 1997. 17. A. Lor´ıa. Cascaded nonlinear time-varying systems: Analysis and design. Lecture notes, Minicourse at ECTS, France, 2004. 18. R. Olfati-Saber and R. M. Murray. Distributed structural stabilization and tracking for formations of dynamic multi-agents. In Proceedings of the 41st IEEE CDC, Las Vegas, Nevada, USA, 2002. 19. F. A. Papoulias. Bifurcation analysis of line of sight vehicle guidance using sliding modes. International Journal of Bifurcation and Chaos, 1(4):849–865, 1991.

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20. W. Ren and R. W. Beard. Formation feedback control for multiple spacecraft via virtual structures. IEE Control Theory and Applications, 151(3):357–368, 2004. 21. C. W. Reynolds. Flocks, herds, and schools: A distributed behavioral model. Computer Graphics, 21(4):25–34, 1987. 22. C. Samson. Path following and time-varying feedback stabilization of a wheeled mobile robot. In Proceedings of the ICARCV’92, Singapore, 1992. 23. R. Skjetne. The Maneuvering Problem. PhD thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2005. 24. R. Skjetne, I.-A. F. Ihle, and T. I. Fossen. Formation control by synchronizing multiple maneuvering systems. In Proceedings of the 6th IFAC MCMC, Girona, Spain, 2003. 25. R. Skjetne, S. Moi, and T. I. Fossen. Nonlinear formation control of marine craft. In Proceedings of the 41st IEEE CDC, Las Vegas, Nevada, USA, 2002. 26. P. Tabuada, G. J. Pappas, and P. Lima. Motion feasibility of multi-agent formations. IEEE Transactions on Robotics, 21(3):387–392, 2005. 27. H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Stable ﬂocking of mobile agents, part i: Fixed topology. In Proceedings of the 42nd IEEE CDC, Maui, Hawaii, USA, 2003. 28. H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Stable ﬂocking of mobile agents, part ii: Dynamic topology. In Proceedings of the 42nd IEEE CDC, Maui, Hawaii, USA, 2003. 29. A. Teel, E. Panteley, and A. Lor´ıa. Integral characterization of uniform asymptotic and exponential stability with applications. Mathematics of Control, Signals, and Systems, 15:177–201, 2002. 30. S. Wolfram. A New Kind of Science. Wolfram Media, 2002.

5 ISS-Based Robust Leader/Follower Trailing Control X. Chen and A. Serrani Department of Electrical and Computer Engineering The Ohio State University 2015 Neil Ave, 205 Dreese Lab Columbus, OH 43221- USA [email protected], [email protected] Summary. The leader/follower scheme is one of the most important building blocks for formation control of autonomous mobile agents. Recently, the concept of input-to-state stability (ISS) has been proposed as a tool to relate the inﬂuence of the motion of the leader to the overall formation error in a leader/follower conﬁguration. The ISS approach allows great ﬂexibility in the formation topology, and yields computable gains relating the formation error to external inputs and disturbances. However, its application in this speciﬁc context requires some attention, due to the peculiar characteristics of the vehicular formation dynamics. In this paper, the problem of letting followers to trail their leader at desired relative position in a planar formation is cast into a stabilization problem by resorting to an equivalent non-holonomic kinematic model. The proposed solution, based on a local version of ISS originally given by Teel, employs saturated controls to enforce ISS of the dynamics, without resorting to input/output linearization techniques, and thus avoiding the need to deal with critically stable or locally asymptotically stable zero-dynamics. The aforementioned techniques- saturated control and ISS with restrictions- are key to robust control design which addresses leader-follower formation implicitly. Furthermore, the analysis also shows that the proposed controller not only can deal with a class of small measurement errors but also can accommodate small control input delays. Simulation results on an illustrative example are presented and discussed.

5.1 Introduction Coordination and control of groups of autonomous agents has been the object of intense research eﬀorts in the last few years. Existing methodologies include Lyapunovbased techniques, the use of artiﬁcial potentials, logic-based control, vision-based, and distributed optimization methods (see [2, 3, 5, 7, 6, 8, 9, 10], to name just a few contributions). In this paper, we are primarily interested in leader-follower trailing control which is the basic building block for several formation control setup. The desired positions of followers are deﬁned relative to the actual state of a physical leader. The group reference trajectory is simply deﬁned by the leader motion, while the stability of the internal shape of the formation can in principle be guaranteed by the stability of individual vehicles. Although a leader-follower formation architecture

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 75–91, 2006. © Springer-Verlag Berlin Heidelberg 2006

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relies heavily on the leader path planning to achieve its goal, it may be desirable for security purposes. In coordinating the motion of a group of vehicles it is important to relate the eﬀect of external stimuli (such as commanded inputs and disturbances) to the shape of the formation, and to quantify the propagation of the position error of each single agent along the interconnection. The tools of string stability [13] and mesh stability [11], for instance, aim at giving conditions under which the eﬀect of disturbances is attenuated while propagating through the formation. Recently, Tanner et al. [14] have proposed a notion of leader-to-formation stability that, building upon the concept of input-to-state stability [12], relates the inﬂuence of the motion of the leader to the formation error. The particular feature that makes the ISS concept appealing in the analysis of formation stability is the invariance of the ISS property under cascade interconnection. This allows bounds on the error as a function of the exogenous inputs to be computed explicitly, and the propagation of the error to be quantiﬁed via nonlinear gains. The aim of the paper is to derive a controller of decentralized type that ensures precise tracking of a desired formation conﬁguration, while guaranteeing robustness with respect to external disturbances and small measurement errors. By resorting to a non-holonomic vehicle model, a leader/follower trailing problem is reduced to a robust stabilization one. Building upon our previous work [1], the proposed solution is based on the version of ISS “with restrictions”, originally developed by Teel [15], and employs saturated controls to enforce ISS of each follower dynamics. The relevant feature of the design is that it is not based on input-output linearization (as the one adopted in [14]) and thus avoids dealing with the presence of a zero-dynamics. Bounds on appropriate norms of formation errors as function of external inputs can also be derived explicitly, in the spirit of the original work [14]. This paper is organized as follows: The non-holonomic model is introduced in Section 5.2, and the saturated controller is proposed in Section 5.3. Proofs are given in Section 5.4, followed by a brief analysis for the case of ﬁxed time delay in Section 5.5. An illustrative simulation is given in Section 5.6, while Section 5.7 oﬀers some conclusions.

5.2 Leader/Follower Trailing Model Consider a leader/follower formation of N autonomous vehicles on a plane. A kinematic model for the motion of the i-th agent relative to its designated leader (labeled as agent 0) is given as follows (see [3]) ρ˙ i = v i cos(ψ i + θ0i ) − v 0 (t) cos(ψ i ) ψ˙ i = ρ1 v 0 (t) sin(ψ i ) − v i sin(ψ i + θ0i ) − ρi ω 0 (t) θ˙0i = ω 0 (t) − ω i , i = 1, . . . , N

(5.1)

where v 0 ∈ R>0 and ω 0 ∈ R are respectively the translational and angular velocities of the leader, ρi is the relative distance between the i-th follower and its leader, ψ i is the angle between the direction of the leader velocity and that of the line segment ρi . The angle θ0i is the orientation of the follower relative to the leader’s moving direction, that is θ0i = φ0 − φi , where φi is the orientation of the i-th agent with

5 ISS-Based Robust Leader/Follower Trailing Control

ψi

ψ i,d

Leader Trajectory

77

Leader Trajectory

φ0

Leader Leader

ρi

ρi,d

θ0i

v i,d x

y

θi

vi

φi Follower Position

(a)

x-axis

y-axis

Virtual Follower at Follower Actual Position Desired Position

(b)

Fig. 5.1. Coordinate frames and notations for non-holonomic kinematics model

respect to a ﬁxed inertial frame. The coordinate frames and notations are shown in Fig. 5.1a. The follower translational and angular velocities (v i , ω i ) are regarded as the control inputs to (5.1). The control task considered in this paper is to let each agent to follow its leader in a desired constant conﬁguration, that is, to regulate each output (ρi , ψ i ) to the setpoint (ρi,d , ψ i,d ). In order to maintain a desired relative position, a corresponding desired control input (v i,d , ω i,d ) must be provided to each follower. By letting ρi ≡ ρi,d and ψ i ≡ ψ i,d in (5.1), the desired virtual velocity input is computed as follows v i,d (t) =

(v 0 (t))2 − 2v 0 (t) sin(ψ i,d )ρi,d ω 0 (t) + (ρi,d ω 0 (t))2 , ω i,d (t) = ω 0 (t) − θ˙0i,d (t), where the desired orientation is given by θ0i,d (t) = arctan[tan(ψ i,d ) −

ρi,d ω 0 (t) ] − ψ i,d . cos ψ i,d v 0 (t)

Note that if the ratio ω 0 /v 0 is constant, then θ0i,d is constant as well, and ω i,d = ω 0 . Dubins [4] has proved that if a vehicle moves forward with a given velocity proﬁle and a bounded turning radius, the optimal shortest paths joining two arbitrary conﬁgurations can be built with at most three pieces of either type ‘C’ (arcs of circle with minimal radius), or type ‘S’ (straight line segments). Suppose that the leader is moving according to some optimal path planning algorithm, then based on the result of [4], it is convenient to consider the constant velocity (v 0 , 0) corresponding to straight line, or (v 0 , ω 0 ) corresponding to arcs of circle as a steady state trajectory for the motion of the leader, and regard a particular time-varying motion (v 0 (t), ω 0 (t)) as a deviation from that. Note that the analysis is not necessarily restricted to these two cases, but they are simply regarded as a “preferred” motion. Let a virtual agent (the dotted one depicted in Fig. 5.1b) be ﬁxed at the exact desired follower position relative to its formation leader. The x-axis of a moving

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frame is aligned along the direction of the follower motion (y-axis perpendicular to x-axis). Since we want to control the follower to converge to and maintain the desired leader/follower formation setpoint, it’s more appropriate to set the origin of the moving frame at the center of mass of the virtual agent than that of the actual follower. As a result, the kinematics of the follower relative to the virtual agent is written as x˙ i = v i − v d cos θi + y i ω i y˙ i = −v d sin θi − xi ω i (5.2) ˙θi = ω d − ω i where (xi ,y i ), as shown in Fig. 5.1, is the actual position of the follower in the moving frame, and θi = φi − φ0 − θ0i,d is the angle between the actual moving direction of the follower and that of the virtual agent. In this setup, a leader/follower trailing problem is reduced to a regulation problem, that is to regulate (xi , y i , θi ) to the origin for the non-holonomic model (5.2). The advantage of this setup lies in its simplicity; the price to pay for this simpliﬁcation, however, is that the computation of θi depends on the leader velocity (v 0 , ω 0 ). For ease of notation and without loss of generality, in what follows we will drop the index i from equations, with the understanding that we are referring to the kinematics of the i-th follower relative to its virtual agent.

5.3 Controller Design Intrinsic to our formulation of the problem, we assume that the angular and translational velocities (v d , ω d ) of virtual agent are not known exactly, and that all measured state variables are aﬀected by small sensor errors. This situation arises because the state variables are calculated based on the position of the virtual agent, and this computation is aﬀected by the uncertainty on the leader velocity. In particular, θ is more vulnerable to this type of indirect measurement errors. Without loss of generality, we assume that available information for feedback control are given by ˆ d = ωd + ω ˜ d, vˆd = v d + v˜d ω x ˆ = x + x˜,

yˆ = y + y˜,

θˆ = θ + θ˜

(5.3)

where the notation (∼) refers to the errors between the true values of the states and their estimates. Before proceeding further, we make the following assumptions: ˜ v˜d , ω Assumption 1 The signals x˜, y˜, θ, ˜ d are continuous and bounded, and their d d ˜ d x), dt (˜ y ), dt (θ) exist almost everywhere and are essentially bounded. derivatives dt (˜ If the bounds of the errors are suﬃciently small, this assumption implies that we do have a relatively good estimation of the virtual agent velocity, and the measurement of the states have not deviated from their true values too much. The assumption on bounded derivatives means that the errors change slowly, excluding high frequency noise from our consideration. However, they do include certain classes of situations, for example, the error introduced by sensor bias.

5 ISS-Based Robust Leader/Follower Trailing Control

79

Assumption 2 The desired translational and angular velocities v d and ω d are d d d bounded. More speciﬁcally, deﬁne ωM := max{|ω d |}, vm := min{v d } , and vM := max{v d }. d The lower bound vm > 0 is needed for well-posedness of the non-holonomic model, d while the upper bound vM > 0 is due to physical restriction; ﬁnally, the upper bound d ωM > 0 not only is due to physical restriction, but also is needed for bearing feasible desired formation while turning, without causing the follower to spin continuously ˆd about the vertical axis. Combining assumptions 1 with 2, it follows that vˆd and ω are also bounded. For the sake of simplicity, with a slight abuse of notation, we use d d d ωM , vm and vM as bounds for vˆd and ω ˆ d as well. We ﬁrst introduce the preliminary input transformation

v = vˆd cos θˆ − yˆω + u1 ,

ω=ω ˆ d − u2 ,

(5.4)

where (u1 , u2 ) is the new control input. The closed-loop system (5.4) and (5.2) can ˆ as follows be written in terms of the measured state (ˆ x, yˆ, θ) x ˆ˙ = u1 + d1 ω d − u2 ) + d2 yˆ˙ = −v d sin θˆ − xˆ(ˆ ˙ θˆ = u2 + d3 ,

(5.5)

where the disturbance input d = (d1 , d2 , d3 ) is deﬁned as v d cos θˆ − v d cos θ) − y˜ω + x˜˙ d1 = (ˆ d2 = v d (sin θˆ − sin θ) + x˜ω + y˜˙ ω d + θ˜˙ d = −˜ 3

A bounded control input pair (u1 , u2 ) is employed to stabilize (5.5) to obey realistic physical limits. This choice, combined with Assumptions 1 and 2, allows us to regard d1 , d2 and d3 as bounded disturbance inputs to the closed-loop system. Next, we adopt the following change of coordinate ˆ ζ = tan θ,

ζ ∗ = L2 σ(

K2 yˆ), L2

ζ˜ = ζ − ζ ∗

for some constants L2 > 0 and K2 > 0 to be chosen later, while σ(·) is a continuously diﬀerentiable saturation function deﬁned in Appendix. The design of the proposed robust stabilizing controller is completed by choosing u1 = −L1 σ(

K1 x ˆ), L1

u2 = −L3 σ(

K3 ˜ ζ) , L3

(5.6)

where the constants K1 , K3 , L1 , and L3 are to be chosen. Substituting (5.6) into (5.5), we obtain the following expression of the closed-loop system 1 ˆ) + d1 x ˆ˙ = −L1 σ( K L1 x

yˆ˙ = −v d

2 ˆ) ζ˜ + L2 σ( K L2 y 2 1 + (ζ˜ + L2 σ( K ˆ)2 ) L2 y

3 ˜ −x ˆ(ˆ ω d + L3 σ( K L3 ζ)) + d2

˙ 2 2 3 ˜ ˆ)2 )][−L3 σ( K ˆ) . ˆ˙ σ( ˙ K ζ˜ = [1 + (ζ˜ + L2 σ( K L2 y L3 ζ) + d3 ] + K2 y L2 y

(5.7)

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The control problem is then cast as that of ﬁnding appropriate values of the saturation levels Li and gains Ki , 1 ≤ i ≤ 3 to obtain input-to-state stability of the closed-loop system (5.7) with respect to the bounded disturbance d.

5.4 Main Result The main result of the paper is stated as follows: Theorem 5.1. The closed-loop system (5.7) is rendered ISS with no restriction on ˜ arbitrary but ﬁxed restrictions ∆1 , ∆3 on d1 , d3 , and nonzero x, yˆ, ζ), initial states (ˆ d restriction ∆2 < vm on d2 respectively, if the saturation levels Li and the gains Ki , 1 ≤ i ≤ 3, are chosen to satisfy the following inequalities: L1 > ∆1 ,

L3 >

d K3 > 20K2 vM ,

d d + ∆2 + ωM L1 /K1 ) + ∆3 2K2 (vM , 1 − 2K2 L1 /K1 d + L3 ∆2 L 1 ωM L2 − 1 + d , > d K1 vm vm 1 + (L2 + 1)2

(5.8)

d v d + ∆2 + L1 /K1 (ωM + L3 ) ∆3 + 2K2 M < 1. K3 K3

Remarks: It is useful to describe how these inequalities can be satisﬁed, and the order in which the selection of the gains and the saturation levels proceeds. The restrictions ∆1 > 0 and ∆3 > 0 can take any ﬁnite arbitrarily large (but ﬁxed) value; however, the bound ∆2 > 0 is required to be strictly smaller than the leader minimal speed d vm , in order for a feasible selection for L2 to exist. First, the saturation level L1 must be chosen to be greater than ∆1 , while K1 is regarded as a free parameter. Then, a value for the gain K2 is ﬁxed, according to the following considerations: a small value for K2 is helpful to reduce the demand on the choice of L3 , while a large value of K2 increases the speed of convergence of y-subsystem to the origin. A tradeoﬀ must be taken in practice. The gain K1 is selected to be much larger than L1 to allow a possibly small value L2 to meet the requirement L2 − 1 1 + (L2 +

1)2

>

d L 1 ωM ∆2 + L3 + d d K1 vm vm

such that the follower remains close to the desired position during transient. The next step is to choose a suitable value for L3 according to the inequality: L3 >

d d + ∆2 + ωM L1 /K1 ) + ∆3 2K2 (vM . 1 − 2K2 L1 /K1

Finally, by increasing the value of K3 to be suﬃciently large, all remaining inequalities can be satisﬁed. Before proceeding to prove Theorem 5.1, we ﬁrst state some lemmas that are instrumental in constructing its proof.

5 ISS-Based Robust Leader/Follower Trailing Control

81

Lemma 5.1. Assume that the trajectory yˆ(t) of (5.7) exists for all t ≥ 0. Then, the x ˆ-subsystem is ISS with no restriction on initial state x ˆ(0) and nonzero restriction ∆1 on d1 , if L1 > ∆1 . Moreover, the asymptotic norm of x ˆ(t) can be made arbitrarily small by increasing the gain K1 . In particular, given ε > 0 and K1 ≥ Lε1 , for any R > 0 there exists a time T0 = T0 (ε, R) such that for any initial condition |ˆ x(0)| ≤ R ˆ(t) and for any disturbance d1 such that d1 ∞ < ∆1 , the corresponding trajectory x satisﬁes |ˆ x(t)| ≤ ε, ∀ t ≥ T0 . Proof. Choose the candidate Lyapunov function V1 = 21 x ˆ2 , and calculate its derivative along (5.7) to obtain K1 K1 |ˆ x|) + x ˆd1 ≤ −|ˆ x|(L1 σ( |ˆ x|) − |d1 |). x|L1 σ( V˙ 1 = −|ˆ L1 L1 1 When the operator σ( K x|) is saturated, the above inequality yields L1 |ˆ

V˙ 1 ≤ −|ˆ x|(L1 − |d1 |) , and the condition L1 > ∆1 guarantees that V˙ 1 is negative deﬁnite. On the other 1 x| < 1, then hand, when K L1 |ˆ V˙ 1 ≤ −|ˆ x|(K1 |ˆ x| − |d1 |) , and thus V˙ 1 < 0 for |ˆ x| > the form

1 K1 |d1 |.

This relation leads to an asymptotic bound [15] of x ˆ

a

≤

1 d1 K1

a

,

with linear gain which can be made arbitrarily small by increasing the value of the x(0)| ≤ R, the trajectory xˆ(t) is captured by the parameter K1 . As a result, for any |ˆ L1 L1 , ] ⊂ [−ε, ε] for K1 ≥ Lε1 in at most T0 time units, where compact set [− K 1 K1 T0 =

K1 L2 (R2 − 12 ) 2L1 (L1 − ∆1 ) K1

Lemma 5.2. The closed-loop system (5.7) is forward complete for L1 > ∆1 . Proof. Let [0, Tmax) be the maximal interval of existence for the solutions of (5.7). From the proof of Lemma 5.1, for any given x ˆ(0), it follows that |x(t)| ≤ L1 ˙ } for all t ∈ [0, T ). As a result, | y ˆ (t)| is bounded from above on max{|ˆ x(0)|, K max 1 t ˙ 2 ˙ K [0, Tmax ). Since |ˆ y(t)| ≤ |ˆ y(0)|+ 0 |yˆ(s)|ds, the trajectories y(t) and K2 yˆ˙ (t)σ( ˆ(t)) L2 y ˜ are bounded over [0, Tmax ). Finally, boundedness of ζ(t) over [0, Tmax ) is easily proved using the diﬀeomorphism ζ˜ → θˆ = arctan(ζ˜ + ζ ∗ ), and noting that in the θˆ coor˜ dinates the dynamics is globally Lipschitz. Since the trajectory (ˆ is x(t), yˆ(t), ζ(t)) bounded over its maximal interval of existence, it follows that Tmax = ∞.

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Lemma 5.3. Fix K2 > 0, and ﬁx K1 and L1 as in Lemma 5.1. For any given d restriction ∆2 < vm on d2 , ∆3 on d3 , and any given constant 0 < ζ¯ < 1, there exist ˜ is captured by the set |ζ| ˜ ≤ ζ¯ for all L3 > 0 and K3 > 0 such that the trajectory ζ(t) K3 ≥ K3 . Proof. Since the closed-loop system is forward complete, without loss of generality, L1 we can assume that t ≥ T0 , and thus |ˆ . Then, from the second equation x| ≤ K 1 in (5.7) it follows that |yˆ˙ (t)| ≤ vˆd +

L1 d (ω + L3 ) , K1 M

∀ t ≥ T0 .

The ζ˜ subsystem can be rewritten as K3 ˜ ˙ ζ) + d3 ] + δ , ζ˜ = −(1 + ζ 2 )[L3 σ( L3

(5.9)

2 ˙ K ˆ)yˆ˙ , after t ≥ T0 , satisﬁes where the bounded perturbation δ = −K2 σ( L2 y

|δ| < 2K2 (v d + d2 +

L1 d L1 d d (ω + L3 )) < 2K2 (vM + ∆2 + (ω + L3 )). K1 M K1 M

2 Note that δ vanishes when σ( K ˆ) is saturated. The derivative of the ISS-Lyapunov L2 y function candidate V3 = 12 ζ˜2 along the solution of (5.9) reads as

K3 ˜ K3 ˜ ˜ ˜ + |ζ||δ| ˜ . ζ) − d3 ) + δ] ≤ −(1 + ζ 2 )[L3 σ( V˙ 3 = ζ[−(1 + ζ 2 )(L3 σ( |ζ|) − ∆3 ]|ζ| L3 L3 ˜ ≥ For all |ζ|

L3 K3 ,

3 ˜ σ( K L3 ζ) is saturated, and thus

L1 d d ˜ V˙ 3 ≤ −|ζ|[(1 + ∆2 + + ζ 2 )(L3 − ∆3 ) − 2K2 (vM (ω + L3 ))] < 0 K1 M as long as

d d + ∆2 + ωM L1 /K1 ) + ∆3 2K2 (vM . 1 − 2K2 L1 /K1 ˜ < L3 , we have On the other hand, for all |ζ| K3

L3 >

(5.10)

L1 d d ˜ 3 |ζ| ˜ − ∆3 ) + |ζ|2K ˜ V˙ 3 ≤ −|ζ|(K (ω + L3 )) , 2 (vM + ∆2 + K1 M and thus the asymptotic bound ζ˜

a

≤

d ∆3 v d + ∆2 + L1 /K1 (ωM + L3 ) + 2K2 M K3 K3

is obtained. By increasing the value of K3 , we can make the right hand side arbitrarily ˜ small. Therefore, similarly to Lemma 5.1, there exists T1 ≥ T0 such that |ζ(t+T 1 )| ≤ ¯ ζ for all t ≥ 0 if the gain K3 is chosen such that d v d + ∆2 + L1 /K1 (ωM + L3 ) ∆3 ¯ + 2K2 M < ζ. K3 K3

5 ISS-Based Robust Leader/Follower Trailing Control

83

x˙ y˙ ζ˜˙ Fig. 5.2. System structure for ISS small gain interconnection

Lemma 5.4. There exists a choice of K2 , K3 , L2 , L3 such that the yˆ − ζ˜ subsystem in 5.7 is a small-gain interconnection, and the loop is an ISS system, with no restriction on the initial condition for yˆ(t), restriction ζ¯ < 1 of the initial condition ˜ ˆ, and nonzero restrictions ∆2 , ∆3 for ζ(t), nonzero restriction L1 /K1 for the input x on d2 , d3 , respectively. ˜ Proof. Without loss of generality, let |ˆ x| ≤ L1 /K1 , and consider the (ˆ y , ζ)-dynamics as the loop interconnection of two ISS subsystem for t ≥ T1 (see Fig. 5.2). The derivative of V2 = 12 yˆ2 along the solution of (5.7) reads as V˙ 2 = yˆ[−ˆ vd

2 ˆ) ζ˜ + L2 σ( K L2 y

1 + ζ2

+ d2 − x ˆ(ˆ ω d + L3 σ(

K3 ˜ ζ))] L3

2 ˆ) is saturated, the parameters L1 , L2 , L3 , K1 must To make V˙ 2 negative when σ( K L2 y satisfy the inequality

L2 − 1 1 + (L2 + 1)2

>

d + L3 ∆2 L 1 ωM + d . d K1 vm vm

The above inequality is always feasible by tuning L2 , provided that ∆2 is smaller d ˆ, ζ˜ than vm . For the y-subsystem, the asymptotic gains with respect to the inputs x and d2 are computed by resorting to the ISS-Lyapunov function V2 . In particular, L1 ˜ , ζ < 1, and |d2 | ≤ ∆2 hold, the inequality assuming that the restrictions |ˆ x| < K 1 ˜ |ζ|

K2 yˆ2 V˙ 2 < −ˆ vd + |ˆ y |d2 + |ˆ y |ˆ vd 1 + ζ2

1 + ζ2

d + |ˆ y |(ωM + L3 )|ˆ x|

implies yˆ

a

≤ 3 max

1 ˜ 2 ζ a, d2 d K2 K2 vm

a

,

d + L3 L2 + 2 ωM x ˆ d K2 vm

a

.

It is worth mentioning that the above asymptotic gains are all linear. Meanwhile, ˜ the ζ-subsystem can be rewritten as

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X. Chen and A. Serrani

K2 yˆ + ζ˜ K2 K3 ˜ K3 ˜ ˙ ˙ yˆ)[ˆ vd + d2 + (ˆ ω d + L3 σ( ζ˜ = −(1 + ζ 2 )[L3 σ( ζ) + d3 ] + K2 σ( ζ))ˆ x] L3 L2 L3 1 + ζ2 which is ISS with Lyapunov function V3 if (5.10) holds. Keeping in mind that the value of ζ˜ is now within the limits of the saturation operator, and recalling the properties of the saturation functions given in Appendix, it it seen that the condition d 3 ˜ + L3 ||ˆ x|, implies that an asymptotic bound of the form x| ≤ |ωM |(ˆ ω d + L3 σ( K L3 ζ))ˆ ˜ ζ

a

≤ 3 max{

d 2K22 vM 4K2 y a, d2 a , d d K3 − 2K2 vM K3 − 2K2 vM d 4 ωM + L 3 d3 a , x a} d d K3 − 2K2 vM K3 − 2K2 vM

d holds for ζ˜ if K3 − 2K2 vM > 0. The loop interconnection of the two ISS subsystems ˜ y and ζ results in an ISS system with respect to the common inputs xˆ and d2 , d3 if a simple contraction between the respective gains hold [15]. In this case, the small-gain condition reads as d 3 2K22 vM ·3 < 1, d K2 K3 − 2K2 vM

which can always be enforced choosing the value of K3 large enough such that d K3 > 20K2 vM .

˜ can be expressed by the asymptotic bound The ISS property of the subsystem (ˆ y , ζ) max{ ˜ ζ a , yˆ a } ≤ max{γ23 · x ˆ

a

, γ23d2 · d2

a

, γ23d3 · d3

a },

where 1 4 4 · }, , d d K2 K3 − 2K2 vM K3 − 2K2 vM 2 d 4 4K2 2K2 vM 1 1 , , }, · = 9 max{ , d d d d K2 K2 vm K3 − 2K2 vM K3 − 2K2 vM K3 − 2K2 vM

γ23d3 = 9 max{ γ23d2

γ23 = 9 max{

d d d d + L3 (ωM + L3 + L3 )2 ωM ωM 2K2 vM , , }. d d d )K K3 − 2K2 vM (K3 − 2K2 vM )K2 (K3 − 2K2 vM 2

Now, we are in the position to prove Theorem 5.1. Proof of Theorem 5.1 Lemma 5.1 proves that the decoupled driving subsystem x ˆ in Fig. 5.2 ISS with respect to disturbance input d1 . Lemma 5.3 and Lemma 5.4 ˜ is ISS with respect to inputs x show that the driven system (ˆ y, ζ) ˆ, d2 and d3 after ˆ(t) to satisfy |ˆ x(t)| ≤ ε. some ﬁnite time T1 which is required for the trajectory x Following [15], the cascade-connected system preserves ISS property assuming that all restrictions on bounded disturbances are satisﬁed and the gains satisfy (5.8). ˜ = (0, 0, 0) is 1-1 mapping preserving the origin of Because the equilibrium (ˆ x, yˆ, ζ)

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ˆ our proposed control stabilizes the original system as the original system (ˆ x, yˆ, θ), ˜ exists for all t > 0, and so does ζ(t). This well. Furthermore, the trajectory of ζ(t) ˆ will not exceed the invariant set (− π , π ) guarantees that the internal dynamic θ(t) 2 2 even during transient. The merit of all previous proofs is that we only need to focus on the system asymptotic behavior, time while pay relatively little attention to the initial transient. However, the restrictions for the magnitude of disturbances corresponding to the given choice of the saturation levels guarantee that the closed-loop system is well behaved even during transients. It should be noted that these conditions may be overly conservative. In practice, it has been observed in simulation that some of the constraints can be relaxed without compromising the validity of the results for a large envelope of disturbance inputs. Since the saturation levels and gains depend essentially on the disturbance bounds, as well as the actual velocity of the virtual agent, in an actual implementation the above bounds should be regarded as guidelines for the choice of the design parameters. The main result states that the proposed controller can achieve and maintain predeﬁned leader/follower trailing speciﬁcation robustly with respect to the errors in the measured state variables and in the estimated velocity of the virtual agent.

5.5 Small Input Time Delay Analysis In practice, the angular and translational velocities of virtual agent as well as all state variables are aﬀected by delays. For example, transmission delays, computation delay and execution delay in the leader-follower trailing control problem can all be investigated in the framework of delay at the input. Here we only analyze brieﬂy the simplest case, i.e. small ﬁxed delay τ , as an application of ISS-design using saturated control, and defer a thorough analysis of the general case to a forthcoming paper. To accommodate the presence of small input delays into the present framework, we let the available information listed in 5.3 be deﬁned as vˆd (t) = v d (t − τ ),

ω ˆ d (t) = ω d (t − τ )

x ˆ(t) = x(t − τ ),

yˆ(t) = y(t − τ ),

ˆ = θ(t − τ ). θ(t)

(5.11)

K1 K1 K1 1 When both x(t) > K L1 and x(t − τ ) > L1 , or both x(t) < − L1 and x(t − τ ) < − L1 , then K1 K1 L1 σ( x(t − τ )) = L1 sgn(x(t)) = L1 σ( x(t)). L1 L1 This means that the stabilizing term of the x ˆ subsystem behaves as if there is no time delay when |x(t)| and |x(t − τ )| are suﬃciently far away from the origin. This is one of the most important advantage in using saturation operator in systems aﬀected by a small delay. Next, we need to verify that Assumption 1 and 2 are still valid. According to the mean-value theorem, the error x ˜(t) satisﬁes

ˆ(t) − x(t) = −τ xˆ˙ (ξ) x ˜(t) = x for some ξ ∈ (t − τ, t), while the derivative x ˜˙ (t) = x(t ˙ − τ ) − x(t) ˙ exists almost ˜ is bounded, x everywhere. If the state (ˆ x, yˆ, ζ) ˜(t) and |x ˜˙ (t)| are bounded and small,

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0

7 6π

5 6π

L

L

3

1

4

1 6π

4

L

2

5

L

3

L

6

Desired Formation

2 1 2L

3π

7 6π

L

Initial Position

L

1 2L 5 6π

1

3L

2 3π

04

2 3π

L

L 4 3π

6

5

Fig. 5.3. Initial Position and Desired Formation

˜ and if the delay τ is suﬃciently small. Similar observations hold for both y and ζ, d Assumption 1 and 2 retain their validity. The crucial restriction ∆2 < vm in Theorem 5.1 can be translated into a requirement on the maximum delay that can be tolerated by the control system. This level of robustness with respect to small time delay is consistent with the result given in the paper [16] on connections between Razumikhlin- type Theorems and the ISS- nonlinear small gain theorem.

5.6 Illustrative Simulation To illustrate the proposed methodology, we present in this section simulation results concerning an example of robust formation control, which is the main motivation for our leader/follower trailing design. Consider seven UAVs evolving on the plane. The ﬁrst group is formed by agents 2 (the leader) 4, 5 and 6, as shown in Fig. 5.3(a). This group is required to join a second group, given by agents 0 (the leader), 1 and 3, by assuming agent 0 as the overall leader, as shown in Fig. 5.3(b). The parameters of the initial formation are given in Table 5.1 while the desired formation parameters are given in Table 5.2. All group formations are deﬁned via the pairwise leader/follower paradigm. In this simulation example, the overall leader 0 of the formation follows the path shown in Fig. 5.4 (a). The task is to converge and maintain a stable formation along the path. Each follower is equipped with the same controller with the gains and saturation levels selected as in Table 5.3. The velocity of the overall leader of the formation is 14[m/s]. To consider a more realistic scenario, slow and small sensor errors has been added to the measurements of the state vector fed to the controller as follows w1 = 0.2 sin(0.1t),

w1 = 0.2 sin(0.1t),

w3 =

π sin(0.1t). 360

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Table 5.1. Parameters of the desired formation, L = 6 [m]. Follower 1

2

3

√

ρid

L

ψdi

2 π 3

π

Leader

0

0

1 L 2

4 5 6

L L L L 4 π 65 π 3

0

π

7 π 6

2 2 2

Table 5.2. Parameters of the initial position, L = 6 [m]. Follower 1 ρi0 ψ0i θ0i

2

3

4

5

6

L 3L L L

1 L 2

L

5 π 6

π

7 π 32 π 6

π

4 π 3

0

1 π 6

0

0

0

0

Table 5.3. Controller gains and saturation levels K1 = 180

K2 = 2 K3 = 140

L1 = 25

L2 = 10 L3 = 20

The results of the simulation show indeed robust behaviors in the presence of the given disturbances. The shape of the formation at four diﬀerent time instants are given in Figure 5.4 (b-e). Subplot (b) shows a snapshot of the initial position at point A, while Subplot (e) shows a snapshot of the ﬁnal state at point D, from which we can see that the formation does converge to desired one as prescribed. Subplot (c) shows a snapshot of the transient behavior at point B, during which agent 2 approaches a small neighborhood along the x direction, while the error with respect to the y- and θ- variables remains relatively large. This kind of two-time scale behavior is typical of systems stabilized by means of saturated controls. Subplot (d) shows a snapshot of the transient at point C, at a point in which, despite the fact that the overall leader 0 is making a large turn, the formation remains well-behaved. Fig. 5.5 shows the state trajectories of agenta 2 and 5 during the initial transient. The state of agent 2 clearly exhibits the convergence scenario outlined in Section 5.4. The state trajectories of agent 5 experience a two-stage transient. First, its states converge fast as the initial condition is close to origin. Then, around the time interval t ∈ [0.4, 0.5] [s], the trajectory (ˆ y 2 (t), ζ˜2 (t)) converges to the origin, while the virtual agent 5 changes its position corresponding to the internal dynamic of agent 2, causing (ˆ x5 (t), yˆ5 (t), ζ˜5 (t)) to move away from the origin initially, and then converging again to a neighborhood of the origin.

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(a) Planer trajectory

600

Leader st

500

1 Follower nd

2

y − position [m]

400

D

Follower

rd

1 Follower

300

4th Follower

200

6th Follower

5th Follower

A B

100

C

0 −100 −100

100

200 300 x − position [m]

(b) formation at time t=0 [s]

10 y − position [m]

5 y − position [m]

0

0 −5

A

0

y − position [m]

y − position [m]

540

15 410

C 415 420 x − position [m]

425

B

−5

30

20

600

5

−10 −5

25

500

(c) formation at time t=0.216 [s]

−10 −30

−20 −10 0 x − position [m] (d) formation at time t=41.3 [s]

400

0 5 10 x − position [m] (e) formation at time t=100 [s]

535 530 525 455

D 460 465 x − position [m]

470

Fig. 5.4. Planar trajectory (a) and Formation shape(b-e)

5.7 Conclusion We presented in this paper an ISS-based approach to the design of a decentralized control architecture for planar non-holonomic vehicle formations of leader/follower type. By expressing the the relative vehicle dynamics with respect to a moving frame oriented along the direction of motion of the follower, and setting the origin of the frame on a virtual agent ﬁxed at the desired follower position, the leader/follwer trailing problem is reduced to a robust stabilization problem. The use of saturated controllers with tunable gains are proposed to take advantage of the ISS property with restriction, allowing to focus on the asymptotic behavior. The controller is shown to provide robustness with respect to small slow varying measurement errors and possibly to certain small time delay in the input channels, if appropriate conditions hold. Simulations results validate the robustness of the proposed controller.

−20

0

0.2

0.4 time [s]

0.6

0.8

10 0 −10

0

0.2

0.4 time [s]

0.6

0.8

theta [rad]

2 1 0

x − position [m]

0

y − position [m]

State variables of Agent 2

20

theta [rad]

y − position [m]

x − position [m]

5 ISS-Based Robust Leader/Follower Trailing Control

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0.2

0.4 time [s]

0.6

0.8

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State variables of Agent 5

5 0 −5

0

0.2

0.4 time [s]

0.6

0.8

0

0.2

0.4 time [s]

0.6

0.8

0

0.2

0.4 time [s]

0.6

0.8

2 0 −2 2 0 −2

Fig. 5.5. State variables for agent 2 and 5 during transient.

5.8 Appendix The class of saturation function employed in this paper is deﬁned as follows: Deﬁnition 5.1. A saturation function is a mapping σ : R −→ R deﬁned as any nondecreasing odd diﬀerentiable function satisfying:

! ! ! !

|σ(s)| ˙ := | dσ(s) ds | ≤ 2 for all s ∈ R, sσ(s) > 0 for all s = 0, σ(0) = 0 σ(s) = sgn(s) for |s| ≥ 1. |s| < |σ(s)| < 1 for |s| < 1.

For example, a candidate saturation function can be chosen as follows     sgn(s) for |s| ≥ 1 σ(s) =    − 1 s3 + 3 s for |s| < 1 . 2 2 The following deﬁnition of ISS with restriction [15] is adopted throughout the paper: Deﬁnition 5.2. Consider the nonlinear system x˙ = f (x, u)

(5.12)

with state x ∈ Rn , input u ∈ Rm , with f (0, 0) = 0 and f (x, u) locally Lipschitz on Rn × Rm . System (5.12) is said to be ISS with restriction X ⊆ Rn on the initial

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state and restriction ∆ > 0 on the input if there exist class K functions γ0 (·) and 0 γ(·) suth that for any input u(·) ∈ Lm ∞ satisfying u ∞ < ∆ and for any x ∈ X, the response x(t) of (5.12) in the initial state x0 satisﬁes x x where the notation x

a

∞ a

≤ max{γ0 ( x0 ), γ( u )∞ }, ≤ γ( u a ) ,

stands for the asymptotic norm of x(·) x

a

lim sup x(t) . t→∞

References 1. X. Chen and A. Serrani. Remarks on ISS and formation control. Proceedings of the 43rd IEEE Conference on Decision and Control, 1:177–182, 2004. 2. A.K. Das, R. Fierro, V. Kumar, J.P. Ostrowski, J. Spletzer, and C.J. Taylor. A visionbased formation control framework. IEEE Transactions on Robotics and Automation, 18(5):813–825, 2002. 3. J.P. Desai, J.P. Ostrowski, and V. Kumar. Modeling and control of formations of nonholonomic mobile robots. IEEE Transactions on Robotics and Automation, 17(6):905– 908, 2001. 4. L. E. Dubins. On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. American Journal of Mathematics, 79:497–516, 1957. 5. W.B. Dunbar and R.M. Murray. Model predictive control of coordinated multi-vehicle formations. Proceedings of the 41st IEEE Conference on Decision and Control, 2002, 4:4631–4636 Vol.4, 2002. 6. M. Egerstedt and Xiaoming Hu. Formation constrained multi-agent control. IEEE Transactions on Robotics and Automation, 17(6):947–951, 2001. 7. V. Gazi and K.M. Passino. Stability analysis of swarms. IEEE Transactions on Automatic Control, 48(4):692–697, 2003. 8. A. Jadbabaie, J. Lin, and A.S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6):988–1001, 2003. 9. N.E. Leonard and E. Fiorelli. Virtual leaders, artiﬁcial potentials and coordinated control of groups. Proceedings of the 40th IEEE Conference on Decision and Control, 2001, 3:2968–2973 Vol.3, 2001. 10. P. Ogren, M. Egerstedt, and X. Hu. A control Lyapunov function approach to multiagent coordination. IEEE Transactions on Robotics and Automation, 18(5):847–851, 2002. 11. A. Pant, P. Seiler, and K. Hedrick. Mesh stability of look-ahead interconnected systems. IEEE Transactions on Automatic Control, 47:403–7, 2002. 12. E.D. Sontag. On the input-to-state stability property. European J. Control, 1:24–36, 1995. 13. D. Swaroop and J.K. Hedrick. String stability of interconnected systems. IEEE Transactions on Automatic Control, 41(3):349–357, 1996.

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14. H.G. Tanner, G.J. Pappas, and V. Kumar. Leader-to-formation stability. IEEE Transactions on Robotics and Automation, 20(3):443–455, 2004. 15. A.R. Teel. A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Transactions on Automatic Control, 41(9):1256–1270, 1996. 16. A.R. Teel. Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Transactions on Automatic Control, 43(7):960–964, 1998.

6 Coordinated Path Following Control of Multiple Vehicles subject to Bidirectional Communication Constraints R. Ghabcheloo1 , A. Pascoal1, C. Silvestre1 and I. Kaminer2 1 2

Institute for Systems and Robotics/Instituto Superior T´ecnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. {reza,antonio,cjs}@isr.ist.utl.pt, Department of Mechanical and Astronautical Engineering, Naval Postgraduate School, Monterey, CA 93943, USA. [email protected]

Summary. The paper addresses the problem of making a set of vehicles follow a a set of given spatial paths at required speeds, while ensuring that they reach and maintain a desired formation pattern. Problems of this kind arise in a number of practical applications involving ground and underwater robots. The paper summarizes and brings together in a uniﬁed framework previous results obtained by the authors for wheeled robots and fully actuated underwater vehicles. The decentralized solution proposed does not require the concept of a leader and applies to a very general class of paths. Furthermore, it addresses explicitly the dynamics of the vehicles and the constraints imposed by the inter-vehicle bi-directional communications network. The theoretical machinery used brings together Lyapunov-based techniques and graph theory. With the set-up proposed, path following (in space) and inter-vehicle coordination (in time) can be viewed as essentially decoupled. Path following for each vehicle is formulated in terms of driving a conveniently deﬁned generalized error vector to zero; vehicle coordination is achieved by adjusting the speed of each vehicle along its particular path, based on information on the position and speed of a number of neighboring vehicles, as determined by the communications topology adopted. The paper presents the problem formulation and summarizes its solution. Simulations with dynamics models of a wheeled robot and an underwater vehicle illustrate the eﬃcacy of the solution proposed.

6.1 Introduction There is growing interest in the problem of coordinated motion control of multiple autonomous vehicles. Applications include aircraft and spacecraft formation ﬂying control [3], [17], [29], [30], coordinated control of land robots [5], [27] and control of multiple surface and underwater vehicles [6], [24], [33], [34], [37]. The work reported in the literature is by now quite vast and addresses a large class of topics that include, among others, leader/follower formation ﬂying, control of the ”center of mass” and radius of dispersion of swarms of vehicles, and reaching a moving formation. In the latter case, the goal is for the vehicles to achieve and maintain desired relative positions and orientations with respect to each other, while evolving at a desired

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 93–111, 2006. © Springer-Verlag Berlin Heidelberg 2006

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formation speed [23]. Central to the problems stated is the fact that each vehicle can only exchange information with a subset of the remaining group of vehicles. Interestingly enough, analogous constraints appear in the vibrant area of networked control systems, from which some of the theoretical tools developed can be borrowed. See for example [38] and the references therein. The problem of coordinated motion control has several unique aspects that are at the root of new theoretical problems. As pointed out in [7] and [8], the following are worth stressing: i) except for some cases in the area of aircraft control, the motion of one vehicle does not directly aﬀect the motion of the other vehicles, that is, the vehicles are dynamically decoupled; the only coupling arises naturally out of the speciﬁcation of the tasks that they are required to accomplish as an ensemble. ii) there are strong practical limitations to the ﬂow of information among vehicles, which may often be severely restricted due to the nature of the underlying communications network. In marine robotics, for example, underwater communications rely on the propagation of acoustic waves which travel at an approximate speed of 1500[m s−1 ]. As is well known, this fact sets tight limitations on the communication bandwidths that can be achieved and introduces unavoidable latencies that depend on the distance between the emitter and the receiver [28]. Thus, as a rule, no vehicle will be able to communicate with the entire formation. Furthermore, a reliable vehicle coordination scheme should exhibit some form of robustness against certain kinds of vehicle failures or temporary loss of inter-vehicle communications. A rigorous methodology to deal with some of the above issues has emerged from the work reported in [7], [8], where the authors address explicitly the topics of information ﬂow and cooperation control of vehicle formations, simultaneously. The methodology proposed builds on an elegant framework that involves the concept of Graph Laplacian (a matrix representation of the graph associated with a given communication network). In particular, the results in [8] show clearly how the Graph Laplacian associated with a given inter-vehicle communication network plays a key role in assessing stability of the behaviour of the vehicles in a formation. It is however important to point out in that work that: i) the dynamics of the vehicles are assumed to be linear, time-invariant, and ii) the information exchanged among vehicles is restricted to linear combinations of the vehicles’ state variables. Motivated by the progress in the ﬁeld, this paper describes a problem in coordinated vehicle control that departs slightly from mainstream work reported in the literature. Speciﬁcally, we consider the problem of coordinated path following where multiple vehicles are required to follow pre-speciﬁed spatial paths while keeping a desired inter-vehicle formation pattern in time. This mission scenario occurs naturally in underwater robotics [28]. Namely, in the operation of multiple autonomous underwater vehicles for fast acoustic coverage of the seabed. In this important case, two or more vehicles are required to ﬂy above the seabed at the same or diﬀerent depths, along geometrically similar spatial paths and map the seabed using copies of the same suite of acoustic sensors. By requesting that the vehicles traverse identical paths so as to make the acoustic beam coverage overlap along the seabed, large areas can be covered in a short time. This imposes constraints on the inter-vehicle

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formation pattern. Similar scenarios can of course be envisioned for land and air vehicles. To the best of our knowledge, previous work on coordinated path following control has essentially been restricted to the area of marine robotics. See for example [6], [33], [24], [25] and the references therein. However, the solutions developed so far for underactuated vehicles are restricted to two vehicles in a leader-follower type of formation and lead to complex control laws. Even in the case of fully actuated vehicles, the solutions presented do not address communication constraints explicitly. There is therefore a need to re-examine this problem to try and arrive at eﬃcient and practical solutions. Preliminary steps in this direction were taken in [10] where the problem of coordinated path following of multiple wheeled robots was solved by resorting to linearization and gain scheduling techniques. The solutions obtained were conceptually simple and allowed for the decoupling of path following (in space) and vehicle synchronization (in time). The price paid for the simplicity of the solutions is the lack of global results, that is, attractivity to so-called trimming paths and to a desired formation pattern can only be guaranteed locally, when the initial vehicle formation is suﬃciently close to the desired one. The above limitation was ﬁrst lifted for the case of wheeled robots in [12] - see also [11] - to yield global results that allow for the consideration of arbitrary paths, formation patterns, and initial conditions under the assumptions that all communications are bidirectional. The solution adopted in the above references is well rooted in Lyapunov-based theory and addresses explicitly the vehicle dynamics as well as the constraints imposed by the topology of the inter-vehicle communications network. The latter are tackled in the framework of graph theory [18], which seems to be the tool par excellence to study the impact of communication topologies on the performance that can be achieved with coordination. Once again, using this setup, path following (in space) and inter-vehicle coordination (in time) are essentially decoupled. Path following for each vehicle amounts to reducing a conveniently deﬁned error variable to zero. In its simplest form, vehicle coordination is achieved by adjusting the speed of each of the vehicles along its path, according to information on the relative positions and speeds of the other vehicles, as determined by the communications topology adopted. No other kinematic or dynamic information is exchanged among the vehicles. This circle of ideas was later extended for wheeled robots in [13] to deal with unidirectional communication links and in [14] for the case of fully actuated marine vehicles. The framework developed seems to hold great potential to be extended to the case of underactuated marine vehicles and to multiple autonomous air vehicles for time-critical applications such as coordinated landing and rendez-vous maneuvers [20]. The paper summarizes and brings together in a uniﬁed framework previous results obtained by the authors for wheeled robots and fully actuated underwater vehicles. The decentralized solution proposed does not require the concept of a leader and applies to a very general class of paths. Furthermore, it addresses explicitly the dynamics of the vehicles and the constraints imposed by the inter-vehicle bi-directional communications network.

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The paper is organized as follows. Section 6.2 introduces the basic notation required, describes the simpliﬁed models of representative wheeled and marine robots, and oﬀers a solution to the problem of path following for a single vehicle. The main contribution of the paper is summarized in Section 6.3, where a strategy for multiple vehicle coordination is proposed that builds on Lyapunov and graph theory. Section 6.4 describes the results of simulations. Finally, Section 6.5 contains the main conclusions and describes problems that warrant further research.

6.2 Path Following This section describes solutions to the problems of path following for a single wheeled robot and for a single fully actuated underwater vehicle. Due to space limitations, the exposition is brief. The notation is standard. For further details, the reader is referred to [13] and [14] for the case of wheeled robots and marine vehicles, respectively. In what follows, to stress the similarities of the problems considered, we ﬁrst tackle the case of the underwater vehicle from which the solution for the wheeled robot follows directly. Consider a fully actuated autonomous underwater vehicle depicted in Figure 1(a), together a spatial path Γ in the x − y plane that must be followed. The vehicle is equipped with propellers and we assume that the propeller arrangement is such that the forces in surge and sway and the torque in yaw can be generated independently. The problem of path following can now be brieﬂy stated as follows: Given a spatial path Γ , develop feedback control laws for the surge and sway forces and (yaw) torque acting on the vehicle so that its center of mass converges asymptotically to the path while its total speed and heading angle track desired temporal proﬁles. To arrive at a formal mathematical formulation of the above problem, consider Figure 1(a) where P is an arbitrary point on the path to be followed and Q is the center of the mass of the vehicle. Associated with P , consider the Serret-Frenet {T}. The signed curvilinear abscissa of P along the path is denoted by s. Clearly, Q can −−→ be expressed either as OQ = (x, y) in the inertial reference frame {U}, or as (xe , ye ) −−→ in {T}. Let OP be the position of P in {U} and deﬁne two frames with their origin at the center of mass of the vehicle: i) the body-ﬁxed frame denoted {B} with its x-axis along the main axis of the body, and ii) the ﬂow frame denoted {F} with U its x-axis along the total velocity vector Vt of the vehicle. Further let U T R, and F R denote the rotation matrices from {T} to {U} and from {F} to {U} respectively, parameterized by ψT and ψF . The yaw angle of the vehicle will be denoted ψB . Deﬁne the variables u and v as the surge and sway linear speeds, respectively and r = ψ˙ B as the angular speed of the vehicle. Let vt denote the vehicle’s speed (that is the signed absolute value of Vt ). From the ﬁgure, it follows that   xe  −−→ −−→ U   OQ = OP + T R   . ye

6 Coordinated Path Following Control

( , )

( , )

{ }

{ }

{ }

97

{ }

(b) Wheeled robot

(a) Underwater vehicle

Fig. 6.1. Frames and error variables

Taking derivatives and expressing the result in frame {U} yields          vt  U  s˙  U  xe  U  x˙ e   = T R   + T R˙   + T R   ,        0 0 ye y˙ e

U  FR

where





 

 vt  F  u    = BR   .     v 0

(6.1)

From the above expression, simple calculations lead to the kinematics of the vehicle in the (xe , ye ) coordinates as     x˙ e = (ye cc (s) − 1)s˙ + vt cos ψe     (6.2) Kinematics y˙ e = −xe cc (s)s˙ + vt sin ψe        ψ˙ e = r − cc (s)s˙ + β˙ where ψe = ψF − ψT is the error angle, β = ψF − ψB is the side-slip angle, and cc (s) is the path’s curvature at P determined by s, that is ψ˙ T = cc (s)s. ˙ Notice how the kinematics are driven by vt , r, and the term s˙ that plays the role of an extra control parameter.

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As shown in [14] the dynamics of the vehicle can be written in the ﬂow frame, in terms of variables (vt , β, r), as v˙ t = fvt (vt , β, r) + τvt β˙ = fβ (vt , β, r) + τβ r˙ = fr (vt , β, r) +

(6.3)

1 mr τr

where fvt , fβ , fr are nonlinear functions of their arguments, τr is the torque control signal, and (τvt , τβ ) are one-to-one functions of the surge and sway forces which we assume can be controlled directly. With this set-up, the problem of path following can be mathematically formulated as follows: Problem 1 [Path following, Marine vehicle] Given a spatial path Γ and desired time proﬁles vd (t) and βd (t) for the vehicle total speed vt and side-slip angle β, respectively, derive a feedback control law for τvt , τβ , τr and s˙ to drive xe , ye , ψe , β − βd and vt − vd asymptotically to 0. Driving the speed vt and β to their desired values is trivial to do with the simple control laws τvt = −fvt + v˙ d − k0 (vt − vd ) and τβ = −fβ + β˙ d − k0 (β − βd ), which make the errors vt − vd and β − βd decay exponentially to zero. Controlling vt and β is therefore decoupled from the control of the other variables, and all that remains is to ﬁnd suitable control laws for τr and for s˙ to drive xe , ye , ψe to zero, no matter what the evolutions of vt (t) and β(t) are. The only technical assumptions required are that the path be suﬃciently smooth and that limt→∞ vt (t) = 0. The main result of this Section is stated next. Proposition 1 [Path following, Marine vehicle]. Let Γ be a path to be followed by a fully actuated underwater vehicle. Further let the kinematic and dynamic equations of motion of the vehicle be given by (6.2) and (6.3), respectively. Assume vt (t) is uniformly continuous on [0, ∞) and limt→∞ vt (t) = 0. Deﬁne ye σ = σ(ye ) = −sign(vt ) arcsin( |yke2|+ε ) 0    1 if ψe = σ δ = δ(ψe , σ) =    sin ψe −sin σ otherwise ψe −σ

(6.4)

φ = cc s˙ + σ˙ − k1 (ψe − σ) − vt ye δ for some k1 > 0, 0 < k2 ≤ 1 and ε0 > 0. Let the control laws for τr and s˙ be given by τr = mr (−fr + φ˙ − k4 (r − φ) − (ψe − σ)) s˙ = vt cos ψe + k3 xe

(6.5) (6.6)

for some k3 , k4 > 0. Then, xe , ye , and ψe are driven asymptotically to zero from any initial condition.

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Due to space limitations, the proof of the above proposition is not detailed here. Instead, we give some intuition for the formal proof in [11]. The proof starts with the Lyapunov function V = 12 x2e + 12 ye2 + 21 (ψe − σ)2 + 21 (r − φ)2 . The ﬁrst two terms capture the distance between the vehicle and the virtual target, which must be reduced to 0. The third term aims to shape the approach angle of the vehicle to the path as a function of the distance ye , by forcing it to follow a desired orientation proﬁle embodied in the function σ [26]. The fourth term is the result of applying backstepping, and is the diﬀerence between the actual yaw rate r and desired yaw rate φ, which must be reduced to zero. Under the above conditions s˙ tends to vt , that is, the speed of the virtual target approaches vt asymptotically. Furthermore, r approaches cc s˙ = cc vt as t increases. Consider now a wheeled robot of the unicycle type depicted in Figure 1(b), together a spatial path Γ in horizontal plane to be followed. The vehicle has two identical parallel, nondeformable rear wheels and a steering front wheel. The contact between the wheels and the ground is pure rolling and non-slipping. Each rear wheel is powered by a motor which generates a control torque. This will in turn generate a control force and a control torque applied to the vehicle. When compared with the marine vehicle, the wheeled robot is underactuated since only the force τvt and the (yaw) torque τr can be manipulated. Notice the obvious fact that the side-slip angle is zero because the total velocity vector vt is aligned with the main axis of the vehicle. In (6.2) and (6.3), making vt = u, v ≡ 0, and β ≡ 0, and taking fvt and fr also as 0, the ﬁnal kinematics and dynamic equations of a simpliﬁed model of the wheeled robot follow immediately [13]. Again, driving the speed u to the desired value is trivial (as explained before for the marine vehicle) and this problem is therefore decoupled from that of controlling the remaining variables. With the simpliﬁcations above, (6.5) and (6.6) lead directly to the required path following ˙ The details are omitted. control laws for τr and s.

6.3 Coordination Equipped with the results obtained in the previous section, we now consider the problem of coordinated path following control. In the most general set-up, one is given a set of n ≥ 2 wheeled robots (or fully actuated marine vehicle) and a set of n spatial paths Γk ; k = 1, 2, ..., n and require that robot k follow path Γk . We further require that the vehicles move along the paths in such a way as to maintain a desired formation pattern compatible with those paths. The speeds at which the vehicles are required to travel can be imposed in a number of ways; for example, by nominating one of the vehicles as a formation leader, assigning it a desired speed, and having the other vehicles adjust their speeds accordingly. Figures 6.2 and 6.3 show the simple cases where 3 vehicles are required to follow straight paths and circumferences Γi ; i = 1, 2, 3, respectively, while keeping a desired ”triangle” or ”inline” formation pattern. In the simplest case, the paths Γi may be obtained as simple parallel translations of a ”template” path Γ t – Figure 6.2. A set of paths can also be obtained by considering the case of scaled circumferences with a common center and diﬀerent radii Ri – Figure 6.3.

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Fig. 6.2. Coordination: triangle formation

Fig. 6.3. Coordination: in-line formation

Fig. 6.4. Along-path distances: straight lines

Fig. 6.5. Along-path distances: circumferences

Assuming that separate path following controllers have been implemented for each vehicle, it now remains to coordinate (that is, synchronize) them in time so as to achieve a desired formation pattern. As will become clear, this will be achieved by adjusting the speeds of the vehicles as functions of the ”along-path” distances among them. To better grasp the key ideas involved in the computation of these distances, consider for example the case of in-line formations maneuvering along parallel translations of straight lines. For each robot i, let si denote the signed curvilinear abscissa of the origin of the corresponding Serret-Frenet frame {Ti } being tracked, as described in the previous section. Since each vehicle ﬂow frame {Fi } tends asymptotically to {Ti }, it follows that the vehicles are (asymptotically) synchronized if (6.7) si,j (t) := si (t) − sj (t) → 0, t → ∞; i = 1, .., n; i < j ≤ n. This shows that in the case of translated straight lines si,j is a good measure of the along-path distances among the robots. Similarly, in the case of scaled circumferences an appropriate measure of the distances among the robots is s¯i,j := s¯i − s¯j ; i = 1, .., n; i < j ≤ n where s¯i = si /Ri . See Figures 6.4 and 6.5.

(6.8)

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Fig. 6.6. A general coordination scheme

Notice how the deﬁnition of s¯i,j relies on a normalization of the lengths of the circumferences involved and is equivalent to computing the angle between vectors li and lj directed from the center of the circumferences to origin of the Serret-Frenet frames {Ti } and {Tj }, respectively. In both cases, we say that the vehicles are coordinated if the corresponding along path distance is zero, that is, si − sj = 0 or s¯i − s¯j = 0. The extension of these concepts to a more general setting requires that each path Γi be parameterized in terms of a parameter ξi that is not necessarily the arc length along the path. An adequate choice of the parameterization will allow for the conclusion that the vehicles are synchronized iﬀ ξi = ξj for all i, j. For example, in the case of two robots following two circumferences with radii R1 and R2 while keeping an in-line formation pattern, ξi = si /Ri ; i = 1, 2. This seemingly trivial idea allows for the study of more elaborate formation patterns. As an example, consider the problem depicted in Figure 6.6 where vehicles 1 and 2 must follow paths Γ1 and Γ2 , respectively, while maintaining vehicle 2 ”to-the-left-and-behind” vehicle 1, that is, along straight lines √ that make an angle of 45 degrees between themselves. Let ξ1 = s1 and ξ2 = s2 2. It is clear that the vehicles are synchronized if ξ1 − ξ2 = 0. Since the objective of the coordination is to synchronize the ξi ’s, we sometimes refer to the latter variables as coordination states. The above considerations motivate the mathematical development that follows. We start by computing the coordination error dynamics, after which a decentralized feedback control law is derived to drive the coordination error to zero asymptotically. In the analysis, graph theory - as the mathematical machinery par excellence to deal with inter-vehicle communication constraints - plays a key role. 6.3.1 Coordination Error Dynamics The dynamics (6.3) of the total speed of the i’th vehicle can be rewritten as v˙ i = Fi where Fi = τvti + fvti .

(6.9)

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As before, we let the path Γi be parameterized by ξi and denote by si = si (ξi ); i = 1, 2, ..., n the corresponding arc length. We deﬁne Ri (ξi ) = ∂si /∂ξi and assume that Ri (ξi ) is positive and uniformly bounded for all ξi . In particular, si is a monotonically increasing function of ξi . We further assume that all Ri (ξi ) is bounded away from 0 and that ∂Ri /∂ξi is uniformly bounded. The symbol Ri (.) is motivated by the nomenclature adopted before for the case of paths that are nested arcs of circumferences. Using equation (6.6), it is straightforward to show that the evolution of ξi is given by 1 ξ˙i = (vi cos ψei + k3i xei ) (6.10) Ri (ξi ) which can be re-written as

1 vi + di Ri (ξi )

(6.11)

1 [(cos ψei − 1)vi + k3i xei ]. Ri (ξi )

(6.12)

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where di =

Notice from the previous section that di → 0 asymptotically as t → ∞, under the assumption that vi is bounded. It can be shown that this assumption is indeed met, see [11]. Suppose one vehicle, henceforth referred to as vehicle L, is elected as a ”leader” and let the corresponding path ΓL be parameterized by its length, that is, ξL = sL . In this case, RL (ξL ) = 1. It is important to point out that L can always be taken as a ”virtual” vehicle that is added to the set of ”real” vehicles as an expedient to simplify the coordination strategy. Let vL = vL (t) be a desired speed proﬁle assigned to the leader in advance, that is ξ˙L = vL , and known to all the other vehicles. Notice now that in the ideal steady situation where the vehicles move along their respective paths while keeping the desired formation, we have ξi − ξL = 0 and therefore ξ˙i = vL for all i = 1, .., n. Thus, vL becomes the desired speed of each of the vehicles, expressed in ξi coordinates. Therefore, one can proceed without having to resort to the concept of an actual or virtual leader vehicle, thus making the coordination scheme truly distributed. We remark that because the coordination states ξi are non-dimensional, vL is simply expressed in [s−1 ]. From (6.11), making di = 0, it follows that the desired inertial velocities of vehicles 1 ≤ i ≤ n equal Ri (ξi )vL (t). This suggests the introduction of the speedtracking error vector (6.13) ηi = vi − Ri (ξi )vL , 1 ≤ i ≤ n. Taking into account the vehicle dynamics (6.9), the derivative of (6.13) yields η˙ i = fi = Fi −

d (Ri (ξi )vL ) . dt

(6.14)

Using (6.11), it is also easy to compute the dynamics of the origin of each SerretFerret frame {Ti } as ξ˙i =

1 Ri ηi

+ vL + di .

(6.15)

To write the above dynamic equations in vector form, deﬁne η = [ηi ]n×1 , ξ = [ξi ]n×1 , f = [fi ]n×1 , d = [di ]n×1 and C = C(ξ) = diag[1/Ri (ξi )]n×n to obtain

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η˙ = f ξ˙ = Cη + vL 1 + d

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(6.16)

where 1 = [1]n×1 . In the above, ||d|| → 0 asymptotically as t → ∞ and matrix C is positive deﬁnite and bounded, that is, 0 < c1 I ≤ C(ξ(t)) ≤ c2 I

(6.17)

for all t, where c1 and c2 are positive scalars and I the identity matrix. Notice that C is state-driven and can therefore be time-varying, thus allowing for more complex formation patterns than those in the motivating examples of the previous section. The objective is to derive a control strategy for f to make ξ1 = ... = ξn or, equivalently, (ξi − ξj ) = 0 for all i, j. At this point, however, two extremely important control design constraints must be taken into consideration. The ﬁrst type of constraints is imposed by the topology of the inter-vehicle communications network (that is, by the types of links available for communication). The second type of constraints arises from the need to drastically reduce the amount of information that is exchanged over the communications network. In this paper, it will be assumed that the vehicles only exchange information on their positions and speeds. A possible coordination control law is of the form fi = fi (ηi , ξi , ηj , ξj : j ∈ Ji )

(6.18)

where Ji is the index set (of the neighbors) that determines what coordination states ξj and speeds ηj ; j = i are transmitted to vehicle i. With this control law, each vehicle i requires only access to its own speed and coordination state and to some or all of the speeds and the coordination states of the remaining vehicles, as deﬁned by the index set Ji . Throughout the paper, we assume that the communication links are bidirectional, that is, if vehicle i sends information to j, then j also sends information to i. Formally, i ∈ Jj ⇔ j ∈ Ji . See [13] for the case of directional communication links where some of the vehicles may only send or receive information. Clearly, the index sets capture the type of communication structure that is available for vehicle coordination. This suggests that the vehicles and the data links among them be viewed as a graph where the vehicles and the data links play the role of vertices of the graph and edges connecting those vertices, respectively. It is thus natural that the machinery of graph theory be brought to bear on the deﬁnition of the problem under study. 6.3.2 Graphs We summarize below some key concepts and results of graph theory that are relevant to the paper. See for example [4], [18], [2], and the references therein. Basic Concepts and Results An undirected graph or simply a graph G(V, E) (abbv. G) consists of a set of vertices Vi ∈ V(G) and a set of edges E(G), where an edge {Vi , Vj } is an unordered

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pair of distinct vertices Vi and Vj in V(G). A simple graph is a graph with no edges from one vertex to itself. In this paper we only consider simple graphs, and will refer to them simply as graphs. As stated before, in the present work the vertices and the edges of a graph represent the vehicles and the data links among the vehicles, respectively. If {Vi , Vj } ∈ E(G), then we say that Vi and Vj are adjacent or neighbors. A path of length N from Vi to Vj in a graph is a sequence of N + 1 distinct vertices starting with Vi and ending with Vj , such that two consecutive vertices are adjacent. The graph G is said to be connected if two arbitrary vertices Vi and Vj can be joined by a path of arbitrary length. The adjacency matrix of a graph G, denoted A, is a square matrix with rows and columns indexed by the vertices, such that the i, j-entry of A is 1 if {Vi , Vj } ∈ E and zero otherwise. The degree matrix D of a graph G is a diagonal matrix where the i, i-entry equals the valency of vertex Vi , that is |Ji | the cardinality of Ji . The Laplacian of a digraph is deﬁned as L = D − A. If the graph is connected L has a single eigenvalue at zero with the eigenvector L1 = 0, and the rest of the eigenvalues are positive. Given any arbitrary vector ξ, if y = Lξ, then the i’th element of y is (ξi − ξj ),

yi = j∈Ji

that is, yi is a linear combination of the terms (ξi − ξj ), where j spans the set Ji of vehicles that i communicates with. This seemingly trivial point plays a key role in the computation of a decentralized coordination control law that takes into consideration the a priori existing communication constraints, as will become clear later. 6.3.3 Coordination. Problem Formulation and Solutions Equipped with the above machinery we now state the coordination problem that is the main focus of this section. First, however we comment on the type of communication constraints considered in the paper. It is assumed that: i) the communications are bidirectional (L is symmetric) and ii) the communications graph is connected. Notice that if assumption (ii) is not veriﬁed, then there are two or more clusters of vehicles and no information is exchanged among the clusters. Clearly, in this situation no coordination is possible. Problem 2 [Coordination]. Consider the coordination system with dynamics (6.16) and assume that d tends asymptotically to 0. Further assume that each of the n vehicles has access to its own state and exchanges information on its path parameter (coordination state) ξi and speed ηi with some or all of the other vehicles deﬁned by index set Ji . Let G be a graph with n vertices and ε edges, where the presence of an edge between vertex i and j signiﬁes that vehicle i and j communicate through a bidirectional link, that is j ∈ Ji or equivalently i ∈ Jj . Determine a feedback control law for f such that limt→∞ η = 0 and limt→∞ (ξi − ξj ) = 0 for all i, j = 1, .., n.

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The next proposition oﬀers a solution to the coordination problem, under the basic assumption that the communications graph G is connected. See [11] for a detailed proof of this result. Proposition 2 [Solution to the coordination problem]. Consider the coordination problem described before and assume that the communications graph G is connected. Let L be the Laplacian of G. Further let A = diag[ai ]n×n and B = diag[bi ]n×n be arbitrary positive deﬁnite diagonal matrices. Then, the control law f = −(A−1 L + A)Cη − B sat(η + A−1 Lξ), where sat is the saturation function     xm x > xm     sat(x) = x |x| ≤ xm        −xm x < −xm

(6.19)

(6.20)

with xm > 0 arbitrary, solves the coordination problem. Namely, the control law meets the communication constraints and the speed tracking errors ηi and coordination errors ξi − ξj tend to zero as d vanishes. The control law (6.19) for vehicle i can be written as fi = −

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( j∈Ji

1 1 1 ηi − ηj ) − bi sat(ηi + Ri Rj ai

(ξi − ξj )).

(6.21)

j∈Ji

We recall that Ji denotes the set of vehicles (vertices in the graph) that communicate with vehicle i. Notice how the control input of vehicle i is a function of its own speed and coordination state as well as of the coordination states and speeds of the other vehicles included in the index set Ji . Clearly, the control law is decentralized and meets the constraints imposed by the communications network, as required. Matrices A and B play the role of tuning knobs aimed at shaping the behavior of the coordination system. Notice that the coordination vector ξ appears inside the sat function. From the form of control law, it is clear that the sat function aﬀords the system designer an extra degree of freedom because as xm increases, the control activity f becomes more ”responsive” to vector ξ (intuitively, as xm increases, the coordination dynamics become ”faster”). Interestingly enough, the introduction of the sat function allows for a simple proof that vt (t) remains bounded when the path following and coordination systems are put together. Remark 1. The control law in (19) can be simpliﬁed to address the case where the vehicles exchange information on the coordination states only. In fact, it is shown in [11] that the control law f = −Aη − BCLξ also solves the coordination problem. However, the proof of this result is far more complex than that of the result stated in Proposition 2.

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Remark 2. The path following and the coordination control laws deﬁned before were developed separately. It remains to show that the overall coordinated path following system that is obtained by applying the two control laws simultaneously is well-posed and yields convergence of all error variables to 0. We eschew the proof of this fact, which is detailed in [16].

6.4 Simulations This section contains the results of simulations that illustrate the performance obtained with the coordinated path following control laws developed in the paper for both wheeled robots and underwater vehicles. Figure 6.7 illustrates the situation where 3 wheeled robots are required to follow paths that consist of parallel straight lines and nested arcs of circumferences (C is piecewise constant). The ﬁgure corresponds to the case of an in-line formation pattern. In the simulation, vehicle 1 is allowed to communicate with vehicles 2 and 3, but the last two do not communicate between themselves directly. The reference speed vL was set to vL = 0.1 [s−1 ]. Notice how the vehicles adjust their speeds to meet the formation requirements. Moreover, the coordination errors ξ12 = ξ1 − ξ2 and ξ13 = ξ1 − ξ3 and the path following errors decay to 0. Figure 6.8 illustrates a diﬀerent kind of coordinated maneuver, also for wheeled robots, in the x − y plane: one robot is required to follow the x−axis, while the other must follow a sinusoidal path as the two maintain an in-line formation along the y−axis. This example captures the situation where a vehicle direct another one to inspect a certain area rapidly while keeping ”line-of-sight contact” along a ﬁxed direction. In this case, C is time varying. Notice in Figure 6.8(b) how vehicle 1 adjusts its speed along the path so as to achieve coordination. As seen in Figures 6.8(c) and 6.8(d), the vehicles converge to the assigned paths and drive the distance between their x−coordinates to 0. Finally, Figure 6.9 corresponds to a simulation where 3 fully actuated underwater vehicles are required to follow 3 straight parallel lines 3 meters apart, in the horizontal plane, while holding their side-slip angles to 10, 0, −10 degrees. Moreover, the vehicles were required to keep an in-line formation pattern. In the simulation, the parameters of the SIRENE AUV were used [1]. As in the ﬁrst simulation, vehicle 1 is allowed to communicate with vehicles 2 and 3, but the last two do not communicate between themselves directly. The reference speed vL was set to vL = 0.2 [s−1 ] and the initial states of the three vehicles to (u, v)(t0 ) = (0.1, 0), (0.1, 0), (0.1, 0)[m/s] (x, y)(t0 ) = (0, 7), (0.5, 2), (0, −5)[m] β(t0 ) = (0, 0, 0); and βd = (10, 0, −10)[deg]. Figure 6.9(a) shows the evolution of the vehicles as they start from the initial points oﬀ the assigned paths and converge to them. Figure 6.9(b) is a plot of the

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vehicle speeds that ensure coordination along the paths. Finally, Figure 6.9(c) shows the coordination errors ξ1 − ξ2 , ξ2 − ξ3 and ξ3 − ξ1 decaying to 0 and Figure 6.9(d) the side-slip angles converging to the desired values. The controller parameters were set to a = 2; b = 1; k1 = 1; k2 = 1; k3 = 1; k4 = 1; ε0 = 0.25 for wheeled robots, and to a = 2; b = 1; k1 = 0.5; k2 = 1; k3 = 0.25; k4 = 0.5; ε0 = 0.25 for marine vehicles.

6.5 Conclusions and Suggestions for Further Research The paper addressed the problem of making a set of vehicles follow a set of given spatial paths at required speeds, while ensuring that they reach and maintain a desired formation pattern. Previous results obtained by the authors for wheeled robots and fully actuated underwater vehicles were summarized and brought together in a uniﬁed framework. The decentralized solution adopted for coordinated path following does not require the concept of a leader and applies to a very general class of paths. Furthermore, it addresses explicitly the dynamics of the vehicles and the constraints imposed by the inter-vehicle bi-directional communications network. The theoretical machinery used brought together Lyapunov-based techniques and graph theory. Simulations illustrated the eﬃcacy of the solution proposed. Further work is required to extend the methodology proposed to air and underactuated underwater vehicles. In the latter case, the challenging problem of achieving coordination in the presence of low data rate acoustic links that are plagued with intermittent failures and inevitable communication delays warrants further research. Recent results on networked control systems are expected to play a key role in this context.

Acknowledgments This research work was partially supported by the Portuguese FCT POSI programme under framework QCA III and by project MAYA-Sub of the AdI. The ﬁrst author beneﬁted from a doctoral scholarship from FCT.

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References 1. Aguiar A. P. (2002). Nonlinear Motion Control of Nonholonomic and Underactuated Systems, Ph.D. Thesis. Dept. Electrical Engineering, IST, Lisbon, Portugal. April, 2002 2. Balakrishnan R. and Ranganathan K. (2000). A Textbook of Graph Theory, SpringerVerlag, New York 3. Beard R., Lawton J., and Hadaegh F. (2001). A coordination architecture for spacecraft formation control. IEEE Trans. Contr. Syst. Technol., Vol. 9, pp. 777 - 790 4. Biggs N. (1993). Algebraic Graph Theory, Second Edition, Cambridge University Press 5. Desai J., Otrowski J., and Kumar V. (1998). Controlling formations of multiple robots. Proc. IEEE International Conference on Robotics and Automation (ICRA), Leuven, Belgium 6. Encarna¸ca ˜o P., and Pascoal A. (2001). Combined trajectory tracking and path following: an application to the coordinated control of marine craft. IEEE Conf. Decision and Control (CDC), Orlando, Florida 7. Fax A. and Murray R. (2002). Information Flow and Cooperative Control of Vehicle Formations. Proc. 2002 IFAC World Congress, Barcelona, Spain 8. Fax A. and Murray R. (2002). Graph Laplacians and Stabilization of Vehicle Formations. Proc. 2002 IFAC World Congress, Barcelona, Spain 9. Fossen T. (1994). Guidance and Control of Ocean Vehicles, John Willey & Sons, Inc., New York

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10. Ghabcheloo R., Pascoal A., Silvestre C., and Kaminer I. (2004). Coordinated Path Following Control of Multiple Wheeled Robots. Proc. 5th IFAC Symposium on Intelligent Autonomous Vehicles, Lisbon, Portugal 11. Ghabcheloo R., Pascoal A., Silvestre C. and Kaminer I.(2005), Coordinated Path Following Control using Nonlinear Techniques. Internal Report CPF02, Institute for Systems and Robotics, Nov., 2005 12. Ghabcheloo R., Pascoal A., and Silvestre C. (2005), Nonlinear Coordinated Path Following Control of Multiple Wheeled Robots with Communication Constraints. International Conference on Advanced Robotics (ICAR), Seattle, WA, USA 13. Ghabcheloo R., Pascoal A., Silvestre C., Kaminer I. (2005), Coordinated Path Following Control of Multiple Wheeled Robots with Directed Communication Links. 44th IEEE Conference on Decision and Control and European Control Conference (CDCECC), Seville, Spain 14. Ghabcheloo R., Carvalho D., Pascoal A., and Silvestre C. (2005), Coordinated Motion Control of Multiple Autonomous Underwater Vehicles, International Workshop on Underwater Robotics (IWUR), Genoa, Italy 15. Ghabcheloo R., Pascoal A., Silvestre C., and Kaminer I. (2006), Coordinated Path Following Control of Multiple Wheeled Robots using Linearization Techniques. To appear in International Journal of Systems Science (IJSS) 16. Ghabcheloo R., Pascoal A., Silvestre C., and Kaminer I. (2006), Nonlinear Coordinated Path Following of Multiple Wheeled Robots with Bidirectional Communication Constraints. Submitted to the International Journal of Adaptive Control and Signal Processing 17. Giuletti F., Pollini L., and Innocenti M. (2000). Autonomous formation ﬂight. IEEE Control Systems Magazine, Vol. 20, pp. 34 - 44 18. Godsil C. and Royle G. (2001). Algebraic Graph Theory. Graduated Texts in Mathematics, Springer-Verlag New York, Inc. 19. Horn R. A., and Johnson C. R. (1985). Matrix Analysis. Cambridge Univ. Press 20. Kaminer I., Yakimenko O., Dobrokhodov V., Lizaraga M., and Pascoal A. (2004). Cooperative Control of Small UAVs for Naval Applications, IEEE Conference on Decision and Control (CDC), Atlantis, Bahamas 21. Kaminer I., Pascoal A. and Yakimenko O. (2005). Nonlinear Path Following Control of Fully Actuated Marine Vehicles with Parameter Uncertainty, 16th IFAC World Congress, Prague, Czech Republic 22. Khalil H. K. (2002). Nonlinear Systems. Third Edition, Prentice Hall 23. Laferriere G., Williams A., Caughman J., and Veerman J. (2005). Decentralized control of vehicle formations. Systems and Control Letters, 54, pp. 899 - 910 24. Lapierre L., Soetanto D., and Pascoal A. (2003). Coordinated motion control of marine robots. Proc. 6th IFAC Conference on Manoeuvering and Control of Marine Craft (MCMC), Girona, Spain 25. Lapierre L., Soetanto D. and Pascoal A. (2003). Nonlinear path following control of autonomous underwater vehicles. Proc. 1st IFAC Conference on Guidance and Control of Underwater Vehicles (GCUV), Newport, South Wales, UK 26. Micaelli A. and Samson C. (1993). Trajectory - tracking for unicycle - type and two steering - wheels mobile robots. Technical Report No. 2097. INRIA, Sophia-Antipolis, France ¨ 27. Ogren, P., Egerstedt, M., and Hu, X. (2002). A control lyapunov function approach to multiagent coordination. IEEE Trans. on Robotics and Automation, Vol. 18, pp. 847 - 851

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28. Pascoal A. et al. (2000). Robotic ocean vehicles for marine science applications: the european ASIMOV Project. Proc. OCEANS’2000 MTS/IEEE, Rhode Island, Providence, USA 29. Pratcher M., D’Azzo J., and Proud A. (2001). Tight formation control. Journal of Guidance, Control and Dynamics, Vol. 24, No. 2, March-April 2001, pp. 246 - 254 30. Queiroz M., Kapila V., and Yan Q. (2000). Adaptive nonlinear control of multiple spacecraft formation ﬂying. Journal of Guidance, Control and Dynamics, Vol. 23, No.3, May-June 2000, pp. 385 - 390 31. Rouche N., Habets P. and Laloy M. (1977). Stability theory by Liapunov’s direct method. Springer-Verlag New York, Inc. 32. Samson C. and Ait-Abderrahim K. (1990). Mobile Robot Control Part 1: Feedback Control of a Nonholonomic Wheeled Cart in Cartesian Space. Unit´e de Recherche INRIA, No. 1288 33. Skjetne R., Moi S., and Fossen T. (2002). Nonlinear formation control of marine craft. Proc. IEEE Conf. on Decision and Control (CDC), Las Vegas, NV 34. Skjetne R., Flakstad I., and Fossen T. (2003). Formation control by synchronizing multiple maneuvering systems. Proc. 6th IFAC Conference on Manoeuvering and Control of Marine Craft (MCMC), Girona, Spain 35. Soetanto D., Lapierre L., and Pascoal A. (2003). Adaptive, Non-Singular Path Following, Control of Dynamic Wheeled Robots Proc. ICAR, Coimbra, Portugal 36. Sontag E. D. and Wang, Y.(1996). New characterizations of input-to-state stability. IEEE Trans. on Automatic Control, Vol.41, Issue 9, Sept. 1996, pp. 1283 - 1294 37. Stilwell D. and Bishop B. (2000). Platoons of underwater vehicles. IEEE Control Systems Magazine, Vol. 20, Issue 6, pp. 45 - 52 38. Xu Y. and Hespanha J. (2004). Optimal Communication Logics for Networked Control Systems, Proc. 43rd Conf. on Decision and Contr., Atlantis, Bahamas

7 Robust Formation Control of Marine Craft Using Lagrange Multipliers I.-A.F. Ihle1 , J. Jouﬀroy1 and T.I. Fossen1,2 1 2

Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Department of Engineering Cybernetics, Norwegian University of Technology and Science, N-7491 Trondheim, Norway E-mail:[email protected], [email protected], [email protected]

Summary. This paper presents a formation modelling scheme based on a set of inter-body constraint functions and Lagrangian multipliers. Formation control for a ﬂeet of marine craft is achieved by stabilizing the auxiliary constraints such that the desired formation conﬁguration appears. In the proposed framework we develop robust control laws for marine surface vessels to counteract unknown, slowly varying, environmental disturbances and measurement noise. Robustness with respect to time-delays in the communication channels are addressed by linearizing the system. Simulations of tugboats subject to environmental loads, measurement noise, and communication delays verify the theoretical results. Some future research directions and open problems are also discussed.

7.1 Introduction Ever since man was able to construct ships that were able to cross the open seas have formations been used to improve safety during travel and for tactical advantages in naval battles. Indeed, the ancient Greeks employed a strategy where light and fast vessels would ram their bow into heavier vessels to disable them, thus making them an easy target [4]. This is an old example but still proves how a formation of smaller, but ﬂexible, vessels can outperform larger and more specialized vessels. Even today, this incentive remains valid for vehicle formations, cooperative robotics, and sensor networks. The motivation has, together with rapid developments in computation, communication, control, and miniaturization abilities, led to a major research interest in cooperative control of mobile robots, satellites, and marine vessels during the last decade–see for example [17] and references therein. Formation control schemes aim to develop decentralized control laws that yield formation stability, and has been pursued using artiﬁcial potentials, virtual structures, leader-follower architecture, and behavioral rules [1, 5, 15, 21, 23, 29]. Some related topics of wide interest are networked control systems, synchronization of dynamic systems (see, e.g., [24]), consensus problems in computer science, and computer graphics applications, e.g., [26].

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 113–129, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Coordinated control of several independent objects in an unknown environment pose a demanding task for the designer. The controllers should be robust to changes in the environment, unknown disturbances, communication/sensor noise, model uncertainties, and communication constraints between formation members. Previously, stability has been addressed for formations with changing topologies and timedelays–see the recent survey paper [25] for an overview–but many of the issues mentioned above have not yet been addressed. An interesting application for marine surface vessels where safety towards perturbations is critical is under-way replenishment operations [18]. One of the principal problems in abeam refuelling is the suction eﬀect, caused by the bow waves, which draws the two vessels together. The magnitude of suction forces can increase rapidly, but they are eﬀectively zero when the vessels are in a certain distance from each other. A reliable control design can improve safety of replenishment operations by designing robust control laws that maintain an inter-vessel distance larger than that threshold. This paper develops control laws for marine surface vessels based on a recent formation control scheme [11, 12]. The vessel models are based on new results in nonlinear ship modelling [7]. The resulting model of the formation is on feedback form such that the designer may apply control designs available in the literature. In particular, we develop control laws that are robust with respect to unknown environmental disturbances, and it is proved that the resulting closed-loop system is Input-to-State Stable [28] with respect to these disturbances. Examples of other possible control designs are also given. A linearization of the system provides information about robustness to time delays. The following sections are organized as follows. In Section 7.2, the formation will be modelled using classical tools from analytical mechanics. Robust control laws for assembling individual ships into a predeﬁned conﬁguration are derived in Section 7.3. A formation of marine surface vessels exposed to unknown environmental disturbances, noise, and time delays are investigated in Section 7.4. Finally, concluding remarks are given in Section 7.5.

7.2 Auxiliary Constraints, Analytical Mechanics, and Formation Setup The modelling approach in this section is motivated by how Lagrange’s “method of the undetermined multiplier” can treat a collection of independent bodies and a set of auxiliary constraints. The method works quite generally for any number of constraints, and has been applied with success, e.g., in computer graphics [2] and multibody dynamics. 7.2.1 Modelling Consider a system of independent bodies on a 2-dimensional surface as in the left part of Figure 7.1. Each body has a kinetic energy Ti and potential energy Ui , and the Lagrangian of the total system is

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Fig. 7.1. Imposed constraints transform a group of independent bodies to a formation.

L=T −U =

Ti − Ui .

(7.1)

i

Let the generalized position and velocity of body i be qi and vi , respectively. A set of functions can relate the position of two bodies i and j to each other as in the right part of Figure 7.1 by using a norm qi − qj = rij

(7.2)

where rij is the desired distance between body i and j. To apply the tools from analytical mechanics, we rewrite the functions as a kinematic relation of all positions q := [. . . , qi , . . .] (7.3) C (q) = 0 and denote it the constraint function. A constraint adds potential energy to the system, as preserving forces are necessary to maintain (7.3), and the modiﬁed Lagrangian becomes [20] L¯ = T − U + λ C (q) (7.4) where λ is the Lagrangian multiplier(s). The equations of motion for system i are then obtained by applying the Euler-Lagrange equations with auxiliary conditions to Eq. (7.4) ∂L ∂C (q) d ∂L − +λ = τi , (7.5) dt ∂ q˙i ∂qi ∂qi where τi is the known generalized external force associated with the i-th body, and the constraint force is given by τconstraint,i = −λ Wi ,

(7.6)

where Wi is the i-th column of the Jacobian of (7.3), Wi = ∂C(q) ∂qi . When the set of constraint functions are applied to the independent bodies, a formation emerges, see Figure 7.1. The conﬁguration is given by the constraint functions, and the equations of motion for all bodies are ∂C (q) d ∂L ∂L − =τ −λ . dt ∂ q˙ ∂q ∂q

(7.7)

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The left hand side of (7.7) is the decoupled Euler-Lagrange equations of motion for the independent bodies, and the term for the constraint force interprets how the bodies interact when connected through constraint functions. This is the Lagrangian approach to multi-body dynamics originally described in J.-L. Lagrange’s book [19] and appears in many textbooks on analytical mechanics, see for example [20]. The Lagrangian λ-factor has a physical interpretation as a measure of the violation of the constraint function (7.3), [20], and is found by combining the constraint function with (7.7). Since we want the position of the bodies to satisfy the constraint functions, neither the velocity nor the acceleration should violate them. To ﬁnd the kinematic admissible velocities, the constraint function is diﬀerentiated with respect to time. Similarly, we diﬀerentiate twice to ﬁnd the acceleration of the constraints. This gives the additional conditions .

C (q) = W (q) q˙ = 0 ..

˙ (q) q˙ = 0 C (q) = W (q) q¨ + W

(7.8)

where W (q) is the Jacobian of the constraint function, i.e., W (q) = ∂C(q) ∂q . An expression for the Lagrangian multiplier is found by obtaining an expression for q¨ in (7.7), insert it into (7.8) and solve for λ. Consider a system of point masses with mass equal to identity: We solve for λ to obtain λ = WW

−1

˙ v + Wτ , W

det W W

= 0.

(7.9)

In order to obtain λ, the product W W must be nonsingular, that is, the Jacobian W must have full row rank. The combination of (7.3) and (7.5) yields a DiﬀerentialAlgebraic Equation (DAE), which we will discuss further in Section 7.3. 7.2.2 Constraint Functions The approach above is valid both for systems where the constraints appear in the model, and when functions are imposed to keep the formation together. An example of systems with model constraints can be molecules consisting of multiple atoms connected by chemical bonds. The conﬁguration of the molecule depends on the properties of the interacting atoms. When the constraints are not physically present in the model, it can be denoted a virtual constraint. Two types of constraint functions for formation control purposes are considered in this paper, but others can be found in [13], the position relative constraint Cl (q, t) = (qi − qj ) (qi − qj ) − rij (t) = 0,

rij (t) ∈ R,

(7.10)

which is equivalent to the Euclidean norm of the distance between two bodies, and the position oﬀset constraint Cl (q, t) = qi − qj − oij (t) = 0,

oij (t) ∈ Rn ,

(7.11)

where qi and qj are the position of body i and j. The scalar rij (t) is the desired distance between qi and qj while the column vector oij (t) describes the oﬀset between qi and qj in each degree of freedom (DOF). We say that two bodies are neighbors if they appear in the same index of the constraint function, and in which case they can access each others information.

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Fig. 7.2. Examples of redundant auxiliary constraints. Consider the position oﬀset constraints (7.11) on the left: when two constraints are given, the third is simply a linear combination of those. For the position relative constraints (7.10) on the right two constraints are either contradictory or redundant. In both cases, the Jacobian W has less than full row rank.

7.2.3 Formation Topology As mentioned above, the Lagrangian multipliers exist if the Jacobian W has full row-rank. This limits the number of constraint functions that can be imposed on the formation. In addition, each imposed function reduces the degrees of freedom of the formation, thus the total number of constraints must not exceed the total degree of freedom. The full-row rank condition of W implies that the given constraint functions cannot be contradictory nor redundant. Suppose the formation has r members, each with an n DOF system. For constraints on the form (7.11), this means that there must be p < r such constraints for W to have full row rank since a new constraint would be a linear combination of the previous. An example of a feasible formation topology is a line. A formation can be subject to p ≤ nr − 3 position relative constraints as long as they neither contradict the existing constraints nor is a linear combination of other constraints. The above conditions yield a Jacobian with full row rank and we can solve (7.9) to obtain the Lagrangian multiplier. The results can be extended to a time-varying formation topology as long as W has full row rank for all t ≥ 0. Illustrations of constraint functions that lead to singularities are shown in Figure 7.2. 7.2.4 Control Plant Ship Model We consider a fully actuated surface vessel i in three degrees of freedom where the surge mode is decoupled from the sway and yaw mode due to port/starboard symmetry. Let an inertial frame be be approximated by the Earth-ﬁxed reference frame called NED (North-East-Down) and let another coordinate frame be attached to the ship as in Figure 7.3. The states of the vessel are then ηi = [xi , yi , ψi ] and νi = [ui , vi , ri ] where (xi , yi ) is the position on the ocean surface, ψi is the heading (yaw) angle, (ui , vi ) are the body-ﬁxed linear velocities (surge and sway), and r is the yaw rate. The Earth-ﬁxed frame is related to the body-ﬁxed frame through a rotation ψi about the zi -axis, expressed in the rotation matrix

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Fig. 7.3. Inertial Earth-ﬁxed frame and body-ﬁxed frame for a ship.





 cos ψi − sin ψi 0       ∈ SO (3) . R (ψi ) =  cos ψ 0 sin ψ   i i     1 0 0 The equations of motion for vessel i are given as (see [6] for details) η˙ i = R (ψi ) νi Mi ν˙ i + Ci (νi ) νi + Di (νi ) νi + g (ηi ) = τi .

(7.12a) (7.12b)

The model matrices Mi , Ci , and Di denote inertia, Coriolis plus centrifugal, and damping, respectively, gi is a vector of gravitational and buoyancy forces and moments, and τi is a vector of generalized control forces and moments. For notational convenience system (7.12) is converted to the Earth-ﬁxed frame using a kinematic transformation [6, Ch. 3.3.1]. The following relationships Mηi (ηi ) = R (ψi ) Mi R (ψi ) Cηi (νi , ηi ) = R (ψi ) Ci (νi ) − Mi R (ψi ) R˙ (ψi ) R (ψi ) Dηi (νi , ηi ) = R (ψi ) Di (νi ) R (ψi ) gηi (ηi ) = R (ψi ) g (ηi ) ni (νi , ηi , η˙ i ) = Cηi (νi , ηi ) η˙ i + Dηi (νi , ηi ) η˙i + gηi (ηi ) give the new equations of motion for vessel i Mηi (ηi ) η¨i + ni (νi , ηi , η˙ i ) = R (ψi ) τi subject to the following assumptions:

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119

Assumption A1: The mass matrix Mi is positive deﬁnite, i.e. Mi = Mi > 0; hence Mηi (ηi ) = Mηi (ηi ) > 0 by [6] and [8]. Assumption A2: The damping matrix can be decomposed in a linear and a nonlinear part: Di (ν) = Dlin,i +Dnonlin,i (ν) where Dnonlin,i is a matrix of nonlinear viscous damping terms for instance quadratic drag. We consider a group of r vessels subject to constraint functions as in (7.10) or (7.11) such that the vector of control forces and moments is the force due to the violation of the constraint function C, τi = τconstraint,i . Furthermore, we collect the vectors into new vectors, and the matrices into new, block-diagonal, matrices by deﬁning η = [η1 , . . . , ηr ] , Mη = diag {Mη1 , . . . , Mηr }, and so on. For i = 1, . . . , r, this results in the model Mη (η) η¨ + n (ν, η, η) ˙ = R (ψ) τconstraint

(7.13)

where the constraint force τconstraint is given by τconstraint = −W λ and the Lagrangian multiplier is found to be λ = W Mη−1 RW

−1

˙ η˙ . −W Mη−1 n + W

(7.14)

When the Jacobian W has full row rank W Mη−1 RW exists since Mη is positive deﬁnite, hence Mη−1 exists and W Mη−1 RW is nonsingular. Furthermore, the constraint force arise pairwise between neighbors and is always present when C = 0.

7.3 Robust Formation Assembling If the vessels are not already in a formation, they must be assembled into the speciﬁc conﬁguration at the start of an operation. Furthermore, the members of the formation should stay assembled in the presence of unknown environmental disturbances, noise, and time delays in the communication channels. This section presents a robust control law that assembles vessels into a desired formation shape and maintain the conﬁguration in the presence of unknown environmental perturbations. 7.3.1 Formation Assembling We want the formation to be conﬁgured as deﬁned by (7.3), that is, we want to stabilize (7.3) as an equilibrium of (7.13). This is a higher-index DAE, which is known to be inherently unstable. Indeed, the procedure in the previous section gives d2 C=0 dt2

(7.15)

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which is unstable – when (7.3) is a scalar function the Laplace-transform of (7.15) yields a transfer function with two poles in the origin. Hence, if the initial conditions of (7.13) do not satisfy the imposed constraint function, or there are measurement noise or external disturbances acting on the system, the trajectories of (7.15) will go to inﬁnity. We stabilize the system by replacing the right hand side of (7.15) with u and consider the stabilization of the constraint function as an ordinary control design problem. By applying negative feedback from the constraint and its derivative, we get a Proportional-Derivative type control law . d2 C = u = −Kp C − Kd C (7.16) 2 dt which corresponds to the Baumgarte technique for stabilization of constraints in dynamic systems [3]. The resulting constraint forces are still given by τconstraint = −W λ, but the Lagrangian multiplier from (7.14) has been replaced with the stabilizing Lagrangian multiplier

λ = W Mη−1 RW

−1

˙ η˙ − u . −W Mη−1 n + W

(7.17)

The control law (7.6) is locally implementable as each λi in (7.17) only requires information about neighboring system and the ith column of the Jacobian, Wi , depends on the position of neighbors. The stabilizing Lagrangian multiplier is then substituted in the constraint force in (7.13) and the formation conﬁguration is stabilized around C = 0. The resulting forces from imposing a constraint between two bodies are found in Figure 7.4. The formation assembling problem is now transformed into a linear control system which can be analyzed with the large number of methods in the literature. An extension of this framework to underactuated vessels is given in [13]. 7.3.2 Extension to Other Control Schemes Coordinated control laws for several independent models in an unknown environment poses challenges to the designer. When the designer has some a priori knowledge of the environmental eﬀects, they can be incorporated into the design of the control law. Due to the structure of (7.15), we are able to use many of the designs in the control literature. Assume that the constraints are of the form (7.10) or (7.11) and satisfy the conditions in Section 7.2.3. Equation (7.15) is basically a double integrator which can be put into an upper. triangular form. Indeed, with φ1 (t) := C (η (t)) and φ2 (t) := C (η (t)) we obtain φ˙ 1 = φ2 φ˙ 2 = u.

(7.18a) (7.18b)

The constraint stabilization problem is then to design a controller u that renders (φ1 , φ2 ) = (0, 0) stable.

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121

Constraint Forces

8

Position Offset Position Relative

7 6 5

Forces

4 3 2 1 0 −1 −2

0

0.5

1

1.5

2

Relative Distance

2.5

3

3.5

4

Fig. 7.4. The resulting constraint forces between two point masses as a function of their relative distance. Position relative constraint function (–) as in (7.10) and position oﬀset constraint functions (- -) as in (7.11).

The structure of (7.18) allows us to take advantage of existing control designs. For example, with a quadratic cost function, the controller can be designed using LQR-techniques. In the presence of unknown model parameters, an adaptive control scheme can be used [14]. The system (7.18) has an upper triangular structure and falls into the class of strict feedback systems. This class of systems has been thoroughly investigated and is frequently used as a basis for systematic and constructive control design as it encompasses a large group of systems. If disturbances or unknown model parameters appear as nonlinearities designs for uncertain systems or adaptive control, for example, in [16], can be applied. Consider the formation of r vessels perturbed by unknown bounded disturbances δ (t) ˙ = R (ψ) τconstraint + Wd∗ δ (t) (7.19) Mη (η) η¨ + n (ν, η, η) where Wd∗ is a smooth, possibly nonlinear, function. These disturbances may represent slowly-varying environmental loads due to second-order wave-induced disturbances (wave drift), currents and mean wind forces. The method in Sections 7.2.1 and 7.2.4 transforms (7.19) to the form of (7.18)

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φ˙ 1 = φ2 φ˙ 2 = u + Wd (φ1 , φ2 ) δ (t) where Wd = W Mη−1 Wd∗ . The goal Input-to-State Stable [28] from the closed-loop system (7.20). This can [16]: Control Design: We deﬁne the

(7.20a) (7.20b)

will now be to render the closed-loop system disturbances with respect to the origin of the be achieved by using a design procedure from error variable as

z (t) = φ2 − α

(7.21)

where α is a virtual control to be speciﬁed later. The time derivative of φ1 is φ˙ 1 = φ2 = z + α We choose Hurwitz design matrices Ai , i = 1, 2, such that Pi = Pi solution of Pi Ai + Ai Pi = −Qi where Qi = Qi > 0. Let the ﬁrst control Lyapunov function be

> 0 is the

V1 (φ1 , t) = φ1 P1 φ1 The time derivative V˙ 1 becomes, with the choice α = A1 φ1 , V˙ 1 = −φ1 Q1 φ1 + 2φ1 P1 z. Diﬀerentiating (7.21) with respect to time gives z˙ = φ˙ 2 − α˙ = u + Wd (φ1 , φ2 ) δ (t) − A1 φ2 . We deﬁne the second control Lyapunov function V2 (φ, t) = V1 + z P2 z with the following time derivative V˙ 2 = −φ1 Q1 φ1 + 2z P2 u + P2−1 P1 φ1 + Wd δ − A1 φ2 and the control law is chosen as u (φ, t) = A2 z − P2−1 P1 φ1 + A1 φ2 + α0

(7.22)

where α0 is a damping term to be determined. Young’s inequality yields 2z P2 Wd δ ≤ 2κz P2 Wd Wd P2 z +

1 δ δ, 2κ

κ>0

and we obtain 1 V˙ 2 ≤ −φ1 Q1 φ1 − z Q2 z + δ δ + 2z P2 α0 + κWd Wd P2 z . 2κ The choice α0 = −κWd Wd P2 z yields

7 Robust Formation Control of Marine Craft

1 V˙ 2 ≤ −φ1 Q1 φ1 − z Q2 z + δ δ 2κ 1 2 2 |δ| < 0, ∀ |y| > ≤ −qmin |y| + 2κ

123

1 |δ| 2κqmin

where qmin = min (λmin (Q1 ) , λmin (Q2 )) and y := [φ1 , z ] . Hence, the control law (7.22) renders the closed-loop system ISS from δ (t) to z. In φ-coordinates the control law (7.22) is written as u = −Kp φ1 − Kd φ2 − κWd Wd P2 (φ2 − A1 φ1 ) where Kd = − (A1 + A2 ) and Kp = A2 A1 − P2−1 P1 so the robust backstepping design encompasses the Baumgarte stabilization technique (7.16). Hence, by exploiting existing design methodologies the formation scheme in Section 7.2 is rendered robust against unknown disturbances. The control scheme can also be extended to include parameter adaptation and to ﬁnd constant unknown biases: Let ϕ ∈ Rx be a vector of constant unknown parameters ˙ = R (ψ) τconstraint + Wa∗ ϕ Mη (η) η¨ + n (ν, η, η) where Wa∗ is a smooth function. Recall that the Lagrangian multiplier λ is still as in (7.17). The transformed model becomes φ˙ 1 = φ2 φ˙ 2 = u + Wa (φ1 , φ2 ) ϕ

(7.23) (7.24)

where Wa is smooth. By adopting the adaptive control design procedure from [16] . or [14] we can ﬁnd a control law that renders the equilibrium points C = C = 0 and . ϕ˜ = ϕ − ϕˆ uniformly globally convergent and guarantees that C, C, ϕ˜ → 0 in the limit as t → ∞.

7.4 Case Study We investigate a formation of three vessels where one vessel track a desired path while the others follow according to the formation constraint function. All vessels are subject to unknown environmental perturbations, measurement noise and the communication channels are aﬀected by time delays. We will use a time-varying constraint function to allow a time-varying conﬁguration. Consider the following functions   2 2 2  (x1 − x2 ) + (y1 − y2 ) − r12      2 2  2 Cf c (η, t) =  (x2 − x3 ) + (y2 − y3 ) − r23 (t)   , Ctt (η, t) = η˜ := η1 − ηd (t)     2 2 2 (x3 − x1 ) + (y3 − y1 ) − r31

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where r23 (t) and ηd (t) are three times diﬀerentiable. The ﬁrst functions are on the constraint function form (7.10), while the last is a constraint function that yields a control law for trajectory tracking [13]. Since the two functions are not conﬂicting we can collect them into the following constraint function    Cf c (η, t)   = 0. C (η, t) =    Ctt (η, t)

(7.25)

Together with the ship model (7.19), where Wd = W Mη−1 R (ψ), the backstepping design in Section 7.3 yields robust control laws for formation control and trajectory tracking with φ = C V (φ, t) = φ P φ

where P = P

> 0.

The closed-loop equations of motion for the three vessels are ˙ = −R (ψ) Wf c λf c − τtt + R (ψ) δ (t) Mη (η) η¨ + n (ν, η, η) where the formation control laws are given as ˙ η˙ R (ψ) Wf c λf c = Wf c Mη−1 Wf c (−Wf c n + W .

(7.26)

.

+ Kp Cf c + Kd C f c + Pf c2 C f c − Af c1 Cf c where Kp , Kd ∈ R3×3 are positive deﬁnite. The trajectory tracking control law is τtt = [R (ψ1 ) λtt , 0, 0] where R (ψ1 ) λtt is the control law for the ﬁrst vessel to track the desired path ηd R (ψ1 ) λtt = −n1 (ν1 , η1 , η˙ 1 ) − Mη1 (¨ ηd − ktp η˜ − ktd η˜˙ − Ptt2 η˜˙ − Att η˜ )

(7.27)

where ktp , ktd ∈ R3×3 are positive deﬁnite. 7.4.1 Linearized Analysis of Robustness to Time-Delays We know that the delay robustness for a single-input-single-output linear system is given by [10] PM (7.28) Tmax = ωgc where Tmax is the maximum delay in the feedback loop that does not destabilize the system, P M is the phase margin, and ωgc is the gain crossover frequency. Thus, increasing the phase margin and/or decreasing the bandwidth improves delay robustness. We linearize by assuming small variations in the constraint functions and heading angle. The loop-gain of the linearized system (about the heading angle ψ ≈ 0) from the disturbance δ to the constraint Cf c1 is found to be

7 Robust Formation Control of Marine Craft

Gcδ (s) =

κ/2 s2 + kd s + kp

125

(7.29)

where kp and kd are the (1, 1)-elements of Kp and Kd , respectively. Using tools from linear systems theory we can adjust the gains to maximize the delay that does not destabilize the system. This has to be done in a trade-oﬀ relation with other performance properties. A critically damped system is desired since it implies no overshoot, and this is achieved for kd = 2 kp . This analysis is no guarantee for stability in the presence of delays, but it gives us an indication. 7.4.2 Simulation Results The control plant model of a fully actuated tugboat in three degrees of freedom (DOF), surge, sway, and yaw, is used in the case study.The model has been developed using Octopus SEAWAY for Windows [27] and the Marine Systems Simulator [22]. SEAWAY is a frequency-domain ship motions PC program based on the linear strip theory to calculate ship motions. The Marine Systems Simulator (MSS) is a Matlab/Simulink library and simulator for marine systems. A nonlinear speed dependent formulation for station-keeping (u = v = r = 0) and maneuvering up to u = 0.35 gLpp = 6.3 m / s (Froude number 0.35) is derived in [7] where Lpp is the length between the perpendiculars. Based on [7], output from SEAWAY were used in MSS to generate a 3 DOF horizontal plane vessel model linearized for cruise speeds around u = 5 m / s with nonlinear viscous quadratic damping in surge. Furthermore, the surge mode is decoupled from the sway and yaw mode due to port/starboard symmetry. The model is valid for cruise speeds in the neighborhood of u = 5 m / s. The model in body-ﬁxed reference frame is then Mi ν˙ i + Di (νi ) νi + Ci νi = τi with the following model matrices 



0   180.3 0     6  Mi =   0 2.436 1.3095  × 10     0 1.3095 172.2   0  0 0     6  Ci =  0 0 8.61   × 10     0 −8.61 0

(7.30)

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I.-A.F. Ihle, J. Jouﬀroy, and T.I. Fossen





0   3.883 × 10−9 − 2.393 × 10−3 |ui | 0     6 Di (νi ) =  0 0.2181 −3.434   × 10 .      0 3.706 26.54 The desired path for vessel 1 is







t  xd (t)           ηd (t) =   yd (t)  =  A sin ωt       atan2 xy˙˙ dd ψd (t)

       

where A = 200 and ω = 0.005, and the unknown environmental disturbances are    103 + 2 · 103 sin (0.1t)       3 δi (t) =   2 · 10 sin (0.1t)  + white noise     3 2 · 10 sin (0.1t)

(7.31)

acting the same on all vessels. The unknown environmental disturbances are seen to be slowly-varying while the ﬁrst-order wave-induced forces (oscillatory wave motion) are assumed to be ﬁltered out of the measurements by using a wave ﬁlter – see [9]. This is a good assumption since a ship control system is only supposed to counteract the slowly-varying motion components of the environmental disturbances to reduce wear and tear of actuators and propulsion system. The desired formation conﬁguration is given by r12 = 70, r31 = 70, and r23 is initially 65 and changes smoothly to 130 at about 1000s . The control gains are ktp = 4I, ktd = 2I, Kp = diag (kpi ), kpi = 3.24, Kd = diag (kdi ), kdi = 6 and κ = 20. The initial values are η1 (0) = 0, 0, π2 , η2 (0) = −45, 25, π2 , η3 (0) = −40, −10, π2 and ν1 (0) = ν2 (0) = ν3 (0) = 0. Figure 7.5 shows the resulting position trajectories and ﬁve snapshots of the vessels during the simulation: the vessels assemble into the desired conﬁguration and vessel 1 tracks the desired path. The position tracking and formation constraints errors due to the disturbances (7.31) were attenuated to less than 1 m, and 5 m, respectively. The time-varying conﬁguration is seen in the third and fourth snapshot as the formation changes from a triangle to a line. For the linearized relation (7.29) the values give a bandwidth of 0.59 rad / s and a phase margin of 85◦ . This corresponds to a maximum time delay of 2.5 s. In the simulation all communication channels are aﬀected by 2.5 s time delays, and simulations show that delays larger than 3 s cause instabilities in the closed-loop system. Thus, the transfer function provides a good estimate of robustness towards time delays.

7 Robust Formation Control of Marine Craft

127

250 200 150 100 50 0 −50 −100 −150 −200 −250 −500

0

500

1000

1500

2000

Fig. 7.5. Position response of vessels during simulation. Vessel 1 follows the desired (dashed) path, and the desired conﬁguration changes from a triangular shape to a line about halfway through the simulation.

7.5 Concluding Remarks and Future Directions This paper has shown how a group of individual vessels can be controlled by imposing formation constraint functions and applying tools from analytical mechanics. Stabilization of these constraints is interpreted as a control design problem and a robust control law that maintains formation conﬁguration in the presence of unknown disturbances has been developed. The theoretical results have been veriﬁed by a simulation where a group of marine surface vessels moves while maintaining the desired conﬁguration. The results demonstrated robustness with respect to unknown disturbances aﬀecting the vessels and time delays in the communication channels. In the current literature, cooperative control has become an umbrella term for feedback applications in sensor networks, vehicle formations, cooperative robotics, and consensus problems. The applications have all something in common: multiple, dynamic, entities share information to accomplish a common task. The design of formation control schemes poses challenges to the engineer, and many are not yet addressed [25]. A real-life implementation for formation control will depend heavily

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on the choice of communication protocols, available hardware, sensors, instrumentation and actuators, onboard computing possibilities, and power consumption. Sharing information is a vital component for cooperation and communication issues such as inconsistent delays, noise, signal dropouts, and possible asynchronous updates should be taken into account. In environments with limited bandwidth a formation control design with a minimal amount of information exchange is desired. Most formation control problems are studied in the context of homogenous groups with double integrator dynamics. A vehicle’s dynamics might change with time, or as the formation moves through its environment. It would be interesting to study formations with more complex or uncertain dynamics, or both, and where a single vehicle drop out of formation or its actuators break down. Future research on formation control should facilitate implementation of theoretical results to be veriﬁed experimentally. An important step would be to address these topics and mathematically guarantee that a group of systems, subject to a wide range of practical challenges, can cooperate to solve a problem that would be out of reach for a single system.

Acknowledgments This project is sponsored by The Norwegian Research Council through the Centre for Ships and Ocean Structures (CeSOS), Norwegian Centre of Excellence at the Norwegian University of Science and Technology (NTNU).

References 1. R. Arkin. Behavior-based robotics. MIT Press, Cambridge, MA, USA, 1998. 2. D. Baraﬀ. Linear-time dynamics using Lagrange multipliers. In Computer Graphics Proceedings, pages 137–146. SIGGRAPH, 1996. 3. J. Baumgarte. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanical Engineering, 1:1–16, 1972. 4. L. Casson. The Ancient Mariners: Seafarers and Sea Fighters of the Mediterranean in Ancient Times. Princeton University Press, Princeton, NJ, USA, 2nd edition, 1991. 5. M. Egerstedt and X. Hu. Formation constrained multi-agent control. IEEE Transactions on Robotics and Automation, 17(6):947–951, 2001. 6. T. I. Fossen. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, 2002. www.marinecybernetics.com. 7. T. I. Fossen. A nonlinear uniﬁed state-space model for ship maneuvering and control in a seaway. International Journal of Bifurcation and Chaos, 15(9), 2005. 8. T. I. Fossen and Ø. N. Smogeli. Nonlinear time-domain strip theory formulation for low-speed maneuvering and station-keeping. Modelling, Identiﬁcation and Control, 25(4):201–221, 2004. 9. T. I. Fossen and J. P. Strand. Passive Nonlinear Observer Design for Ships Using Lyuapunov Methods: Full Scale Experiments with a Supply Vessel. Automatica, 35(1):3–16, 1999.

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10. G. F. Franklin, J. D. Powell, and A. Emami-Naeini. Feedback Control of Dynamic Systems. Addison-Wesley, Boston, MA, USA, 4th edition, 2002. 11. I.-A. F. Ihle, J. Jouﬀroy, and T. I. Fossen. Formation control of marine craft using constraint functions. In IEEE Marine Technology and Ocean Science Conference Oceans05, Washington D.C., USA, 2005. 12. I.-A. F. Ihle, J. Jouﬀroy, and T. I. Fossen. Formation control of marine surface craft using lagrange multipliers. In Proc. 44rd IEEE Conference on Decision & Control and 5th European Control Conference, pages 752–758, Seville, Spain, 2005. 13. I.-A. F. Ihle, J. Jouﬀroy, and T. I. Fossen. Formation control of marine surface craft: A Lagrangian approach. 2006. Submitted. 14. P. A. Ioannou and J. Sun. Robust Adaptive Control. Prentice Hall, Inc., 1996. (Out of print in 2003), Electronic copy at ¡http://www-rcf.usc.edu/ ioannou/Robust Adaptive Control.htm¿. 15. A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6):988–1101, 2002. 16. M. Krsti´c, I. Kanellakopoulos, and P. V. Kokotovi´c. Nonlinear and Adaptive Control Design. John Wiley & Sons Ltd, New York, 1995. 17. V. Kumar, N. Leonard, and A. S. Morse (Eds.). Cooperative Control. Lecture Notes in Control and Information Sciences. Springer-Verlag, Heidelberg, Germany, 2005. 18. E. Kyrkjebø and K. Y. Pettersen. Ship replenishment using synchronization control. In Proc. 6th IFAC Conference on Manoeuvring and Control of Marine Crafts, pages 286–291, Girona, Spain, 2003. 19. J. L. Lagrange. M´ecanique analytique, nouvelle ´edition. Acad´emie des Sciences, 1811. Translated version: Analytical Mechanics, Kluwer Academic Publishers, 1997. 20. C. Lanczos. The Variational Principles of Mechanics. Dover Publications, New York, 4th edition, 1986. 21. N. E. Leonard and E. Fiorelli. Virtual leaders, artiﬁcial potentials and coordinated control of groups. In Proc. 40th IEEE Conference on Decision and Control, pages 2968–2973, Orlando, FL, USA, 2001. 22. MSS. Marine systems simulator, 2005. Norwegian University of Science and Technology, Trondheim, Norway ¡www.cesos.ntnu.no/mss¿. 23. R. Olfati-Saber and R. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9):1520– 1533, 2004. 24. A. Pogromsky, G. Santoboni, and H. Nijmeijer. Partial synchronization: From symmetry towards stability. Physica D Nonlinear Phenomena, 172:65–87, 2002. 25. W. Ren, R. W. Beard, and E. M. Atkins. A survey of consensus problems in multi-agent coordination. In Proc. American Control Conference, pages 1859–1864, Portland, OR, USA, 2005. 26. C. W. Reynolds. Flocks, herds, and schools: A distributed behavioral model. Computer Graphics Proceedings, 21(4):25–34, 1987. 27. SEAWAY. Octopus seaway, 2005. Amarcon B.V., The Netherlands www.amarcon.com. 28. E. D. Sontag. Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control, 34(4):435–443, 1989. 29. P. K. C. Wang. Navigation strategies for multiple autonomous robots moving in formation. Journal of Robotic Systems, 8(2):177–195, 1991.

8 Output Feedback Control of Relative Translation in a Leader-Follower Spacecraft Formation R. Kristiansen1 , A. Lor´ıa2, A. Chaillet2 and P.J. Nicklasson1 1 2

Department of Computer Science, Electrical Engineering and Space Technology, Narvik University College, Lodve Langesgt. 2, N-8505, Norway {rayk,pjn}@hin.no CNRS-LSS, Sup´elec, 3 rue Joliot Curie, 91192 Gif s/Yvette, France, {loria,chaillet}@lss.supelec.fr

Summary. We present a solution to the problem of tracking relative translation in a leader-follower spacecraft formation using feedback from relative position only. Three controller conﬁgurations are presented which enables the follower spacecraft to track a desired reference trajectory relative to the leader. The controller design is performed for diﬀerent levels of knowledge about the leader spacecraft and its orbit. The ﬁrst controller assumes perfect knowledge of the leader and its orbital parameters, and renders the equilibrium points of the closed-loop system uniformly globally asymptotically stable (UGAS). The second controller uses the framework of the ﬁrst to render the closed-loop system uniformly globally practically asymptotically stable (UGPAS), with knowledge of bounds on some orbital parameters, only. That is, the state errors in the closed-loop system are proved to converge from any initial conditions to a ball in close vicinity of the origin in a stable way, and this ball can be diminished arbitrarily by increasing the gains in the control law. The third controller, based on the design of the second, utilizes adaptation to estimate the bounds that were previously assumed to be known. The resulting closed-loop system is proved to be uniformly semiglobally practically asymptotically stable (USPAS). The last two controllers assume boundedness only of orbital perturbations and the leader control force. Simulation results of a leader-follower spacecraft formation using the proposed controllers are presented.

8.1 Introduction 8.1.1 Background Spacecraft ﬂying in formation are revolutionizing our way of performing space-based operations, and brings out several advantages in space mission accomplishment, as well as new opportunities and applications for such missions. The concept makes the way for new and better applications in space industry, such as monitoring of the Earth and its surrounding atmosphere, geodesy, deep-space imaging and exploration and even in-orbit spacecraft servicing and maintenance. Replacing large and complex spacecraft with an array of simpler micro-satellites introduces a multitude of advantages regarding mission cost and performance. However, the advantages of

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using spacecraft formations come at a cost of increased complexity and technological challenges. Formation ﬂying introduces a control problem with strict and time-varying boundaries on spacecraft reference trajectories, and requires detailed knowledge and tight control of relative distances and velocities for participating spacecraft. As in other applications of cooperative control, the control problem for the follower simpliﬁes as the knowledge about the leader and its orbit increases. However, complete knowledge of the leader is hard to achieve. Another feature of the problem that stymies its solution is that the spacecraft parameters change during its lifetime, by fuel consumption and body deformations due to external radiation and particle collisions. The orbital parameters must often be changed to achieve mission goals, both as planned changes in orbit acquisition and unexpected, necessary changes during operation. Such changes lead to modiﬁcations in the system parameters, which can be hard to communicate to the follower during operation. In addition, equipment for determining position and velocity is costly, heavy and computationally demanding, and therefore the follower spacecraft must often rely on measurements of the position of the leader spacecraft only. Hence, the challenge lies in synchronized control of the formation, with as little exchange of information between the spacecraft as possible. 8.1.2 Previous Work Position feedback control of leader-follower spacecraft formations has received some attention during the last years. The ﬁrst solution to this control problem was presented in [1], and use of the nonlinear control law results in global uniform ultimate boundedness of position and velocity tracking errors. The solution includes a ﬁltering scheme to allow for use of knowledge of relative velocity in the controller equations. A similar result was also presented in [2], providing the same stability properties to the closed-loop system. Nonlinear adaptive tracking control was developed in [3], a result which ensures global asymptotic position tracking errors. This latter result was however based on a circular orbit assumption. Later, in [4], a nonlinear tracking controller for both translation and rotation was presented, including an adaptation law to account for unknown mass and inertia parameters of the spacecraft. The controller ensures global asymptotic convergence of position and velocity errors, and the stability result was proved using a Lyapunov framework and the principle of certainty equivalence. Based on the latter two references, semiglobal asymptotic convergence of relative translation errors was proved in [5] for an adaptive output feedback controller using relative position only, with a similar ﬁltering scheme as in [2]. This result was extended to a similar result for both relative translation and rotation in [6], tracing the steps of [4]. 8.1.3 Contribution The purpose of this study is to provide a solution to the spacecraft formation control problem with as little knowledge about the leader spacecraft as possible. This relieves the necessity for communication between the spacecraft, and the leader spacecraft

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can change its orbital parameters without communicating such changes to every other spacecraft in the formation. This is desirable especially for largely populated formation, to diminish the overall communication load. We present a solution to the problem of tracking relative translation in a leader-follower spacecraft formation using feedback from relative position only. Assuming the case of a follower spacecraft performing in-orbit surveillance of a leader spacecraft, three controller conﬁgurations are presented which enables the follower spacecraft to track a desired reference trajectory relative to the leader. The controller design is performed for diﬀerent levels of knowledge about the leader spacecraft and its orbit. The ﬁrst controller assumes perfect knowledge of the leader and its orbital parameters and that the orbital perturbations working on the follower are known, and renders the equilibrium points of the closed-loop system uniformly globally asymptotically stable (UGAS), using measurements of relative position only. An additional ﬁlter, similar to the one in [7], is included, using the method of approximate diﬀerentiation, or ”dirty derivatives”, to provide suﬃcient knowledge about the relative velocity to solve the control problem. The second controller uses the framework of the ﬁrst to render the closed-loop system uniformly globally practically asymptotically stable (UGPAS), with knowledge of bounds on orbital parameters, orbital perturbations, and leader control force only. That is, the state errors in the closed-loop system are proved to converge from any initial conditions to a ball in close vicinity of the origin in a stable way, and this ball can be diminished arbitrarily by increasing the gains in the control law. Hence, the controller gains can be tuned based on a payoﬀ between the acceptable position error between the leader and the follower spacecraft and the fuel consumption. The third controller, based on the design of the second, utilizes adaptation to estimate some of the bounds that were previously assumed to be known. The resulting closedloop system is proved to be uniformly semiglobally practically asymptotically stable (USPAS), meaning that the domain of attraction can be made arbitrarily large by picking convenient gains. The rest of the paper is organized as follows: Section 8.2 contains a description of the notation, together with some mathematical preliminaries that the main results rest on. In Sect. 8.3 the model of relative translation in a leader-follower formation is presented. The tracking control laws are presented in Sect. 8.4. In Sect. 8.5 we present simulation results using the proposed controllers. Finally, some conclusions are contained in Sect. 8.6.

8.2 Mathematical Preliminaries 8.2.1 Notation In the following, we denote by x˙ the time derivative of a vector x, i.e. x˙ = dx/dt. ¨ = d2 x/dt2 . We denote by x (t, t0 , x0 ) the solution to the nonlinear Moreover, x diﬀerential equation x˙ = f (t, x) with initial conditions (t0 , x0 ). We denote by |·| the Euclidian norm of a vector and the induced L2 norm of a matrix. We use the notation H(δ, ∆) := {x ∈ Rn | δ ≤ |x| ≤ ∆}. We denote by Bδ the closed ball in Rn of radius δ, i.e. Bδ := {x ∈ Rn | |x| ≤ δ}. Moreover, for such a ball we denote

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|z|δ = inf |z − x| . x∈Bδ

A continuous function α : R≥0 → R≥0 is said to be of class K (α ∈ K) if it is strictly increasing and α (0) = 0. Moreover, α is of class K∞ (α ∈ K∞ ) if, in addition, α (s) → ∞ as s → ∞. A continuous function σ : R≥0 → R≥0 is of class L (σ ∈ L), if it is strictly decreasing and σ (s) → 0 as s → ∞. When the context is suﬃciently explicit, we may omit to write arguments of a function. 8.2.2 Stability with Respect to Balls For a general nonlinear system x˙ = f (t, x)

(8.1)

where f (t, x) : R≥0 × Rn → Rn is locally Lipschitz in x and piecewise continuous in t, and the origin is an equilibrium point of (8.1), we use the following deﬁnitions of robust stability. Let δ and ∆ be positive numbers such that ∆ > δ and let them generate closed balls Bδ and B∆ as previously deﬁned. Deﬁnition 8.1 (Uniform stability of a ball [8]). For the system (8.1) the ball Bδ is said to be uniformly stable (US) on B∆ if there exists a class K∞ function α such that the solutions of (8.1) from any initial state x0 ∈ B∆ and initial time t0 ≥ 0 satisfy |x (t, t0 , x0 )|δ ≤ α (|x0 |) ,

∀t ≥ t0 .

Deﬁnition 8.2 (Uniform attractivity of a ball [8]). The closed ball Bδ is said to be uniformly attractive (UA) on B∆ if there exists a class L function σ such that the solutions of (8.1) from any initial state x0 ∈ B∆ and initial time t0 ≥ 0 satisfy |x (t, t0 , x0 )|δ ≤ σ (t − t0 ) ,

∀t ≥ t0 .

Deﬁnition 8.3 (Uniform asymptotic stability of a ball [8]). For the system (8.1) the ball Bδ is said to be uniformly asymptotically stable (UAS) on B∆ if it is uniformly stable on B∆ with respect to Bδ , and Bδ is uniformly attractive on B∆ . 8.2.3 Practical Stability For parameterized nonlinear systems of the form x˙ = f (t, x, θ)

(8.2)

where f (t, x, θ) : R≥0 × Rn × Rm → Rn is locally Lipschitz in x, piecewise continuous in t, we have the following stability deﬁnitions: Deﬁnition 8.4 (Uniform semiglobal practical asymptotic stability [8]). Let Θ ⊂ Rm be a set of parameters. The system (8.2) is said to be uniformly semiglobally practically asymptotically stable (USPAS) on Θ if, given any ∆ > δ > 0, there exists θ ∈ Θ such that for x˙ = f (t, x, θ ) the ball Bδ is uniformly asymptotically stable on B∆ .

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Furthermore, we say that (8.2) is USPAS if the set of initial states and the set with respect to which the system is UAS can be arbitrarily enlarged and diminished, respectively, by a convenient choice of the parameters θ. From Deﬁnition 8.4, if δ = 0, we recover the notion of uniform semiglobal asymptotic stability (USAS). Also, if the system has the property of Deﬁnition 4 with ∆ = +∞ we say that the ball Bδ is uniformly globally practically asymptotically stable (UGPAS). Finally, if δ = 0 and ∆ = +∞ we recover the deﬁnition of uniform global asymptotic stability (UGAS). As it will be clear from our main results, in a number of concrete control problems the parameter θ corresponds to the values of the control gains. The following result, which is a corollary of [8, Proposition 2], applies to the semiglobal practical stability analysis of systems presenting a Lyapunov function that can be upper and lower bounded by a polynomial function, as this situation arises very often in concrete applications. Typically, the parameter θ contains the gain matrices that can be freely tuned in order to enlarge and diminish the domain of attraction and the vicinity of the origin to which solutions converge. To ﬁx the ideas, the functions σi (θ) refer often to the minimal eigenvalues of these gain matrices. So, roughly speaking, we impose that the dependency of these minimum eigenvalues in the radii ∆ and 1/δ be polynomial and of an order not greater that the one of the bounds on V . Corollary 8.1. Let σi : Rm → R≥0 , i ∈ {1, ..., N }, be continuous functions, positive over Θ, and let a, a and q be positive constants. Assume that, for any θ ∈ Θ, there exists a continuously diﬀerentiable Lyapunov function V : R≥0 × Rn → R≥0 satisfying, for all x ∈ Rn and all t ≥ 0, a min {σi (θ)} |x|q ≤ V (t, x) ≤ a max {σi (θ)} |x|q .

(8.3)

Assume also that, for any ∆ > δ > 0, there exists a parameter θ (δ, ∆) ∈ Θ and a class K function αδ,∆ such that, for all |x| ∈ [δ, ∆] and all t ≥ 0, ∂V ∂V (t, x) + (t, x)f (t, x, θ ) ≤ −αδ,∆ (|x|) . ∂t ∂x

(8.4)

Assume also that for all i ∈ {1, ..., N } and for every ﬁxed ∆ > 0, lim σi (θ (δ, ∆))δ q = 0

δ→0

and

lim σi (θ (δ, ∆)) = 0

(8.5)

lim σi (θ (δ, ∆)) = 0 .

(8.6)

δ→0

and, for every ﬁxed δ > 0, lim

∆→∞

σi (θ (δ, ∆)) =0 ∆q

and

∆→∞

Then, the system x˙ = f (t, x, θ) is USPAS on the parameter set Θ. Moreover, when ∆ = ∞ and the parameter θ is independent of ∆, i.e. θ = θ (δ), the conditions in (8.6) are no longer required, and the system x˙ = f (t, x, θ) is UGPAS on the parameter set Θ. It should be underlined that the assumptions of Corollary 8.1 are much more conservative than the ones of the original result [8, Proposition 2]. The previous statement

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has, however, the important advantage of being more easily applicable and ﬁts to many concrete applications. The following example illustrates how to verify the conditions of Corollary 8.1. This example is also representative of the method of proof that we employ in our main result – cf. Section 8.4.3. Example 8.1. Consider a mechanical system in closed loop with a given controller and under external disturbances, i.e. ¨ = − θ1 x − θ2 x˙ + a (t) x˙ + b (t) x

(8.7)

where θ1 and θ2 are control gains hence, design parameters, and a (t) and b (t) are continuous functions satisfying |a (t)| ≤ a ¯ and |b (t)| ≤ ¯b. Using the Lyapunov function V =

1 2 θ1 2 x˙ + x + εxx˙ 2 2

(8.8)

with ε ≥ 0 as a design variable, where 1 3 min {1, θ1 } |x|2 ≤ V ≤ max {1, θ1 } |x|2 , 4 2 ˙ 2 /2, and letting δ designate any together with the property |x||x| ˙ ≤ λ|x|2 + λ1 |x| positive constant satisfying δ ≤ |x| and δ ≤ |x|, ˙ we ﬁnd εθ1 εθ1 2 θ2 2 εθ2 ε¯b ε¯ a V˙ ≤ − |x| − |x| ˙ − − − − 2 2 2 2 δ 2 1−ε ε¯ a ¯b − ¯− − θ2 − a |x| ˙2 2 2 δ

|x|2

along the trajectories of system (8.7). By choosing θ1 = θ2 +

2¯b +a ¯ δ

and

θ2 =

1 (1 − ε)

2¯ a + ε¯ a+

2¯b δ

(8.9)

with ε < 1 we ensure that V˙ < 0, and from (8.9) we see that lim θ1 (δ)δ 2 = 0 and

δ→0

lim θ1 (δ) = 0 .

δ→0

Therefore, the conditions (8.3)-(8.5) are satisﬁed, and (8.6) follows trivially, since in our case, ∆ = ∞. Hence, the system (8.7) is UGPAS.

8.3 Relative Translational Motion Having established the mathematical framework for our main result, let us formulate the satellite formation problem that we study in this paper. The general orbit equation for two point masses m1 and m2 (cf. [9])

8 Output Feedback Control of a Leader-Follower Spacecraft Formation

µ ¨r + 3 r = 0 r

137

(8.10)

where r is the relative position of masses, r = |r|, and µ = G (m1 + m2 ), G being the universal constant of gravity, is the equation describing the uncontrolled orbit dynamics for a spacecraft under ideal conditions. This equation can be generalized to include force terms due to control input vectors from onboard actuators, aerodynamic disturbances, gravitational forces from other bodies, solar radiation, magnetic ﬁelds and so on. Accordingly, (8.10) can be expressed for the leader and follower spacecraft as ul fdl µ + rl + rl3 ml ml µ fdf uf ¨rf = − 3 rf + + rf mf mf ¨rl = −

where fdl , fdf ∈ R3 are the disturbance force terms due to external perturbation eﬀects and ul , uf ∈ R3 are the actuator forces of the leader and follower, respectively. In addition, spacecraft masses are assumed to be small relative to the mass of the Earth Me , so µ = GMe . Taking the second order derivative of the relative position vector p = rf − rl , and using the true anomaly ν (t) of the leader, which is the orbit plane angle measured in the center of the Earth between the orbit perigee point and the leader spacecraft center of mass, the relative position dynamics can be written as (cf. [10]) ¨ + C (ν) ˙ p˙ + D (ν, ˙ ν¨, rf ) p + n (rl , rf ) = U + Fd mf p where





 0 −ν˙ 0       C (ν) ˙ = 2mf  ν˙ 0 0   ∈ SS (3)     0 0 0 is a skew-symmetric matrix;





   D (ν, ˙ ν¨, rf ) p = mf    

µ rf3

− ν˙

2

ν¨ 0

−¨ ν µ rf3

− ν˙ 2 0

0    0  p  

µ rf3

may be viewed as a time-varying potential force; n (rl , rf ) = mf µ

1 rl − 2 , 0, 0 rf3 rl

;

(8.11)

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the composite disturbance force Fd is given by Fd = fdf −

mf fdl ml

and the relative control force U is given by U = uf −

mf ul . ml

For control design, we introduce the more convenient notation ¯ C (ν) ˙ =2mf ν˙ C µ ¯ ¯ + mf ν¨C D (ν, ˙ ν¨, rf ) =mf 3 I + mf ν˙ 2 D rf where

 0 −1 0      ¯ = 1 0 0 C       0 0 0

(8.13)









(8.12)

 −1 0 0      ¯ =  0 −1 0  . D       0 0 0

and

The rate of the true anomaly of the leader spacecraft is given by ν˙ (t) =

nl (1 + el cos ν (t))2

(8.14)

3

(1 − e2l ) 2

where nl = µ/a3l is the mean motion of the leader, al is the semimajor axis of the leader orbit, and el is the orbit eccentricity. Diﬀerentiation of (8.14) results in the rate of change of the true anomaly, ν¨ (t) =

3

−2n2l el (1 + el cos ν (t)) sin ν (t) (1 −

3 e2l )

.

When the leader spacecraft is revolving the Earth in an elliptical orbit, the true anomaly rate ν˙ (t) and true anomaly rate of change ν¨ (t) are bounded by constants, i.e. αν˙ ≤ ν˙ (t) ≤βν˙ |¨ ν (t)| ≤βν¨

∀ t ≥ t0 ≥ 0 ∀ t ≥ t0 ≥ 0

(8.15) (8.16)

where αν˙ , βν˙ , βν¨ > 0. The latter is a reasonable standing assumption that we make. Within the framework previously described we establish three results that, for the sake of clarity, are presented in order of increasing complexity: ﬁrstly, we assume that the leader’s true anomaly ν(t), true anomaly rate ν(t) ˙ and the orbital perturbations Fd are known; secondly, we relax these hypotheses by assuming that only bounds

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on ν(t), ν(t) ˙ and Fd are known; thirdly, we propose an adaptation algorithm to estimate these bounds. While the third case captures the most realistic situation and constitutes our main result, the ﬁrst two are interesting in their own right: from a practical viewpoint we make it clear that the property of asymptotic stability is lost due to the lack of measurements and, from a technical viewpoint, the proofs of the ﬁrst two cases allow to present that of the main result in a clearer and structured way.

8.4 Controller Design 8.4.1 Known Orbital Parameters Under the assumptions that the leader spacecraft is controlled to overcome external disturbances in an elliptic orbit, and the follower spacecraft has available measurements of relative position p, leader true anomaly rate ν˙ (t), true anomaly rate of change ν¨ (t) and orbital perturbations fdf , we have the following proposition: Proposition 8.1. Assuming that the desired relative position p∗ (t), desired relative ¨ ∗ (t) are all bounded functions, the velocity p˙ ∗ (t) and desired relative acceleration p origin of the system (8.11), in closed loop with the control law ¨∗ uf = − kp e − kd ϑ + n − fdf + D (ν, ˙ ν¨, rf ) p + C (ν) ˙ p˙ ∗ + mf p p˙ c = − aϑ ϑ =pc + be

(8.17) (8.18) (8.19)

where e = p − p∗ , and kp , kd , a and b are suﬃciently large constants, is uniformly globally asymptotically stable (UGAS). Proof of Proposition 8.1 , the closed-loop dynamics of the Denoting the state vector as x = e , e˙ , ϑ system in (8.11) and the controller (8.17)-(8.19) are ¨ = A (t, x) mf e

(8.20)

A (t, x) := −C (ν) ˙ e˙ − kp e − kd ϑ .

(8.21)

where

Diﬀerentiating (8.19) and inserting (8.18) results in ϑ˙ = p˙ c + be˙ = −aϑ + be˙ .

(8.22)

˙ ϑ) = (0, 0, 0) of the closed-loop system, the To prove UGAS of the origin (e, e, Lyapunov function candidate V1 (x) =

1 x P1 x 2

(8.23)

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is used, where





0   kp ε1 mf      P1 :=   ε1 mf mf −ε1 mf      kd 0 −ε1 mf b with ε1 ≥ 0 as a design variable. Evaluating the eigenvalues of the matrix P1 we obtain that V1 (x) is positive deﬁnite if ε21 ≤ min

kp kd , 2mf 2bmf

(8.24)

which holds for suﬃciently small ε1 . The Lyapunov function candidate also satisﬁes pm |x|2 ≤ V1 (x) ≤ pM |x|2

(8.25)

where pm and pM are the smallest and largest eigenvalue of P1 , respectively. The derivative of V1 (x) along the trajectories of (8.20) and (8.22) is kd ˙ ¨ + e kp e˙ + ϑ V˙ 1 (x) = (e˙ + ε1 e − ε1 ϑ) mf e ϑ + ε1 mf e˙ b

e˙ − ϑ˙

and insertion of (8.20) and (8.22) results in 1 V˙ 1 (x) = − x Q1 (ν˙ (t)) x 2 where



   Q1 (ν) ˙ :=    

 2ε1 kp I

ε1 C (ν) ˙

−ε1 C (ν) ˙

2ε1 mf (b − 1) I

−ε1 [kd − kp ] I

−ε1 C (ν) ˙ − ε1 mf aI

ε1 [kd − kp ] I

   ε1 C (ν) ˙ − ε1 mf aI  .   a 2 b − ε1 kd I

Using (8.15), the skew-symmetry property of C (ν) ˙ and Schur’s complement on the ˙ we obtain that the latter is positive deﬁnite when submatrices in Q1 (ν), kp (b − 1) ≥ 4mf βν2˙ a − ε1 (b − 1) ≥ ε1 4mf βν2˙ + mf a2 kd b a 2 kp kd ≥ ε1 (kd − kp ) + kp kd , b

(8.26) (8.27) (8.28)

hence, V˙ 1 (t, x) is negative deﬁnite. Moreover, in view of the boundedness of ν(t), ˙ there exists q1,m > 0 such that |Q1 (ν(t))| ˙ ≥ q1,m for all t ≥ 0; indeed, if the conditions (8.26)-(8.28) hold, we have q1,m > αq > 0, where αq is independent of

8 Output Feedback Control of a Leader-Follower Spacecraft Formation

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ν. ˙ The conditions (8.24) together with (8.26)-(8.28) guarantee that V is positive deﬁnite and V˙ is negative deﬁnite if ε1 ≤ min

kp 2mf

1 2

,

kd 2bmf

1 2

,

a akd 1 − 1b b kp kd , 2 2 (b−1) kd + 4mf βν˙ + mf a (kd −kp )2+ kp kd

and b≥1+

4mf βν2˙ . kp

Accordingly, when a, kp , kd > 0, then for any βν˙ > 0 there exists a b (βν˙ , mf , kp ) such that for any ν˙ (t) such that |ν˙ (t)| ≤ βν˙ for all t ≥ t0 ≥ 0, V1 (x) is positive deﬁnite and V˙ 1 (x) is negative deﬁnite. Therefore, the origin of the nonlinear system (8.11) in closed loop with the controller given by in (8.17)-(8.19) is UGAS (cf. [11, Theorem 4.9]). 8.4.2 Known Bounds on Leader True Anomaly We now relax the assumption that we know the values ν˙ (t) and ν¨ (t), and rather assume that we know the values of αν˙ , βν˙ and βν¨ on the leader true anomaly as given in (8.15) and (8.16). In addition, we relax the assumptions that orbital perturbations fdf are known, and rather assume that the perturbation term is bounded as |fdf | ≤ βf . Similarly, we relax the requirement on leader spacecraft control, and assume that the sum of forces working on the leader due to control thrust and external perturbations are bounded, such that |fdl + ul | ≤ βl . Finally, we also assume that the follower spacecraft has available measurements of relative position p. For these assumptions, we have the following proposition: Proposition 8.2. Assuming that the desired relative position p∗ , desired relative ¨ ∗ are all bounded functions, the system velocity p˙ ∗ and desired relative acceleration p (8.11), in closed loop with the control law given by (8.18), (8.19) and uf = −kp e −kd ϑ +n+ mf

µ ¯ + βν¨ C ¯ p + 2mf βν˙ C ¯ p˙ ∗ +mf p ¨∗ I + βν2˙ D rf3

(8.29)

where e = p − p∗ , and kp , kd , a and b are suﬃciently large constants, is uniformly globally practically asymptotically stable (UGPAS). Proof of Proposition 8.2 The closed-loop dynamics of the system in (8.11) and the controller (8.18), (8.19) and (8.29) are ¯ + δν¨ C ¯ (e + p∗ ) + C (ν) ¯ p˙ ∗ ¨ − mf δν˙ 2 D ˙ e˙ − 2mf δν˙ C mf e mf + (fdl + ul ) − fdf + kp e + kd ϑ = 0 ml

(8.30)

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where the denotations δν˙ = βν˙ − ν, ˙ δν¨ = βν¨ − ν¨ and δν˙ 2 = βν2˙ − ν˙ 2 have been used. Note that this closed-loop system is the same as (8.20) with an additional perturbation term G (t, e) := G1 (t, e) + G2 (p∗ (t) , p˙ ∗ (t))

(8.31)

consisting of the vanishing perturbation G1 (t, e) and the non-vanishing perturbation G2 (p∗ (t) , p˙ ∗ (t)), given by ¯ e ¯ + δν¨ C G1 (t, e) := mf δν˙ 2 D ¯ p˙ ∗ − ¯ p∗ + 2mf δν˙ C ¯ + δν¨ C G2 (p∗ (t) , p˙ ∗ (t)) := mf δν˙ 2 D

mf (fdl +ul )+ fdf . ml

Accordingly, the closed-loop system can be written as ¨ = A (t, x) + G (t, e) . mf e

(8.32)

By assumption, the desired relative position p∗ , relative velocity p˙ ∗ , follower orbital perturbations fdf and leader forces fdl + ul are all bounded, and using |p∗ | ≤ βp∗ , |p˙ ∗ | ≤ βp˙ ∗ , |fdf | ≤ βf , |fdl + ul | ≤ βl , and (8.15)-(8.16), we ﬁnd that |G| ≤ βG1 |e| + βG2 , where βG1 =mf [(βν˙ 2 − αν˙ 2 ) + 2βν¨ ] βG2 =mf [(βν˙ 2 − αν˙ 2 ) + 2βν¨ ] βp∗ + 2mf (βν˙ − αν˙ ) βp˙ ∗

mf βl + βf . + ml

(8.33) (8.34)

The vanishing part of the perturbation in (8.31) can be assimilated in A (t, x). To analyse the stability of the closed-loop system (8.32) we use the Lyapunov function (8.23), i.e. V2 (x) := V1 (x). The total derivative of V2 (x) along the trajectories of (8.22) and (8.30) yields ∂V1 1 V˙ 2 (x) = − x Q1 (ν) G (t, e) ˙ x+ 2 ∂x 1 ≤ − x Q2 (ν) ˙ x + Q0 (ν, ˙ p∗ , p˙ ∗ ) x 2 where ˙ = [qij ] , Q2 (ν)

i, j = 1, 2, 3

(8.35)

with submatrices given by ¯ q11= 2ε1 kp I−mf δν˙ 2 D q22= 2ε1 mf (b − 1) I a − ε1 kd I q33= 2 b and

¯ ¯ − δν¨ C q12= q21 = ε1 C (ν) ˙ − mf δν˙ 2 D ¯ ¯ ν¨ C q13= q = ε1 (kd −kp ) I+mf δν˙ 2 D−δ

(8.37)

q23= q32 = −ε1 C (ν) ˙ − ε1 mf aI

(8.38)

31

(8.36)

8 Output Feedback Control of a Leader-Follower Spacecraft Formation



143



ε1 mf [(δν˙ 2 + δν¨ ) p∗ + 2δν˙ p˙ ∗ ]       m   f Q0 (ν, ˙ p∗ , p˙ ∗ ) =  mf [(δν˙ 2 + δν¨ ) p∗ + 2δν˙ p˙ ∗ ] − ml (fdl + ul ) + fdf      −ε1 mf [(δν˙ 2 + δν¨ ) p∗ + 2δν˙ p˙ ∗ ]

.

(8.39)

In view of (8.33),(8.34) and the assumptions on the external forces fdl , ul and fdf it follows that there exists q0 > 0 independent of the control gains and the states, ˙ p∗ (t), p˙ ∗ (t))| ≤ q0 . On the other hand, the conditions for positive such that |Q0 (ν(t), deﬁniteness of Q2 are (8.27); and

ε21 (b − 1) (kp + mf δν˙ 2 ) ≥ 4ε21 mf βν2˙ + 4ε1 mf βν˙ δν¨ + mf δν2˙2 + δν2¨

(8.40)

a − ε1 kd (kp + mf δν˙ 2 ) ≥ ε1 m2f δν2¨ + (kd − kp + mf δν˙ 2 )2 . b

(8.41)

Thus, V˙ 2 (x) is negative deﬁnite if (8.24), (8.27), (8.40) and (8.41) hold and q0 |x| ≥ 2 (8.42) q2,m where q0 ≥ |Q0 (ν˙ (t) , p∗ (t) , p˙ ∗ (t))| and q2,m ≤ |Q2 (ν˙ (t))| for all t ≥ 0. Furthermore, to verify the conditions of Corollary 8.1 we exhibit a quadratic upper-bound on −x Q2 x. To that end, we use the formula 2|ab| ≤ a2 + b2 for any a, b ∈ R, to obtain x Q2 x ≥(q11,m − q12,M − q13,M )|x1 |2 + (q22,m − q12,M − q23,M )|x2 |2 + (q33,m − q13,M − q23,M )|x3 |2 , where qij,m and qij,M denote, respectively, lower and upper bounds on the induced norm of the submatrices qij . Choosing the gains kp , kd and b large enough so that q11,m ≥2(q12,M + q13,M ) q22,m ≥2(q12,M + q23,M ) q33,m ≥2(q13,M + q23,M ) , which is always possible due to the structure of the sub-matrices qij , we obtain x Q2 x ≥

1 (q11,m |x1 |2 + q22,m |x2 |2 + q33,m |x3 |2 ) . 2

That is, we can choose q2,m ≥ 21 min{q11,m , q22,m , q22,m }. Note that each of these terms can be arbitrarily enlarged by an appropriate choice of kp , kd and b. Thus, q q2,m can be enlarged accordingly and the attractive ball Bδ with δ := 2 0 , can be q2,m

arbitrarily diminished. We see that condition (8.4) holds. Furthermore, in view of (8.25) the Lyapunov function V2 (x) = V1 (x) also satisﬁes (8.3) and (8.4). Finally, since all feedback gains are scalar, conditions (8.5) and (8.6) also hold. We conclude that for the system (8.11), in closed loop with the control law (8.18), (8.19) and (8.29) Bδ is UGPAS.

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Remark 8.1. Note that if the leader spacecraft follows a circular orbit we have that ν˙ (t) is constant, and hence βν˙ = αν˙ = ν˙ in (8.15) and βν¨ = 0 in (8.16). Accordingly, the perturbation term in (8.31) is reduced to external perturbations and leader actuator force only. 8.4.3 Adaptation of True Anomaly Bounds We present now our main result. We relax the assumption that we know the bounds on the leader spacecraft variables ν˙ (t) and ν¨ (t). That is, assuming that the follower spacecraft has available measurements of relative position p, we have the following proposition: Proposition 8.3. Assuming that the desired relative position p∗ , desired relative ¨ ∗ are all bounded functions, consider velocity p˙ ∗ and desired relative acceleration p the control law given by (8.18), (8.19), uf = −kp e −kd ϑ +n+ mf

µ ¯ p˙ ∗ +mf p ¯ p + 2mf βˆν˙ C ¯ + βˆν¨ C ¨∗ I + βˆν2˙ D rf3

(8.43)

where e = p − p∗ , kp , kd , a and b are suﬃciently large, the update law ˆ˙ = H (t, x) = −ε2 Γ B (e, p∗ , p˙ ∗ ) (e − ϑ) φ

(8.44)

where Γ > 0 satisﬁes γm I ≤ Γ ≤ γM I, ε2 ≥ 0, and ¯ p˙ ∗ C ¯ (e + p∗ ) ¯ (e + p∗ ) 2C B (e, p∗ , p˙ ∗ ) = mf D

.

(8.45)

˜ 0 (t) := B (0, p∗ (t) , p˙ ∗ (t)) and let γM be suﬃciently small. Under these Deﬁne B conditions, the closed-loop system is uniformly semiglobally practically asymptotically stable (USPAS) if, moreover, B0 is persistently exciting, i.e. there exists µ,T > 0 such that for all t ≥ 0, t+T t

˜ 0 (τ ) B ˜ 0 (τ ) dτ ≥ µI . B

Remark 8.2. We stress that persistency of excitation has been shown to be also necessary for asymptotic stability for certain adaptive linear systems ( cf. [12]) and nonlinear systems (cf. [13]). Proof of Proposition 8.3 The closed-loop dynamics of the system in (8.11) and the controller structure (8.18)(8.19) and (8.43) are

8 Output Feedback Control of a Leader-Follower Spacecraft Formation

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¯ ¯ ¯ p˙ ∗ + mf (fdl +ul ) − fdf ¨ −mf δν˙ 2 D+δ mf e ˙ e˙ − 2mf δν˙ C ν ¨ C (e + p∗ ) + C (ν) ml ¯ + β˜ν¨ C ¯ (e + p∗ ) − 2mf β˜ν˙ C ¯ p˙ ∗ = 0 + kp e + kd ϑ − mf β˜ν2˙ D where we have introduced the adaptation errors β˜ν˙ = βˆν˙ − βν˙ , β˜ν2˙ = βˆν2˙ − βν2˙ and β˜ν¨ = βˆν¨ − βν¨ . Similar to (8.32), we now have ˜ ¨ = A (t, x) + G (t, e) + B (e, p∗ , p˙ ∗ ) φ mf e

(8.46)

where φ = βν2˙ , βν˙ , βν¨ . The proof consists in constructing a Lyapunov function that is positive deﬁnite and has a total derivative that is negative deﬁnite on compact sets of the state, H (δ, ∆). In other words, we aim at verifying Corollary 8.1. For the construction of such a Lyapunov function, we rely on the proofs of Propositions 8.1 and 8.2, as well as on [14]. The rationale is the following: First, notice that the ˜ = 0 corresponds to (8.32); hence, we use V2 as part closed-loop system (8.46) with φ of the new Lyapunov function. Also, note that (8.46) with the adaptation law (8.44) and G (t, e) = 0 has the structure of systems considered in [15, 14], where UGAS of the origin was proved. In particular, in the latter reference a Lyapunov function with negative deﬁnite derivative on compact sets was proposed. Thus, the Lyapunov function that we use to prove USPAS of (8.46) and (8.44) consists of V2 (x) plus a function inspired by [14], i.e. following the lines of the proof in [14], we deﬁne ˜ and consider the function W : R≥0 × Rq → R≥0 , given as χ = [x, φ] 1˜ ˜ − ε2 ˜ − ε2 mf e˙ B ˜ 0 (t) φ W (t, χ) = φ mf Γ −1 φ 2

∞ t

˜ 2 dτ . ˜ 0 (t) φ| e(t−τ ) |B

(8.47)

In order to exhibit some useful properties, for further developments, of the function W (t, χ) we stress the following: Note that since B (e, p∗ , p˙ ∗ ) is continuously diﬀer¨ ∗ are bounded by assumption, then there exists bM > 0 entiable, and p∗ , p˙ ∗ and p such that for all t ≥ 0 we have that ˜ 0 (t)|, | max |B

˜ 0 (t) dB | dt

≤ bM .

(8.48)

From (8.45), (8.21), (8.31) and (8.44) it follows that there exist constants ρ1 , ρ2 and ρ3 such that, for all t ≥ 0 and all x ∈ Rn , we have ˜ 0 (t)| ≤ ρ1 |e| ≤ ρ1 |x| |B (e, p∗ (t) , p˙ ∗ (t)) − B

(8.49) 2

|H (t, x)| ≤ ε2 γM ρ1 [|e| + |ϑ|] |e| + ε2 γM bM [|e|+|ϑ|] ≤ γM ρ2 |x| +|x|

(8.50)

|A (t, x) + G (t, e)| ≤ ρ3 |x|

(8.51)

where γM is the largest eigenvalue of Γ in (8.44). Note that in (8.50) we have assumed, without much loss of generality, that ε2 ≤ 1. From the assumption in Proposition 8.3 that B0 is persistently exciting, we obtain that ε2

∞ t

˜ 0 (τ ) φ| ˜ 2 dτ ≥ ε2 e(t−τ ) |B

t+T t

˜ 0 (τ ) φ| ˜ 2 dτ ≥ ε2 e−T µ|φ| ˜ 2. e(−T ) |B

(8.52)

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Thus, using all the above properties, we obtain that 1 ˜2 ˜ 2 + mf bM |x||φ| ˜ ≤ W (t, χ) ≤ 1 |φ| ˜ 2 + ε2 mf bM |x||φ| ˜ |φ| −ε2 b2M |φ| 2γM 2γm where γm is the smallest eigenvalue of Γ in (8.44). Accordingly, using (8.25), we have that for suﬃciently small ε2 there exist α1 > 0 and α2 > 0 such that α1 |χ|2 ≤ V2 (x) + W (t, χ) ≤ α2 |χ|2 . The derivative of (8.47) along the trajectories of the closed-loop system (8.46) and (8.44) is ˜T mf Γ −1 H (t, x) − ε2 (A (t, x) + G (t, e))T B0 φ ˜ ˙ (t, χ) =φ W ˜ − B0 φ ˜ − ε 2 Bφ − ε2

∞ t

T

˜ − ε2 mf e˙ T B0 φ

˜ 0 (τ )φ| ˜ 2 dτ − ε2 e(t−τ ) |B

∞ t

dB0 ˜ φ + B0 H (t, x) dt ˜T B0 (τ )T H(t, x)dτ . e(t−τ ) 2φ

Using V3 (t, χ) = V2 (x) + W (t, χ) we ﬁnd that 1 ∂V2 ˜ ˙ (t, χ) V˙ 3 (t, χ) = − x Q2 (ν) ˙ x + Q0 (ν, ˙ p∗ , p˙ ∗ ) x + Bφ + W 2 ∂x2 1 ˜ ˜ Γ B e| ˙ + ε2 bM ρ3 |x||φ| ≤ − q2,m |x|2 + q0 |x| + |φ 2 ˜ 2 + ε2 mf bM |e| ˜ + γM ρ2 |x|2 + γM ρ2 |x| ˙ |φ| + ε2 bM ρ1 |e||φ| ˜ 2 + 2ε2 bM γM ρ2 |φ| ˜ |x|2 + |x| − ε2 e−T µ|φ| where we have used (8.48)-(8.52). Moreover, using (8.48) and (8.49) we obtain ˜ Γ B −B ˜ Γ B e| ˜ ˙ ≤|φ |φ 0 ˜ + ˙ φ| ≤γM ρ1 |e||e||

˜ ΓB ˜ e| ˙ + |φ e| 0 ˙

γM b M ˜ 2 γM b M 2 |φ| + |x| . 2 2

Accordingly, we get 1 ˜ 2 + γM bM |x|2 ˜ + γM bM |φ| ˙ φ| V˙ 3 (t, χ) ≤ − q2,m |x|2 + q0 |x| + γM ρ1 |e||e|| 2 2 2 ˜ + ε2 bM ρ1 |e||φ| ˜ 2 + 2ε2 bM γM ρ2 |φ| ˜ |x|2 + |x| + ε2 bM ρ3 |x||φ| ˜ + γM ρ2 |x|2 + γM ρ2 |x| − ε2 e−T µ|φ| ˜2 ˙ |φ| + ε2 mf bM |e| Restricting the states to a ball B∆ , so that |χ| ≤ ∆, and collecting terms, we obtain 1 γM b M λ1 −γM (ρ1 + 2ε2 bM ρ2 + ε2 mf bM ρ2 ) ∆ − ε2 bM ρ3 V˙ 3 (t, χ) ≤− q2,m − 2 2 2 λ3 2 2 λ2 − ε2 bM ρ1 ∆ − mf bM |x|2 + [q0 + (2 + mf ) ε2 bM γM ρ2 ∆] |x| 2 2 1 b 1 ε2 γ M M ˜2 − − ε2 e−T µ − ε 2 b M ρ3 − ε2 bM ρ1 ∆ − 2 |φ| 2 2λ1 2λ2 2λ3

8 Output Feedback Control of a Leader-Follower Spacecraft Formation

147

where we have used 2|a||b| ≤ λ|a|2 + λ1 |b|2 . Similarly, assuming that δ ≤ |χ|, and choosing ε2 ≤ 1/∆, we get 1 V˙ 3 (t, χ) ≤ − q2,m |x|2 − η1 (q2,m , γM , δ, ∆) |x|2 4 ε2 ˜ 2 − η2 (γM , ∆) |φ| ˜2 − e−T µ|φ| 2

(8.53)

where 1 γM b M λ1 − γM ρ1 ∆ + (2 + mf ) bM ρ2 − b M ρ3 η1 (q2,m , γM , δ, ∆) := q2,m − 4 2 2∆ λ2 λ3 1 − bM ρ1 − m2f b2M + [q0 + (2 + mf ) bM γM ρ2 ] (8.54) 2 2 δ and η2 (γM , ∆) :=

1 1 1 −T γM b M 1 e µ− − b M ρ3 − b M ρ1 − . 2∆ 2 2λ1 ∆ 2λ2 2λ3 ∆2

(8.55)

Let q2,m (δ, ∆) be suﬃciently large so that η1 (q2,m , γM , δ, ∆) ≥ 0, and γM ≤ ε2 e−T

µ bM

so that η2 (γM , ∆) ≥ 0. Then, ε2 1 ˜2 ˜ ∈ [δ, ∆] V˙ 3 (t, χ) ≤ − q2,m |x|2 − e−T µ|φ| ∀ |x| ∈ [δ, ∆] , |φ| 4 2 √ which also holds for |χ| ∈ 2δ, ∆ . Notice that, as before, the matrix Q2 (ν), ˙ presented earlier in (8.35) and (8.36)-(8.38), contains all the controller gains, and q2,m can be enlarged arbitrarily. Proceeding as in Example 8.1, we can choose q2,m = c0 + c1 ∆ +

c2 δ

(8.56)

where c0 =2γM bM + 4 (2 + mf ) bM ρ2 + 2λ2 bM ρ1 + 2λ3 m2f b2M c1 =4γM ρ1 c2 =4q0 + 4 (2 + mf ) bM γM ρ2 + 2λ1 bM ρ 3 with use of ρ3 = ρ 3 ∆/δ to show the dependency of the terms in ρ3 on δ and ∆ through (8.21). This results in η1 (q2,m , γM , δ, ∆) ≥ 0, and from (8.56) it is seen that the conditions (8.5) and (8.6) hold; hence, from Corollary 8.1 we conclude that the closed-loop system (8.46) and (8.44), is USPAS.

8.5 Simulations In this section, simulation results for a leader-follower spacecraft formation are presented to illustrate the performance of the presented control laws. The leader spacecraft is assumed to be following an elliptic orbit with eccentricity e = 0.6. Both

148

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50

er eθ eh

0

−50

0

10

20

30

40

50

60

70

80

90

Vel error [m]

4

er eθ eh

2 0 −2 −4

0

10

20

30

40

50

60

70

80

90

400 Vel filter

100

100 ϑ1 ϑ2 ϑ3

200 0 −200 −400

0

10

20

30

40

50 Time [s]

60

70

80

90

100

Fig. 8.1. Position error, velocity error and velocity ﬁlter output for the case when the leader true anomaly rate and rate of change are known

spacecraft have mass m = 100 kg. The follower spacecraft is assumed to have available continuous thrust in all directions, limited to 27 N. The follower has initial values p = [20, 10, −20] , and is further commanded to track sinusoidal trajectories around the leader, given as p∗ (t) = −10 cos

3π t , 10 sin To

4π t , 5 cos To

5π t To

where To is the orbital period of the leader spacecraft. A possible scenario for this motion is in-orbit inspection, where the follower moves in orbit around the leader. In all simulations performed, we used the controller gains kp = 3, kd = 5, a = 1 and b = 5. In the adaptive controller we used ε1 = 0.3 and Γ = diag([3.3 · 10−8 , 4.5 · 10−6 , 3.3 ·10−8]). Orbital perturbation forces due to gravitational perturbations and aerodynamic drag are included in the simulations. 8.5.1 Results The result from simulating the system (8.11) in closed loop with the controller (8.17)(8.19) is shown in Fig. 8.1. This is the case where the leader spacecraft true anomaly and rate of change are known to the follower spacecraft. The follower settles and tracks the desired trajectory without errors in relative position and relative velocity. The results for the case where only the upper bounds on the leader true anomaly rate and rate of change are known, are presented in Fig. 8.2. The UGPAS property of the closed-loop system is seen in the ﬁgure as persistent oscillations around the origin. However, it should be noted that the magnitude of the oscillations can be arbitrarily diminished by increasing the controller gains. Figure 8.3 shows the performance of the adaptive control law (8.18),(8.19) and (8.43) with the parameter update law

8 Output Feedback Control of a Leader-Follower Spacecraft Formation Pos error [m]

50

0

−50

Vel error [m]

er eθ eh

0

0

20

40

60

80

4

100

er eθ eh

2 0

−0.05 2000 −5 x 10 5

3000

4000

5000

er eθ eh

0

−2 −4

0

20

40

60

80

400

100

−5 2000

0

−200

−0.05 0

20

40 60 Time [s]

80

4000

5000 ϑ1 ϑ2 ϑ3

0.05

0

−400

3000

0.1

ϑ1 ϑ2 ϑ3

200

Vel filter

0.05

er eθ eh

149

100

−0.1 2000

3000

Time [s]

4000

5000

Fig. 8.2. Position error, velocity error and velocity ﬁlter output for the case when only the bounds on the leader true anomaly rate and rate of change are known Pos error [m]

50 0 −50

0

20

40

60

80

Vel error [m]

4 0

er eθ eh

0

100

er eθ eh

2

−0.05 2000 −5 x 10 5

3000

4000

5000

er eθ eh

0

−2 −4

0

20

40

60

80

400

Vel filter

0.05

er eθ eh

100 ϑ1 ϑ2 ϑ3

200 0

−5 2000 0.1

3000

4000

5000 ϑ1 ϑ2 ϑ3

0

−200 −400

Adap error

4

−3 20 0 x 10

40

60

80

100

φ˜1 φ˜2 φ˜3

2 0

−0.1 2000 −4 x 10 2

3000

4000

5000 φ˜1 φ˜2 φ˜3

0

−2 −4

0

20

40 60 Time [s]

80

100

−2 2000

3000

Time [s]

4000

5000

Fig. 8.3. Position error, velocity error, velocity ﬁlter output and adaptation error for the case when adaptation has been used for the bounds on the leader true anomaly rate and rate of change

(8.44). The follower spacecraft settles at the desired trajectory and proceeds to track the trajectory. The parameter errors converge to a small ball around zero. Note that the same controller gains have been used in Figs. 8.2 and 8.3, however, the errors in position and velocity are smaller when adaptation is used.

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8.6 Conclusion We have presented a solution to the problem of tracking relative translation in a leader-follower spacecraft formation using feedback from relative position only, using an approximate diﬀerentiation ﬁlter of the position error to provide suﬃcient knowledge about the velocity error. In addition, the problem with knowledge of the leader spacecraft true anomaly has been solved using a scheme based on constant upper bounds and adaptation of these bounds. Three diﬀerent controller conﬁgurations have been presented for diﬀerent levels of knowledge about the leader orbit, and the resulting stability properties of the closed-loop systems left by these controller conﬁgurations have been derived.

Acknowledgment This work was partially supported through a European Community Marie Curie Fellowship, and in the framework of the CTS Fellowship Program, under contract number HPMT-CT-2001-00278.

References 1. M. de Queiroz, Q. Yan, G. Yang, and V. Kapila, “Global output feedback tracking control of spacecraft formation ﬂying with parametric uncertainty,” in Proceedings of the IEEE Conference on Decision and Control, Phoenix, AZ, 1999. 2. Q. Yan, G. Yang, V. Kapila, and M. de Queiroz, “Nonlinear dynamics and output feedback control of multiple spacecraft in elliptical orbits,” in Proceedings of the American Control Conference, Chicago, Illinois, 2000. 3. M. de Queiroz, V. Kapila, and Q. Yan, “Adaptive nonlinear control of multiple spacecraft formation ﬂying,” AIAA Journal of Guidance, Control and Dynamics, vol. 23, no. 3, pp. 385–390, 2000. 4. H. Pan and V. Kapila, “Adaptive nonlinear control for spacecraft formation ﬂying with coupled translational and attitude dynamics,” in Proceedings of the Conference on Decision and Control, Orlando, FL, 2001. 5. H. Wong, V. Kapila, and A. G. Sparks, “Adaptive output feedback tracking control of spacecraft formation,” International Journal of Robust & Nonlinear Control, vol. 12, no. 2-3, pp. 117–139, 2002. 6. H. Wong, H. Pan, and V. Kapila, “Output feedback control for spacecraft formation ﬂying with coupled translation and attitude dynamics,” in Proceedings of the American Control Conference, Portland, OR, 2005. 7. R. Kelly, “A simple set-point robot controller by using only position measurements,” in Proceedings of the IFAC World Congress, Sydney, Australia, 1993. 8. A. Chaillet and A. Lor´ıa, “Uniform semiglobal practical asymptotic stability for non-autonomous cascaded systems and applications,” 2006, http://arxiv.org/PS cache/math/pdf/0503/0503039.pdf. 9. R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, ser. AIAA Education Series. Reston, VA: American Institute of Aeronautics and Astronautics, 1999.

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10. R. Kristiansen, E. I. Grøtli, P. J. Nicklasson, and J. T. Gravdahl, “A 6DOF model of a leader-follower spacecraft formation,” in Proceedings of the Conference on Simulation and Modeling, Trondheim, Norway, 2005. 11. H. K. Khalil, Nonlinear Systems, third edition. Upper Saddle River, New Jersey, USA: Pearson Education, 2002. 12. K. Narendra and A. Annaswamy, Stable Adaptive Systems. New Jersey: Prentice-Hall, Inc., 1989. 13. R. Ortega and A. L. Fradkov, “Asymptotic stability of a class of adaptive systems,” International Journal of Adaptive Control and Signal Processing, vol. 7, pp. 255–260, 1993. 14. A. Lor´ıa, R. Kelly, and A. R. Teel, “Uniform parametric convergence in the adaptive control of mechanical systems,” European Journal of Control, vol. 11, pp. 1–14, 2005. 15. A. Lor´ıa, E. Panteley, D. Popovi´c, and A. Teel, “δ-persistency of excitation: A necessary and suﬃcient condition for uniform attractivity,” in Proceedings of the 41th Conference on Decision and Control, Las Vegas, CA, USA, 2002.

9 Coordinated Attitude Control of Satellites in Formation T.R. Krogstad and J.T. Gravdahl Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway [email protected], [email protected] Summary. In this chapter we derive a coordinated control scheme to control relative attitude of a formation of satellites using methods from nonlinear control theory. We ﬁrst develop control laws for the leader satellite, and then propose an adaptive synchronizing control law for the follower. Global convergence of the synchronization errors are proven mathematically and the resulting controllers are simulated in presence of environmental disturbances and measurement noise.

9.1 Introduction Formation ﬂying missions and missions involving the coordinated control of several autonomous vehicles have been areas of increased interest in later years. This is due to the many inherent advantages the distributed design adds to the mission. By distributing payload on several spacecraft, redundancy is added to the system, minimizing the risk of total mission failure, several cooperating spacecraft can solve assignments which are more diﬃcult and expensive, or even impossible to do with a single spacecraft, and the launch costs may be reduced since the spacecrafts may be distributed on more inexpensive launch vehicles. The disadvantage is the requirement for a fully autonomous vehicle, as controlling the spacecraft in close formation is only possible using control. This results in stringent requirements on the control algorithms and measurement systems. Several control objectives can be deﬁned depending on the speciﬁc mission of the formation. Missions may be divided into Earth observation and Space observation. Earth observation missions include missions such as Synthetic Aperture Radar (SAR) missions, where the use of a formation increases the achievable resolution of the data. In SAR missions the control objective is usually to point the payload at the same location on Earth, involving keeping the relative attitude either constant or tracking a time-varying signal depending on formation conﬁguration and satellite orbit. Examples of planned missions are TanDEM-X [1] and SAR-Lupe [2]. Space observation missions focus on astronomical and astrophysical research outside our solar-system. The control objective usually involves keeping a constant absolute at-

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 153–170, 2006. © Springer-Verlag Berlin Heidelberg 2006

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titude in an inertial stellar system and to keep relative attitudes ﬁxed. Examples of planned missions are XEUS [3] and DARWIN [4]. Noticeable contributions on formation control may be divided in to three separate approaches; leader-follower, behavioural and virtual structure. In the leader-follower strategy, one spacecraft is deﬁned as the leader of the formation while the rest are deﬁned as followers. The control objective is to enable the followers to keep a ﬁxed relative attitude with respect to the leader [5, 6, 7, 8]. The behavioural strategy views each vehicle of the formation as an agent and the control action for each agent is deﬁned by a weighted average of the controls corresponding to each desired behaviour for the agent. This approach eases the implementation of conﬂicting or competing control objectives, such as tracking versus avoidance. It is however diﬃcult to enforce group behaviour, and to mathematically guarantee stability and formation convergence. In addition, unforseen behaviour may occur when goals are conﬂicting. This strategy is widely reported for use on mobile robots [9, 10, 11], and was also applied to spacecraft formations in [12]. In the virtual structure approach, the formation is deﬁned as a virtual rigid body. In this approach the problem is how to deﬁne the desired attitude and position for each member of the formation such that the formation as a whole moves as a rigid body. In this scheme it is easy to prescribe a coordinated group behaviour and to maintain the formation during maneuvers. It is however dependent on the performance of the individual control systems of each member. This approach was used on mobile robots in [13] and more recently on spacecraft formations in [14, 15]. In this chapter we design a coordinated control scheme, referred to as external synchronization, based on theory derived by [16]. This may be viewed as a version of the leader-follower approach, where one designs interconnections, virtual or physical, between designated leaders and followers. We visualize and evaluate the performance of the controllers by applying them to a satellite formation consisting of two micro-satellites. The satellites are actuated by means of reaction wheels and magnetic torquers for 3-axis attitude control and use thrusters for position control. Attitude is assumed measured at all times, with an accuracy of 0.001 degrees in all axis. The measurement signal is assumed to be noise contaminated. The angular velocity is assumed to be either estimated or measured using a gyroscope.

9.2 Modelling In this section we derive the equations of motion for a satellite actuated by means of reaction wheels and magnetic torquers, using the notation of [20, 21] and [22]. 9.2.1 Reference Frames Equations of motion will be expressed in three diﬀerent reference frames, illustrated in Fig. 9.1. A general reference frame will be denoted as F with a subscript corresponding to a given frame.

9 Coordinated Attitude Control of Satellites in Formation

155

ECI - Earth-centered inertial frame This reference frame has its origin in the center of the Earth with the xi -axis pointing in the vernal equinox direction, Υ . This is in the direction of the vector from the center of the Sun through the center of the Earth during vernal equinox. The yi -axis points 90◦ east, spanning the equatorial plane together with the xi -axis. The zi -axis points through the North Pole, completing the right-hand system. In the following this frame will be denoted by Fi Orbit-ﬁxed reference frame This frame, denoted Fo , has its origin in the satellite’s center of gravity. The zo -axis points in the nadir direction. The yo -axis points in the direction of the negative orbit normal. The xo -axis is chosen as to complete a right-hand coordinate system. Body-ﬁxed reference frame As the Fo frame, this reference frame also has its origin in the satellite’s center of gravity, with the axes pointing along the satellites principal axes of inertia. The frame is denoted Fb . In the control design we denote the body frame of the leader and follower satellites as Fl and Ff respectively.

yo yb

Orbit

zb zo yi +

xi Earth center Fig. 9.1. Illustration of reference frames. The completing axis of each system points out of the paper.

9.2.2 Kinematics We describe the attitude kinematics in the form of Euler parameters, η and ε, which may be deﬁned from the angle-axis parameters θ and k as θ η = cos , 2

θ ε = k sin , 2

and which has the corresponding rotation matrix

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R(η, ε) = I3×3 + 2ηε× + 2ε× ε×, ×

(9.1)

×

where denotes the vector cross product operator, and ε is skew-symmetric. The choice of Euler parameters is motivated by their nonsingular properties. To describe a rotation between frames Fa and Fb , we use the notation ηab and εab . From the properties of the rotation matrix, it can be shown that the kinematic diﬀerential equation is i × i b × ˙ ib = (ωib ) Rb = Rib (ωib ) , (9.2) R b where ωib is the angular velocity of the body frame Fb with respect to the inertial frame Fi , with coordinates given in Fb and Rib is the rotation matrix from Fb to Fi Using (9.1) and (9.2) the kinematic diﬀerential equations b η˙ ib = − 21 εTib ωib

ε˙ib =

1 2 [ηib I3×3

(9.3a) +

b ε× ib ]ωib ,

(9.3b)

can be derived. Given the quaternion vector   qib

ηib   ,   εib

we may write the (9.3) in compact form b , q˙ib = 12 Q(qib )ωib

where Q(qib )

(9.4)

   

 −εTib ηib I3×3 +

ε× ib

  

(9.5)

Euler angles, or roll-pitch-yaw angles, have been applied in the visualization of results, since these are easier to relate to physical motion. Fig. 9.2 illustrates a rotation from Fa to Fb in Euler angles. 9.2.3 Kinetics The kinetic diﬀerential equations, relates the change of angular momentum to the applied control and disturbance torques. In this section we derive the kinetic diﬀerential equations for a rigid body satellite actuated by means of redundant reaction wheels and magnetic torquers, using vectorial mechanics and the Newton-Euler formulation. This system of a rigid body and the wheels may be referred to as a gyrostat [21], that is a mechanical system that consists of a rigid body R and one or more (symmetrical) rotors whose axes are ﬁxed in R, and which are only allowed to rotate with respect to R about their axes of symmetry. We start by writing the total angular momentum of the gyrostat in Fb and the total axial angular momentum of the wheels b + AIs ωs hb = Jωib

(9.6a)

b + Is ωs , ha = Is AT ωib

(9.6b)

9 Coordinated Attitude Control of Satellites in Formation

157

za

zb φ θ

xa ψ

yb θ

φ

xb

ψ ya

Fig. 9.2. The ﬁgure shows how the rotation between two frames can be interpreted in form of Euler angles.

where A ∈ R3×4 is a matrix of wheel axes in Fb given by, [23], Is ∈ R4×4 a diagonal matrix of wheel axial inertias, ωs ∈ R4 a vector of wheel velocities and J ∈ R3×3 the total moment of inertia. The A-matrix is dependent on the geometric placement of the wheels. In [23], four wheels in tetrahedron conﬁguration were employed. In that case, the A-matrix is given by      A=   

1 3 2 3

0

1 3 2 3

− 0

1 3

−

−

0 −

2 3

1 3

  0  ,  

(9.7)

2 3

which will also be used here. Assuming that the body system coincides with the center of mass, we may obtain an expression for the change of angular momentum as ˙ b = (hb )× J ¯ −1 (hb − Aha ) + τ b h e ˙ a = τa , h

(9.8a) (9.8b)

¯ ∈ R3×3 is an inertia-like matrix deﬁned as where J ¯ J

J − AIs AT

and τeb is the resultant external torques due to magnetic control and environmental disturbances b + τgb + τdb , (9.9) τeb = τm b and τgb are magnetic where τdb is a vector of unknown disturbance torques, and τm control torques and gravitational disturbance torques, respectively, given by

158

T.R. Krogstad and J.T. Gravdahl b b × ) Jzo3 τgb = 3ω02 (zo3 b τm

b ×

(9.10)

b

= (m ) B (t),

(9.11)

b where ω0 is the orbital rotation rate, zo3 is the Earth pointing vector, mb is the magnetic moment generated by the actuators and B b (t) is the local geomagnetic ﬁeld vector. Magnetic control torques are employed to desaturate the reaction wheels. This is a required part of a control system based on reaction wheels, as angular momentum will build up due to non-cyclic environmental torques over the course of an orbit. This leads to an steady increase of wheel speeds, until the saturation level is reached and no more control is achieved. See [24] for an example of how this can be done. The gravitational torque model (9.10) assumes a circular orbit. Equation (9.8) may also be expressed in terms of angular velocities as b b × b b = (Jb ωib ) ωib + (AIs ωs )× ωib − Aτa + τe Jb ω˙ ib

Is ω˙ s = τa −

b Is Aω˙ ib

(9.12a) (9.12b)

9.3 External Synchronization Design In this synchronization scheme the leader satellite is controlled separately, either by a tracking or a stabilizing controller. The goal is to design feedback interconnections from the leader to the follower, in such a way that the follower synchronizes its orientation with that of the leader. The design is done by ﬁrst designing controllers for the leader, then designing a synchronizing controller for the follower. 9.3.1 Leader Controller Three diﬀerent controllers are proposed for the control of the leader; set-point control in Fi , set-point control in Fo and ﬁnally trajectory tracking in Fi . Set-point in Fi If the leader is to point in a speciﬁc direction in inertial space, as in the case of space-based interferometry missions as DARWIN and XEUS, a set-point stabilizing controller is suﬃcient. We deﬁne the desired satellite orientation in the inertial frame, as the quaternion as qid and the desired angular velocity as zero, resulting in the error variables qe

−1 ⊗ qil qdl = qid

ωe

l ωdl

=

ωill

−

l ωid

(9.13) =

ωill

−0=

ωill .

(9.14)

By writing the dynamics as q˙il = 12 Q(qil )ωill ,

Jl ω˙ ill = (Jl ωill + AIs,l ωs,l )× ωill − Aτa,l + τg,l ,

(9.15a) (9.15b)

9 Coordinated Attitude Control of Satellites in Formation

159

the error-dynamics may be written q˙e = 21 Q(qe )ωe

(9.16a)

Jb ω˙ e = (Jb ωe + AIs,l ωs,l )× ωe − Aτa,l + τg,l

(9.16b)

Proposition 9.1. The leader satellite, with dynamics (9.15b)-(9.15a), and errordynamics (9.16a)-(9.16b), with control law given by τa,l = −A† −τg,l − kd ωe + kp

dH(ηe ) εe , dηe

(9.17)

where H(·) is a scalar function satisfying

! H(·) : [−1; 1] → R+ (non-negative) ! H(−1) = 0 or/and H(1) = 0 ! H(·) is assumed C 1 , with bounded derivatives which in the following this function is chosen as H(ηe ) = 1 − |ηe |, with derivative −

dH (ηe ) = sgn(ηe ) = dηe

1, −1,

(9.18) if x ≥ 0 , if x < 0

(9.19)

has a globally asymptotically stable (GAS) origin (ωe , y) = (0, 0), where y = col(1 − |ηe |, εe ). Proof. To prove the proposition we choose the Lyapunov function candidate V = 12 ωeT Jl ωe + 2kp H(ηe ),

(9.20)

which is positive deﬁnite, zero in the origin and radially unbounded. The timederivative along the trajectories is given by dH(ηe ) V˙ = ωeT Jl ω˙ e + 2kp η˙ e dηe

(9.21)

= ωeT (Jl ωill + AIs,l ωs,l )× ωill − Aτa,l + τg,l − kp = ωeT −Aτa,l + τg,l − kp

dH(ηe ) εe dηe

dH(ηe ) T εe ω e dηe

(9.22) (9.23)

If we now select the control input (9.17), we get V˙ = −kd ωeT ωe ≤ −kd ωe

2 2

≤ 0.

(9.24)

Since this is a time-invariant system we have fulﬁlled the criteria of LaSalle’s invariance principle [17], and have convergence to the region E = x ∈ Ωc |V˙ = 0 ,

(9.25)

where Ωc = {x ∈ R7 |V ≤ c} and x = col(ωe , 1 − |ηe |, εe ). We can show that the largest invariant set M in E is the origin, and hence we have global asymptotic stability.

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Set-point in Fo If the leader satellite is supposed to be stabilized to a ﬁxed attitude in Fo rather than Fi , as is the case in some Earth observation missions, we have to solve the problem as a tracking problem in the inertial frame. We now deﬁne the desired attitude and angular velocities as qid

qio (t) ⊗ qod

(9.26)

l ωid

o Rlo ωio ,

(9.27)

o where qod is the desired oﬀset from nadir, ωio is the orbit angular velocity and qio (t) is the quaternion describing the orientation of Fo in Fi . Assuming a circular orbit this quaternion is periodic in time, with period equal to the orbit-period. This results in the error-variables

−1 −1 (t) ⊗ qil ⊗ qio qe = qod

ωe =

ωill

−

(9.28)

o , Rlo ωio

(9.29)

and error-dynamics l × l o Jl ω˙ e = (Jl ωill + AIs,l ωs,l )× ωill − Aτa,l + τg,l + Jl (ωol ) Ro ωio

q˙e = 12 Q(qe )ωe

(9.30a) (9.30b)

Proposition 9.2. The satellite leader, with dynamics (9.15b)-(9.15b) and errordynamics (9.30a)-(9.30b), where the control is given by o l × l o − Jl (ωol ) Ro ωio τa,l = −A† − τg,l − (Jl ωill + AIs,l ωs,l )× Rlo ωio

+kp

dH(ηe ) εe − kd ωe dηe

(9.31)

has a uniformly globally asymptotically stable (UGAS) origin (ωe , y) = (0, 0), where y = col(1 − |ηe |, εe ), under the assumption of a ﬁxed desired attitude in the orbit frame. Proof. We prove the proposition by choosing the Lyapunov function candidate (9.20). Taking the time-derivative along the trajectories of the solution to (9.30) and selecting the control as (9.31), we obtain V˙ = ωeT (Jl ωill + AIs,l ωs,l )× ωe − kd ωe = kd ωeT ωe ≤ 0,

(9.32)

where we have used property aT (b)× a = 0. Since the error-dynamics (9.30a)-(9.30b) is periodic, and the Lyapunov function candidate is periodic with the same period, V is positive deﬁnite and radially unbounded Krasovskii-LaSalle’s theorem [25] is applicable and the origin of the error-dynamics is proven to be UGAS.

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Trajectory tracking in Fi We now show stability of a nonlinear state feedback controller for tracking in the inertial frame. That is, given a smooth trajectory qid (t) ∈ C 2 , such that q˙id (t), q¨id (t) are well deﬁned for t ≥ 0, we obtain uniform global asymptotic stability of the tracking error. Let the desired angular velocity and acceleration be given by d ωid = 2Q(qid )T q˙id

d = 2QT (qid )q¨id . and ω˙ id

(9.33)

Further, we deﬁne the tracking errors as ωe

d l = ωill − Rld ωid ωdl

(9.34)

qe

−1 qid (t)

(9.35)

qdl =

⊗ qil ,

such that the error-dynamics can be written q˙e = 12 Q(qe )ωe . Jl ω˙ e =

(Jl ωill

(9.36a) ×

+ AJs,l ωs,l )

ωill

×

− Aτa,l + τg,l + Jl (ωe )

d Rld ωid

−

d Jl Rld ω˙ id

(9.36b)

Proposition 9.3. Given the smooth continuous trajectory qid (t) ∈ C 2 , errordynamics (9.36), and control input l d − Jl (ωe )× Rld ωid τa,l = −A† − τg,l − (Jl ωill + AIs,l ωs,l )× Rld ωid d +Jl Rld ω˙ id − kp sgn(ηe )εe − kd ωe ,

(9.37)

the origin (ωe , y) = (0, 0), where y = col(1 − |ηe |, εe ), is uniformly globally asymptotically stable, UGAS. Proof. Since the trajectory is an explicit function of time, we now have to deal with a nonlinear time-varying system (NLTV). This adds some diﬃculties to the analysis of the error-dynamics, as the invariance principle due to LaSalle [17] is no longer applicable in the case of a semi-deﬁnite Lyapunov derivative. A common solution to this is to use the convergence theorem due to Barbalat, but one cannot conclude asymptotic stability. We instead prove the proposition using the generalized Matrosov theorem [18], as given in [19], where four assumptions have to be satisﬁed in order to conclude uniform global asymptotic stability. The theorem may be summarized as Theorem 9.1 (Generalized Matrosov’s theorem). Under the following assumptions the origin of the system x˙ = f (t, x) is UGAS. Assumption 9.1. The origin of the system is UGS.

(9.38)

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Assumption 9.2. There exist functions Vi and φ such that max{|Vi (t, x)|, |φ(t, x)|} ≤ µ, V˙ i (t, x) ≤ Yi (x, φ(t, x)).

(9.39)

Assumption 9.3. Each Yi ≤ 0 where the previous Yi s are zero. Assumption 9.4. The only point where all the Yi (z, ψ)s are zero is the origin. Where (z, ψ) ∈ B(∆) × B(µ). Satisfying Assumption 9.1 Choosing the Lyapunov function V = 21 ωeT Jl ωe + kp y T y,

(9.40)

with time derivative V˙ = ωeT Jl ω˙ e + kp sgn(ηe )εTe ωe =

(9.41)

kp sgn(ηe )εe − (Jl ωill + AIs,l ωs,l )× ωill d d Jl (ωe )× Rld ωid , + Jl Rld ω˙ id

ωeT −

+ Aτa − τg,l

(9.42) (9.43)

and inserting for (9.37), results in V˙ = −kd ωeT ωe ≤ 0,

(9.44)

which guarantees uniform global stability (UGS) for the error-dynamics, satisfying Assumption 1 in [19]. Remark 9.1. From this result it is possible to show asymptotic convergence as in [26], by using Barbalat’s Lemma and showing that convergence of ωe leads to convergence of εe . Satisfying Assumption 9.2 Since the origin is UGS, ω˙ e , ωe , y are bounded functions of time. For i = 1 we choose V1

V

(9.45)

φ1

0

(9.46)

Y1

−β ωe ≤ 0

(9.47)

V1 is continuously diﬀerentiable and bounded, φ1 is continuous and bounded, and ﬁnally Y1 is continuous and hence Assumption 2 in [19] is satisﬁed for i = 1. For i = 2, we choose V2

ωeT Ib εe ηe

φ2

ω˙ e

Y2

ηe φT2 Ib εe

(9.48) (9.49) +

ηe ωeT Ib ε˙ e

+

η˙e ωeT Ib εe

(9.50)

9 Coordinated Attitude Control of Satellites in Formation

163

Since ω˙ e , ωe , y, η˙e are bounded functions of time, V2 ,φ2 and Y2 are bounded. Moreover, V2 is continuously diﬀerentiable, and φ2 and Y2 are continuous in their arguments. Hence, Assumption 2 in [19] is satisﬁed for i = 2. Satisfying Assumption 9.3 Y1 ≤ 0 for all ωe ∈ R3 , satisfying Assumption 3 in [19] for i = 1. Moreover, Y1 = 0 ⇒ ωe = 0 ⇒ Y2 = ηe φT2 Ib εe

(9.51)

Inserting for φ2 and ωe = 0, gives Y2 = −kp ηe sgn(ηe )εTe εe = −kp |ηe |εTe εe ≤ 0.

(9.52)

Thus, Assumption 9.3 has been satisﬁed for both i ∈ {1, 2}. Satisfying Assumption 9.4 It can now be seen that {Y1 = 0, Y2 = 0} ⇒ ωe = 0, εe = 0 ⇒ 1 − |ηe | = 0,

(9.53)

satisfying Assumption 4 in [19] for i ∈ {1, 2}. Remark 9.2. This hold as long as ηe is diﬀerent from zero. Using UGS property of Assumption 9.1 and that ηe = 0 is an unstable equilibrium point when using the given deﬁnition of signum, as shown in [26], the condition is met by requiring ηe to initially be diﬀerent from 0. The Assumptions of Matrosov’s Theorem are satisﬁed, and we may conclude uniform global asymptotic stability.

9.3.2 Adaptive Synchronizing Controller In this section we derive a synchronizing controller, to synchronize the attitude of the satellites in the formation. We have considered the two-satellite formation problem, as it appears in applications such as SAR missions and XEUS. We assume that the absolute angular velocity is measured relative to Fi and that the relative attitude is measured with the required accuracy for mission speciﬁcations. For XEUS the speciﬁcations requires relative attitude knowledge with such accuracy that normal methods using star-trackers and inertial navigation is insuﬃcient, instead some form of laser-metrology is required. In some cases the inertia matrix may be unknown or poorly known, and it may also change over time due to mass expulsion when ﬁring thrusters. Assuming the parameters are constant or slowly varying, we may remedy the lack of information using an adaptive controller. We ﬁrst assume perfect model knowledge and design a controller for this scenario. This controller is extended with a parameter update law in the following step. The design method is based on vectorial integrator backstepping as was done in [27] for ships and in [28] for a satellite actuated by means of thrusters. A similar approach was also designed in [29] using passivity arguments.

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We ﬁrst deﬁne the synchronization measure s as a linear parametrization of the angular velocity synchronization error and the quaternion synchronization error, s

ωse + λe,

(9.54)

f and e = εlf are deﬁned by where ωse = ωlf

qlf

f f and ωlf = ωif − ωilf .

qil−1 ⊗ qif

Further, we deﬁne ωr as

ωilf − λe,

(9.56)

f s = ωif − ωr .

(9.57)

ωr

such that we may write

(9.55)

ωr may be viewed as a virtual reference trajectory. Deﬁning the parametrization f f × )θ = Jf ω˙ r − (Jf ωif ) ωr , Y(ω˙ r , ωr , ωif

(9.58)

where, if Jf is diagonal, 



 ω˙ r1 −ω2 ωr3 ω3 ωr2      f  )= Y(ω˙ r , ωr , ωif ω1 ωr3 ω˙ r2 −ω3 ωr1  ,     ω1 ωr2 −ω2 ωr1 ω˙ r3

(9.59)

is the so called regressor matrix, with ω˙ ri , ωri and ωi being the components of ω˙ r , f and ωr and ωif θ = [ixx , iyy , izz ]T

(9.60)

the parameter vector containing the diagonal elements of the inertia matrix. A differential equation in terms of s and qe may be deﬁned as

Jf s˙ −

f (Jf ωif

q˙e = 12 Q(qe )(s − λεe ) ×

(9.61a) ×

+ AIs,f ωs,f ) s = −Aτa,f + τg,f + (AIs,f ωs,f ) ωr − Yθ. (9.61b)

Proposition 9.4. Given the dynamics (9.61a)-(9.61b), and the control τa,f selected as f )θ − (AIs,f ωs,f )× ωr − τg,f − Kd s − εe . τa,f = A† −Y(ω˙ r , ωr , ωif

(9.62)

Then, the origin (s, εe ) = (0, 0) is uniformly globally exponentially stable (UGES). This implies exponential convergence of ωe to 0 and to the convergence of ηe to 1.

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Proof. To prove UGES of the origin we start with the subsystem (9.61a). The control Lyapunov function for the ﬁrst subsystem is chosen as V1 = εTe εe + (1 − ηe )2 .

(9.63)

Calculating the time-derivative of (9.63) along the solution trajectories of (9.61a) results in V˙ 1 = 2εTe ε˙e − 2η˙ e + 2ηe η˙ e = =

(9.64)

εTe (ηe I3×3 − (εe )× )(s −λεTe εe + sT εe .

− λεe ) −

εTe (s

− λεe ) −

ηe εTe (s

− λεe )

(9.65) (9.66)

In the next step a Control Lyapunov function is selected as V2 = 21 sT Jl s + V1 ,

(9.67)

with time-derivative along the solution trajectories of (9.61b) given by V˙ 2 = sT Jf s˙ + V˙ 1 T

=s

f (Jf ωif

(9.68) ×

×

− AIs,f ωs,f ) s + (AIs,f ωs,f ) ωr − Aτa,f + τg,f

f , ωs,f )θ − λεTe εe + sT εe . − Y(ω˙ r , ωr , ωif

(9.69) (9.70)

Selecting the control as f τa,f = A† −Y(ω˙ r , ωr , ωif )θ − (AIs,f ωs,f )× ωr − τg,f − Kd s − εe ,

(9.71)

equation (9.70) may be written as V˙ 2 = −sT Kd s − λεTe εe T

= −s Kd s − λ(1 − T

≤ −s Kd s − λ(1 −

(9.72) t)εTe εe t)εTe εe

− tλ(1 −

ηe2 )

(9.73) 2

− tλ(1 − |ηe |) < 0

(9.74)

Since V2 fulﬁlls the requirements of uniform global exponential stability as given in [17], with the squared two norm, D = Rn and the constants deﬁned as k1 = λmin (P) k2 = λmax (P)

(9.75) (9.76)

k3 = λmin (Q),

(9.77)

where P=diag(Jf , I3×3 , 1), Q=diag(Kd , I3×3 , 1), and λmin (·) and λmax (·) is the minimum and maximum eigenvalue respectively. This shows that both s and εe converges to zero exponentially, which implies exponential convergence of ωe to zero and of ηe to 1.

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Remark 9.3. Though the synchronizing controller was proven UGES to the origin (ωe , εe ) = (0, 0), indicating UGES of the scalar quaternion error to either 1 or 1, inspection of the equilibrium points show that ηe = −1 is actually an unstable equilibrium point. This is a known fact for controllers using the quaternion error feedback, and was pointed out both in [26] and [29]. In [26], feedback e = sgn(ηe )εe was used, which resulted in two stable equilibrium points. This was also used in the tracking and stabilizing controllers derived in the previous section. Since the error-dynamics (9.61a)-(9.61b), satisﬁes the matching condition[30]: The terms containing the unknown parameters are in the span of the control, that is, they can be directly cancelled by τa,f when the parameters are known, an adaptive control law may be deﬁned by exchanging the real parameter vector with estimated parameters, and deﬁning a parameter estimate update law. Proposition 9.5. Given the dynamics (9.61a)-(9.61b), choosing the control τa as f τa,f = A† −Y(ω˙ r , ωr , ωif , ωs,f )θˆ − τg,f − Kd s − εe ,

(9.78)

with the parameter estimate update law given by ˆ˙ = −Γ−1 YT s θ

(9.79)

renders the origin (s, εe ) = (0, 0) globally convergent. This implies convergence of ωe to 0 and to the convergence of ηe to 1. Proof. The ﬁrst part of the proof follows directly from the proof of Proposition 9.4, and is not repeated. We now select V2 = 12 sT Jf s + 21 θ˜T Γθ˜ + V1 .

(9.80)

If the control law and parameter update law is given by (9.78) and (9.79) respectively, the time-derivative V2 . V˙ 2 = −sT Kd s − λεTe εe ≤ 0 (9.81) Since V2 can be lower bounded, V˙ 2 is negative semi-deﬁnite and uniformly continuous in time, the conditions of Barbalat’s Lemma [17] are satisﬁed and we have convergence of (s, εe ) to (0, 0) globally, which as in the proof of Proposition 9.4, leads to ωe → 0, ηe → 1, as t → ∞. (9.82) Remark 9.4. Though we can not guarantee convergence of the parameter estimation error, we know that the error will stay bounded. To obtain true parameter estimation it is necessary for the input to the adaptive update law to be persistently exciting (PE). In this case the input is the synchronization error, thus the PE property will only be possible during transients. This is observed during simulations, as the convergence of the parameter estimate stops when the synchronization error has reached zero.

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9.4 Simulations In this section we present simulations of the synchronizing controller. The model used is based on realistic values for a cubic small-size satellite, and a summary of model parameters is given in Table 9.1 and the simulation initial conditions and parameters are summarized in Table 9.2. Table 9.1. Model parameters Parameter

Value

Inertia matrix

diag{4, 4, 3} [kgm2 ]

Wheel inertia

8 · 10−3 [kgm2 ]

Max magnetic moment

40 [Am2 ]

Max wheel torque

0.2 [N m]

Max wheel speed

400 [rad/s]

Table 9.2. Simulation parameters Parameter

Value

Controller gains

kp = 1, kd = 5

Orbit angular velocity

1.083 · 10−3 [rad/s]

Initial leader attitude

[30, −30, −10]T [◦ ]

Initial follower attitude

[20, −20, 10]T [◦ ]

Initial leader angular velocity 10−3 · [1, 3, 1]T [rad/s] Initial follower angular velocity 10−3 · [1, 3, 1]T [rad/s] Desired attitude

! " 2π 45 sin 500 t % &◦ 2π % #25 sin 500 t& $[ ] 0

In Fig. 9.3(a) the transient synchronization error is presented, clearly indicating convergence of the errors. The ﬁnal synchronization error is dependent on factors such as actuator bandwidth, measurement accuracy, actuator saturation, and so on. In the simulations actuator saturation and noise contaminated measurements were employed to model the uncertainties. An estimate of the achievable control accuracy was found to be about 0.1 ◦ . The tracking error of the leader satellite is displayed in Fig. 9.3(d), and the previous statement is also valid for this result. In Fig. 9.3(c) we see that we have convergence of the estimates, but as commented in Remark 9.4, the convergence is dependent on the excitation level of the synchronization error, and as the plot show we do not have convergence to the actual inertia parameters.

T.R. Krogstad and J.T. Gravdahl 20

φlf θlf ψlf

Angle[◦ ]

10

5

Angle[◦ ]

Angle[◦ ]

15

Angle[◦ ]

168

0

−5 0

20

40

60

80

100

50

φid φil φif

0 −50 0 50

20

40

60

80

0 −50 0 10

20

40

60

80

100

ψid ψil ψif

0 −10 0

100

θid θil θif

20

40

60

80

100

Time [sec] Time [sec] (a) Synchronization error transient, qlf , (b) Simulation results showing the attivisualized using Euler angles tude of the leader and follower versus the desired 40

θˆ1 θˆ2 θˆ3

10 8 6

φdl θdl ψdl

30 20

Angle[◦ ]

Moment of inertia [kgm2 ]

12

10 0 −10

4 2 0

−20

20

40

60

Time [sec]

80

100

(c) Plot of the convergence of the estimated inertia estimates

−30 0

20

40

60

80

Time [sec] (d) Tracking error of the leader

100

Fig. 9.3. Simulation results

Finally, Fig. 9.3(b) shows the how the attitude of follower satellite tracks that of the leader.

9.5 Conclusions We have in this paper presented the design of an adaptive synchronizing controller, for a satellite actuated by means of redundant wheels in a tetrahedron composition. The proposed controller was proved to be uniformly globally exponentially stable for the case of parameter knowledge, and globally convergent when the parameters where estimated using a parameter update law. Simulations have been utilized to support the propositions, showing that the control system performs to speciﬁcations also when no knowledge of the parameters are available.

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References 1. A. Moreira, G. Krieger, I. Hajnsek, M. Werner, S. Riegger, and E. Settelmeyer. TandemX: a Terrasar-X add-on satellite for single-pass SAR interferometry. In Proceedings of the 2004 IEEE International Geoscience and Remot Sensing Symposium, volume 2, pp. 1000–1003, 2004. 2. OHB System AG. SAR-lupe: The innovative program for satellite-based radar reconnaissance. Brochure. 3. ESA. XEUS CDF report. Technical report, ESA, 2004. 4. C.V.M. Fridlund. Darwin - the infrared space interferometry mission. ESA Bulletin, 100:20–25, 2000. 5. H. Pan and V. Kapila. Adaptive nonlinear control for spacecraft formation ﬂying with coupled translational and attitude dynamics. In Proceedings of the 40st IEEE Conference on Decision and Control, 2001. 6. W. Kang and H. Yeh. Co-ordinated attitude control of multi-satellite systems. International Journal of Robust and Nonlinear Control, 12:185–205, 2002. 7. P.K.C Wang, F.Y. Hadaegh, and K. Lau. Synchronized formation rotation and attitude control of multiple free-ﬂying spacecraft. Journal of Guidance, Control and Dynamics, pp. 28–35, 1999. 8. H. Nijmeijer and A. Rodriguez-Angeles. Synchronization of Mechanical Systems. World Scientiﬁc, 2003. 9. M.J. Mataric. Minimizing complexity in controlling a mobile robot population. In 1992 IEEE Int. Conf. on Robotics and Automation Proceedings, volume 1, pp. 830–835, May 1992. 10. T. Balch and R.C. Arkin. Behavior-based formation control for multirobot teams. IEEE Transactions on Robotics and Automation, 14(6), December 1998. 11. A. D. Mali. On the behavior-based architectures of autonomous agency. IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, 32(3), August 2002. 12. J.R.T. Lawton. A Behavior-Based Approach to Multiple Spacecraft Formation Flying. PhD thesis, Bringham Young University, Dept. of El. and Comp. Engineering, November 2000. 13. M.A. Lewis and K. Tan. High precision formation control of mobile robots using virtual structures. Autonomous Robots, 4:pp. 387–403, 1997. 14. R.W. Beard, J. Lawtond, and F.Y. Hadaegh. A coordination architecture for formation control. IEEE Transactions on Control Systems Technology, 9:pp. 777–790, 2001. 15. W. Ren and R.W. Beard. Formation feedback control for multiple spacecraft via virtual structures. In 2004 IEEE Proceedings of Control Theory Application, 2004. 16. Alejandro Rodriguez-Angeles. Synchronization of Mechanical Systems. PhD thesis, Techniche Universiteit Eindhoven, 2002. 17. H.K. Khalil. Nonlinear Systems. Prentice Hall, 3 edition, 2002. 18. V. M. Matrosov. On the stability of motion. Journal of Appl. Math. Mech., 26:pp. 1337–1353, 1962. 19. A. Loria, E. Panteley, D. Popovic, and A.R. Teel. An extension of matrosov’s theorem with application to stabilization of nonholonomic control systems. In Proceedings of the 41st IEEE Conference on Decision and Control, volume 2, pp. 1528–1533, December 2002. 20. O. Egeland and J.T. Gravdahl. Modeling and Simulation for Automatic Control. Marine Cybernetics, 2001. 21. P. C. Hughes. Spacecraft attitude dynamics. John Wiley & Sons, 1986.

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22. C.D. Hall. Spinup dynamics of gyrostats. Journal of Guidance, Control, and Dynamics, 18(5):1177–1183, Sptember-October 1995. 23. R. Wisniewski. Lecture notes on modeling of a spacecraft. [Available online:www.control.auc.dk/ raf/ModellingOfMech/Rep8sem.ps, Last accessed: 05.06.2005], May 2000. 24. T.R. Krogstad, J.T. Gravdahl, and R. Kristiansen. Coordinated control of satellites. In Proceedings of the 2005 International Astronautical Congress, October 2005. 25. M. Vidyasagar. Nonlinear Systems Analysis. Prentice Hall, 1993. 26. O.-E. Fjellstad. Control of unmanned underwater vehicles in six degrees of freedom. PhD thesis, Norwegian University of Science and Technology, 1994. 27. T.I. Fossen. Marine Control Systems: Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, 2002. 28. A. K. Bondhus, K.Y. Pettersen, and J.T. Gravdahl. Leader/follower synchronization of satellite attitude without angular velocity measurements. In Proceedings of the 44th IEEE Conf. on Decision and Control, 2005. Submitted. 29. O. Egeland and J.-M. Godhavn. Passivity-based adaptive attitude control of a rigid spacecraft. IEEE Transactions of Automatic Control, 39(4):pp. 842 – 846, April 1994. 30. M. Kristic, I. Kanellakopoulos, and P. Kokotovic. Nonlinear and Adaptive Control Design. Wiley Interscience, 1995.

10 A Virtual Vehicle Approach to Underway Replenishment E. Kyrkjebø1 , E. Panteley2 , A. Chaillet2 and K.Y. Pettersen1 1

2

Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. [email protected]; [email protected] C.N.R.S, UMR 8506, Laboratoire de Signaux et Syst`emes, 91192 Gif s/Yvette, France. [email protected]; [email protected]

Summary. The problem of coordinated control of a leader-follower system is solved using a virtual vehicle approach to alleviate information requirements on the leader. The virtual vehicle stabilizes to a shifted reference position/heading deﬁned by the leader, and provides a reference velocity for the synchronization control law of the follower. Only position/heading measurements are available from the leader, and the closed-loop errors are shown to be uniformly globally practically asymptotically stable.

10.1 Introduction Underway replenishment at sea requires a close coordination of two vessels, and has up to now been conducted using manual control together with control ﬂags to exchange instructions between the vessels. Recent advances in control theory and measurement systems, in particular the introduction of the Global Positioning System (GPS)([1]) and the Automatic Identiﬁcation System (AIS)([2]), have allowed automatic control approaches for replenishment purposes. These autopilots are faced with the goal of suppressing eﬀects of external disturbances due to wind, waves and currents, while achieving the accuracy demands of the operation using a reduced set of measurements. The introduction of autopilots expand the range of operating conditions for safe replenishment in terms of increased manoeuvrability in close waters or in the proximity of other vessels, and in the robustness towards environmental disturbances. Control approaches used in [3] and [4] are based on the assumption that a complete mathematical model of both vessels is available, and thus autopilots for both vessels can be designed to suppress the eﬀects of external disturbances. However, in a practical leader-follower replenishment operation, the follower may have limited access to information of the control input, model and states of the leader. Therefore, to alleviate the information requirements on the leader vessel, [5] proposed a synchronization observer-controller scheme where the only information available from the main (leader) ship were position and heading measurements. Due to the inherent coupling between the observer and the controller in the proposed observer-controller

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 171–189, 2006. © Springer-Verlag Berlin Heidelberg 2006

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design of [5], the stability analysis and practical tuning of the scheme is involved. In order to facilitate both tuning of the controller and the stability analysis in a leader-follower synchronization scheme, we propose in the following a virtual vehicle approach to the underway replenishment operation. The proposed virtual vehicle approach requires no additional information on leader states or model parameters. Synchronization is found both as a natural phenomenon in nature like in the ﬂashing of ﬁreﬂies, choruses of crickets and musical dancing, as well as the controlled synchronization of a pacemaker or a transmitter-receiver system. It can be seen as a type of state cooperation among two or more (sub)systems, and synchronization in control was pioneered by the work on vibromechanics in [6], and has since received an increasing attention in the control community (cf. [7, 8]). Synchronization has been utilized in maritime application by [4], [5] and [9]. [4] and [9] expanded on traditional tracking methods with predeﬁned paths, and introduced a synchronization feedback from the actual position of a vessel (subject to disturbances) to the other vessels through a path parameterization variable. All vessels have predeﬁned paths with individual tracking controllers requiring knowledge of model parameters and control inputs for all vessels, and the synchronization is in terms of progression along the path. Thus, disturbances aﬀecting tracking performance along the path are cancelled, but cross-track errors due to any diﬀerence in disturbances are not. Based on the results of [7] for synchronization of mechanical systems, [5] proposed a leader-follower synchronization observer-controller scheme for underway replenishment. Experimental results on this scheme were presented in [10] addressing practical tuning issues and performance. There is no need for a predeﬁned path with known derivatives or model parameter information for the leader vessel in [5], and the coordination of the vessels is achieved using a controller that synchronizes the position and velocity of the follower to the leader based on position measurements only, through the design of state observers. The virtual vehicle approach has been utilized both as an abstraction vehicle (cf. [11, 12]) and as an intermediate level between the desired trajectories of a system and the controller. In a way, it can be considered as a low-level controller in a two-level control structure (cf. [13, 14]), and was used in [15] as the mapping of a physical vehicle on an entry-ramp on a main lane in order to do merging control of autonomous mobile robots, and in [16] to control a reference point on a planned path. The latter approach has been utilized in [17] to combine the task of path following and obstacle avoidance, and in [18] with a modiﬁed goal point to improve practical robustness to path diversity. The virtual vehicle approach of [15] for mobile robots has its parallel in the approach of [9] for marine vessels in the use of a path parameterization variable mapping the leader vessel on the desired path of the follower, while the mobile platform control approach of [16] can be related to the manoeuvring approach of [19] applied to marine vessels where the tracking problem is subdivided into a geometric and a dynamic task with a path variable, which can be viewed as virtual vehicle. In this text, we propose a virtual vehicle approach to the underway replenishment problem to impose a cascaded structure of the systems, as opposed to the controller-observer approach proposed in [5] where the observers and controller are closely interconnected. The virtual vehicle is designed to follow the behaviour of the

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leader based on position feedback, and provides a velocity output through the controller design. The states of the virtual vehicle can thus be used in a synchronization controller to control the follower to the virtual vehicle. We have made the additional assumption on the problem of [5] in that the velocity of the follower is assumed known to focus our treatment on the interplay between the virtual vehicle and the follower, and are able to extend the stability results from semi-global uniform ultimate boundedness of the closed-loop errors in [5] to uniform global practical asymptotic stability. The stability of the closed-loop errors is valid globally through the availability of velocity information of the follower. I addition, through the use of a virtual vehicle approach, the ultimate boundedness property is replaced by the more interesting property of practical stability, meaning asymptotic stability to a ball which can be made arbitrarily small by tuning some gains. We will ﬁrst present the general vessel model and the diﬀerent vehicles and coordinate frames used in the development in Sect. 10.2, and then the virtual vehicle design in Sect. 10.3. The synchronization controller for underway replenishment is presented in Sect. 10.4, while the overall stability is presented in Sect. 10.5. Simulations are presented in Sect. 10.6, while some ﬁnal remarks and conclusions are reported in Sect. 10.7.

10.2 Preliminaries We will consider underway replenishment operations for surface vessels described by a 3 degrees-of-freedom (3DOF) model, and we will assume that the only available measurements are position/heading from the leader, and position/heading and their velocities for the follower vessel. First, we will introduce some deﬁnitions on stability before presenting the reference frames and vehicles used in the text with dynamic and kinematic model properties. In the following, the minimum and maximum eigenvalue of a positive deﬁnite matrix M will be denoted √ as Mm and MM , respectively. The norm of a vector x is deﬁned as x = xT x and the induced norm of a matrix M is M = max x =1 Mx . We denote the by Bδ the closed ball in Rn of radius δ centered at the origin, i.e. Bδ := {x ∈ Rn | x ≤ δ}. We deﬁne x δ := inf z∈Bδ x − z and designate by N≤N the set of all nonnegative integers less than or equal to N . 10.2.1 Practical Stability In order to prove that the control system we propose is stable, we will utilize some deﬁnitions and propositions on practical stability from [20]. Consider x˙ = f (t, x)

(10.1)

where x ∈ Rn , t ∈ R≥0 and f : R≥0 × Rn → Rn is piecewise continuous in t and locally Lipschitz in x uniformly in t. Deﬁnition 10.1 (UGAS of a ball). Let δ be a positive number. The ball Bδ is Uniformly Globally Asymptotically Stable for the system (10.1) if there exists a class

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KL function β such that the solution of (10.1) from any initial state x0 ∈ Rn and initial time t0 ∈ R≥0 satisﬁes x (t, t0 , x0 )

δ

≤ β ( x0 , t − t0 ) ,

∀t ≥ t0

Remark 10.1. While the deﬁnition given above implies the property of ultimate boundedness, it actually constitutes a stronger property. Notably, the transient is guaranteed to remain arbitrarily near Bδ for any suﬃciently small initial state, which is a stability property not covered by the notion of ultimate boundedness. For nonlinear time-varying systems of the form x˙ = f (t, x, θ)

(10.2)

where x ∈ Rn , t ∈ R≤0 , θ ∈ Rm is a constant parameter vector, and f : R≥0 × Rn × Rm → Rn is locally Lipschitz in x and piecewise continuous in t, we can deﬁne the following property Deﬁnition 10.2 (UGPAS). Let Θ ⊂ Rm be a set of parameters. The system (10.2) is said to be Uniformly Globally Practically Asymptotically Stable on Θ if, given any positive δ, there exists θ ∈ Θ such that Bδ is UGAS for the system x˙ = fθ (t, x) := f (t, x, θ). For systems where the Lyapunov function candidate can be upper and lower bounded by polynomial functions, the parameter θ typically contains the gain matrices that can be freely tuned in order to enlarge or diminish the domain of attraction and the ball around the origin to which the solutions converge. The following corollary is derived from [20, Proposition 1] as Corollary 10.1 ([20]). Let σi : Rm → R≥0 , i ∈ N≤N be continuous functions positive over Θ, and am , aM and q be positive constants. Assume that, for any θ ∈ Θ, there exists a continuously diﬀerentiable Lyapunov function V : R≥0 × Rn → R≥0 satisfying, for all x ∈ Rn and all t ∈ R≥0 , am min {σi (θ) : i ∈ N≤N } x

q

≤ V (t, x) ≤ aM max {σi (θ) : i ∈ N≤N } x

q

(10.3)

Assume also that, given any positive δ, there exists θ (δ) ∈ Θ and a class K function αδ such that, for all x ∈ Rn \ Bδ and all t ∈ R≥0 , ∂V ∂V (t, x) + (t, x) f (t, x, θ) ≤ −αδ ( x ) ∂t ∂x

(10.4)

Assume also that for all i ∈ N≤N , lim σi (θ (δ)) δ q = 0,

δ→0

and

lim σi (θ (δ)) = 0.

δ→0

Then the system (10.2) is UGPAS on the parameter set Θ. The functions σi (θ) refer in most cases to the minimum eigenvalues of the gain matrices in θ, and thus, roughly speaking, we impose that the dependency of these minimum eigenvalues in the radius 1/δ should be polynomial and of a lower order than the bounds on V .

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10.2.2 Vehicles Deﬁnitions and Reference Frames In the development of the underway replenishment control scheme a number of reference frames, intermediate vehicles and dynamic and kinematic models will be used. A brief introduction to these concepts is given in this section, but for a more elaborate discussion see [21]. The control problem studied is as follows: Given the position (x, y) and heading angle ψ of a leader vessel, we want the follower vessel to follow the leader with its position shifted by a distance d at an angle γm relative to the leader. For this purpose we will utilize the concepts of a reference vehicle and a virtual vehicle, and we designate the following vehicles as illustrated in Fig. 10.1. yn

Vv

Vr

Vs d

γm

xm ψm

pn r pn m

Vm

ym

xn

Fig. 10.1. Vehicles and coordinate frames T

Vm The leader (main) vessel where the position xm = (x, y, ψ) is available as output. Vr A reference vehicle shifted a distance d in the direction given by γm relative to the position of the leader vehicle. Vv A virtual vehicle controlled to track the reference vehicle Vr through a kinematic model approach. Vs The follower (supply) vessel synchronizing to the leader vessel. The position x and velocity x˙ is available for control design, and the parameters of the dynamic model are known. Note that the physical vehicles in the control scheme are the leader vessel Vm and the follower vessel Vs synchronizing to the leader. The reference vehicle is a mathematical reference constructed by shifting the position of the leader vessel to the desired position of the follower vessel in relation to the leader, and the virtual vehicle is a virtual reference vehicle controlled to this shifted position through a kinematic control law. Through the use of a virtual vehicle as an intermediate control vehicle in the scheme, we can control the physical follower vessel to the leader using the known velocity of the virtual reference. Note also that although we derive the control scheme for one follower vessel, it can be easily extended to any number of followers providing the introduction of a collision avoidance scheme.

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Vessel kinematics and dynamics can be expressed in diﬀerent reference frames, and we deﬁne the two essential reference frames used in this text as (cf. [21]) NED This is a ﬁxed reference frame deﬁned relative to the Earth’s reference ellipsoid, where the xn -axis points toward true North, the y n -axis toward East, and the z n -axis points downwards normal to the Earth’s surface. BODY p This is a body-ﬁxed moving reference frame where the origin is chosen in the centre of gravity of the vehicle p, and the axes coincide with the principal axes of inertia. Due to vessel symmetry, we can choose the xb -axis along the axis of inertia in the forward direction of the vessel, the y b -axis directed to the right and the z b -axis to complete the right-handed coordinate system pointing downwards. In the 3DOF case considered here, the vector of vessel generalized coordinates xn = T [x, y, ψ] is deﬁned in the NED frame, where (x, y) is the position with respect to n the x - and y n -axis, and ψ is the heading angle of the vessel about the z n -axis. T The velocities νpb = [u, v, r] in the surge, sway and yaw directions are deﬁned in the BODY p frame of the vehicle p. Superscripts n and b will be dropped from the notation when the reference frame is evident from the context. Subscripts p ∈ {m, r, v, s} on these vectors will indicate their vehicle of origin (main, reference, virtual, supply). 10.2.3 Ship Model and Properties The marine vessel equations of motions can be written in vectorial form in the BODY frame of the vessel as ([21]) x˙ = J (x) ν Mν ν˙ + Cν (ν) ν + Dν (ν) ν + gν (x) = τν

(10.5) (10.6)

where Mν is a constant positive deﬁnite inertia matrix including added mass effects, Cν (ν) is a skew-symmetric matrix of Coriolis and centripetal forces (Cν (ν)+ CTν (ν) = 0), Dν (ν) is a non-symmetric damping matrix, and gravitational/buoyancy forces in gν (x) can be ignored for surface vessels. J (x) is a Jacobian-like transformation matrix from the BODY frame to the NED frame, and in a 3DOF surface application where pitch and roll motion are negligible, the matrix J (x) reduces to a simple rotation matrix around the z n -axis as    cos ψ − sin ψ   J (x) =   sin ψ cos ψ   0 0

0   0    1

(10.7)

Inserting the kinematic equation (10.5) and its derivative in the dynamics (10.6) yields the dynamic model in the NED frame ¨ + C (x, x) ˙ x˙ + D (x, x) ˙ x˙ + g (x) = τ M (x) x

(10.8)

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where M (x) = J−T (x) Mν J−1 (x) ˙ = J−T Cν J−1 (x) x˙ − Mν J−1 (x) J˙ −1 (x) J−1 (x) C (x, x) ˙ = J−T (x) Dν J−1 (x) x˙ J−1 (x) D (x, x)

(10.9)

−T

g (x) = J (x) gν (x) τ = J−T (x) τν The inertia matrix M (x) is positive deﬁnite but no longer constant, and the Coriolis ˙ is deﬁned in terms of Christoﬀel symbols. and centripetal matrix C (x, x) The dynamic model (10.8) in the NED frame satisﬁes a number of properties similar to those of robotics systems (see for example [22]) such as P1 The positive deﬁnite inertia matrix satisfy 0 < Mm ≤ M (x) ≤ MM < ∞, where Mm and MM are positive constants. ˙ (x) − 2C (x, x) ˙ y= P2 The inertia matrix M (x) is diﬀerentiable in x and yT M 0, ∀ x, y ∈ R3 . ˙ = We will also assume the following property of the dissipation vector d (x, x) ˙ x˙ for a marine vessel (cf. [23]) D (x, x) ˙ = D (x, x) ˙ x˙ is continuously diﬀerentiable in x P3 The dissipation vector d (x, x) and x˙ and satisﬁes for some Dm > 0 yT

˙ ∂d (x, x) y ≥ Dm y ∂ x˙

2

,

˙ y ∈ R3 ∀ x, x,

(10.10)

and for a continuous function DM (s) : R≥0 → R≥0 ˙ ∂d (x, x) ≤ DM ( x˙ ) . ∂ x˙

(10.11)

10.2.4 Reference Vehicle Kinematics As a ﬁst step in order to assure a safe replenishment operation, we design a reference position for the follower vessel at some distance from the leader. This reference position will be in the form of a reference vehicle with a kinematic model. We will in this section develop a general reference vehicle model for an arbitrary heading assignment, i.e. the heading angle of the reference vehicle ψr can be diﬀerent from the heading angle of the leader vessel ψm , and in Sect. 10.2.5 for the particular case of parallel motion where the heading angles are equal; ψr = ψm . The kinematic model (10.5) of the leader vehicle Vm with the position/heading vector xm can be written as x˙ m = J (xm ) νm and by using (10.7) takes the form

(10.12)

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Vr

d

γm

xm

γm

ψr d

α

xm Vr

ψm

ψm xr ψr

Vm y

m

xn

Vm y

r

ym xn

Fig. 10.2. Reference vehicle at the distance d and angle γm from the leader vessel

x˙ m = um cos ψm − vm sin ψm y˙ m = um sin ψm + vm cos ψm ψ˙ m = rm

(10.13)

Let d (= const) be the desired (required) distance between the leader and the follower vessel, and γm (= const) the angle between the xb -axis of the leader and the vector d, positive counterclockwise, as shown in Fig. 2. We can calculate the ”position” χ of the leader in the BODY m -frame1 of the leader T χm m = J (xm ) xm

(10.14)

Superscripts denote the reference frames, and subscripts denote which vehicle position/heading vector that is described. We will use χ to denote position/heading in the BODY -frames with no immediate physical interpretation, and x as the position/heading of vehicles in the NED-frame. Thus, xm is the position vector of the leader vehicle m in the NED frame, while χm m is the position vector of the leader vehicle in its own BODY m frame. We also have that the position vector of the reference vehicle in the NED frame can be written as xr = J (xr ) χrr

(10.15)

The position vector of the reference vehicle in the BODY m frame of the leader can be written as m m χm r = χ m + dr

(10.16)

where 1

Note that the position of the vehicle in! the body-frame does not have any immediate t physical interpretation as the integral 0 ν b dt, but its mathematical representation is still valid.

10 A Virtual Vehicle Approach to Underway Replenishment



dm r



 d cos γm       =  d sin γm      α

179

(10.17)

with the distance d and rotation α separating the two frames. The angle α is the desired diﬀerence in heading between the leader vessel and the reference vehicle, and deﬁned in the BODY m -frame. Expressed in the NED frame, (10.16) becomes m xr = J (xm ) χm r = xm + J (xm ) dr

(10.18)

Taking the time derivative through J˙ (xm ) = J (xm ) S (rm ), where S (rm ) is the skew-symmetric matrix    0 −rm 0       S (rm ) =   rm 0 0      0 0 0

(10.19)

˙m x˙ r = x˙ m + J (xm ) S (rm ) dm r + J (xm ) dr

(10.20)

we get

In component form this is equivalent to x˙ r = um cos ψm − vm sin ψm − drm cos γm sin ψm − drm sin γm cos ψm y˙ r = um sin ψm + vm cos ψm + drm cos γm cos ψm − drm sin γm sin ψm (10.21) ψ˙ r = rm + α˙ By investigating (10.21) more closely, we can rewrite this as x˙ r = cos ψm (um − drm sin γm ) − sin ψm (vm + drm cos γm ) =

2

(10.22)

2

(um − drm sin γm ) + (vm + drm cos γm ) (cos ψm cos α − sin ψm sin α)

where we have cos α = sin α =

um − drm sin γm 2

2

(um − drm sin γm ) + (vm + drm cos γm ) vm − drm cos γm

(um − drm sin γm )2 + (vm + drm cos γm )2

Similarly, we can rewrite (10.21) as

(10.23) (10.24)

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y˙ r = sin ψm (um − drm sin γm ) + cos ψm (vm + drm cos γm ) =

2

(10.25)

2

(um − drm sin γm ) + (vm + drm cos γm ) (sin ψm cos α + cos ψm sin α)

and by using that tan α =

sin α um − drm sin γm = cos α vm + drm cos γm

(10.26)

we get α˙ =

(u˙ m −dr˙m sin γm )(vm +drm cos γm )−(um −drm sin γm )(v˙ m +dr˙m cos γm ) (vm + drm cos γm )

2

cos2 α

(10.27)

and thus the total system can be written as 2

2

2

2

x˙ r =

(um − drm sin γm ) + (vm + drm cos γm ) cos (ψm + α)

y˙ r =

(um − drm sin γm ) + (vm + drm cos γm ) sin (ψm + α)

(10.28)

ψ˙ r = ψ˙ m + α˙ 10.2.5 Reference Kinematics in an Underway Replenishment Operation The STREAM (Standard Tension Alongside Replenishment Method) is currently the preferred underway replenishment conﬁguration at sea (cf. [24]). In the underway replenishment scenario, it is desirable that the reference vehicle is placed at a constant distance d orthogonally oﬀ one of the sides of the leader with γm = ± π2 , and thus the kinematic equations (10.28) of the reference vehicle can be greatly simpliﬁed. This conﬁguration corresponds to a replenishment operation where the supply ship moves in parallel with the leader at a ﬁxed distance and with the same heading angle. In this conﬁguration, the supply ship will always be at a right angle to the replenished ship, and the tension on the replenishment rig is at a minimum. With this particular choice of γm , the parallel motion suggests that J (xr ) = J (xm ), and the position of the reference vehicle in the NED frame becomes xr = J (xr ) χrr = xm + J (xm ) dm r

(10.29)

Diﬀerentiating (10.29) we obtain x˙ r = x˙ m + J (xm ) S (rm ) dm r

(10.30)

π since the vector dm r is now constant. Taking γm = − 2 , we obtain the component form for (10.30) as

x˙ r = x˙ m + drm cos ψm y˙ r = y˙ m + drm sin ψm ψ˙ r = rm

(10.31)

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Substituting x˙ m from (10.13) in (10.30), and deﬁning ur = um + drm , vr = vm and rr = rm gives x˙ r = ur cos ψm − vr sin ψm y˙ r = ur sin ψm + vr cos ψm ψ˙ r = rr

(10.32)

It is easy to see from (10.32) that in this particular case only the reference forward velocity is changed (ur = um + drm ) with respect to that of the leader vessel. Note that this is necessary for the follower vessel to maintain its position parallel to the leader vessel during turns due to the diﬀerence in turn radius. The kinematic model of the reference vehicle can now be written as x˙ r = J (xm ) νr

(10.33)

T

where νr = [um + drm , vm , rm ] .

10.3 Virtual Vehicle Design The only measurement available from the leader vessel is the position/heading measurement xm , and since we have no information on the parameters of the mathematical model or of the control input to the leader vessel, we are precluded from designing a model-based observer for the leader states. An alternative approach is to utilize the time ﬁltered derivatives from the position measurements at the expense of robustness under noisy conditions, but in order to reduce noise sensitivity we propose instead to design a virtual vehicle as an intermediate controlled vessel Vv stabilizing to the reference vehicle Vr based on position measurement feedback only. As in [14], on the ﬁrst step (kinematic level) we consider the velocities νv of the virtual vehicle as the control inputs, and design them in such a way that we ensure convergence of the virtual trajectories to the reference trajectories. In a way, we can consider the trajectories xv and velocities νv as estimates of xr and νr , that is, the virtual vehicle is a form of kinematic estimator of the leader states through the position feedback loop. The virtual vehicle is deﬁned by its kinematic model x˙ v = J (xv ) νv

(10.34)

Based on practical considerations, we assume that the velocity and acceleration of the leader vessel are bounded, and thus the velocity and acceleration of the reference vehicle satisfy sup νr = VM < ∞

(10.35)

sup ν˙ r = AM < ∞

(10.36)

t t

with known constants VM and AM . The kinematics of the reference vehicle in (10.33) is given by

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x˙ r = J (xm ) νr

(10.37)

and the virtual vehicle tracking errors are deﬁned as e˙ v = x˙ v − x˙ r = J (xv ) νv − J (xm ) νr

ev = xv − xr ,

(10.38)

We propose the following control law for the virtual vehicle νv = −J−1 (xv ) L1 ev − J−1 (xv ) L2 z

(10.39)

where L1 and L2 are symmetric positive gain matrices, and z˙ = ev

(10.40)

The closed-loop equations can be written in the following form e˙ v = −L1 ev − L2 z − J (xm ) νr

(10.41)

Consider the following Lyapunov function candidate 1 T 1 1 ev ev + zT L2 z + zT ev 2 2 2 Diﬀerentiating along the closed-loop trajectories we get Vv (z, ev ) =

(10.42)

1 1 1 1 L1 − I ev − zT L2 z − zT L1 ev − eTv + zT 2 2 2 2

V˙ v (z, ev ) = −eTv

J (xm ) νr

(10.43) Using (10.35) and the relation 2|ab| ≤ (λa2 + b2 /λ) for any real a, b and any positive λ, it follows that 1 V˙ v (z, ev ) ≤ − 2

1 3VM L1,M − z 2 2λ 2 (e, z) 3VM 1 λ − L1,m − − L1,M − ev 2 4 2 (ev , z) L2,m −

2

,

(10.44)

where λ designates any positive constant and Li,m (resp. Li,M ) designates the minimum (resp. maximum) eigenvalue of Li . We design the gain matrices L1 and L2 in such a way that Li,M ≤ Li,m , i ∈ {1, 2}, for some > 0. Then, letting λ = 2/ and δv be any given positive constant, we can see that any gain matrices satisfying 3VM δv 2 3 2 + 1+ = 2+ 4 4

L1,m = 3 + L2,m

(10.45) 3VM 2δv

(10.46)

generate the following bound of the derivative of Vv : ev

2

+ z

2

≥ δv2

⇒

V˙ v (z, ev ) ≤ − ev

2

− z

2

.

(10.47)

Note that Vv is positive deﬁnite and radially unbounded for this choice of gains. Due to the linear dependency of L1,m and L2,m in 1/δv , we conclude with Corollary 1 that (10.40)-(10.41) is uniformly globally practically asymptotically stable, meaning roughly that the region to which solutions converge –from any initial condition– can be reduced at will by enlarging L1,m and L2,m .

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10.4 Follower Vehicle Design Using the velocity information from the virtual vehicle design of Sect. 3, we can design a synchronization controller for the follower vessel Vs to follow the virtual vehicle Vv of (10.34). Note that the body-ﬁxed velocity νv is now known through the deﬁnition of the control law of (10.39), and through the kinematic relationship of (10.34) we can obtain the velocity x˙ v of the virtual vehicle in the NED frame. Furthermore, due to our design of the virtual velocity controller, we can also obtain an expression for the acceleration of the virtual vehicle which will be partially available for control purposes. In our synchronization approach, we will assume that the velocity of the follower vessel is known. The variables available from the virtual vehicle design to the synchronization controller are x˙ v = J (xv ) νv = −L1 ev − L2 z ¨ v = −L1 e˙ v − L2 ev = L21 − L2 ev + L1 L2 z + L1 J (xm ) νr x

(10.48) (10.49)

Deﬁne the synchronization errors as e = x − xv ,

e˙ = x˙ − x˙ v ,

¨=x ¨−x ¨v e

(10.50)

Using the sliding surface from [25] as a passive ﬁltering of the virtual vehicle states, we can design a virtual reference trajectory as y˙ v = x˙ v − Λe ¨v = x ¨ v − Λe˙ y

(10.51) (10.52)

where Λ > 0. Let us denote ¨ v = L21 − L2 ev + L1 L2 z − Λe˙ y

(10.53)

¨v = y ¨ v + L1 J (xm ) νr y

(10.54)

and thus

¨ v is available for control design. Deﬁning where y s = x˙ − y˙ v = e˙ + Λe

(10.55)

as a measure of tracking, and using the relationship x˙ = s − y˙ v we can rewrite (10.8) as ˙ s − D (x, x) ˙ s +τ − M (x) y ¨ v − C (x, x) ˙ y˙ v − D (x, x) ˙ y˙ v − g (x) M (x) s˙ = −C (x, x) (10.56) We propose the following control law ˙ y˙ v + D (x, x) ˙ y˙ v + g (x) − Kd s − Kp e ¨ v + C (x, x) τ = M (x) y

(10.57)

where Kp and Kd are symmetric positive gain matrices. Consider the following Lyapunov function candidate

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Ve (e, s) =

1 T 1 s M (x) s + eT Kp e 2 2

(10.58)

Diﬀerentiating along the closed-loop trajectories we get ˙ + Kd ] s − eT ΛT Kp e − sT M (x) L1 J (xm ) νr (10.59) V˙ e (e, s) = −sT [D (x, x) Let δe be any given positive constant. Then, it holds that, for all e 1 V˙ e (e, s) ≤ Dm + Kd,m − MM L1,M 2δe

s

2

− Λm Kp,m −

2

+ s

1 MM L1,M 2δe

2

≥ δe2 , e

2

(10.60) Proceeding as in Section 3, we can invoke Corollary 1 by observing that the choice of Kd,m and Kp,m can be made as an aﬃne function of 1/δe , and conclude uniform global practical asymptotic stability.

10.5 Stability Analysis of the Overall System The control law of the follower synchronizes the follower vessel to the virtual vehicle based on a computed virtual reference velocity from the virtual vehicle controller, and the virtual vehicle is in turn stabilized to the reference vehicle parallel to the leader vessel. Theorem 10.1. Consider the vessel model (10.8) satisfying Properties 1-3, the virtual vehicle control law (10.39) and the synchronization controller (10.57). Under assumptions (10.35-10.36), the overall closed-loop system is uniformly globally practically asymptotically stable. Proof. Take as a positive deﬁnite Lyapunov function candidate V (η) = where

1 T η Pη 2

  Kp 0    0 M (x)  P=   0 0    0 0

(10.61) 

0 0    0 0     1  L2 2 I    1 2I I

(10.62)

is a composition of the Lyapunov functions (10.42) and (10.58) of Sect. 3 and 4. Diﬀerentiating along trajectories yields V˙ (η) = −η T Q η + β (s, ev , z, νr )

(10.63)

10 A Virtual Vehicle Approach to Underway Replenishment

where



185

 T

0  Λ Kp    0 D (x, x) ˙ + Kd  Q=   0 0    0 0

0

0

0

0

1 2 L2

1 4 L1

1 4 L1

L1 − 21 I

          

(10.64)

and 1 β (s, ev , z, νr ) = −sT M (x) L1 J (xm ) νr − zT J (xm ) νr − eTv J (xm ) νr (10.65) 2 Let δ be any given positive constant, and we have the following property η ≥δ

⇒

β (s, ev , z, νr ) ≤

VM δ

MM L1,M s

2

+

z 2

2

+

ev 2

2

.

Consequently, in view of (10.44) and (10.60), and repeating a similar reasoning while choosing the minimum eigenvalue of the gain matrices Kp , Kd , L1 and L2 large enough, it holds that V˙ (η) ≤ − η

2

,

∀ η ≥ δ.

Since the dependency on the bound on β (and on the gain matrices) in 1/δ is again aﬃne, uniform global practical asymptotic stability follows from Corollary 1.

10.6 Simulation Study The underway replenishment scheme presented above was tested in a simulation environment in MATLAB using the surface ship model of Cybership II from [26]. In the simulations, the distance between the ships was given by d = 2 m with γm = − π2 , and the model matrices in the body frame were   0   25.8 0      M=  0 33.8 1.0115      0 1.0115 2.76

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

0

 0

−33.8v − 1.0115r        C (ν) =  0 0 25.8u       33.8v + 1.0115r −25.8u 0 



2

0 0   0.72+1.33|u|+5.87u      D (ν) =  0 0.8896+36.5|v|+0.805|r| 7.25+0.845|v|+3.45|r|       0 0.0313+3.96|v|−0.130|r| 1.90−0.080|v|+0.75|r| T

where ν = [u, v, r] are the body ﬁxed velocities in surge, sway and yaw, respectively. Controller gains were chosen as Kp = diag [70, 140, 70], Kd = diag [100, 100, 50], L1 = diag [0.8, 1.6, 1.6], L2 = diag [0.55, 0.55, 0.55], and Λ = diag [0.3, 0.3, 0.3]. In the simulations, the leader ship tracks a sine wave reference trajectory sin (ωt) with frequency ω = 1/15 with heading angle ψm along the tangent line. T T Initial states were chosen as x (0) = [0, 0, 0] for the follower, xv (0) = 1, 0.5, π4 T for the virtual vehicle and as xm (0) = [2, 4, 0] for the leader ship to illustrate stability in all degrees of freedom as illustrated in the upper plot of Fig. 10.3. From Fig. 10.4 we see that the virtual vehicle control errors ev = xv − xr , the synchronization errors e = x − xv and the overall control errors x − xr are practically asymptotically stable. We observe small oscillations, especially in the velocity errors, due to the unknown velocity of the leader ship. However, due to the practical stability property of the closed-loop system, the magnitude of these oscillations can be arbitrarily reduced within control saturation limits by enlarging the gains.

10.7 Conclusions and Future Work We have presented a control design approach to the underway replenishment case, where the use of a virtual vehicle separates the leader state estimator from the synchronization controller design. The virtual vehicle is an intermediate vehicle placed between the leader and the follower to provide a velocity estimate to the synchronization controller. The overall control scheme is shown to be uniformly globally practically asymptotically stable. Future work includes investigating the stability by exploiting the cascaded structure of the control design in the stability analysis to provide a basis for individual tuning of the virtual vehicle and the synchronization controller. Also, the requirements of velocity measurements of the follower can be relaxed by introducing a velocity ﬁlter to reconstruct the state based on the position measurements. Furthermore, the scheme should be tested experimentally to investigate the behaviour and performance in a realistic environment.

10 A Virtual Vehicle Approach to Underway Replenishment 5

Supply ship Virtual vehicle Reference vehicle

4 Trajectories [m],[rad]

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3 2 1 0 −1

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Position Supply ship Virtual vehicle Reference vehicle

2 1.5 t = 10s

1 0.5 0 −0.5 −1 −1

0

1

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x−position [m]

3

4

5

6

Fig. 10.3. Trajectories of the follower x, the virtual vehicle xv and the reference vehicle xr in the upper plot, and the xy-plot of the vehicles with special marks at initial states and at time t = 10s in the lower.

Acknowledgments This work was partially supported through a European Community Marie Curie Fellowship, and in the framework of the CTS Fellowship Program, under contract number HPMT-CT-2001-00278.

References 1. B. W. Parkinson and J. J. Spilker, Eds., Global Positioning System: Theory and Applications. American Institute of Aeronautics and Astronautics, Inc., 1995. 2. I. Harre, “AIS adding new quality to VTS systems,” Journal of Navigation, vol. 53, no. 3, pp. 527 – 539, September 2000. 3. S. Fu, C. Cheng, and C. Yin, “Nonlinear adaptive tracking control for underway replenishment process,” in Proc. 2004 IEEE Int. Conf. on Networking, Sensing and Control, no. 2, Taipei, Taiwan, March 2004, pp. 707 – 712.

E. Kyrkjebø et al. Position

0.5

Virtual vehicle error [m],[rad] Synchronization error [m],[rad]

1 [m/s],[rad/s]

0 −0.5 x

−1

r

yr

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0

5

10 time [s]

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−2

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1

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15

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2 1

0 −0.5 xv

−1

ψv 0

5

10 time [s]

15

−2

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1

0 −1

yv

−1.5

0

5

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2

0.5

1

0 −0.5 xs

−1

ys

−1.5 −2

0 −1

ψr

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−2

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2

[m/s],[rad/s]

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1

[m/s],[rad/s]

188

−1

ψs 0

5

10 time [s]

15

0

20

−2

0

5

10 time [s]

Fig. 10.4. The total errors x − xr in the upper row, the virtual vehicle control errors xv − xr in the middle row, and the synchronization errors x − xv in the lower row, with positions on the left and velocities on the right. 4. R. Skjetne, I.-A. F. Ihle, and T. I. Fossen, “Formation control by synchronizing multiple maneuvering systems,” in Proc. 6th IFAC Conf. on Manoeuvring and Control of Marine Craft, Girona, Spain, September 2003, pp. 280–285. 5. E. Kyrkjebø and K. Pettersen, “Ship replenishment using synchronization control,” in Proc. 6th IFAC Conf. on Manoeuvring and Control of Marine Craft, Girona, Spain, September 2003, pp. 286–291. 6. I. Blekhman, Synchronization of Dynamical Systems. Nauka, Moscow: in Russian, English translation in ASME Press, New York: Synchronization in Science and Technology, 1971. 7. H. Nijmeijer and A. Rodriguez-Angeles, Synchronization of Mechanical Systems. World Scientiﬁc Series on Nonlinear Science, Series A, 2003, vol. 46. 8. A. L. Fradkov, H. Nijmeijer, and A. Y. Pogromsky, Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, 2000, ch. Adaptive observer-based synchronization, pp. 417 – 438. 9. P. Encarnacao and A. Pascoal, “Combined trajectory tracking and path following: An application to the coordinated control of autonomous marine craft,” in Proc. 40th IEEE Conf. on Decision and Control, Orlando, FL, USA, December 2001, pp. 964 – 969.

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10. E. Kyrkjebø, M. Wondergem, K. Y. Pettersen, and H. Nijmeijer, “Experimental results on synchronization control of ship rendezvous operations,” in Proc. IFAC Conf. On Control Applications in Marine Systems, Ancona, Italy, 2004, pp. 453 – 458. 11. J. Crowley, “Asynchronous control of orientation and displacement in a robot vehicle,” in Proc. 1989 IEEE Int. Conf. on Robotics and Automation, Scottsdale, AZ, USA, May 1989, pp. 1277 – 1282. 12. M. Salichs, E. Puente, D. Gachet, and L. Moreno, “Trajectory tracking for a mobile robot - an application to contour following,” in Proc. Int. Conf. on Industrial Electronics, Control and Instrumentation, Kobe, November 1991, pp. 1067 – 1070. 13. A. Fradkov, S. Gusev, and I. Makarov, “Robust speed-gradient adaptive control algorithms for manipulators and mobile robots,” in Proc. 30th IEEE Conf. on Decision and Control, Brighton, England, December 1991, pp. 3095 – 3096. 14. S. Gusev, I. Makarov, I. Paromtchik, V. Yakubovich, and C. Laugier, “Adaptive motion control of a noholonomic vehicle,” in Proc. 1998 IEEE Int. Conf. on Robotics and Automation, 1998, pp. 3285 – 3290. 15. T. Sakaguchi, A. Uno, and S. Tsugawa, “Inter-vehicle communications for merging control,” in Proc. IEEE Int. Vehicle Electronics Conf., September 1999, pp. 365 – 370. 16. M. Egerstedt, X. Hu, and A. Stotsky, “Control of mobile platforms using a virtual vehicle approach,” IEEE Trans. on Automatic Control, vol. 46, no. 11, pp. 1777 – 1782, November 2001. 17. X. Hu, D. Alarcon, and T. Gustavi, “Sensor-based navigation coordination for mobile robots,” in Proc. 42nd IEEE Conf. on Decision and Control, 2003., December 2003, pp. 6375 – 6380. 18. G. Cheng, J. Gu, T. Bai, and O. Majdalawieh, “A new eﬃcient control algorithm using potential ﬁeld: extension to robot path tracking,” in Canadian Conf. on Electrical and Computer Engineering, 2004., May 2004, pp. 2035 – 2040. 19. R. Skjetne, T. I. Fossen, and P. Kokotovic, “Robust output maneuvering for a class of nonlinear systems,” Automatica, no. 40, pp. 373 – 383, March 2004. 20. A. Chaillet and A. Lor´ıa, “Uniform global practical asymptotic stability for nonautonomous cascaded systems,” July 2006, submitted to the 17th Int. Symposium on Mathematical Theory of Networks and Systems. 21. T. Fossen, Marine Control Systems: Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles. Trondheim, Norway: Marine Cybernetics, 2002. 22. R. Ortega and M. W. Spong, “Adaptive motion control of rigid robots: A tutorial,” Automatica, vol. 25, no. 6, pp. 877 – 888, November 1989. 23. M. J. Paulsen and O. Egeland, “Tracking controller and velocity observer for mechanical systems with nonlinear damping terms,” in Proc. 3rd European Control Conf., Rome, Italy, September 1995. 24. M. Miller and J. A. Combs, “The next underway replenishment system,” Naval Engineers Journal, vol. 111, no. 2, pp. 45–55, March 1999. 25. J.-J. E. Slotine and W. Li, “Adaptive manipulator control a case study,” in Proc. 1987 IEEE Int. Conf. on Robotics and Automation, March 1987, pp. 1392 – 1400. 26. R. Skjetne, Ø. Smogeli, and T. I. Fossen, “Modelling, identiﬁcation and adaptive maneuvering of cybership II: A complete design with experiments,” in Proc. IFAC Conf. on Control Applications in Marine Systems, July 2004, pp. 203 – 208.

11 A Study of Huijgens’ Synchronization: Experimental Results W.T. Oud, H. Nijmeijer and A.Yu. Pogromsky Eindhoven University of Technology, Department of Mechanical Engineering, Eindhoven, The Netherlands [email protected], [email protected], [email protected] Summary. The paper presents experimental results aimed at better understanding of the Huygens’ phenomenon. The experimental setup consists of two metronomes attached to a bar that can move horizontally. Diﬀerent types of oscillations that can be observed during the experiments are discussed.

11.1 Introduction One of the ﬁrst scientiﬁcally documented observations of synchronization is by the Dutch scientist Christiaan Huygens. In the 17th century maritime navigation called for more accurate clocks in order to determine the longitude of a ship. Christiaan Huygens’ solution for precise timekeeping was the invention of the pendulum clock [15] with cycloidal-shaped plates to conﬁne the pendulum suspension. Those plates resulted in isochronous behavior of the pendulum independent of the amplitude and were a genius invention of that time. For some period Huygens was bound to his home due to illness he observed that two pendulum clocks attached to the same beam supported by chairs would swing in exact opposite direction after some time [8, 9, 10]. A drawing made by Christiaan Huygens is given in ﬁgure 11.1. Disturbances or diﬀerent initial positions did not aﬀect the synchronous motion which resulted after about half an hour. This eﬀect which Huygens called “sympathie des horloges” is nowadays known as synchronization and is characterized by [12] as “an adjustment of rhythms of oscillating objects due to their weak interaction”. The oscillating objects in Huygens’ case are two pendulum clocks and are weakly coupled through translation of the beam. Many more cases of synchronization have been identiﬁed in nature and technology around us [14]. Two striking examples in biology are the synchronized ﬂashing of ﬁreﬂies [5] or synchronization of neurons in the brain when performing perceptual tasks. Synchronization is also found in technology, for example the frequency synchronization of triode generators. These generators were the basic elements of early radio communication systems [3, 6]. Using synchronization it is possible to stabilize the frequency of high power generators and there are more applications we are unable to mention in this paper.

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 191–203, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Fig. 11.1. Drawing by Christiaan Huygens of two pendulum clocks attached to a beam which is supported by chairs. Synchronization of the pendula was observed by Huygens in this setup. From [9]

Three centuries later the phenomenon of synchronizing driven pendula is, to our best knowledge, repeated twice experimentally. In the ﬁrst research by Bennett, Schatz, Rockwood and Wiesenfeld [4], one has tried to accurately reproduce the ﬁndings of Huygens in an experimental setup consisting of two pendulum clocks attached to a free moving cart. The results of this experiment conﬁrm the documented observations of Christiaan Huygens. A rather simple but interesting experiment is described by Pantaleone [11], where the synchronization of two metronomes is discussed, which are coupled by a wooden board rolling on soda cans. The metronomes in this setup would synchronize most of the time with in phase oscillations. The research presented in this paper is inspired by the observations of Christiaan Huygens, the work of Bennett et al. and Pantaleone. The main objective is to perform and analyze synchronization experiments with a setup consisting of driven pendula. Particular attention is paid to diﬀerent synchronization regimes that can be observed in this situation : anti-phase synchronization, observed by Huygens, inphase synchronization, observed by Blekhman and explained with the van der Pol equation for each pendulum, and possible intermediate regimes. The paper is organized as follows. First the design of the setup and the measurement methods are discussed and the mathematical model describing the setup is introduced. Then the synchronization experiments are discussed. Finally, conclusions are drawn and recommendations for further research are given.

11.2 Experimental Setup The experimental setup consists of two standard metronomes coupled by a platform which can translate horizontally. The metronomes are made by Wittner, type Maelzel (series 845). The platform is suspended by leaf springs, which allows a frictionless

11 A Study of Huijgens’ Synchronization: Experimental Results

193

Fig. 11.2. Photograph of the setup.

horizontal translation. A photograph of the experimental setup is given in ﬁgure 11.2. 11.2.1 Metronomes The metronomes which are normally used for indicating a rhythm for musicians consist of a pendulum and a driving mechanism, called the escapement. The energy lost due to friction is compensated by this escapement. The escapement consists of a spring which loads on a toothed wheel. These teeth push alternately one of the two cams on the axis of the pendulum. Each time the teeth hit a cam a tick is produced, the typical sound of mechanical metronomes. The frequency of the metronomes can be adjusted with a contra weight attached to the upper part of the pendulum. Variation of the frequency between 2.4 rad/s and 10.8 rad/s is possible with the weight attached, without it the frequency of the metronomes increases to 12.3 rad/s. The amplitude of the metronome’s oscillations cannot be inﬂuenced, however at increasing frequencies the amplitude decreases. 11.2.2 Platform The platform does not only act as a support for the metronomes but because of its horizontal translation it couples the dynamics of both metronomes as well. In order to keep the equations of motion of the total system simple a suspension with linear stiﬀness and damping is desired. As long as the translation of the platform is not too large (mm range) the use of leaf springs makes a frictionless translation possible with linear stiﬀness and damping properties [13]. In order to calculate the necessary dimensions for leaf springs the following estimates have been used. The stiﬀness of

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a leaf spring can be estimated by assuming it behaves as two bars clamped at one side. For small deﬂections the stiﬀness of bar clamped at one side is given by [7] k=

Ehb3 , 4L3

(11.1)

where E is the elastic modulus, h the width, b the thickness and L the length of the bar. The leaf spring has a stiﬀness equal to (11.1) since both halves of the leaf spring take half the deﬂection and half the force, thus equal to the stiﬀness of a single sided clamped beam. The eigenfrequency of the platform in rad/s is estimated by Ω=

g Ehb3 + 4(l/2)3M l

(11.2)

where l is the length of the leaf spring, M the mass of the platform and g the constant of gravity. For the setup the following dimensions for the leaf springs are chosen: l=80 mm, b=0.4 mm and h=15+15+10 mm (three leaf springs). The constants are E = 200 · 109 kg/m/s2 and g=9.81 m/s2 . The platform has a mass of 2.35 kg which results in an eigenfrequency of 31.2 rad/s. The mass can be varied, that allows to study inﬂuence of the system parameters on the diﬀerent synchronization regimes. 11.2.3 Measurements In the setup the angle of the metronomes and the translation of the platform are of interest. Since alteration of the dynamics of both the metronomes and the platform should be avoided, contactless measurement methods have been chosen. All signals are recorded using a Siglab data acquisition system, model 20-42. First the measurement of the metronomes is discussed, secondly that of the translation of the platform. The angle of the metronomes is measured using a sensor based on the anisotropic magnetoresistance (AMR) principle [1], [2]. The resistance of AMR materials changes when a magnetic ﬁeld is applied. Above a minimal ﬁeld strength the magnetization of the material aligns with the external ﬁeld and the following relation holds for the resistance R (11.3) R ∼ cos2 θ where θ is the angle between the magnetic ﬁeld and the current through the resistor. By combining four AMR resistors in a bridge of Wheatstone a change in resistance is converted to a voltage diﬀerence. Two of these bridges of Wheatstone are located in the sensor, but are rotated 45◦ degrees with respect to each other. As a result the voltage diﬀerence of bridges A and B can be written as ∆VA = Vs S sin 2θ , ∆VB = Vs S cos 2θ

(11.4)

where Vs is the voltage supplied to the bridges and S is the AMR material constant. The angle θ can be calculated from these signals by

11 A Study of Huijgens’ Synchronization: Experimental Results

θ=

1 2

arctan(∆VA /∆VB )

195

(11.5)

regardless of the value of voltage Vs and constant S. Due to manufacturing tolerances the bridges will show an oﬀset when no magnetic ﬁeld is applied. This oﬀset can be corrected in software when both signals are recorded. The velocity of the platform is measured using a Polytec Vibrometer, type OFV 3000 with a OFV 302 sensorhead. The measurement is based on the Doppler shift of a laser beam reﬂected on the platform.

11.3 Mathematical Model of the Setup Assuming the setup consists of rigid bodies the equations of motion can be derived using Lagrangian mechanics. The generalized coordinates are chosen as q T = [θ1 , θ2 , x]

(11.6)

which are the angles of the pendula from the vertical and the translation of the platform. The kinetic energy T (q, q) ˙ of the system can be expressed as 1 T (q, q) ˙ = 12 m1 r˙ 1 · r˙ 1 + 21 m2 r˙ 2 · r˙ 2 + M r˙ 3 · r˙ 3 2

(11.7)

where r1 , r2 and r3 are respectively the translation of the center of mass of pendulum 1, 2 and the platform, m1 and m2 are the mass of pendulum 1 and 2, l1 and l2 the lengths of the center of mass to the pivot point of pendulum 1 and 2, M is the mass of the platform and r1 = (x + l1 sin θ1 ) · e1 − l1 cos θ1 · e2 r2 = (x + l2 sin θ2 ) · e1 − l2 cos θ2 · e2

(11.8a) (11.8b)

r 3 = x · e1

(11.8c)

The potential energy V (q) of the system is given by V (q) = m1 gl1 (1 − cos θ1 ) + m2 gl2 (1 − cos θ2 ) + 12 kx

(11.9)

where g is the constant of gravity and k is the spring stiﬀness of the platform. The generalized forces Qnc include viscous damping in the hinges of the pendula and the platform and the torque f i (θi , θ˙i ) exerted by the escapement mechanism on the pendula and can be written as   ˙ ˙  f 1 (θ1 , θ1 ) − d1 θ1      nc  ˙ ˙ (11.10) Q =  f 2 (θ2 , θ2 ) − d2 θ2       −d3 x˙

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where di are the viscous damping constants of respectively the two pendula and the platform. With Lagrange’s equations of motions d T,q˙ − T,q + V,q = Qnc dt

T

(11.11)

the equations of motion for the system become m1 l1 2 θ¨1 + m1 l1 g sin θ1 + m1 l1 cos(θ1 )¨ x + d1 θ˙1 = f 1 (θ1 , θ˙1 ) 2¨ m2 l2 θ2 + m2 l2 g sin θ2 + m2 l2 cos(θ2 )¨ x + d2 θ˙2 = f 2 (θ2 , θ˙2 ) n

Mx ¨ + d3 x˙ + kx +

(11.12)

mi li θ¨i cos θi − θ˙i2 sin θi = 0.

i=1

These equations for the metronomes can be simpliﬁed by dividing all terms by mi li 2 , which give for i = 1, 2 1 1 di ˙ θ¨i + ωi 2 sin θi + cos(θi )¨ f i (θi , θ˙i ) x+ θi = li mi l i 2 mi l i 2

(11.13)

where ωi = g/li . The equations of motion can be written in dimensionless form using the following transformations. The dimensionless time is deﬁned as τ = ωt and the position of the platform as y = x/l = xω 2 /g, where ω is the mean angular frequency of both pendula. The derivatives of the angles with respect to the dimensionless time are written as dθ dθ dτ d2 θ = = ωθ , = ω2θ . dt dτ dt dt2 The equations of motion now become θi + γi2 cos θi y + γi2 sin θi + δi θi = εi f(θi , θi ), y + 2Ωξy + Ω 2 y +

2 i=1

βi γi−2 cos θi θi − sin θi θi

2

(11.14)

= 0,

i with coupling parameter βi = m M , scaled eigenfrequency of the metronomes γi = dωi 2 k ωi /ω, damping factor δi = mi g , eigenfrequency of the platform Ω 2 = Mω 2 and d3 √ damping ratio of the platform ξ = 2 kM .

11.3.1 Escapement So far the escapement has been indicated by the function f(θ, θ ). A close inspection of the metronomes shows that the escapement gives the pendulum a push when going upward. Without deriving an accurate mechanical model of the escapement mechanism this torque is approximated by the following normalized expression: f(θ, θ ) = 0, if θ < φ ∨ θ > φ + ∆φ

(11.15)

11 A Study of Huijgens’ Synchronization: Experimental Results

f(θ, θ ) =

1 − cos(2π θ−φ ∆φ ) 2∆φ

197

,

if φ ≤ θ ≤ φ + ∆φ ∧ θ > 0 where θ1 and θ2 are angles between which the mechanism works and φ and ∆φ are the parameters. In ﬁgure 11.3 the torque of the escapement is plotted versus time when the pendulum would follow a periodic trajectory.

Fig. 11.3. The torque exerted by the escapement on the pendulum is plotted versus time together with the angle and velocity of the pendulum. The vertical axis is scaled in order to illustrate the torque more clearly.

11.4 Experimental Results Several experiments are performed in order to gain experience with the dynamics of the system. Parameters which can be varied in the experiment are the mass of the platform, the mass and frequency of the metronomes and the amount of damping in the system. Converted to the dimensionless parameters the inﬂuence of the physical parameters is: ∆ = (ω1 − ω2 )/ω ,

ω = (ω1 + ω2 )/2

β = m/M k/M /ω √ ξ = d/(2 kM )

Ω=

The experiments show diﬀerent phenomena, ﬁrst of all when the damping of the platform is too small (< 2.0 kg/s) the pendula hit the frame. These experiments are

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discarded since we want to avoid collisions in the experiments. Apparently, without enough damping the platform and consequently the metronomes are excited too much. For larger damping synchronization with diﬀerent phase diﬀerences is observed. Three diﬀerent types of responses can be identiﬁed, anti phase synchronization, intermediate (neither anti nor in) regime with a large amplitude diﬀerence of the angles of the metronomes and ﬁnally in phase synchronization. 11.4.1 Anti-phase Synchronization Experiments have been performed with the setup to show synchronization of the metronomes. The parameters that can be varied in the experimental setup are the frequency of the metronomes and the mass of the platform. The frequency of the metronomes is chosen as high as possible in order to reduce the time experiments will take. By adjusting the counterweights the frequency diﬀerence between both metronomes is minimized, since a too large frequency diﬀerence will make synchronization impossible [4]. The second parameter that can be changed is the mass of the platform. Changing this inﬂuences the dynamics of the system in three ways. First, the coupling between the metronomes changes since the coupling parameter β is the ratio between the mass of the metronome’s pendulum and the total mass of the platform. Secondly the relative eigenfrequency of the platform changes, increasing the mass of the platform lowers the eigenfrequency. Finally the relative damping factor of the platform depends on the mass. In the experiments the frequency of metronome I is set to 10.565 rad/s and that of metronome II to 10.553 rad/s, which results in a mean frequency of ω = 10.559 rad/s and a relative frequency diﬀerence of ∆ = 1.1 · 10−3 . The mass of the platform is varied between 2.35 kg and 8.18 kg in ﬁve steps, resulting in a coupling parameter β varying between 3.8 ·10−3 and 1.1 ·10−3. For each choice of mass the experiments are started with several initial conditions. Since the metronomes need to be started by hand, reproducing the initial conditions exactly between experiments is impossible. When the mass of the platform is varied, the following observations can be made from the experiments. For small mass the setup synchronizes to approximately constant phase for all initial conditions. When the mass of the platform increases the metronomes do not always synchronize anymore. When they do the phase diﬀerence is comparable to that in experiments with small mass of the platform. A typical example of the experiments, in which synchronization is observed, is depicted in ﬁgure 11.4, where the diﬀerence in phase of the metronomes, their amplitudes and the dimensionless velocity of the platform are plotted. The mass of the platform in this experiment is 2.35 kg, resulting in a coupling factor β = 3.77 · 10−3, dimensionless eigenfrequency Ω = 2.1 and dimensionless damping factor ξ = 7.9 · 10−4 . The resulting diﬀerence in phase between the metronomes, when they are synchronized, is approximately 0.8π with a variation of 0.1π. The amplitude of the oscillations of metronome II is larger than that of metronome I, which is also the case, and with comparable magnitude, when the metronomes run uncoupled. The diﬀerence is approximately 0.05 rad and the amplitude of metronome I and II is respectively about 0.80 and 0.85 rad. As the metronomes do not synchronize in exact anti-phase due to the frequency diﬀerence ∆, the platform keeps oscillating. The amplitude of the dimensionless velocity of the platform is approximately 0.01.

11 A Study of Huijgens’ Synchronization: Experimental Results 2

1

1.8

1.4 1.2 1

4 3

I

2 dy/dτ [×0.001]

amplitude θ [rad]

∆ phase [π rad]

1.6

5

II

0.8

0.6

0.4

1 0 −1 −2

0.2

0.8

199

−3 −4

0

1000 2000 3000 4000 τ

−5

1000 2000 3000 4000 τ

1000 2000 3000 4000 τ

Fig. 11.4. A synchronization experiment in which phase synchronization can be observed. The dimensionless parameters are as follows in the experiment, ∆ = 7.9 · 10−4 , β = 3.77 · 10−3 , Ω = 2.1 and ξ = 7.8 · 10−4 . The mean phase diﬀerence between both metronomes is 0.80π after τ = 2000, but a variation of about 10% can be seen around this value.

When the mass of the platform increases, the system does not always synchronizes to a constant phase diﬀerence. An example of such experiment is plotted in ﬁgure 11.5, where M = 5.17 kg and accordingly β = 1.7 · 10−3 , Ω = 1.6 and ξ = 6.8 · 10−4 . After approximately τ = 2000 the system looses synchrony and the amplitude of the metronomes start oscillating. A similar phenomenon occurs in the experiment plotted in ﬁgure 11.6 with the same parameters but diﬀerent initial conditions. In this experiment the metronomes seem to synchronize around τ = 1000, then diverge, but synchronize again after τ = 2600. However the length of the experiment is too short to be sure whether the metronomes will not desynchronize again. 7

1

6

4 3 2

3 I

0.6

0.4

1 0 −1 −2

0.2

1

2 dy/dτ [×0.001]

amplitude θ [rad]

∆ phase [π rad]

4

0.8

5

0

5

II

−3 −4

1000 2000 3000 4000 τ

0

1000 2000 3000 4000 τ

−5

1000 2000 3000 4000 τ

Fig. 11.5. Experiment with β = 1.7 · 10−3 in which synchronization is lost. When this happens the amplitudes of the metronomes diverge and start oscillating.

W.T. Oud, H. Nijmeijer and A.Yu. Pogromsky 4

1

3.5 0.8 amplitude θ [rad]

3 ∆ phase [π rad]

5

2.5 2 1.5 1

3

I

0.6

0.4

2 1 0 −1 −2

0.2

−3

0.5 0

4

II

dy/dτ [×0.001]

200

−4 1000 2000 3000 4000 τ

0

1000 2000 3000 4000 τ

−5

1000 2000 3000 4000 τ

Fig. 11.6. After losing synchronization around τ = 2000 the system synchronizes with a phase diﬀerence of 0.8π. The coupling factor in this experiment is β = 1.7 · 10−3 .

In the performed experiments phase synchronization of two metronomes is visible, however the inﬂuence of disturbances in the system are clearly visible in the diﬀerence in phase of the metronomes. One of the disturbances acting on the system is the irregular operation of the escapement. Due to this the amplitude of the uncoupled metronomes also show a variation of about 10% when oscillating. 11.4.2 In- and Anti-phase Synchronization In a slightly changed experimental setup more types of synchronization can be observed. Instead of leaf springs of 0.4 mm thickness, more ﬂexible leaf springs with 0.1 mm thickness are used. With this change in the experimental setup the value of Ω is approximately 1. A major drawback of this value is that resonance of the platform is possible since the frequency of the platform matches that of the metronomes. When oscillations of the platform become too large, the metronomes will hit the frame. To prevent this, damping of the platform is increased using magnetic damping. The counterweights of the metronomes are removed in these experiments, as a result the frequency of the metronomes is increased, as well as the relative frequency diﬀerence compared to the previous experiments. For a small coupling parameter the system synchronizes with approximate antiphase, as can be seen in ﬁgure 11.7. The mass of the platform is 2.35 kg in this experiment, resulting in the following dimensionless parameters, β = 9.6 · 10−3 , Ω = 0.96 and ξ = 7.8%. A similar behavior can be also observed via computer simulation. The results of computer simulation are presented in ﬁgure 11.8. When the mass of the platform increases anti-phase synchronization does not longer occur, instead the metronomes synchronize in two diﬀerent ways depending on the initial conditions. In ﬁgure 11.9 in-phase synchronization is obtained after starting the metronomes with approximate equal angles and in-phase. If the metronomes are started anti-phase, the system synchronizes to a constant phase difference of about 0.65 rad and a large diﬀerence in amplitude between metronome I

11 A Study of Huijgens’ Synchronization: Experimental Results

201

and II, this is shown in ﬁgure 11.10. In both ﬁgures the parameters of the system are β = 4.7 · 10−3 , Ω = 0.94 and ξ = 4.2%. 2

1

3 0.8

1.4 1.2

I

2 dy/dτ [×0.01]

1.6

amplitude θ [rad]

∆ phase [rad]

1.8

0.6

0.4

1 0 −1 −2

0.2

1 0.8 0

4

II

−3 1000

τ

0 0

2000

1000

τ

−4 0

2000

1000

τ

2000

Fig. 11.7. Experiment in which the metronomes synchronize with approximate anti-phase.

2

1

1.8

1.4 1.2 1

2 I

dy/dτ [×0.01]

amplitude θ [rad]

∆ phase [rad]

3

0.8

1.6

0.6

0.4

1 0 −1 −2

0.2

0.8 0

4 II

−3 1000 τ

2000

0 0

1000 τ

2000

−4 0

1000 τ

2000

Fig. 11.8. Results of computer simulation for the same parameters as in the previous ﬁgure.

11.5 Conclusions This paper presents some experimental results on synchronization of two metronomes attached to a common beam that can move in horizontal direction. From those experiments it becomes evident that diﬀerent synchronization regimes can (co-)exist depending on the system parameters. It is worth mentioning that we have observed from those experiments some intermediate seemingly chaotic regimes of oscillations. Further research will be devoted towards theoretical studies of those oscillations.

W.T. Oud, H. Nijmeijer, and A.Yu. Pogromsky 2.05

1

1.95 1.9 1.85

I

0.6

II

0.4

0.2

1.8 1.75 0

2

0.8 amplitude θ [rad]

∆ phase [rad]

2

3

dy/dτ [×0.01]

202

1000

τ

0 0

2000

1 0 −1 −2

1000

τ

−3 0

2000

1000

τ

2000

Fig. 11.9. For large enough mass of the platform and with the initial conditions close to in-phase synchronization, the metronomes synchronize in-phase. 1.2

1

3 II

0.6

0.4

0.6

0.4

I

0.2

1000

τ

2000

0 0

dy/dτ [×0.01]

0.8

0.2 0

2

0.8 amplitude θ [rad]

∆ phase [rad]

1

1 0 −1 −2

1000

τ

2000

−3 0

1000

τ

2000

Fig. 11.10. The metronomes synchronize with a constant phase but a large diﬀerence in amplitude for equal parameters of the system as when in-phase synchronization is observed, the initial conditions diﬀer however.

Acknowledgement This work was partially supported by the Dutch-Russian program on interdisciplinary mathematics “Dynamics and Control of Hybrid Mechanical Systems” (NWO grant 047.017.018) and the HYCON Network of Excellence, contract number FP6IST-511368.

References 1. Applications of magnetic position sensors. Technical report, Honeywell, 2002. 2. Linear / angular / rotary displacement sensors. Technical report, Honeywell, 2003. 3. E.V. Appleton. The automatic synchronization of triode oscillators. Proc. Cambridge Phil. Soc. (Math. and Phys. Sci.), 21:231–248, 1922.

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4. Matthew Bennett, Michael Schatz, Heidi Rockwood, and Kurt Wiesenfeld. Huygens’s clocks. Proc. R. Soc. Lond. A, 458(2019):563–579, jan 2002. 5. J. Buck. Synchronous rhythmic ﬂashing of ﬁreﬂies. ii. Quaterly review of biology, 63(3):265–289, 1988. 6. B. Van der Pol. Theory of the amplitude of free and forced triod vibration. Radio Rev., 1:701–710, 1922. 7. Roger T. Fenner. Mechanics of solids. Blackwell Scientiﬁc, 1989. 8. Christiaan Huygens. Oeuvres compl´etes de Christiaan Huygens, volume 5. Martinus Nijhoﬀ, 1893. Includes works from 1665. 9. Christiaan Huygens. Oeuvres compl´etes de Christiaan Huygens, volume 17. Martinus Nijhoﬀ, 1932. Includes works from 1651-1666. 10. Christiaan Huygens. Christiaan Huygens’ the pendulum or geometrical demonstrations concerning the motion of pendula as applied to clocks (translated by R. Blackwell). Iowa State University Press, Ames, Iowa, 1986. 11. James Pantaleone. Synchronization of metronomes. American Journal of Physics, 70(10):992–1000, oct 2002. 12. Arkady Pikosvky, Michael Rosenblum, and J¨ urgen Kurths. Synchronization. Cambridge University Press, 2001. 13. P.C.J.N Rosielle and E.A.G. Reker. Constructieprincipes 1. Technical report, Technische Universiteit Eindhoven, 2000. 14. Steven Strogatz. Sync. Hyperion, New York, 2003. 15. Joella G. Yoder. Unrolling Time. Cambridge University Press, 1988.

12 A Study of Controlled Synchronization of Huijgens’ Pendula A.Yu. Pogromsky1, V.N. Belykh2 and H.Nijmeijer1 1 2

Eindhoven University of Technology, Department of Mechanical Engineering, Eindhoven, The Netherlands [a.pogromsky, h.nijmeijer]@tue.nl Volga State Transport Academy, Department of Mathematics, N. Novgorod, Russia [email protected]

Summary. In this paper we design a controller for synchronization problem for two pendula suspended on an elastically supported rigid beam. A relation to Huijgens’ experiments as well as the practical motivation are emphasized.

12.1 Introduction The large number of scientiﬁc contributions in the ﬁeld of synchronization reﬂects the importance of this subject. Synchronization is a fundamental nonlinear phenomenon which was discovered already in the XVIIth century: the Dutch scientist Christian Huijgens discovered that a couple of mechanical clocks hanging from a common support were synchronized [1]. In the ﬁeld of electrical engineering the burst of interest towards synchronization phenomenon was initiated by pioneering works due to van der Pol [3, 4] who studied frequency locking phenomenon in externally driven nonlinear generator as well as mutual synchronization of two coupled oscillators. The achievements of modern electronics would be impossible without a solid synchronization theory developed in last 60 years [5, 6]. In the ﬁeld of mechanical engineering some of the applications are mentioned in a book on the subject [2]. One of the recent directions in this ﬁeld is to employ control theory to handle synchronization as a control problem. For example, in robotics the problem of synchronization is usually referred to as coordination, or cooperation [7, 8, 9, 10]. The problem appears when two or more robot-manipulators have to operate synchronously, especially in situations when some of them operate in hazardous environment, while others (that serve as reference) may be guided by human operation. In this paper we study a controlled synchronization problem for two pendula attached to an elastically ﬁxed horizontal beam. The problem of synchronization is formulated as a controller design problem. The control goal is twofold: ﬁrst, two pendula should be swung up to the desired level of energy and, second they have to move synchronously in opposite directions. This problem has a practical motivation: during a start-up procedure of various vibrational transporters and mills the

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 205–216, 2006. © Springer-Verlag Berlin Heidelberg 2006

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synchronous motion of the rotors allows to avoid resonance oscillations and to reduce energy consumption. The system behaviour considered in this paper is closely related to the phenomenon observed by Huijgens and this relation is emphasized. This paper is an extended version of our previously published conference paper [11]. The paper is organized as follows. First we formulate the problem statement. Next we analyze the behaviour of the uncontrolled system. The controller is proposed and then investigated in Section 12.4. Section 12.5 deals with the local stability analysis of Huijgens phenomenon. Conclusions are formulated in the last section.

12.2 Problem Statement Consider the system schematically depicted in Fig. 12.1. The beam of mass M can move in horizontal direction with viscous friction with damping coeﬃcient d. One side of the beam is attached to the wall through a spring with elasticity k. The beam supports two identical pendula of length l and mass m. The torque applied to each pendulum is the control input.

M

l

k d

u1

u2

x

l

m

g

m Fig. 12.1. The setup

The system equations can be written using Euler-Lagrange equations: x cos φ1 + mgl sin φ1 = u1 ml2 φ¨1 + ml¨ ml2 φ¨2 + ml¨ x cos φ2 + mgl sin φ2 = u2 2

(M + 2m)¨ x + ml

(12.1)

φ¨i cos φi − φ˙ 2i sin φi = −dx˙ − kx,

i=1

where φi ∈ S are the angles of the pendula, x ∈ R1 is the horizontal displacement of the beam, and u1 , u2 are the control inputs. Introduce the Hamiltonian function of each pendulum: 1

ml2 φ˙ 2i + mgl(1 − cos φi ). H(φi , φ˙ i ) = 2 The problem which we are going to address in this paper is to design a controller to swing the pendula up to the desired energy level H∗ in such a way that the pendula

12 A Study of Controlled Synchronization of Huijgens’ Pendula

207

move in opposite directions. Therefore the control objective can be formalized by the following relations: lim H(φi (t), φ˙ i (t)) = H∗ ,

t→∞

i = 1, 2,

lim (φ1 (t) + φ2 (t)) = 0.

(12.2) (12.3)

t→∞

The ﬁrst relation implies that the periods of oscillations of each pendulum are identical (frequency synchronization), while the second relation indicates a particular case of phase synchronization, so called anti-phase synchronization. Although the problem statement formulated above looks rather artiﬁcial it can ﬁnd an important practical motivation. The system we are going to study is a typical system which models vibrational transporters and mills of several kind [2, 12]. When the rotors of those setups are operating at nominal speed they can demonstrate synchronous phenomenon [2, 12]. At nominal speed the setup does not consume signiﬁcant power compared to that during the start-up mode. To decrease the energy consumption during the start-up procedure one can swing up the imbalanced rotors like oscillating pendula until they gain enough potential energy to operate in the rotational mode. One of the main diﬃculties in applying this technique is the vibration of the beam. This is due to the fact that during the start-up procedure the frequency of the external force applied to the beam 2

F := ml

φ¨i cos φi − φ˙ 2i sin φi

i=1

can be in resonance with the eigenfrequency of the beam that results in a high level of energy dissipation by the beam. This eﬀect is known to everybody who observed vibrations of the washing machine during the ﬁrst seconds of wringing. On the other hand, if the two (identical) rotors move synchronously in opposite directions the resulting force applied to the beam is zero. Therefore, the controller able to achieve the objectives (12.2,12.3) can be utilized during the start-up procedure to pass through the resonance and to reduce the power consumption. The main beneﬁt in this case is an opportunity to install lighter and less expensive motors than that used in uncontrolled start-up procedure. A problem of this kind was considered for example in [13, 14, 15, 16].

12.3 Analysis of the Uncontrolled System We begin our study with the analysis of motion of the uncontrolled system taking ui ≡ 0 in the system equations. To analyze the limit behaviour of this system, consider the total energy as a Lyapunov function candidate: V =

m 2

2 i=1

2

M x˙ 2 kx2 x˙ 2 + l2 φ˙ 2i + 2x˙ φ˙ i l cos φi + + mgl . (1 − cos φi ) + 2 2 i=1

Clearly, V ≥ 0. Calculating the time derivative of V along the solutions of the uncontrolled system yields

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A.Yu. Pogromsky, V.N. Belykh, and H. Nijmeijer

V˙ = −dx˙ 2 ≤ 0, so the system can be analyzed by La Salle’s invariance principle. The last inequality implies that V is a bounded function of time, so are x(t), x(t), ˙ φ˙ i (t). Since φi (t) ∈ S 1 every trajectory has a nonempty ω-limit set. On this set V˙ = 0 and, therefore, x ¨ = 0, x˙ = 0 and x = x∗ = const. Since d2 (sin φ1 (t) + sin φ2 (t)) = dt2

2

φ¨i (t) cos φi (t) − φ˙ 2i (t) sin φi (t)

i=1

from the third equation of (12.1) it follows that on the ω-limit set the following relation is true: kx∗ d2 = const (sin φ1 (t) + sin φ2 (t)) = − dt2 ml and hence kx∗ 2 sin φ1 (t) + sin φ2 (t) = − t + c1 t + c2 (12.4) 2ml for some constants c1 , c2 . The left handside of the last equation is a bounded function of time, thus x∗ = 0 and c1 = 0. From the ﬁrst two equations of (12.1) and (12.4) it follows that on the ω-limit set g (12.5) φ¨1 (t) + φ¨2 (t) = − c2 l and therefore g φ˙ 1 (t) + φ˙ 2 (t) = − c2 t + c3 l for some constant c3 . However, φ˙ i (t) is bounded and therefore necessarily c2 = 0. Finally from (12.4) and (12.5) we have for the ω-limit set of any trajectory x(t) = 0,

sin φ1 (t) + sin φ2 (t) = 0,

φ¨1 (t) + φ¨2 (t) = 0

The second identity implies that either φ1 (t) = −φ2 (t) or

φ1 (t) − φ2 (t) = ±π

The latter implies that φ¨1 (t)− φ¨2 (t) = 0. However on the ω-limit set φ¨1 (t)+ φ¨2 (t) = 0 therefore the two pendula should be at rest. Finally, we have proved that all the system trajectories tend to the set where φ1 = −φ2 , φ˙ 1 = −φ˙ 2 , x = 0, x˙ = 0. From this observation one can make some important conclusions. First, the uncontrolled system can exhibit synchronous behavior. Clearly, it follows that the Hamiltonian of each pendulum tends to a common limit. However, due to energy dissipation the limit value depends on the initial conditions and particularly, if one initializes the pendula from an identical point, the oscillations will decay. Therefore

12 A Study of Controlled Synchronization of Huijgens’ Pendula

209

the uncontrolled system exhibits a behavior which is very close to the desired one, and there is one thing left to the controller – to maintain the energy level for each pendulum. To demonstrate the behavior of the uncontrolled system we made a computer simulation for the following system parameters: M = 10 kg, m = 1 kg, g = 9.81 m/sec2 , l = 1 m, d = 20 N · sec/m, k = 1 N/m, φ1 (0) = 0.5 rad, φ2 (0) = ˙ = 0. The simulations conﬁrm that the sys0, φ˙ 1 (0) = φ˙ 2 (0) = 0, x(0) = 0, x(0) tem trajectories tend to synchronous mode where the pendula oscillate in anti-phase, while the oscillations of the beam decay.

12.4 Controlled Synchronization In this section we propose a controller to solve the synchronization problem. To this end we assume that all the state variables are available for measurements. The controller basically consists of two loops. The ﬁrst loop makes the synchronization manifold globally asymptotically stable, while the second loop swings the pendula up to the desired energy level. To analyze the system behaviour in the coordinates transverse to the desired synchronous mode one can add the two ﬁrst equations of (12.1). Then, to make the synchronous regime asymptotically stable one can try to cancel the terms which depend on x ¨ by means of appropriate feedback. It gives a simple hint how to derive the following controller:      s1   u1    = (I2 + Σ)−1       s2 u2 where



  Σ =K  K=

(12.6)

2

cos φ1

cos φ1 cos φ2  ≥0  2 cos φ1 cos φ2 cos φ2 m M + m sin φ1 + m sin2 φ2 2

and s1 = −γ φ˙ 1 H(φ1 , φ˙ 1 ) + H(φ2 , φ˙ 2 ) − 2H∗ + w cos φ1 + v s2 = −γ φ˙ 2 H(φ1 , φ˙ 1 ) + H(φ2 , φ˙ 2 ) − 2H∗ + w cos φ2 + v where γ is a positive gain coeﬃcient and mg (sin 2φ1 + sin 2φ2 ) 2 + ml(φ˙ 21 sin φ1 + φ˙ 22 sin φ2 ) − dx˙ − kx

w=K

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A.Yu. Pogromsky, V.N. Belykh, and H. Nijmeijer

and

λ2 λ1 mgl (sin φ1 + sin φ2 ) − sin(φ1 + φ2 ) − (φ˙ 1 + φ˙ 2 ) 2 2 2 with positive λ1 , λ2 . v=

0.5 0.04

0.4

φ (t) 1

0.03

φ2(t)

0.3 0.2

0.02

0.1

0.01

x(t)

0

0

−0.1

−0.01

−0.2

−0.02

−0.3

−0.03

−0.4 −0.04

−0.5

0

5

10

15

20

25

30

35

0

5

10

15

20

25

30

35

Fig. 12.2. Synchronization of two pendula in the uncontrolled system. a - oscillations of the pendula; b - oscillations of the beam.

The analysis of the closed loop system then can be performed in two steps. First, one can prove that the set φ1 = −φ2 , φ˙ 1 = −φ˙ 2 is globally asymptotically stable. It follows from the equations of the closed loop system with respect to the variable φ1 + φ2 : ml2 (φ¨1 + φ¨2 ) + λ1 (φ˙ 1 + φ˙ 2 ) + λ2 sin(φ1 + φ2 ) = −γ(H(φ1 , φ˙ 1 ) +H(φ2 , φ˙ 2 ) − 2H∗ )(φ˙ 1 + φ˙ 2 ). Therefore, if λ1 > 2γH∗ the set φ˙ 1 = −φ˙ 2 , φ1 = −φ2 is globally asymptotically stable. There is only one invariant subset of this set, namely, x ≡ 0, and hence, the limit dynamics of each pendulum is given by the following equation ˙ ˙ − H∗ ] φ) ml2 φ¨ + mgl sin φ = −2γ φ[H(φ, and therefore the control objective ˙ lim H(φ(t), φ(t)) = H∗

t→∞

is achieved for almost all initial conditions [17]. The above arguments can be summarized as the following statement: Theorem 12.1. Suppose λ1 > 2γH∗ , then in the closed loop system (12.1,12.6) the control goal (12.2,12.3) is achieved for almost all initial conditions. To demonstrate the ability of the controller to achieve the control objective we carried out computer simulation for the same system parameters as before with the

12 A Study of Controlled Synchronization of Huijgens’ Pendula

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following parameters of the controller: H∗ = 9.81, λ1 = 20, λ2 = 4.905, γ = 0.045. The results of the simulation are presented in ﬁgure 12.3 for the following initial conditions φ1 (0) = 0.1, φ2 (0) = 0.2, φ˙ 1 (0) = φ˙ 2 (0) = 0, x(0) = 0, x(0) ˙ = 0. As predicted by the theorem there is a set of zero Lebesgue measure of exceptional initial conditions for which the control objective can not be achieved. For example, if one initiates the system at a point where φ1 = φ2 , φ˙ 1 = φ˙ 2 , the oscillations of both pendula and the beam will decay. However, from a practical point of view, it is not diﬃcult to modify the controller to handle this problem. −3

1.5

3

x 10

2.5

1 φ (t) 2

2

φ (t) 1

0.5

1.5 0 1 −0.5

x(t) 0.5

−1

−1.5

0

0

5

10

15

20

25

−0.5

0

5

10

15

20

25

Fig. 12.3. Synchronization of two pendula in the controlled system. a - oscillations of the pendula; b - oscillations of the beam.

As we have mentioned, the controlled synchronization can be utilized to avoid resonance oscillations during the start-up procedure of speeding up the rotors installed on a common support. To demonstrate this eﬀect we carried out the next simulation. In this case the desired energy level H∗ is larger than the critical level 2mgl, H∗ = 5mgl. We also decreased the damping coeﬃcient d to make the motion of the beam oscillatory: d = 5. The gains γ, λ1 are decreased as well, γ = 0.005, λ = 6. The orther parameters are the same as in the previous simulation. The results are plotted in ﬁgure 12.4. It is seen that during the start-up procedure the rotors oscillate synchronously and the amplitude of the beam oscillations is small. In the beginning of this section we assumed that all the state variables are available for measurements. This assumption allows to design a simple controller with a relatively simple stability analysis. From a practical point of view, one can impose some additional constraints on the controller, i.e. to avoid the measurements of the beam position/velocity. This problem is deﬁnitely feasible but requires a bit more sophisticated analysis, which will not be given here.

12.5 Huijgens’ Phenomenon The results of the previous sections are immediately related to the experiment described by Christian Huijgens in 1665. He detected that a couple of pendulum clocks

212

A.Yu. Pogromsky, V.N. Belykh, and H. Nijmeijer −3

50

14

40

12

30

10

x 10

8

20 φ1(t)

10

x(t)

6 4

0

2

−10 φ2(t) −20

0

−30

−2

−40

−4

−50

0

5

10

15

20

25

30

35

40

45

50

−6

0

5

10

15

20

25

30

35

40

45

50

Fig. 12.4. Synchronization of two imbalanced rotors during the start-up procedure. a oscillations of the rotors; b - oscillations of the beam.

hanging from a common support had synchronized, i.e. their oscillations coincide perfectly and the pendula moved in opposite directions. Huijgens described in detail such coupled clocks: “In these clocks the length of the pendulum was nine inches and its weight one-half pound. The wheels were rotated by the force of weights and were enclosed together with weights in a case which was four feet long. At the bottom of the case was added a lead weight of over one hundred pounds so that the instrument would better maintain a perpendicular orientation when suspended in the ship. Although the motion of the clock was found to be very equal and constant in these experiments, nevertheless we made an eﬀort to perfect it still further in another way as follows ... the result is still greater equality of clocks than before.” Some of the original Huijgens’ drawings are reproduced in ﬁgures 12.6, 12.7. Particularly, Huijgens described what is now called “frequency synchronization”, i.e. being coupled two oscillators with nonidentical frequencies demonstrate synchronous oscillations with a common frequency. The frequency synchronization of clocks was observed, for example in [2], where the clocks were modelled by Van der Pol equations with slightly diﬀerent periods of oscillations. Via an averaging technique it was shown that the system of interconnected oscillators possesses an asymptotically stable periodic solution. In [18] the authors studied synchronization of two clocks from theoretical and experimental points of view. To perform a theoretical analysis they assumed a rather simple escapement mechanism and then derived the Poincare map from the nonlinear dynamics. In this section we show how to realize the phase synchronization for a very simple yet illustrative model of clocks. The pendulum clocks can be modelled in diﬀerent ways, see, e.g. [19]. In our clock model we combine together two simple ideas: ﬁrst, the oscillations of the clock pendulum should be described by equations of the free pendulum with a given level of energy; second, the model should take into account an escapement mechanism to sustain this level. Then, the simplest model of the pendulum clock is given by the following equation: ˙ ˙ − H∗ ], γ > 0. φ) ml2 φ¨ + mgl sin φ = −γ φ[H(φ,

12 A Study of Controlled Synchronization of Huijgens’ Pendula

213

This equation has an orbitally stable periodic solution which corresponds to the motion of the free pendulum with the energy equal to H∗ . This limit cycle attracts almost all initial conditions as can be seen from the following relation for the Hamil˙ tonian function H(φ, φ): H˙ = −γ φ˙ 2 (H − H∗ ). The model of the two clocks hanging from a beam as shown in ﬁgure 12.1 can thus be derived as the system (12.1) with ui = −γ φ˙ i [H(φi , φ˙ i ) − H∗ ], i = 1, 2.

(12.7)

From the equations of this model it follows that the system has at least two invariant sets Ω1 := {φ˙ 1 = −φ˙ 2 , φ1 = −φ2 , x = 0, x˙ = 0} and Ω2 which is a some subset of the set {φ˙ 1 = φ˙ 2 , φ1 = φ2 }. Computer simulation shows that both of them can be stable provided the constant H∗ is relatively small, while for large values of H∗ the system can demonstrate erratic behaviour. Thus at least three diﬀerent synchronization regimes can be observed for such a system depending on the system parameters and/or initial conditions: anti-phase synchronization, in-phase synchronization and another oscillatory regime which is neither in-phase nor antiphase synchronization (see Fig. 12.5. It is interesting to note that similar oscillatory regimes can be observed for an experimental setup that consists of two metronomes attached to a common beam that can move in the horizontal direction [20]. Motivated by this fact one can try to use the model (12.1) with the escapement mechanism (12.7) as the model to describe the Huijgens’ phenomenon. This model is simpler than that used in [18], yet the global analysis of that model is quite involved because of diﬀerent co-existing attractors. Therefore, we perform a local stability analysis for the set Ω1 assuming that H∗ is a small parameter. Under this assumption the system equations are ml2 φ¨1 + ml¨ x + mglφ1 = u1 ml2 φ¨2 + ml¨ x + mglφ2 = u2 (M + 2m)¨ x + ml(φ¨1 + φ¨2 ) = −dx˙ − kx with

ui = −γ φ˙ i [H(φi , φ˙ i ) − H∗ ], i = 1, 2

and

ml2 ˙ 2 mgl 2 φ + φ . 2 2 Linearizing this system around the set H(φ1 , φ˙ 1 ) = H(φ2 , φ˙ 2 ) = H∗ yields the following equation (written with a little abuse in notations): ˙ = H(φ, φ)

x + mgl(φ1 + φ2 ) = 0 ml2 (φ¨1 + φ¨2 ) + 2ml¨ (M + 2m)¨ x + ml(φ¨1 + φ¨2 ) = −dx˙ − kx and the local stability of the set Ω1 follows. Therefore the synchronous motion of clocks’ pendula is asymptotically stable.

A.Yu. Pogromsky, V.N. Belykh, and H. Nijmeijer 1

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250 t

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Fig. 12.5. Diﬀerent oscillatory modes for system (12.1) with the escapement model (12.7). a-anti-phase synchronization, M = 10 kg, m = 1 kg, g = 9.81 m/sec2 , l = 1 m, d = 20 N · sec/m, k = 1 N/m, γ = 0.145, H∗ = 3, φ1 (0) = 0.1 rad, φ2 (0) = −0.1005, φ˙ 1 (0) = ˙ = 0 , b-in-phase synchronization, the same parameφ˙ 2 (0) = 0, x(0) = 0, x(0) ters as in a, except for φ2 (0) = 0.1005, c-erratic oscillations for M = 10 kg, m = 1 kg, g = 9.81 m/sec2 , l = 1 m, d = 50 N · sec/m, k = 1 N/m, γ = 0.045, H∗ = 9, ˙ =0 φ1 (0) = 0.1 rad, φ2 (0) = 0.101, φ˙ 1 (0) = φ˙ 2 (0) = 0, x(0) = 0, x(0)

12.6 Conclusions In this paper we considered the problem of controlled synchronization of two pendula hanging from a common support. This problem can ﬁnd an important practical application - to avoid resonance vibration during the start up procedure of speeding up two imbalanced rotors. We proposed a controller which is able to solve the synchronization problem in such a way that the pendula reach the desired level of energy and they move synchronously in opposite directions. In this case the oscillations of the support beam can be avoided. It is worth mentioning that the solution proposed in this paper is based on the synchronization phenomenon experimentally observed by C. Huijgens in 1665. This example demonstrates that (non)linear control can be utilized in various applied problems related to synchronization.

12 A Study of Controlled Synchronization of Huijgens’ Pendula

215

Fig. 12.6. The Huijgens drawing describing his synchronization experiment, from [1]

Fig. 12.7. The Huijgens drawing describing his synchronization experiment, from [1]

Acknowledgments This work was supported by the Dutch-Russian program on interdisciplinary mathematics “Dynamics and Control of Hybrid Mechanical Systems” (NWO grant 047.017.018).

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References 1. C. Hugenii, Horoloquim Oscilatorium, Apud F. Muguet, Parisiis, France, 1673; English translation: The pendulum Clock, 1986, Iowa State University Press, Ames. 2. I.I. Blekhman, Synchronization in science and technology, ASME, New York, 1998. 3. B. van der Pol “On oscillation hysteresis in a triode generator with two degrees of freedom”, Phil. Mag., vol. 6(43), p. 700, 1922. 4. B. van der Pol, “Forced oscillations in a circuit with non-linear resistance,” Phil. Mag., vol. 3, p. 64–80, 1927. 5. W. Lindsey, Synchronization Systems in Communication and Control, 1972, PrenticeHall, NJ. 6. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Science, Cambridge University Press, Cambridge, 2001. 7. A. Rodrigues-Angeles, and H. Nijmeijer, “Coordination of two robot manipulators based on position measurements only,” Int. J. Control 74,1311-1323(2001). 8. A. Rodriguez-Angeles, H. Nijmeijer, “Mutual Synchronization of Robots via Estimated State Feedback: A Cooperative Approach”, IEEE Trans. on Control Systems Techn., 12(4), 542–554, (2004) 9. Y.-H. Liu, Y. Xu, and M. Bergerman, “Cooperation control of multiple manipulators with passive joints,” IEEE Trans. Robotics Autom. 15,258-267(1999) 10. P. Ogren, E. Fiorelli, N.E. Leonard, “Cooperative control of mobile sensor networks:Adaptive gradient climbing in a distributed environment,” IEEE Trans. Aut. Control, vol. 49, 1292–1302, 2004 11. A. Pogromsky, V. Belykh, H. Nijmeijer, “Controlled synchronization of Huijgens’ pendula”, in Proc. of Conference on Decision and Control’2003, Hawaii, United States, 4381–4386, (2003). 12. I.I. Blekhman, Vibrational Mechanics, World Scientiﬁc, Singapore, 2000. 13. I.I. Blekhman, Yu.A. Bobtsov, A.L. Fradkov, S.V. Gavrilov, V.A. Konoplev, B.P. Lavrov, V.M. Shestakov, O.P. Tomchina, “Modelling and control of the mechanical vibrational unit”, Proceedings of 5th Symp. Nonl. Contr. Sys., NOLCOS’2001, St. Petersburg, Russia. 14. O.P. Tomchina, “Passing through resonance in vibratory actuators by speed-gradient control and averaging”, Proc. Int. Conf. Control of Oscillationas ansd Chaos, St. Petersburg, 1997, v.1, 138–141. 15. O.P. Tomchina, V.M. Shestakov, K.V. Nechaev, “Start-up mode control for electrical drives of vibrational units”, Proc. 2nd Intern. IEEE-IUTAM Conf. Control of Oscillations and Chaos, St. Petersburg, 2000, 509–512. 16. I.I. Blekhman, A.L. Fradkov, O.P. Tomchina and D.E. Bogdanov, “Self-synchronization and controlled synchronization: general deﬁnition and example design”, Mathematics and Computers in Simulation, vol. 58(4), 2002, 367–384. 17. A.L. Fradkov, “Swinging control of nonlinear oscillations”, Int. J. Control, vol. 64(6), 1189–1202, 1996. 18. M. Bennett, M.F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’s clocks”. Proceedings of the Royal Society A vol.458, pp. 563-579, 2002. 19. N.N. Bautin, The dynamical Theory of Clocks, Nauka, Moscow, 1989, in Russian. 20. W. Oud, H. Nijmeijer, A. Pogromsky, “Experimental results on Huygens synchronization”, submitted.

13 Group Coordination and Cooperative Control of Steered Particles in the Plane R. Sepulchre1 , D.A. Paley2 and N.E. Leonard2 1

2

Electrical Engineering and Computer Science Universit´e de Li`ege Institut Monteﬁore B28, B-4000 Li`ege, Belgium [email protected] Mechanical and Aerospace Engineering Princeton University Princeton, NJ 08544, USA [email protected], [email protected]

Summary. The paper overviews recent and ongoing eﬀorts by the authors to develop a design methodology to stabilize isolated relative equilibria in a kinematic model of identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with ﬁxed relative spacing or to circular motion of all particles about the same center with ﬁxed relative headings.

13.1 Introduction Feedback control laws that stabilize collective motions of particle groups have a number of engineering applications including unmanned sensor networks. For example, autonomous underwater vehicles (AUVs) are used to collect oceanographic measurements in formations that maximize the information intake, see e.g. [6] and the references therein. In this paper, we consider a kinematic model of identical (pointwise) particles in the plane [2]. The particles move at constant speed and are subject to steering controls that change their orientation. In recent work [11], see also [10, 9], we proposed a Lyapunov design to stabilize isolated relative equilibria of the model. Isolated relative equilibria correspond to either parallel motion of all particles with ﬁxed relative spacing or to circular motion of all particles about a common center with ﬁxed relative headings. The stabilizing feedbacks were derived from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results in [11] assume an all-to-all communication topology, that is, the feedback control applied to one given particle uses information about the (relative) heading and position of all other particles. The objective of the present paper is to relax the all-to-all assumption on the communication topology in diﬀerent ways. We show how the Lyapunov design of stabilizing control laws can be extended to any constant, bidirectional, and connected

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 217–232, 2006. © Springer-Verlag Berlin Heidelberg 2006

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communication topology. We provide a uniﬁed interpretation of all the Lyapunov functions considered in earlier work as quadratic forms induced by the Laplacian of the graph associated to the communication topology. We then address the more challenging situation of time-varying and unidirectional communication topologies. We brieﬂy review recent results in the literature that address the stabilization of parallel motions and we propose a new control law that stabilizes circular motions with time-varying and unidirectional communication topologies. The model assumptions are recalled in Section 13.2. Section 13.3 introduces the quadratic functions induced by the communication topology. The main Lyapunov functions considered in [11] are then reinterpreted and generalized in Section 13.4. Section 13.5 provides a further analysis of the phase potentials used for the design. In Section 13.6, we address the situation of time-varying and unidirectional communication topologies. A short discussion concludes the paper in Section 13.7.

13.2 Particle Model and Control Design We consider a continuous-time kinematic model of N > 1 identical particles (of unit mass) moving in the plane at unit speed [2]: r˙k = eiθk θ˙k = uk ,

(13.1)

where k = 1, . . . , N . In complex notation, the vector rk = xk + iyk ∈ C ≈ R2 denotes the position of particle k and the angle θk ∈ S 1 denotes the orientation of its (unit) velocity vector eiθk = cos θk + i sin θk . We refer to θk as the phase of particle k. We use the boldface variable without index to denote the corresponding N -vector, e.g. θ = (θ1 , . . . , θN )T . We also let eiθ ∈ CN where r˙ = eiθ = (eiθ1 , . . . , eiθN )T . The conﬁguration space consists of N copies of the group SE(2). In the absence of steering control (θ˙k = 0), each particle moves at unit speed in a ﬁxed direction and its motion is decoupled from the other particles. We study the design problem of choosing feedback controls that stabilize a prescribed collective motion. The feedback controls are identical for all the particles and depend only on relative orientation and relative spacing, i.e., on the variables θkj = θk −θj and rkj = rk −rj , j, k = 1, . . . , N . Consequently, the closed-loop vector ﬁeld is invariant under an action of the symmetry group SE(2) and the closed-loop dynamics evolve on a reduced quotient manifold (shape space). Equilibria of the reduced dynamics are called relative equilibria and can be only of two types [2]: parallel motions, characterized by a common orientation for all the particles (with arbitrary relative spacing), and circular motions, characterized by circular orbits of the particles around the same ﬁxed point.

13 Group Coordination and Cooperative Control

13.3 Communication Topology and Laplacian Quadratic Forms The feedback control laws are further restricted by a limited communication topology. The communication topology is deﬁned by an undirected graph G(V, E) with N vertices in V = {1, . . . , N } and e edges (k, j) ∈ E whenever there exists a communication link between particle k and particle j. We denote by N (k) = {j | (j, k) ∈ E} the set of neighbors of k, that is, the set of vertices adjacent to vertex j. The control uk is allowed to depend on rkj and θkj only if j ∈ N (k). Consider the (undirected) graph G = (V, E) and let dk be the degree of vertex k. The Laplacian L of the graph G is the matrix deﬁned by     dk , if k = j,     Lk,j = −1, if (k, j) ∈ E, (13.2)        0, otherwise. The Laplacian matrix plays a fundamental role in spectral graph theory [1]. Only basic properties of the Laplacian are used in this paper. First, L1 = 0, where 1 = (1, . . . , 1)T ∈ RN , and the multiplicity of the zero eigenvalue is the number of connected components of the graph. As a consequence, the Laplacian matrix of a connected graph has one zero eigenvalue and N − 1 strictly positive eigenvalues. We denote by < ·, · > the standard inner product in CN . The quadratic form Q(z) =< z, Lz >, where L is the Laplacian of a connected graph, vanishes only when z = 1z0 . It deﬁnes a norm on the shape space CN /C induced by the action of the group of rigid displacements z → z + 1z0 . Consider the valence matrix D = diag(d), the adjacency matrix A, and the incidence matrix B ∈ RN ×e associated to the graph G. One easily shows that L = D − A. Using the property L = BB T for a bidirectional graph, an alternative expression for the quadratic form Q(z) is |zk − zj |2 .

Q(z) = (k,j)∈E

In words, Q(z) is thus the sum of the squared lengths of the edges connecting communicating vertices zk . Example 13.1. Let 1 = (1, . . . , 1)T ∈ RN and I = diag(1). An all-to-all communication topology with N vertices corresponds to a complete graph, KN , with the Laplacian L = N I − 11T = N P where P = I − N1 11T is the projector orthonormal to the vector 1. Using the property P 2 = P , the quadratic form Q(z) then takes the form Q(z) = N P z 2 , which is (N times) the sum of the squared distances of vertices zk , k = 1, . . . , N , to N their centroid N1 j=1 zj .

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Example 13.2. A meaningful generalization of the all-to-all communication topology is the topology corresponding to a circulant graph in which each node is connected to n other nodes, for a ﬁxed n where 2 ≤ n ≤ N − 1. Each column of the Laplacian is a cyclic shift of the vector v with     n, k = 1,     (13.3) vk = −1, k = 2, . . . , n + 1,        0, k = n + 2, . . . , N.

13.4 Collectives Speciﬁed by Synchrony 13.4.1 Parallel and Circular Motion Under the constant control uk = ω0 , ω0 = 0, the particle k rotates on a circle of radius ρ0 = 1/|ω0 | centered at ck = rk + iω0−1 eiθk . Achieving a circular formation amounts to synchronizing all the particle centers. This prompts us to deﬁne sk = iω0 ck = −eiθk + iω0 rk

(13.4)

and to specify a circular formation by the synchronization condition s = 1s0 for an arbitrary s0 ∈ C, or, equivalently, (I −

1 T 11 )s = 0. N

(13.5)

Note that for ω0 = 0, we have sk = −eiθk and the condition (13.5) thus speciﬁes the phase synchrony θ = 1θ0 . One concludes that the sync condition (13.5) either speciﬁes a parallel motion (ω0 = 0), or a circular motion of radius ρ0 = |ω0 |−1 . The dynamics (13.6) s˙ k = −ieiθk (uk − ω0 ) shows that each sk is invariant under the constant control uk = ω0 . The Lyapunov function U (s) = KQ(s) = K < s, Ls >, K > 0, where L is the Laplacian of a connected graph, thus has the property of reaching its global minimum (only) when the sync condition (13.5) is satisﬁed. U (s) is invariant under the constant control uk = ω0 : U˙ = −2K

N

< ieiθk , Lk s > (uk − ω0 ).

(13.7)

k=1

The (dissipation) control uk = ω0 + K < ieiθk , Lk s >,

K >0

(13.8)

ensures that U evolves monotonically along the closed-loop solutions since U˙ = − ∂U 2 ≤ 0. Moreover, it satisﬁes the restrictions imposed by the communication ∂θ topology.

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Proposition 13.1. Let L be the Laplacian matrix of a bidirectional, connected graph. Consider the model (13.1) with the shape control law (13.8). Then for ω0 < 0 (resp. ω0 > 0), the set of clockwise (resp. counter-clockwise) circular motions of radius |ω0 |−1 is globally asymptotically stable and locally exponentially stable. For ω0 = 0, the set of parallel motions is locally exponentially stable; moreover, each solution converges to the set of critical points of U (eiθ ). Proof. This is proved by a straightforward adaptation of the corresponding proof in [11] where it was applied to the particular case L = N (I − N1 11T ). Using the equality L = D − A = BB T , one has several equivalent expressions for Q(s) with ω0 = 0: Q(s)|ω0 =0 = Q(˙r) = Q(eiθ ) = < eiθ , Leiθ > = trD− < eiθ , Aeiθ > = trD − 21Te cos(B T θ).

(13.9) (13.10) (13.11)

Likewise, one has the equivalent expressions for the derivative of Q(˙r), ∂Q = 2B sin(B T θ) ∂θ and

∂Q =2 ∂θk

(13.12)

sin(θk − θj ).

(13.13)

j∈N (k)

The quadratic function Q(r) ˙ reaches its minimum when r˙ = 1eiθ0 , that is, when all phases synchronize, which corresponds to a parallel motion. The control u = −B sin(B T θ) is proposed in [4] to achieve synchronization in the phase model θ˙ = u. It generalizes to arbitrary communication topologies the all-to-all sinusoidal coupling encountered in the Kuramoto model [5]. For the KN topology (all-to-all communication), the quadratic function Q becomes Q(r) ˙ =N

P eiθ

2

= N 2 (1 − |

1 N

N

eiθk |2 ).

(13.14)

k=1

Up to a constant and a change of sign, it coincides with the phase potential |pθ |2 N iθk used in [11], where pθ = N1 denotes the centroid of particle phases, or k=1 e N equivalently, the average linear momentum R˙ = N1 k=1 r˙k . The parameter |pθ | is a classical measure of synchrony of the phase variables θ [5, 12]. It is maximal when all phases are synchronized (identical). It is minimal when the phases balance to result in a vanishing centroid. In the particle model (13.1), synchronization of the phases corresponds to a parallel formation: all particles move in the same direction. In contrast, balancing of the phases corresponds to collective motion with a ﬁxed center of mass.

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13.4.2 Isolated Circular Relative Equilibria For ω0 = 0, the sync condition (13.5) speciﬁes a circular relative equilibrium but the phase arrangement of the particles is arbitrary. A given phase arrangement θ = θ¯ + 1(ω0 t + θ0 ) (with θ0 ∈ S 1 arbitrary) can be speciﬁed by means of the additional sync condition (I −

1 T i(θ−θ) ¯ 11 )e = 0. N

(13.15)

To enforce this additional sync condition by feedback, we use the augmented Lyapunov function ¯

¯

V (r, θ) = K < s, Ls > +K1 < ei(θ−θ) , Lei(θ−θ) > with K > 0 and K1 ≥ 0 arbitrary positive constants, and, accordingly, the augmented feedback control ¯

¯

uk = ω0 + K < ieiθk , Lk s > −K1 < iei(θk −θk ) , Lk ei(θ−θ) >

(13.16)

Proposition 13.2. Let L be the Laplacian matrix of a bidirectional, connected graph. Then the shape feedback control (13.16) exponentially stabilizes the isolated relative equilibrium determined by a circular motion of radius |ω0 |−1 (ω0 = 0) with a phase arrangement satisfying the synchrony condition (13.15). Moreover, every ¯ solution of the closed-loop system converges to a critical point of Q(ei(θ−θ) ). The convergence under control law (13.16) is illustrated in Figure 13.1. Simulations are shown in the case of N = 12 particles with a circulant interconnection topology where each particle communicates with four others. In the case K1 = 0 the particles converge to a circular formation with arbitrary phase arrangement. In the case K1 = 0.01 convergence is to a circular formation with phases dictated by ¯ here, θ¯ is chosen for uniform distribution of the particles around the circle, also θ; known as a splay state. 13.4.3 Isolated Parallel Relative Equilibria For ω0 = 0, the sync condition (13.5) speciﬁes a parallel relative equilibrium but the distance between particles is arbitrary. A ﬁxed vector r¯ ∈ CN speciﬁes an isolated parallel relative equilibrium via the condition r = r¯eiθ0 + 1r0 (t)

(13.17)

where r0 (t) = r0 + teiθ0 and where the constants θ0 ∈ S 1 and r0 ∈ C are arbitrary. Motivated by the previous sections, the following proposition speciﬁes this parallel equilibrium as a synchrony condition. Proposition 13.3. Let L be the Laplacian matrix of a bidirectional, connected graph. Let r¯ be an arbitrary vector in CN such that Lk r¯ = −1, k = 1, . . . , N . Deﬁne the vector

13 Group Coordination and Cooperative Control 30

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Fig. 13.1. Simulations of the circular control law (13.16) for a 4-circulant topology connecting N = 12 particles. The particle positions and headings are indicated by black circles and arrows. Both simulations use ω0 = 0.1, K = 0.04, and were simulated for the dura2π . (a) Control (13.16) with K1 = 0; (b) control (13.16) with K1 = 0.01 and θ¯ tion 10 ω 0 corresponding to a splay state).

¯ −1 (Lr + eiθ ), t = (I + D)

¯ = diag(Lr¯). D

(13.18)

Then the isolated relative equilibrium (13.17) is uniquely determined by the synchrony conditions Lt = 0 Le

iθ

= 0.

(13.19) (13.20)

Proof. The synchrony conditions (13.19) and (13.20) impose t = 1t0 and eiθ = 1eiθ0 for some ﬁxed t0 and eiθ0 . By deﬁnition of t, this yields ¯ 0 = Lr + 1eiθ0 . (I + D)1t

(13.21)

¯ = 1T L¯r = 0. Left multiplication of both sides by 1T yields t0 = eiθ0 since 1T D1 ¯ iθ0 = Lr¯eiθ0 , that is, But then (13.21) implies Lr = D1e r(t) = r¯eiθ0 + 1r0 (t). ˙ Diﬀerentiating both sides, we obtain r(t) = eiθ = 1r˙0 (t) which implies r˙0 = eiθ0 under the synchrony assumption (13.20). This concludes the proof. To enforce the conditions (13.19) and (13.20), we choose the Lyapunov function V (s, θ) = K < t, Lt > +(1 + K) < eiθ , Leiθ > .

(13.22)

The importance of specifying the parallel equilibrium through the synchrony conditions (13.19) and (13.20) is that the Lyapunov function (13.22) can be rendered nonincreasing along the closed-loop solutions by means of a feedback control that satisﬁes the required communication topology. Proposition 13.4. Consider the particle model (13.1) with the control law

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20 y

10 0 0

20

40

x

60

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Fig. 13.2. Simulation of the parallel control law (13.23) for a complete graph connecting N = 3 particles. The particle positions and headings are indicated by black circles and ¯ arrows. The simulation uses K = 1 and r¯ = 2eiθ where θ¯ is a splay state.

uk = −(1 + K) < ieiθk , Lk eiθ > −K < Lk t, (1 + Lk r¯)−1 ieiθk >,

K > 0. (13.23)

The parallel relative equilibrium deﬁned by (13.17) is Lyapunov stable and a global minimum of the Lyapunov function (13.22). Moreover, for every K > 0, there exists an invariant set in which the Lyapunov function is nonincreasing along the solutions. In this set, solutions converge to a parallel equilibrium that satisﬁes < (1 + Lk ¯r)Lk t, ieiθ0 >= 0

(13.24)

for some θ0 ∈ S 1 and for k = 1, . . . , N . Proof. This is proved by a straightforward adaptation of the corresponding proof in [11] where it was applied in the particular case L = N (I − 11T ). Convergence under the control law (13.23) is illustrated in Figure 13.2 with a simulation of N = 3 particles. The phase control (13.8) stabilizes the set of parallel equilibria, which is of dimension 2(N − 1) in the shape space. Away from singularities, the N algebraic constraints (13.24) are independent. As a result, the control law (13.23) isolates a subset of parallel equilibria of dimension N − 2 in the shape space. However, it does not isolate the desired parallel equilibrium for N > 2. A simple calculation indeed shows that the Jacobian linearization of (13.1) at the parallel equilibrium (13.17) possesses N − 2 uncontrollable spatial modes with zero eigenvalue. This means that the Jacobian linearization of the closed-loop system will possess N − 2 zero eigenvalues for any smooth static state feedback. For N > 2, no smooth static state feedback can achieve exponential stability of an isolated relative parallel equilibrium.

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13 Group Coordination and Cooperative Control

13.4.4 Symmetries of the Closed-Loop Vector Field All the control laws in this paper are shape control laws, i.e. they result in a closedloop vector ﬁeld that is invariant under the action of the group SE(2). For an all-toall communication topology, we have shown in [11] how to break this symmetry with the help of an extra virtual particle that acts as a leader. This result easily extends to the ﬁxed, bidirectional, and connected topologies considered in this paper and the details are omitted. The control law (13.8) is further invariant under the (discrete) group of permutations, that is, there is no diﬀerentiation among particles. This symmetry property might be a desirable feature of the design but is lost for the control laws (13.16) and (13.23) that aim at stabilizing isolated relative equilibria of the model. An issue of interest is whether isolated relative equilibria can be stabilized with control laws that retain the permutation symmetry. In [9, 11], we have addressed this question for the stabilization of isolated circular relative equilibria with certain symmetric phase arrangements. The results assume an all-to-all topology and make use of a linear combination of higher-harmonics potentials U (eimθ ), m ∈ {1, . . . , N2 }. The generalization of such results to arbitrary communication topologies remains to be addressed.

13.5 Critical Points of Phase Potentials Several results in the previous section provide global convergence results to a critical point of the phase potential Q(eiθ ). In this section we further investigate the structure of the critical points of these phase potentials. Lemma 13.1. Let L be the Laplacian matrix of a bidirectional, connected graph. If θ¯ is a critical point of the phase potential Q(eiθ ) =< eiθ , Leiθ > then there exists a nonnegative, real vector α ∈ RN such that ¯

and

(L − diag(α))eiθ = 0

(13.25)

¯

(13.26)

αT eiθ = 0.

The Hessian of the phase potential evaluated at these critical points is given by

˜ where L(θ) is given by

¯ ¯ − diag(α)) ˜ θ) H(eiθ ) = 2(L(

(13.27)

˜ kj = Lkj < eiθk , eiθj > . L

(13.28)

Proof. Critical points of the phase potential are characterized by ¯

¯

< ieiθk , Lk eiθ >= 0. If θ¯ is a critical point, then

(13.29)

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R. Sepulchre, D.A. Paley, and N.E. Leonard ¯

¯

Lk eiθ = αk eiθk , αk ∈ R,

(13.30)

for k = 1, . . . , N . The condition (13.25) is the matrix form of (13.30). Leftmultiplying (13.25) by 1T and using 1T L = 0 gives the second condition (13.26). In ¯ words, the αk are the weights for which the weighted centroid of the phasors, eiθk , vanishes. To see that α is nonnegative, we solve (13.30) for αk to obtain ¯

¯

< eiθk , eiθj > .

αk = dk −

(13.31)

j∈Nk

Equation (13.31) gives the bounds 0 ≤ αk ≤ 2dk . The Hessian of Q(eiθ ) is determined by ∂ 2Q = 2(dk − < eiθk , Lk eiθ >) ∂θk2 and, for j = k,

   −2 < eiθk , eiθj >, if j ∈ Nk ,

∂2Q =  ∂θj ∂θk  

0,

(13.32)

(13.33)

otherwise.

Using (13.32) and (13.33) we can express the Hessian as ˜ − diag(< eiθ1 , L1 eiθ >, . . . , < eiθN , LN eiθ >)) H(eiθ ) = 2(L(θ)

(13.34)

˜ where L(θ) is given by (13.28). Evaluating (13.34) at a critical point and using (13.30), we obtain (13.27), which completes the proof. There are three special sets of critical points of the phase potential Q(eiθ ). If α ∈ span{1}, then α = λ1 where λ is an eigenvalue of the graph Laplacian with the ¯ ¯ eigenvector eiθ . Synchronized critical points correspond to λ = 0 and eiθ = eiθ0 1, 1 where θ0 ∈ S . The set of synchronized critical points exists for any graph and is a global minimum of the phase potential. Balanced critical points correspond to ¯ α = λ1 where λ > 0 and 1T eiθ = 0. A suﬃcient condition for balanced critical points to exist is that the graph is circulant, that is, the Laplacian is a circulant matrix. The components of the eigenvectors of any circulant matrix form symmetric patterns on the unit circle centered at the origin of the complex plane. The only critical points for which α ∈ / span{1} that we have identiﬁed are unbalanced (2, N )-patterns. These patterns have two phase clusters that are separated by ¯ π and contain an unequal number of phases. If eiθ is an unbalanced (2, N )-pattern, T iθ¯ T i2θ¯ then 1 e = 0 and 1 e = N . Using (13.13), we observe that the unbalanced (2, N )-patterns are critical points of the phase potential for any Laplacian since sin θ¯kj = 0 for all (j, k) ∈ E. Lemma 13.2. An equivalent expression to (13.27) for the Hessian of the phase potential Q(eiθ ) is the weighted Laplacian, ¯ = BW (θ)B ¯ T. H(θ)

(13.35)

¯ = 2diag(cos(B T θ)) ¯ ∈ Re×e . W (θ)

(13.36)

The weight matrix is deﬁned by

13 Group Coordination and Cooperative Control

Proof. Using (13.27) and (13.31), we obtain   ¯ ¯   2 j∈Nk < eiθk , eiθj >, if j = k,     Hkj = 2 < eiθ¯k , eiθ¯j >, if j ∈ Nk ,       0 otherwise.

227

(13.37)

Equation (13.37) is equivalent to the weighted Laplacian (13.35). Let f ∈ {1, . . . , e} be the index of the edge (j, k) ∈ E. The corresponding weight is Wf f = 2 < ¯ ¯ eiθk , eiθj >= 2 cos θ¯kj in agreement with (13.36). Next, we give suﬃcient conditions for asymptotic stability and instability of critical points of the phase potential that are isolated in the shape manifold T N /S 1 . Proposition 13.5. Let L be the Laplacian matrix of a bidirectional, connected graph. The potential Q(eiθ ) =< eiθ , Leiθ > reaches its global minimum when θ = θ0 1, θ0 ∈ S 1 (synchronization). The gradient control θ˙ = K1 ∂Q ∂θ forces convergence of all solutions to the critical set of Q(eiθ ). If θ¯ is a critical point that is isolated in the shape manifold T N /S 1 , then a suﬃcient condition for asymptotic ¯ < 0 where W (θ) ¯ is given by (13.36). If KW (θ) ¯ > 0, then stability of θ¯ is K1 W (θ) ¯ θ is unstable. Proof. Pursuant to Lemma 13.2, if L = BB T is the Laplacian of a connected graph ¯ is deﬁnite, then H(θ) ¯ has rank N − 1 with the zero eigenvector 1. The and W (θ) stability result follows from the fact that the Jacobian of the gradient control is ¯ implies that all equal to the Hessian of the phase potential. Deﬁniteness of W (θ) ¯ other than λ = 0 are positive. eigenvalues of H(θ) Proposition 13.5 addresses phase arrangements in which cos θkj has the same sign for all (j, k) ∈ E. Thus, synchronized critical points are asymptotically stable for K1 < 0 and unstable otherwise since W (eiθ0 1) is the identity matrix in Re×e . The weight matrix is deﬁnite for some balanced critical points as well. For circulant matrices, it is suﬃcient to check that all cos θ1j have the same sign for j ∈ N1 . A complete characterization of balanced critical points and the unbalanced (2, N )patterns is the subject of ongoing work and will be presented in a separate paper.

13.6 Time-Varying and Unidirectional Topologies The stabilization results of the previous section require a communication graph which is time-invariant, undirected, and connected. It is of interest, both from the theoretical and practical viewpoint, to investigate which of these assumptions can be relaxed. In the rest of the paper, L(t) denotes the Laplacian matrix of a time-varying, directed graph G(t). The simplest situation to analyze is the parallel control (13.8), with ω0 = 0, which only involves the phase dynamics

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θ˙k = uk =< ieiθk , Lk (t)eiθ >

(13.38)

Several synchronization results for pure integrator dynamics have recently appeared in the literature [3, 4, 8, 7], both for continuous-time and discrete-time models. Convergence results are obtained for time-varying and undirected communication graphs, under relaxed connectedness assumptions. These convergence results are not global: they require all the initial phase diﬀerences to be in the open interval (−π/2, π/2). Under this assumption, the phase dynamics (13.38) can be mapped to the Euclidean state-space RN through a change of coordinates. Then, a key observation for the convergence analysis in [7] is that the convex hull of the states can only contract along the solutions. We now propose an extension of the results in [7] for the stabilization of circular motions (ω0 = 0) by considering the synchronization of the “centers” (13.4) which obey the dynamics s˙ k = −ieiθk (uk − ω0 ) (13.39) for k = 1, . . . , N . Following [7], we adopt co{s} as a set-valued Lyapunov function and modify the control (13.8) such that this function is nonincreasing. Using (13.4), one can interpret this modiﬁcation as contracting the convex hull of the centers of the particles’ circular orbits. Denote by conhull(Nk (t)) the conic hull spanned by the neighbors of sk at time t, that is, conhull(Nk (t)) = { βj (sj − sk ) | j ∈ Nk (t), βj ≥ 0} j

Sk+ (t)

and consider the sets = sk + conhull(Nk (t)), Sk− (t) = sk − conhull(Nk (t)), and + − Sk (t) = Sk (t) ∪ Sk (t). To ensure that co{s} contracts along the solutions of the particle model, we modify the circular control (13.8) as uk = ω0 + Kk (θk , Sk (t))|Lk (t)s|

(13.40)

with the gain Kk (θk , Sk (t)) a Lipschitz continuous function that satisﬁes     > 0 if − ieiθk ∈ Sk+ (t)     Kk (θk , Sk (t)) < 0, if − ieiθk ∈ S − (t) k        0, otherwise

(13.41)

Indeed, if s˙ k ∈ Sk+ (t) (resp. s˙ k ∈ Sk− (t)), then s˙ k (resp. −s˙ k ) points inwards co{s} at sk . As a consequence, the control law (13.40) renders the convex hull co{s} invariant, from which we obtain the following result. Proposition 13.6. Let L(t) be the time-dependent Laplacian matrix of a timevarying directed graph. Assume that L(t) is T -periodic and that the following condition is satisﬁed for all s = 1s0 : ∃ θ0 ∈ S 1 ,

k ∈ {1, . . . , N } : −iei(ω0 t+θ0 ) ∈ C\Sk (t)

∀t > 0.

(13.42)

13 Group Coordination and Cooperative Control

229

60 200

50

150

40

100

30

50 y

y

0

20 10

−50

0

−100

−10

−150

−20

−200 −200

−100

0

x

100

200

300

−30

−40

−20

0 x

20

40

Fig. 13.3. Simulations of a unidirectional, connected 2-circulant topology. The particle positions and headings are indicated by black circles and arrows. Both simulations have . (left) identical initial conditions, use ω0 = 0.1, and were simulated for the duration 30 2π ω0 Control (13.8) with K = 0.1; (right) control (13.40) with gain (13.41).

Then the shape feedback control (13.40) uniformly asymptotically stabilizes the set of circular motions of radius |ω0 |−1 (ω0 = 0). Proof. By design, the set-valued Lyapunov function co{s} is nonexpanding with the control (13.40). Moreover, it is strictly contracting when uk −ω0 = 0 for some k. This means that the limit set of each solution is a set where u ≡ 1ω0 . In this set, s˙ = 0 and θ˙ = 1ω0 . Invariance of the limit sets implies that all solutions must converge to a constant s that satisﬁes ∀ t : Kk (θ0 + ω0 t, Sk (t)) = 0, k = 1, . . . , N 1

for some θ0 ∈ S . This set reduces to 1s0 under the condition (13.42). The result of Proposition 13.6 holds without the periodicity assumption on L(t) but the proof is more technical and not detailed in the present paper. The convergence result is illustrated by the simulations shown in Figure 13.3 and Figure 13.4. In the simulations of Figure 13.3, we used a ﬁxed, unidirectional, connected, 2-circulant topology (each particle is connected to two neighbors). The simulation illustrates a situation where the control (13.8) that stabilizes the circular motion with a ﬁxed bidirectional, connected topology fails to achieve the same stabilization when the communication becomes unidirectional. By contrast, the modiﬁed gain (13.41) achieves stabilization. In the simulations of Figure 13.4, we used a unidirectional, weakly connected topology in which the neighbor sets, Nk , k = 1, . . . , 12, have identical cardinality, dk = 4, but whose members are chosen randomly and switched with ω0 . The simulation illustrates a situation where the control (13.8) that frequency 2π stabilizes the circular motion with a ﬁxed, bidirectional, connected topology fails to achieve the same stabilization when these assumptions are relaxed. It is of interest to note that, for the simulation described in Figure 13.4, the control (13.8) is not stabilizing for K = 0.1 but does stabilize the circular formation for smaller values of the gain, e.g. K = 0.01. The convergence for suﬃciently weak coupling gain K = ε > 0 suggests to apply averaging analysis to the closed-loop system

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y

60

60

50

50

40

40

30

30

20

y

10

20 10

0

0

−10

−10

−20 −30

−20 −40

−20

0 x

20

−30

40

−40

−20

0 x

20

40

Fig. 13.4. Simulations of a unidirectional, weakly connected topology in which the neighbor sets, Nk , k = 1, . . . , 12, have identical cardinality, dk = 4, but whose members are 0 . The particle positions and headings are chosen randomly and switched with frequency ω 2π indicated by black circles and arrows. Both simulations have identical initial conditions and randomly switching Laplacian L(t), use ω0 = 0.1, and were simulated for the duration . (left) Control (13.8) with K = 0.1; (right) control (13.40) with gain (13.41). 10 2π ω0

˜

˜

s˙ k = −εiei(ω0 t+θk ) < iei(ω0 t+θk ) , Lk (t)s > ˜ θ˜˙k = ε < iei(ω0 t+θk ) , Lk (t)s >

(13.43)

where we have used the coordinate change θ˜k = θk − ω0 t. For a switching (piecewiseconstant) communication topology with switching times synchronized with the rota2 ˙ tion period 2π ω0 , the averaged system is s = −(εI + 0(ε ))L(t)s and the convergence results in [7] apply, that is, synchronization is achieved under a weak connectedness assumption on L(t).

13.7 Discussion The results summarized in this paper focus on a core collective stabilization problem, that is, the stabilization of relative equilibria of the kinematic model (13.1), which are of two types: parallel or circular. These results can be combined and extended in various ways in order to provide a versatile design methodology that can be eﬀectively used for the coordination of motions in speciﬁc engineering applications. In [6], an illustration is provided in the context of an oceanographic application in which autonomous underwater vehicles (AUVs) are used to collect oceanographic measurements in formations that maximize the information intake. One eﬀective way to use the results of the present paper in the design of group formations is to deﬁne a set of primitives on the basis of the relative equilibria: parallel motion, circular motion with a given radius and a given phase arrangement. Simple symmetry-breaking control laws allow to further specify the direction of parallel motion and the center location of the circular motion. Switching between these primitives oﬀers a way to specify reference trajectories for the group that can

13 Group Coordination and Cooperative Control

231

be tracked by feedback control laws that only use relative information and a limited communication topology. Details are provided in [11]. Figure 13.5 illustrates a situation detailed in [6] that combines several variants of the control laws discussed in the present paper: a group of N = 12 vehicles is stabilized in a formation that combines all-to-all communication (the potential uses the Laplacian of a complete graph) and block all-to-all communication (the potential uses a block diagonal Laplacian of three identical complete subgraphs): the phase arrangement is a splay-state for the entire group but also within each of the subgroups. The spacing control is also block all-to-all and uses a symmetry-braking control law to specify diﬀerent centers of rotation for each of the three subgroups.

40 20 y

0

−20 −40

−40

−20

0

x

20

40

60

Fig. 13.5. A group formation that combines all-to-all coupling and block all-to-all coupling for the phase control so that the phase variables are in a splay state formation both in the entire group and in the three subgroups. The spacing control stabilizes the three circular subformations around three diﬀerent ﬁxed beacons at (R10 , R20 , R30 ) = (−30, 0, 30).

The proposed control laws can be further extended to generalize the circular shape of the formations to any closed convex curve. This work is in progress and will be presented in a forthcoming paper. The formation illustrated in Figure 13.5 could for instance be generalized to a phase-locked motion of the particles along three identical ellipses centered at diﬀerent locations. All these diﬀerent group formations are characterized by two properties: the formation is speciﬁed by a few geometric parameters and the design provides Lyapunovbased feedback control laws that asymptotically stabilize the speciﬁed formation. In applications where one wishes to shape the group trajectory in order to optimize some dynamic performance criterion, this low-order but versatile parametrization of stabilizable collectives oﬀers an interesting alternative to the often prohibitive cost of optimizing the individual trajectories.

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Acknowledgments This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Oﬃce. The scientiﬁc responsibility rests with its authors. D.A. Paley was supported by the National Science Foundation Graduate Research Fellowship, the Princeton University Gordon Wu Graduate Fellowship, and the Pew Charitable Trust grant 2000-002558. N.E. Leonard was partially supported by ONR grants N00014–02–1–0861 and N00014– 04–1–0534.

References 1. Fan Chung, Spectral graph theory, no. 92, Conference Board of the Mathematical Sciences, 1997. 2. E.W. Justh and P.S. Krishnaprasad., Steering laws and continuum models for planar formations, 42nd IEEE Conf. on Decision and Control (Maui, Hawaii-USA), 2003. 3. A. Jadbabaie, J. Lin, and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automatic Control 48(6) (2002), 988–1001. 4. A. Jadbabaie, N. Motee, and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. American Controls Conference, 2004. 5. Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer-Verlag, 1984. 6. N. E. Leonard, D. Paley, F. Lekien, R. Sepulchre, D. Frantantoni, and R. Davis, Collective motion, sensor networks and ocean sampling, accepted for publication in Proceedings of the IEEE (2005). 7. L. Moreau, Stability of multiagent systems with time-dependent communication links, IEEE Trans. Automatic Control 50(2) (2005), 169–182. 8. R. Olfati-Saber and R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. on Automatic Control 49(9) (2004), 1520– 1533. 9. D. Paley, N. Leonard, and R. Sepulchre, Oscillator models and collective motion: Splay state stabilization of self-propelled particles, 44th IEEE Conf. on Decision and Control (Seville, Spain), 2005. 10. R. Sepulchre, D. Paley, and N. Leonard, Collective motion and oscillator synchronization, Cooperative control, V. Kumar, N. Leonard, A. Morse (eds.), vol. 309, SpingerVerlag, London, 2005, pp. 189–205. , Stabilization of planar collective motion, part I: All-to-all communication, pro11. visionally accepted for publication in IEEE Transactions on Automatic Control (2005). 12. S. H. Strogatz, From Kuramoto to Crawford : exploring the onset of synchronization in populations of coupled oscillators, Physica D 143 (2000), 1–20.

14 Coordinating Control for a Fleet of Underactuated Ships A. Shiriaev1 , A. Robertsson, L. Freidovich1 and R. Johansson2 1

2

Department of Applied Physics and Electronics University of Ume˚ a, SE-901 87 Ume˚ a, Sweden; [email protected], [email protected] Department of Automatic Control, LTH, Lund University PO Box 118, SE-221 00 Lund, Sweden; [email protected], [email protected]

Summary. We consider design of coordinating controllers for a ﬂeet of underactuated ships. Each ship is modeled as a mechanical system with 3 degrees of freedom and 2 control inputs, which are the propeller thrust and the angle of the rudder; presence of lumped environmental forces is assumed. The proposed motion planning procedure relies on properties of an auxiliary explicitly constructed second order dynamical system. We propose a procedure for motion planning for the whole ﬂeet and for design of a coordinating feedback controller, which stabilizes the motion. In the case when geometric paths for ship motions are given, following our approach, we obtain a description of all feasible motions of the ﬂeet along these paths. In addition, it is shown how to design controllers to counteract environmental forces while remaining on the paths. As a side result, we provide some modiﬁcations and give some insights into the weather optimal positioning control design, introduced in [1] for a fully actuated ship model.

14.1 Introduction This paper is devoted to the problems of motion planning and feedback stabilization for a ﬂeet of ships, based on an underactuated dynamical model. We use the model with three degrees of freedom and two control inputs. It is taken from [2], where it is suggested to describe the behavior of a ship by three independent variables: the angle of the ship relative to the direction to the north and two coordinates that deﬁne the position of the ship in the horizontal plane, which are usually taken as the north-east coordinates. It is important to notice that the dynamics of the ship diﬀer from the dynamics of a rigid body on a plane. The diﬀerence are due to hydrodynamical eﬀects (behavior of ambient water) and due to the presence of friction terms from the motion in the water with both linear and quadratic velocity dependencies. It is assumed that the ship has two control inputs that correspond to two generalized forces acting on the ship. They are the propeller thrust and the angle of the rudder. The absence of the third control input makes the ship underactuated.

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 233–250, 2006. © Springer-Verlag Berlin Heidelberg 2006

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In this paper we consider a formation of several ships, described by such models. We assume that the speciﬁcations for desired behavior for each ship or a class of ships is given by a geometric path equipped with an orientation along the path. In general, the speciﬁcations can be updated oﬀ-line or/and on-line. They are seen as implicit instrumental tools for further controller designs. Projecting the dynamics of each ship onto such a geometric path, we obtain a description of all feasible motions of the ship formation along such paths without their explicit time parametrization and independent from the control input. Similar compact representation of the ship behavior along a path has been successfully but implicitly used before, see for instance [1, 7]. Here the representation is given; moreover, integrability and other structural properties of the projected dynamics are further brieﬂy discussed. For the dynamic positioning problem, the search of equilibria of the projected dynamics on such path(s) and the analysis of their stability (instability) are of clear interest. As shown below, the structural properties of the projected dynamics enable such analysis and result in constructive suﬃcient (and almost necessary) conditions for stability of stationary points for each ship on the path. The outline of the paper is as follows: In Section 14.2 we give preliminary results related to motion planning and feedback design problems for a single underactuated ship. Here we parametrize its feasible motions via the method of virtual holonomic constraints [3]. We consider the dynamic positioning problem for a ﬂeet of ships in Section 14.3, come to the motion planning problem in Section 14.4, and brieﬂy discuss the presented results in Section 14.5. Section14.6 concludes the paper.

14.2 Preliminaries: Motion Planning for a Single Underactuated Ship Here we present two results from [6]. They are instrumental for the main result given in the following subsections. Let us describe a solution for a path planning problem for a ship with 3 degrees of freedom and 2 control inputs. Our solution is illustrated by an example, with a ship model taken from [2, Example 10.1, p. 410], representing a high speed container ship of length L = 175 m and displacement volume 21.222 m3 . It is assumed that the ship is actuated by one rudder and a thrust propeller. Let the vector T η = [n, e, ψ] (14.1) denote the north-east positions of the ship and its yaw angle, and T

ν = [u, v, r]

(14.2)

be the velocity vector in the body frame, representing the surge, sway and yaw rates. The kinematics equation can be written as d η = R(ψ)ν, dt

ν = RT (ψ)

d η, dt

(14.3)

14 Coordinating Control for a Fleet of Underactuated Ships

where

235



  cos ψ − sin ψ   R(ψ) =   sin ψ cos ψ   0 0

0   0    1

is the rotation matrix in yaw. Assuming that the surge speed equation and the steering equations (sway and yaw) are decoupled, the dynamics of the ship in the body frame can be described by M ν˙ + N (ν)ν = B(ν)τ + RT (ψ)w,

(14.4)

T

where τ = [T, δ] is the control vector input, T is the propeller thrust, δ is the angle of the rudder, w = [w1 , w2 , 0]T is the vector of environmental disturbances,    m − Xu˙ 01×2  , M =   02×1 I2×2  −Xu − |u|X|u|u   N (ν) =    02×1 −U L    (1 − td )   B(ν) =  0    0



(14.5)  

01×2

 a11   a21 L

   La12  ,   a22

(14.6)

0

   , U2 L b11    U2 L2 b21

(14.7)

X(·) , aij , and bij are the hydrodynamic coeﬃcients3 , td is the thrust deduction √ number (td ∈ (0, 1)), and U = u2 + v 2 is the total speed. We assume that the vector w of the environmental disturbance in (14.4) is constant. Next statement reveals an implicit parametrization of all motions of the ship, provided its control inputs T and δ are chosen to preserve a given geometric path and orientation invariant. Theorem 14.1 ([6]). Given the path and the yaw angle as the C 2 -smooth functions of a new independent variable θ : 3

The numerical values for the example [2, p. 410] are m = 21.2 · 106 , Xu˙ = −6.38 · 105 , Xu = X|u|u = −4.226·104 , a11 = −0.7072, a12 = −0.286, a21 = −4.1078, a22 = −2.6619, b11 = −0.2081, b21 = −1.5238.

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n = φ1 (θ),

e = φ2 (θ),

ψ = φ3 (θ),

(14.8)

assume that there exists a control input τ ∗ = [T ∗ , δ ∗ ]T , which makes the relations (14.8) invariant for the ship dynamics (14.4). Then, along this path, θ must satisfy the diﬀerential equation ˙ θ| ˙ + γ(θ) = 0, α(θ)θ¨ + β1 (θ)θ˙2 + β2 (θ)θ|

(14.9)

where the explicit expressions for the functions α(θ), β1 (θ), β2 (θ), and γ(θ) are given below by Eqs.(14.13) – (14.16). Proof. Diﬀerentiating the identities (14.8) we obtain dn dθ = φ1 (θ) , dt dt and

de dθ = φ2 (θ) , dt dt

d2 n d2 θ = φ1 (θ) 2 + φ1 (θ)θ˙2 , 2 dt dt

dψ dθ = φ3 (θ) dt dt

d2 e d2 θ = φ2 (θ) 2 + φ2 (θ)θ˙2 , 2 dt dt

d2 ψ d2 θ = φ (θ) + φ3 (θ)θ˙2 , 3 dt2 dt2 which can be rewritten in matrix form as ˙ η˙ = Φ (θ)θ,

η¨ = Φ (θ)θ¨ + Φ (θ)θ˙2 ,

(14.10)

where T

Φ (θ) = [φ1 (θ), φ2 (θ), φ3 (θ)] ,

Φ (θ) = [φ1 (θ), φ2 (θ), φ3 (θ)]

T

The dynamics of the ship (14.4) is given by η + R˙ T (ψ)η˙ + N RT (ψ)η˙ RT (ψ)η˙ = B RT (ψ)η˙ τ ∗ + RT (ψ)w M RT (ψ)¨ Exploiting (14.10), we rewrite it as the following diﬀerential equation with respect to the θ-variable d M RT (φ3 (θ)) Φ (θ)θ¨ + Φ (θ)θ˙2 +M RT (φ3 (θ)) Φ (θ)θ˙2 + dθ + N RT (φ3 (θ))Φ (θ)θ˙ R(φ3 (θ))T Φ (θ) θ˙ = B RT (φ3 (θ))Φ (θ)θ˙ τ ∗ + RT (φ3 (θ))w

(14.11)

It is easy to see that the constant matrix 1 B ⊥ = 0, − b21 , b11 L

(14.12)

14 Coordinating Control for a Fleet of Underactuated Ships

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is an annihilator for B(·), that is B ⊥ B(ν) = 01×2 . Hence B ⊥ B(ν)τ ∗ = 0 irrespective of choice of the control input τ ∗ . Premultiplying both sides of Eq. (14.11) by B ⊥ , we obtain the system (14.9) with α(θ) = B ⊥ M R(φ3 (θ))T Φ (θ),

(14.13)

d R(φ3 (θ))T Φ (θ), β1 (θ) = B ⊥ M R(φ3 (θ))T Φ (θ) + B ⊥ M dθ

(14.14)

β2 (θ) = B ⊥N R(φ3 (θ))T Φ (θ) R(φ3 (θ))TΦ (θ),

(14.15)

γ(θ) = −B ⊥ R(φ3 (θ))T w =

b21 w1 sin(φ3 (θ)) − w2 cos(φ3 (θ)) L

(14.16)

This completes the proof. The expression of the function β2 (·) is of special interest. As seen in (14.15), the friction forces in surge do not contribute at all to the dynamics of the θ-variable. This is due to the fact that the vector B ⊥ is orthogonal to the surge direction and that the surge speed equation and the steering equations (sway and yaw) are assumed to be decoupled. It is important to notice that the dynamical system (14.9) does not possess a ˙ : θ˙ = 0} term, linear in velocity. Furthermore, there is the switching line {[θ, θ] that could lead to presence of sliding modes and to the loss of uniqueness of the solutions. As a result, the standard tools, such as linearizing dynamics around the equilibria, are not applicable, and the qualitative and quantitative analysis of the system (14.9) is nontrivial. A test for determining asymptotic stability and instability of the equilibria is given in Theorem 14.2 below. Clearly, there are at least two isolated equilibria in (14.9), that is isolated solutions of the equation γ(θ0 ) = 0, provided that the vector of environmental disturbances is not zero and the range of possible orientations of the ship is suﬃciently rich. Theorem 14.2 ([6]). Let θ0 be an isolated equilibrium of the system (14.9). Suppose that the functions α(θ), β1 (θ), β2 (θ), and γ(θ) are such that ω0 = and that the inequality

d γ(θ) dθ α(θ)

θ=θ0

=

γ (θ) α(θ)

β2 (θ0 )α(θ0 ) > 0

θ=θ0

>0

(14.17)

(14.18)

holds. Then, the equilibrium θ = θ0 of (14.9) is asymptotically stable. Moreover, if the sign of the inequality (14.18) is reversed, then the equilibrium is unstable. Proof for Theorem 14.2 is given in Appendix A. Comments Some remarks are in order.

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Both full actuation and underactuation, deﬁned in terms of the relation between numbers of degrees of freedom of a system and available independent actuators, are concepts of the mechanical origin. It is commonly known that controlling an underactuated nonlinear mechanical system is a more diﬃcult task then controlling a fully actuated one. At the same time, if, for instance, dynamics of an underactuated system is linear and controllable, then such an apparent diﬃculty is absent, and feedback design does not necessarily rely on the degree of actuation. This shows that underactuation might have no direct relevance to performance of a feedback controlled system. In Theorem 14.1, a parametrization of feasible motions for an underactuated ship model is suggested, provided a controller preserves both a pre-planned geometric path and an orientation invariant. Such a path description becomes an element of the control design procedure and could be modiﬁed. It is important to notice: if for a given path the projected dynamics of the ship model, Eq. (14.9), is nontrivial, then objectives to follow a prescribed path for underactuated and fully actuated ships are diﬀerent in nature. In the underactuated case, one cannot choose a velocity proﬁle along the prescribed path arbitrarily, the velocity has to be in agreement with solutions of (14.9) and with (14.8). Hence, a choice of a target geometric path may result in a signiﬁcant diﬀerence in achievable performance for fully actuated and underactuated ships irrespective of the feedback controller design. Following the preceding arguments, one could be interested to know: is there a geometric path, along which the velocity proﬁle for the underactuated ship model can be shaped arbitrarily? The answer to this question is clearly linked to the properties of the system (14.9). Indeed, if for some path the functions α(·), β1 (·), β2 (·) and γ(·) of (14.9) are all zero functions, then there is no limitation imposed on the internal variable θ. In such a case, dynamics of the underactuated ship model along the path could be freely chosen.

We are now ready to address a problem of coordinated control for a ﬂeet of underactuated ships. Below, two cases: when Eq. (14.9) is trivial and when it is not are considered. We start with the second one, which illustrates possible extensions for the weather optimal control strategies, reported in [2]. After that, we proceed with the ﬁrst case and show an approach for controlling motion of ships along paths, comprised from straight lines.

14.3 Steady-State-Conﬁguration Control for a Fleet of Underactuated Ships Here we present one of possible extensions of the weather optimal control strategy, which was originally elaborated for one ship to remain on a circular path in [1] and to remain on an arbitrary smooth path in [6], for controlling a ﬂeet of ships. The feedback design is based on an assumption of invertibility of a coupling matrix along a target path and is implemented exploiting the partial feedback linearization technique.

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14.3.1 Decentralized Control Design Based on Partial Feedback Linearization Suppose a ﬂeet consists of N underactuated ships, modeled by Eqs. (14.1)-(14.7), and for each ship there are three scalar C 2 -smooth functions φi1 (θi ) – φi3 (θi ), describing its path speciﬁcations in the inertial frame ni = φi1 (θi ),

ei = φi2 (θi ),

ψi = φi3 (θi ),

(14.19)

where i ∈ {1, . . . , N } represents the ship number and θi is the scalar variable, which deﬁnes the location of the i-th ship on its path. Given the N paths (14.19), let us introduce new (error) coordinates yi1 = ni − φi1 (θi ),

yi2 = ei − φi2 (θi ),

yi3 = ψi − φi3 (θi ).

(14.20)

For each ship i ∈ {1, . . . , N }, yi1 , yi2 , yi3 , together with the scalar variable θi constitute excessive coordinates for the ship dynamics (14.4). Therefore, one of them could always be locally resolved as a function of the others. Suppose that this is the case for yi3 , i.e. there are smooth functions hi such that yi3 = hi (yi1 , yi2 , θi ). The vector of velocities of the i-th ship in the body frame can be rewritten in the new coordinates as follows. νi = RT (ψi )η˙ i





= R φi3 (θi ) + hi (yi1 , yi2 , θi ) 



T

 1    0   

0 1

∂hi ∂hi ∂yi1 ∂yi2

(14.21)

φi1 (θi )

  y˙ i1         φi2 (θi )   y˙ i2      ∂hi ˙ θi ∂θi + φi3 (θi )

 y˙ i1       = Li (yi1 , yi2 , θi )   y˙ i2      θ˙i

(14.22)

where Li in (14.22) is a 3 × 3 matrix function. In turn, the time derivative of νi can be computed as follows    y¨i1      ˙  ν˙ i = Li (yi1 , yi2 , θi )  y¨i2   + Si yi1 , yi2 , θi , y˙ i1 , y˙ i2 , θi ,     θ¨i where Si (·) is a vector function, quadratically dependent on the velocities.

(14.23)

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Substituting the expressions (14.20)-(14.23) into the dynamics (14.4), we obtain the component-wise model of the i-th ship, written in terms of yi1 , yi2 , and θi :         y¨i1   y˙ i1   y˙ i1                        Mi νi = Mi Li (·)  y¨i2  + Mi Si (·) + Ni Li (·)  y˙ i2  Li (·)  y˙ i2                 ¨ ˙ ˙ θi θi θi       y˙ i1     Ti      y˙  = Bi  L (·) (14.24)  + RT φi3 (θi ) + hi (·) wi . i  i2          δi θ˙i Let us assume that the 3 × 3 matrix Mi Li (yi1 , yi2 , θi )

(14.25)

has full rank in a vicinity of (14.19), where θi is in a vicinity of θi0 , parameterizing the equilibrium. Then, multiplying both sides of the system (14.24) by the 2 × 3 matrix −1 Gi (yi1 , yi2 , θi ) = I2 , 02×1 Mi Li (yi1 , yi2 , θi ) from the left, we obtain









 Ti   y¨i1   = Ki (·)   + Qi (·)      δi y¨i2

(14.26)

where Qi (·) is a function independent on control inputs of the i-th ship model and      y˙ i1         y˙  . L Ki (·) = Gi (yi1 , yi2 , θi ) Bi  (y , y , θ ) i i1 i2 i   i2        θ˙i

(14.27)

Let us assume, in addition to invertability of the 3 × 3 matrix in (14.25), the full rank of the 2 × 2 matrix Ki (·) in the same vicinity of the path (14.19). Under these assumptions, we can rewrite the nonlinear equations (14.26) in an equivalent linear form. Indeed, after the feedback transformation        vi1   Ti    = K −1 (·)   − Qi (·) i      vi2 δi

(14.28)

14 Coordinating Control for a Fleet of Underactuated Ships

241

from the control inputs Ti and δi to the new one vi , we obtain y¨i1 = vi1 ,

y¨i2 = vi2 .

(14.29)

As expected, not all dynamical equations of an underactuated ship model could be feedback linearized. It is straight forward to check that the rest (nonlinear equation that complements the linear part (14.29) to an equivalent ship model) is represented by the equation    y¨i1       Bi⊥ Mi νi = Bi⊥ Mi Li (·)   y¨i2      θ¨i

(14.30) 









 y˙ i1   y˙ i1              ⊥ ⊥      + Bi Mi Si (·) + Bi Ni Li (·)  y˙ i2  Li (·)  y˙ i2  =           ˙θi ˙θi        y˙ i1     T    i  T ⊥    = Bi⊥ Bi (·)  Li (·)  y˙ i2   + Bi (·)R (φi3 (θi ) + hi (·)) wi    δ    i θ˙ = Bi⊥ RT φi3 (θi ) + hi (·) wi

(14.31)

Here Bi⊥ is the annihilator for Bi (·), given by (14.12). Equations (14.29) allows elimination of y¨i1 and y¨i2 from (14.31)            y˙ i1   y˙ i1     vi1                    ⊥ ⊥ ⊥          Bi Mi νi = Bi Mi Li  vi2  + Si  + Bi Ni Li  y˙ i2  Li  y˙ i2                     ˙ ˙ ¨ θi θi θi = Bi⊥ RT φi3 (θi ) + hi (·) wi

(14.32)

(14.33)

so that (14.32), (14.33) can be rewritten as a scalar diﬀerential equation of second order w.r.t. θi , which reduces to αi (θi )θ¨i + β1i (θi )θ˙i2 + β2i (θi )θ˙i |θ˙i | + γi (θi ) = 0, in the case when vi1 = vi2 = 0, yi1 = yi2 = 0, and y˙ i1 = y˙ i2 = 0.

(14.34)

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The main advantage of transforming the ships dynamics into the partly linearized form (14.29), (14.32), (14.33) is that it can be deduced from this description that a nonlinear decentralized PD-controller stabilizes the target conﬁguration of the ﬂeet. Theorem 14.3. Given geometric constraints (14.19), representing pre-described paths for N ships in the inertial frame, suppose that for each ship, i.e., for each i ∈ {1, . . . , N }, 1. The system (14.34) possesses an asymptotically stable equilibrium4 at θi0 , 2. The matrices (14.25) and (14.27) are both nonsingular in a vicinity of the path (14.19) with θi in a neighborhood of θi0 . 3. The 2 × 2 matrices Kip and Kid are chosen so that the feedback controller        vi1   yi1   y˙ i1    = −Kip   − Kid         vi2 yi2 y˙ i2

(14.35)

stabilizes the double integrator systems (14.29). Then, the equilibrium ni0 = φi1 (θi0 ),

ei0 = φi2 (θi0 ),

ψi0 = φi3 (θi0 )

(14.36)

of the closed loop system (14.3), (14.4), (14.20), (14.28), and (14.35) is asymptotically stable. The proof follows from the fact that the zero dynamics of the closed-loop system is asymptotically stable. Therefore, feedback stabilization of the regulated outputs yi1 and yi2 implies local asymptotic stability of the equilibrium (14.36). 14.3.2 Illustration for Centralized Control Design Based on Partial Feedback Linearization An alternative approach to deﬁning all N paths in the inertial frame, as have been done above, is a speciﬁcation of relative locations (distances) of some ships in a ﬂeet conﬁguration. The control objective given in terms of relative locations, could be more natural in many situations, especially when not all ships are equipped with accurate global positioning sensors, while local ones are available. Hence, although quite often the relative locations of ships could be rewritten in terms of global coordinates, this may not be practical despite the fact that we are interested only in the steady-state conﬁguration of a ﬂeet. Suppose now, instead of (14.19), we have ni = φi1 (θ), 4

ei = φi2 (θ),

For a suﬃcient condition see Theorem 14.2 above.

ψi = φi3 (θ),

(14.37)

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where θ = [θ1 , . . . , θN ]T and φij (·) are functions of a certain problem related structure. Instead of rigorously describing the allowable structure, we illustrate one possibility via an example. Let us consider a possible solution for building a control system for automatic relocation of pontoons shaping a chain along a straight line. The motivation for this example is the problem of control design for automatic reallocation with bias correction in a chain of pontoons, used for shaping a landing runway on sea. As a ﬁrst step in the design, we need to deﬁne a target path for each pontoon. In Figs. 14.1 (a)–(d), we illustrate the process of shaping such paths: part (a) shows a target geometric path and orientation for the ﬁrst pontoon in a chain given in the inertial frame, part (b) shows the target allocation of the second pontoon relative to the ﬁrst one etc. 1500

1000

direction to North in inertial frame

1000

direction N in inertial frame

500

0

−500

500

0

−500

−1000 −1000

−1000

−500

0 500 1000 direction E in inertial frame

1500

2000

−1500

−1000

−500

0 500 1000 direction to East in inertial frame

1500

1500

1000

1000

500

0

−500

500

0

−500

−1000

−1000

−1500 −1000

2000

(b)

direction to North in inertial frame

direction to North in inertial frame

(a)

1500

−500

0

500 1000 1500 direction to East in inertial frame

(c)

2000

2500

−1500 −1000

−500

0

500 1000 1500 direction to East in inertial frame

2000

2500

(d)

Fig. 14.1. (a) A target geometric path and orientation for the ﬁrst pontoon in a chain given in the inertial frame. (b)–(d) Paths for the other ships to form a chain behind the ﬁrst one. The geometric constraint for the second ship, for the third one, for the fourth one etc. is chosen as a circle centered at the position of the center of mass of the ship previous in the chain.

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It is not hard to see that the stability proof, given in the previous section is not valid anymore due to essential couplings in the zero dynamics. However, the whole control design procedure is still applicable without any change. To prove a mathematical result, analogous to Theorem 14.3, one would have to exploit transversality of the vector ﬁelds to the sliding surface, the special structure of the functions in (14.37) (i.e., topology of the interconnections) and of the integrals of motions, computed for each ship when the other ships are at the corresponding equilibrium positions. Informally speaking, for the example it is obvious that the ﬁrst ship will approach the equilibrium, but we have to show that the second one cannot escape during the transient time; after that, we can follow inductive arguments for the other ships. The proof is to be presented elsewhere. So, as soon as the target paths are chosen, we can use the nonlinear controller (14.35), proposed in Theorem 14.3. For our example at hand, the controller would automatically turn the chain of pontoons dependent on the direction of the lumped environmental forces w. Note that this controller is applicable provided that both conditions (1)–(2) of Theorem 14.3 hold. It can be veriﬁed that this is the case for the choice of paths illustrated above. In Fig. 14.2 we show the transient motion from an initial conﬁguration until the formation reaches the stable equilibrium shaped by the environmental force w = w1 , acting from the west. Suppose that after some time the environmental forces w

1500

1000

500

w

1

0

−500

−1000

−1500 −1000

−500

0

500

1000

1500

2000

2500

Fig. 14.2. The formation reaches the stable equilibrium, shaped by the environmental force w acting from the west.

14 Coordinating Control for a Fleet of Underactuated Ships

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1500

1000

500

0

−500

−1000

w2

−1500 −1500

−1000

−500

0

500

1000

1500

2000

2500

Fig. 14.3. After the direction of the environmental force has been abruptly changed, the chain of ships drifts to the new stable equilibrium.

abruptly change the direction, to w = w2 , acting from the south. The transition of the chain of ships to the new stable equilibrium is depicted in Fig. 14.3.

14.4 Motion Planning for an Underactuated Ship, Moving Along a Straight Line The solution for the dynamical positioning problem, described in the previous section, does not involve a procedure of planning motions for ship models as some functions of time. Actually, we have mainly considered the appropriate steady-state behavior of ships. It has been proved, see Theorem 14.1, that all feasible motions for an underactuated ship that follows a chosen geometric path, can be parameterized by the dynamics of the reduced system (14.9). Here we show that if a target path in the inertial frame is linear and if the eﬀect of environmental forces is negligible, then the reduced system (14.9) becomes trivial. In other words, a parametrization of dynamics of the ship projected on a straight line has no limitations due to underactuation of ship dynamics. Proposition 14.1. Suppose that the target path for the underactuated ship, described by (14.1)-(14.7), is the straight line in the inertial frame (compare to (14.8)) n = kθ,

e = θ,

ψ = arctan(1/k).

(14.38)

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and either the direction of the lumped environmental force w is collinear with the path or w = 0. Then, all the coeﬃcients of the system (14.9) α(θ),

β1 (θ),

β2 (θ),

γ(θ)

are zero functions. Proof consist of straightforward computations. Indeed, following the formulae (14.13)(14.16), we obtain α(θ) = B ⊥ M RT (φ3 (θ))Φ (θ)

  cos ψ − sin ψ    m − Xu˙ 01×2     sin ψ cos ψ    02×1 I2×2   0 0 

1 = 0, − b21 , b11 L

=−

T   0   0    1

k     1       0

b21 (cos ψ − k sin ψ) = 0 L

β1 (θ) = B ⊥ M RT (φ3 (θ)) Φ (θ) +B ⊥ M =0

d RT (φ3 (θ)) Φ (θ) = 0 dθ =0

β2 (θ) = B ⊥ N RT (φ3 (θ))Φ (θ) RT (φ3 (θ))Φ (θ)    1 = 0, − b21 , b11 L



 k cos ψ + sin ψ   k cos ψ + sin ψ           = 0     N 0 0             0 0

b21 (w2 cos ψ − w1 sin ψ) = 0 L The last equality holds iﬀ w = [w1 , w2 , 0]T is co-linear with [k, 1, 0]T or w = 0. γ(θ) = −B ⊥ RT (φ3 (θ))w = −

14.5 Discussion Let us discuss the main results !

The presented results show inherent diﬃculties in controlling underactuted ships if one is interested to pre-plan a geometrical paths for all ships of the ﬂeet. Even disregarding important issues (collision avoidance, limitations in inter-ships communications, ﬂeet topology etc.), there are almost always constraints on ships motions along the pre-planned paths. Such motions can be completely parameterized so that the choice of (relative) geometric paths (locations) of ships becomes a dominant feedback design step prior to the choice of a feedback controller.

14 Coordinating Control for a Fleet of Underactuated Ships

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The presented results on dynamic positioning and extension of weather-optimal control strategy, are obviously dependent on the presence of environmental forces acting on the ship. Intuitively they could be interpreted as potential forces in the loop, responsible for stability (instability) of equilibriums of the ﬂeet conﬁguration. It can be seen from Theorem 2 that such intuitive reasonings could be misleading in some cases, since stability and instability of an equilibrium also depend on the chosen geometrical paths to follow, mass distribution and friction coeﬃcients in the water. Our results are applicable and useful also for the case of fully (or over-) actuated ship models, in a situation when some actuators are of limited authority or power. They could play a role of environmental forces shaping potential forces in the loop, i.e., modifying the function γ(θ) of (14.9) to meet stability conditions. The interpretation of Proposition 14.1 is intuitively clear: if the center of mass of the ship is already on the desired straight line with the proper orientation, then one can manipulate by propeller thrust shaping ship behavior as one likes, provided that the vector of environmental forces is zero or acting along the path. Although such a conclusion may be trivial for ship operation, it is important to notice that it has been obtained for our mathematical model of the ship and should be considered as a partial model validation result. There is no procedure for feedback controller design to ensure stabilization of the motion along a straight line path with a desired velocity proﬁle in Proposition 14.1. It is tempting to re-use partial feedback transformation technique as it is done in the previous section. Unfortunately, it is not possible. It is not hard to check that the coupling matrix K(·), see (14.27), is equal to   b11 2 1 − td 2 cos ψ − k sin ψ − n˙ + e˙ (sin ψ + k cos ψ)   L   m − Xu˙     b21 2 2 n ˙ + e ˙ 0 L2 and degenerates along the straight line path with the corresponding orientation. Indeed, the block in the upper left corner vanishes, thus prohibiting conversion of this subsystem into a linear form. However, there are stabilizing controllers for a motion on the path with a desired velocity proﬁle. For example, if one is interested in uniform motion of the ship with constant non-zero velocity along the straight line, straightforward calculations show that the linearized model of ship along this particular trajectory is completely controllable. Correspondingly, local feedback controller designs could be easily deduced. Further details are to be published elsewhere.

14.6 Conclusions In this paper, we have discussed the problem of motion planning and feedback control design for formations of ships and stand along ships, described by the nonlinear threedegrees-of-freedom models with two actuators, taken from [2]. The proposed motion

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planning procedure relies on properties of certain second order dynamical systems; the detailed investigation of such dynamical systems being done. As a side result, we provide some modiﬁcations and give some insights into the weather optimal positioning control design, introduced in [1] for a fully actuated ship model.

References 1. Fossen T.I. and J.P. Strand. “Nonlinear Passive Weather Optimal Positioning Control for Ships and Rigs: Experimental Results,” Automatica, 37: 701–715, 2001. 2. Fossen T.I. Marine Control Systems. Marine Cybernetics, 2002. 3. Shiriaev, A., J.W. Perram and C. Canudas-de-Wit. “ Constructive Tool for Orbital Stabilization of Underactuated Nonlinear Systems: Virtual Constraints Approach,” IEEE Trans. on Automatic Control, 50(8), 1164–1176, 2005 4. Shiriaev A., J. Perram, A. Robertsson and A. Sandberg “Explicit Formulae for General Integrals of Motion for a Class of Mechanical Systems Subject to Virtual Constraints,” in Proc. 43rd IEEE Conference on Decision and Control, pp. 1158-1163, 2004. 5. Shiriaev A.S., A. Robertsson, J. Perram and A. Sandberg. ‘Motion Planning for Virtually Constrained (Hybrid) Mechanical Systems, in the Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, pp. 4035-4040, 2005. 6. Shiriaev A.S., A. Robertsson, P. Pacull and T.I. Fossen. “Motion planning and feedback stabilization for underacted ships: virtual constraints approach,” in the Proc. 16th IFAC World Congress, Prague, 2005. 7. Skjetne R., T.I. Fossen and P.V. Kokotovi´c. “Robust Output Maneuvering for a Class of Nonlinear Systems,” Automatica, 40: 373–383, 2004.

A Proof of Theorem 14.2 ˙ : θ˙ = 0}. The vector ﬁeld of (14.9) on the The sliding surface of (14.9) is Γ = {[θ, θ] sliding surface Γ is transversal to Γ except for points where γ(θ) = 0. However, these points are isolated equilibria of (14.9), see (14.16). Hence, any solution of (14.9), if exists, is unique. Let us introduce two dynamical systems α(θ)θ¨ + β1 (θ) + β2 (θ) θ˙2 + γ(θ) = 0

(14.39)

α(θ)θ¨ + β1 (θ) − β2 (θ) θ˙2 + γ(θ) = 0

(14.40)

The lack of nontrivial sliding motions on Γ allows us to state that the phase plane is a union of two sets U1 and U2 deﬁned by ˙ : a solution of (14.9) with origin at [θ0 , θ˙0 ] locally U1 = [θ, θ] coincides with a solution of (14.39) originated from [θ0 , θ˙0 ] ˙ : a solution of (14.9) with origin at [θ0 , θ˙0 ] locally U2 = [θ, θ] coincides with a solution of (14.40) originated from [θ0 , θ˙0 ]

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Fig. 14.4. (a) An example of the phase portrait of the system (14.39); (b) An example of the phase portrait of the system (14.40). Both systems have the centers around the equilibrium θ0 ≈ 4.7; Examples of phase portraits of the non-smooth system (14.9): (c) Here the dynamics on U1 coincides with shown on Figure 14.4(a), and on U2 coincides with shown on Figure 14.4(b); (d) An opposite situation: the dynamics on U1 coincides with shown on Figure 14.4(b), and on U2 coincides with shown on Figure 14.4(a). Two qualitatively diﬀerent behaviours: asymptotic stability of the equilibrium for case (c) and instability for case (d) are shown.

Roughly speaking, the set U1 is a union of an open upper half plane of the phase ˙ : θ˙ > 0}, and some subintervals of the sliding line Γ , while the plane, i.e., {[θ, θ] ˙ : θ˙ < 0}, and set U2 , in opposite, consists of an open lower half plane, i.e., {[θ, θ] complementary parts of the sliding surface Γ . Due to (14.17), both systems (14.39) and (14.40) have centers at the equilibrium point θ0 , see [4, 5]. Examples of the phase portraits for (14.39) and (14.40) around their corresponding equilibra are shown on Fig. 14.4(a) and (b). Using continuity, the solution of the original non-smooth system (14.9) can be obtained concatenating the solutions of the systems (14.39) and (14.40). For example, if a solution starts in U1 then it coincides with the solution of (14.39) with the same

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origin until it attains the switching surface Γ . Then, it enters into the area U2 and follows the solution of the system (14.39), and so on. This behaviour is illustrated in Figure 14.4. As seen in Figure 14.4, although the systems (14.39) and (14.40) have centers at the equilibrium θ0 , the non-smooth system (14.9) could have this equilibrium either asymptotically stable or unstable dependent on the character of the periodic motions of (14.39) and (14.40). Figure 14.4(c) shows an example where the dynamics of (14.9) in the area U1 coincides with the one shown on Figure 14.4(d); and the dynamics in the area U2 coincides with the one shown on Figure 14.4(b). As seen, for such a non-smooth dynamical system the equilibrium at θ0 is asymptotically stable. If one changes the dynamics on U1 to coincide with the one shown on Figure 14.4(b) and on U2 to coincide with the one shown on Figure 14.4(a), then the equilibrium becomes unstable. To prove Theorem 14.2 we need to show that the validity of condition (14.18) implies that the non-smooth system (14.9) has the phase portrait of the form shown on Figure 14.4(c). It is easy to check that the case shown on Figure 14.4(c) takes place only if the next statement is true. Proposition 14.2. Let θ0 be an equilibrium of (14.9) and the inequality (14.17) be valid. Consider the solution θ+ (t) of (14.39) and the solution θ− (t) of (14.40), initiated at θ± (0), θ˙± (0) = [θ0 − ε, 0] where ε > 0. Denote T+ the smallest positive time instant when θ˙+ (T+ ) = 0 and T− the smallest positive time instant when θ˙− (T− ) = 0. Both T+ and T− are the functions of the parameter ε, T+ = T+ (ε), T− = T− (ε) . Then the equilibrium θ0 is asymptotically stable if and only if for all suﬃciently small positive ε the inequality θ+ (T+ (ε)) < θ− (T− (ε))

(14.41)

is valid. Furthermore, the inequality (14.41) is valid for all suﬃciently small ε > 0, provided the inequality (14.18) holds.

15 Decentralized Adaptation in Interconnected Uncertain Systems with Nonlinear Parametrization I. Tyukin and C. van Leeuwen RIKEN Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan. [email protected], [email protected] Summary. We propose a technique for the design and analysis of decentralized adaptation algorithms in interconnected dynamical systems. Our technique does not require Lyapunov stability of the target dynamics and allows nonlinearly parameterized uncertainties. We show that for the considered class of systems, conditions for reaching the control goals can be formulated in terms of the nonlinear L2 -gains of target dynamics of each interconnected subsystem. Equations for decentralized controllers and corresponding adaptation algorithms are also explicitly provided.

15.1 Notation According to the standard convention, R deﬁnes the ﬁeld of real numbers and R≥c = {x ∈ R|x ≥ c}, R+ = R≥0 ; symbol Rn stands for a linear space L(R) over the ﬁeld of reals with dim{L(R)} = n; x denotes the Euclidian norm of x ∈ Rn ; C k denotes the space of functions that are at least k times diﬀerentiable; K denotes the class of all strictly increasing functions κ : R+ → R+ such that κ(0) = 0. By Lnp [t0 , T ], where T > 0, p ≥ 1 we denote the space of all functions f : R+ → Rn T

1/p

f (τ ) p dτ < ∞; f p,[t0 ,T ] denotes the Lnp [t0 , T ]such that f p,[t0 ,T ] = 0 n norm of f (t). By L∞ [t0 , T ] we denote the space of all functions f : R+ → Rn such that f ∞,[t0 ,T ] = ess sup{ f (t) , t ∈ [t0 , T ]} < ∞, and f ∞,[t0 ,T ] stands for the Ln∞ [t0 , T ] norm of f (t). A function f (x) : Rn → Rm is said to be locally bounded if for any x < δ there exists a constant D(δ) > 0 such that the following inequality holds: f (x) ≤ D(δ). Let Γ be an n × n square matrix, then Γ > 0 denotes a positive deﬁnite (symmetric) matrix, and Γ −1 is the inverse of Γ . By Γ ≥ 0 we denote a positive semi-deﬁnite matrix, x 2Γ to denotes the quadratic form: xT Γ x, x ∈ Rn . The notation | · | stands for the modulus of a scalar. The solution of a system of diﬀerential equations x˙ = f (x, t, θ, u), x(t0 ) = x0 , u : R+ → Rm , θ ∈ Rd for t ≥ t0 will be denoted as x(t, x0 , t0 , θ, u), or simply as x(t) if it is clear from the context what the values of x0 , θ are and how the function u(t) is deﬁned.

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 251–270, 2006. © Springer-Verlag Berlin Heidelberg 2006

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ˆ and time t. Let u : Rn × Rd × R+ → Rm be a function of state x, parameters θ, Let in addition both x and θˆ be functions of t. Then in case the arguments of u are ˆ clearly deﬁned by the context, we will simply write u(t) instead of u(x(t), θ(t), t). [t , T ]→ The (forward complete) system x˙ = f (x, t, θ, u(t)), is said to have an Lm 0 p Lnq [t0 , T ], gain (T ≥ t0 , p, q ∈ R≥1 ∪ ∞) with respect to its input u(t) if and only if x(t, x0 , t0 , θ, u(t)) ∈ Lnq [t0 , T ] for any u(t) ∈ Lm p [t0 , T ] and there exists a function γq,p : Rn × Rd × R+ → R+ such that the following inequality holds: x(t) q,[t0 ,T ] ≤ γq,p (x0 , θ, u(t) p,[t0,T ] ). The function γq,p (x0 , θ, u(t) p,[t0 ,T ] ) is assumed to be nondecreasing in u(t) p,[t0 ,T ] , and locally bounded in its arguments. For notational convenience when dealing with vector ﬁelds and partial derivatives we will use the following extended notion of the Lie derivative of a function. Let x ∈ Rn and assume x can be partitioned as follows x = x1 ⊕ x2 , where x1 ∈ Rq , x1 = (x11 , . . . , x1q )T , x2 ∈ Rp , x2 = (x21 , . . . , x2p )T , q + p = n, and ⊕ denotes the concatenation of two vectors. Deﬁne f : Rn → Rn such that f (x) = f1 (x) ⊕ f2 (x), where f1 : Rn → Rq , f1 (·) = (f11 (·), . . . , f1q (·))T , f2 : Rn → Rp , f2 (·) = (f21 (·), . . . , f2p (·))T . Then Lfi (x) ψ(x, t), i ∈ {1, 2} denotes the Lie derivative of the function ψ(x, t) with respect to the vector ﬁeld fi (x, θ): dim xi ∂ψ(x,t) Lfi (x) ψ(x, t) = j ∂xij fij (x, θ).

15.2 Introduction We consider the problem how to control the behavior of complex dynamical systems composed of interconnected lower-dimensional subsystems. Centralized control of these systems is practically ineﬃcient because of high demands for computational power, measurements and prohibitive communication cost. On the other hand, standard decentralized solutions often face severe limitations due to the deﬁciency of information about the interconnected subsystems. In addition, the nature of their their interconnections may vary depending on conditions in the environment. In order to address these problems in their most general setup, decentralized adaptive control is needed. Currently there is a large literature on decentralized adaptive control which contains successful solutions to problems of adaptive stabilization [6, 8], tracking [7, 8, 17, 18], and output regulation [9, 23] of linear and nonlinear systems. In most of these cases the problem of decentralized control is solved within the conventional framework of adaptive stabilization/tracking/regulation by a family of linearly parameterized controllers. While these results may be successfully implemented in a large variety of technical and artiﬁcial systems, there is room for further improvements. In particular, when the target dynamics of the systems is not stable in the Lyapunov sense but intermittent, meta-stable, or multi-stable [1, 15, 19] or when the uncertainties are nonlinearly parameterized [2, 3, 4, 11], and no domination of the uncertainties by feedback is allowed. In the present article we address these issues at once for a class of nonlinear dynamical systems. Our contribution is that we provide conditions ensuring forwardcompleteness, boundedness and asymptotic reaching of the goal for a pair of interconnected systems with uncertain coupling and parameters. Our method does not

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require availability of a Lyapunov function for the desired motions in each subsystem, nor linear parametrization of the controllers. Our results can straightforwardly be extended to interconnection of arbitrary many (but still, a ﬁnite number of) subsystems. Explicit equations for corresponding decentralized adaptive controllers are also provided. The paper is organized as follows. In Section 15.2 we provide a formal statement of the problem, Section 15.3 contains necessary preliminaries and auxiliary results. In Section 15.4 we present the main results of our current contribution, and in Section 15.5 we provide concluding remarks to our approach.

15.3 Problem Formulation Let us consider two interconnected systems Sx and Sy : Sx : x˙ = f (x, θx ) + γy (y, t) + g(x)ux Sy : y˙ = q(y, θy ) + γx (x, t) + z(y)uy

(15.1) (15.2)

where x ∈ Rnx , y ∈ Rny are the state vectors of systems Sx and Sy , vectors θx ∈ Rnθx , θy ∈ Rnθy are unknown parameters, functions f : Rnx × Rnθx → Rnx , q : Rny × Rnθy → Rny , g : Rnx → Rnx , z : Rny → Rny are continuous and locally bounded. Functions γy : Rny × R+ → Rn , γx : Rnx × R+ → Rny , stand for nonlinear, non-stationary and, in general, unknown couplings between systems Sx and Sy , and ux ∈ R, uy ∈ R are the control inputs. In the present paper we are interested in the following problem Problem 15.1. Let ψx : Rnx × R+ → R, ψy : Rny × R+ → R be the goal functions for systems Sx , Sy respectively. In the other words, for some values εx ∈ R+ , εy ∈ R+ and time instant t∗ ∈ R+ , inequalities ψx (x(t), t)

∞,[t∗ ,∞]

≤ εx , ψy (y(t), t)

∞,[t∗ ,∞]

≤ εy

(15.3)

specify the desired state of interconnection (15.1), (15.2). Derive functions ux (x, t), uy (y, t) such that for all θx ∈ Rnθx , θy ∈ Rnθy 1) interconnection (15.1), (15.2) is forward-complete; 2) the trajectories x(t), y(t) are bounded; 3) for given values of εx , εy , some t∗ ∈ R+ exists such that inequalities (15.3) are satisﬁed or, possibly, both functions ψx (x(t), t), ψy (y(t), t) converge to zero as t → ∞. Function ux (·) should not depend explicitly on y and, symmetrically, function uy (·) should not depend explicitly on x. The general structure of the desired conﬁguration of the control scheme is provided in Figure 15.1. In the next sections we provide suﬃcient conditions, ensuring solvability of Problem 15.1 and we also explicitly derive functions ux (x, t) and uy (y, t) which satisfy requirements 1) – 3) of Problem 15.1. We start with the introduction of a new class of adaptive control schemes and continue by providing the input-output characterizations of the controlled systems. These results are given in Section 15.4. Then, using these characterizations, in Section 15.5 we provide the main results of our study.

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Fig. 15.1. General structure of interconnetion

15.4 Assumptions and Properties of the Decoupled Systems Let the following system be given: x˙ 1 =f1 (x) + g1 (x)u, x˙ 2 =f2 (x, θ) + g2 (x)u, where

(15.4)

x1 = (x11 , . . . , x1q )T ∈ Rq ; x2 = (x21 , . . . , x2p )T ∈ Rp ; x = (x11 , . . . , x1q , x21 , . . . , x2p )T ∈ Rn

θ ∈ Ωθ ∈ Rd is a vector of unknown parameters, and Ωθ is a closed bounded subset of Rd ; u ∈ R is the control input, and functions f1 : Rn → Rq , f2 : Rn × Rd → Rp , g1 : Rn → Rq , g2 : Rn → Rp are continuous and locally bounded. The vector x ∈ Rn is the state vector, and vectors x1 , x2 are referred to as uncertainty-independent and uncertainty-dependent partition of x, respectively. For the sake of compactness we will also use the following description of (15.4): x˙ = f (x, θ) + g(x)u, where

(15.5)

g(x) = (g11 (x), . . . , g1q (x), g21 (x), . . . , g2p (x))T , f (x) = (f11 (x), . . . , f1q (x), f21 (x, θ), . . . , f2p (x, θ))T .

As a measure of closeness of trajectories x(t) to the desired state we introduce the error or goal function ψ : Rn × R+ → R, ψ ∈ C 1 . We suppose also that for the chosen function ψ(x, t) satisﬁes the following: Assumption 15.1 (Target operator). For the given function ψ(x, t) ∈ C 1 the following property holds: x(t)

∞,[t0 ,T ]

≤ γ˜ x0 , θ, ψ(x(t), t)

∞,[t0 ,T ]

(15.6)

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

where γ˜ x0 , θ, ψ(x(t), t) its arguments.

∞,[t0 ,T ]

255

is a locally bounded and non-negative function of

Assumption 15.1 can be interpreted as a sort of unboundedness observability property [10] of system (15.4) with respect to the “output” function ψ(x, t). It can also be viewed as a bounded input - bounded state assumption for system (15.4) along the constraint ψ(x(t, x0 , t0 , θ, u(x(t), t)), t) = υ(t), where the signal υ(t) serves as a new input. If, however, boundedness of the state is not explicitly required (i.e. it is guaranteed by additional control or follows from the physical properties of the system itself), Assumption 15.1 can be removed from the statements of our results. Let us specify a class of control inputs u which can ensure boundedness of solutions x(t, x0 , t0 , θ, u) for every θ ∈ Ωθ and x0 ∈ Rn . According to (15.6), boundedness of x(t, x0 , t0 , θ, u) is ensured if we ﬁnd a control input u such that ψ(x(t), t) ∈ L1∞ [t0 , ∞]. For this objective consider the dynamics of system (15.5) with respect to ψ(x, t): ∂ψ(x, t) , ψ˙ = Lf (x,θ)ψ(x, t) + Lg(x)ψ(x, t)u + ∂t

(15.7)

−1

Assuming that the inverse Lg(x) ψ(x, t) exists everywhere, we may choose the control input u in the following class of functions: ˆ ω, t) = u(x, θ,

1 Lg(x) ψ(x, t)

−Lf (x,θ) ˆ ψ(x, t) − ϕ(ψ, ω, t) −

∂ψ(x, t) ∂t

ϕ : R × Rw × R+ → R

(15.8)

where ω ∈ Ωω ⊂ Rw is a vector of known parameters of the function ϕ(ψ, ω, t). Denoting Lf (x,θ) ψ(x, t) = f (x, θ, t) and taking into account (15.8) we may rewrite equation (15.7) in the following manner: ˆ t) − ϕ(ψ, ω, t) ψ˙ = f (x, θ, t) − f (x, θ,

(15.9)

For the purpose of the present article, instead of (15.9) it is worthwhile to consider the extended equation: ˆ t) − ϕ(ψ, ω, t) + ε(t), ψ˙ = f (x, θ, t) − f (x, θ,

(15.10)

where, if not stated overwise, the function ε : R+ → R, ε ∈ L12 [t0 , ∞] ∩ C 0 . One of the immediate advantages of equation (15.10) in comparison with (15.9) is that it allows us to take the presence of coupling between interconnected systems into consideration. Let us now specify the desired properties of the function ϕ(ψ, ω, t) in (15.8), (15.10). The majority of known algorithms for parameter estimation and adaptive control [12, 13, 14, 16] assume global (Lyapunov) stability of system (15.10) for ˆ In our study, however, we refrain from this standard, restrictive requirement. θ ≡ θ. ˆ Instead we propose that ﬁnite energy of the signal f (x(t), θ, t) − f (x(t), θ(t), t), 1 deﬁned for example by its L2 [t0 , ∞] norm with respect to the variable t, results in ﬁnite deviation from the target set given by the equality ψ(x, t) = 0. Formally this requirement is introduced in Assumption 15.2:

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Assumption 15.2 (Target dynamics operator). Consider the following system: ψ˙ = −ϕ(ψ, ω, t) + ζ(t),

(15.11)

where ζ : R+ → R and ϕ(ψ, ω, t) is deﬁned in (15.10). Then for every ω ∈ Ωω system (15.11) has L12 [t0 , ∞] → L1∞ [t0 , ∞] gain with respect to input ζ(t). In other words, there exists a function γ∞,2 such that ψ(t)

∞,[t0 ,T ]

≤ γ∞,2 (ψ0 , ω, ζ(t)

2,[t0 ,T ] ),

∀ ζ(t) ∈ L12 [t0 , T ]

(15.12)

In contrast to conventional approaches, Assumption 15.2 does not require global asymptotic stability of the origin of the unperturbed (i.e for ζ(t) = 0) system (15.11). When the stability of the target dynamics ψ˙ = −ϕ(ψ, ω, t) is known a-priori, one of the beneﬁts of Assumption 15.2 is that there is no need to know a particular Lyapunov function of the unperturbed system. So far we have introduced basic assumptions on system (15.4) and the class of feedback considered in this article. Let us now specify the class of functions f (x, θ, t) in (15.10). Since general parametrization of function f (x, θ, t) is methodologically diﬃcult to deal with, but solutions provided for nonlinearities with convenient linear re-parametrization often yield physically implausible models and large number of unknown parameters, we have opted for a new class of parameterizations. As a candidate for such a parametrization we suggest nonlinear functions that satisfy the following assumption: Assumption 15.3 (Monotonicity and Growth Rate in Parameters). For the given function f (x, θ, t) in (15.10) there exists function α(x, t) : Rn × R+ → Rd , α(x, t) ∈ C 1 and positive constant D > 0 such that ˆ t) − f (x, θ, t))(α(x, t)T (θˆ − θ)) ≥ 0 (f (x, θ,

(15.13)

ˆ t) − f (x, θ, t)| ≤ D|α(x, t)T (θˆ − θ)| |f (x, θ,

(15.14)

This set of conditions naturally extends from systems that are linear in parameters to those with nonlinear parametrization. Examples and models of physical and artiﬁcial systems which satisfy Assumption 15.3 (at least for bounded θ, θˆ ∈ Ωθ ) can be found in the following references [2, 3, 4, 5, 11]. Assumption 15.3 bounds the growth rate of ˆ t)| by the functional D|α(x, t)T (θˆ−θ)|. In addition, the diﬀerence |f (x, θ, t)−f (x, θ, ˆ t)| from below, as it might also be useful to have an estimate of |f (x, θ, t) − f (x, θ, speciﬁed in Assumption 15.4: Assumption 15.4. For the given function f (x, θ, t) in (15.10) and function α(x, t), satisfying Assumption 15.3, there exists a positive constant D1 > 0 such that ˆ t) − f (x, θ, t)| ≥ D1 |α(x, t)T (θˆ − θ)| |f (x, θ,

(15.15)

In problems of adaptation, parameter and optimization estimation, eﬀectiveness of the algorithms often depends on how ”good” the nonlinearity f (x, θ, t) is, and how predictable is the system’s behavior. As a measure of goodness and predictability usually the substitutes as smoothness and boundedness are considered. In our study, we distinguish several of such speciﬁc properties of the functions f (x, θ, t) and ϕ(ψ, ω, t). These properties are provided below.

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

257

H 1 The function f (x, θ, t) is locally bounded with respect to x, θ uniformly in t. H 2 The function f (x, θ, t) ∈ C 1 , and ∂f (x, θ, t)/∂t is locally bounded with respect to x, θ uniformly in t. H 3 The function ϕ(ψ, ω, t) is locally bounded in ψ, ω uniformly in t. Let us show that under an additional structural requirement, which relates properties of the function α(x, t) and vector-ﬁeld f (x, θ) = f1 (x, θ) ⊕ f2 (x, θ) in (15.4), (15.5), there exist adaptive algorithms ensuring that the following desired property holds: ˆ t) ∈ L12 [t0 , ∞] x(t) ∈ Ln∞ [t0 , ∞]; f (x(t), θ, t) − f (x, θ(t),

(15.16)

Consider the following adaptation algorithms: ˆ t) = Γ (θˆP (x, t) + θˆI (t)); Γ ∈ Rd×d , Γ > 0 θ(x, θˆP (x, t) = ψ(x, t)α(x, t) − Ψ (x, t)

(15.17)

ˆ˙ I = ϕ(ψ(x, t), ω, t)α(x, t) + R(x, θ, ˆ u(x, θ, ˆ t), t), θ ˆ u(x, θ, ˆ t), t) : Rn × Rd × R × R+ → Rd in (15.17) is given where the function R(x, θ, as follows: ˆ t), t) = ∂Ψ (x, t)/∂t − ψ(x, t)(∂α(x, t)/∂t + Lf α(x, t)) R(x, u(x, θ, 1 ˆ t) + Lf Ψ (x, t) − (ψ(x, t)Lg α(x, t) − Lg Ψ (x, t))u(x, θ, 1

1

(15.18)

1

and function Ψ (x, t) : Rn × R+ → Rd , Ψ (x, t) ∈ C 1 satisﬁes Assumption 15.5. Assumption 15.5. There exists a function Ψ (x, t) such that ∂Ψ (x, t) ∂α(x, t) − ψ(x, t) =0 ∂x2 ∂x2

(15.19)

Additional restrictions imposed by this assumption will be discussed in some details after we summarize the properties of system (15.4), (15.8), (15.17), (15.18) in the following theorem. Theorem 15.1 (Properties of the decoupled systems). Let system (15.4), (15.10), (15.17), (15.18) be given and Assumptions 15.3, 15.4, 15.5 be satisﬁed. Then the following properties hold P1) Let for the given initial conditions x(t0 ), θˆI (t0 ) and parameters vector θ, interval [t0 , T ∗ ] be the (maximal) time-interval of existence of solutions of the closed loop system (15.4), (15.10), (15.17), (15.18). Then ˆ f (x(t), θ, t) − f (x(t), θ(t), t)) Df (θ, t0 , Γ, ε(t)

2,[t0 ,T ∗ ] )

=

2,[t0 ,T ∗ ]

≤ Df (θ, t0 , Γ, ε(t)

D ˆ 0) θ − θ(t 2

2 Γ −1

0.5

+

2,[t0 ,T ∗ ] );

D ε(t) D1

(15.20)

2,[t0 ,T ∗ ]

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I. Tyukin and C. van Leeuwen

ˆ θ − θ(t)

2 Γ −1

ˆ 0) − θ ≤ θ(t

2 Γ −1

+

D ε(t) 2D12

2 2,[t0 ,T ∗ ]

In addition, if Assumptions 15.1 and 15.2 are satisﬁed then P2) ψ(x(t), t) ∈ L1∞ [t0 , ∞], x(t) ∈ Ln∞ [t0 , ∞] and ψ(x(t), t)

∞,[t0 ,∞]

≤ γ∞,2 (ψ(x0 , t0 ), ω, D)

D = Df (θ, t0 , Γ, ε(t)

2,[t0 ,∞] )

+ ε(t)

(15.21)

2,[t0 ,∞]

P3) if properties H1, H3 hold, and system (15.11) has L12 [t0 , ∞] → L1p [t0 , ∞], p > 1 gain with respect to input ζ(t) and output ψ then ε(t) ∈ L12 [t0 , ∞] ∩ L1∞ [t0 , ∞] ⇒ lim ψ(x(t), t) = 0 t→∞

(15.22)

If, in addition, property H2 holds, and the functions α(x, t), ∂ψ(x, t)/∂t are locally bounded with respect to x uniformly in t, then P4) the following holds ˆ t) = 0 lim f (x(t), θ, t) − f (x(t), θ(t),

(15.23)

t→∞

The proof of Theorem 15.1 and subsequent results are given in Section 6. Let us brieﬂy comment on Assumption 15.5. Let α(x, t) ∈ C 2 , α(x, t) = col(α1 (x, t), . . . , αd (x, t)), then necessary and suﬃcient conditions for existence of the function Ψ (x, t) follow from the Poincar´e lemma: ∂ ∂x2

ψ(x, t)

∂αi (x, t) ∂x2

∂ ∂x2

=

ψ(x, t)

∂αi (x, t) ∂x2

T

(15.24)

This relation, in the form of conditions of existence of the solutions for function Ψ (x, t) in (15.19), takes into account structural properties of system (15.4), (15.10). Indeed, consider partial derivatives ∂αi (x, t)/∂x2 , ∂ψ(x, t)/∂x2 with respect to the vector x2 = (x21 , . . . , x2p )T . Let ∂ψ(x, t) = ∂x2

0 0 ··· 0 ∗ 0 ··· 0

,

∂αi (x, t) = ∂x2

0 0 ··· 0 ∗ 0 ··· 0

(15.25)

where the symbol ∗ denotes a function of x and t. Then condition (15.25) guarantees that equality (15.24) (and, subsequently, Assumption 15.5) holds. In case ∂α(x1 ⊕ x2 , t)/∂x2 = 0, Assumption 15.5 holds for arbitrary ψ(x, t) ∈ C 1 . If ψ(x, t), α(x, t) depend on a single component of x2 , for instance x2k , k ∈ {0, . . . , p}, then conditions (15.25) hold and the function Ψ (x, t) can be derived explicitly by integration Ψ (x, t) =

ψ(x, t)

α(x, t) dx2k ∂x2k

(15.26)

In all other cases, existence of the required function Ψ (x, t) follows from (15.24). In the general case, when dim{x2 } > 1, the problems of ﬁnding a function Ψ (x, t) satisfying condition (15.19) can be avoided (or converted into one with an already known solutions such as (15.24), (15.26)) by the embedding technique proposed in

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

259

[21]. The main idea of the method is to introduce an auxiliary system that is forwardcomplete with respect to input x(t) ξ˙ = fξ (x, ξ, t), ξ ∈ Rz

(15.27)

hξ = hξ (ξ, t), Rz × R+ → Rh such that f (x(t), θ, t) − f (x1 (t) ⊕ hξ (t) ⊕ x2 (t), θ, t)

2,[t0 ,T ]

≤ Cξ ∈ R+

(15.28)

for all T ≥ t0 , and dim{hξ }+dim {x2 } = p. Then (15.10) can be rewritten as follows: ˆ t) − ϕ(ψ, ω, t) + εξ (t), ψ˙ = f (x1 ⊕ hξ ⊕ x2 , θ, t) − f (x1 ⊕ hξ ⊕ x2 , θ,

(15.29)

where εξ (t) ∈ L12 [t0 , ∞], and dim{x2 } = p − h < p. In principle, the dimension of x2 could be reduced to 1 or 0. As soon as this is ensured, Assumption 15.5 will be satisﬁed and the results of Theorem 15.1 follow. Suﬃcient conditions ensuring the existence of such an embedding in the general case are provided in [21]. For systems in which the parametric uncertainty can be reduced to vector ﬁelds with low-triangular structure the embedding is given in [22].

15.5 Main Results Without loss of generality let us rewrite interconnection (15.1), (15.2) as follows : x˙ 1 = f1 (x) + g1 (x)ux x˙ 2 = f2 (x, θx ) + γy (y, t) + g2 (x)ux

(15.30)

y˙ 1 = q1 (y) + z1 (y)uy y˙ 2 = q2 (y, θy ) + γx (x, t) + z2 (y)uy

(15.31)

Let us now consider the following control functions ux (x, θˆx , ωx , t) = (Lg(x) ψx (x, t))−1 −Lf (x,θˆx) ψx (x, t) − ϕx (ψx , ωx , t) −

∂ψx (x, t) ∂t

, ϕx : R × Rw × R+ → R

uy (y, θˆy , ωy , t) = (Lz(y) ψy (y, t))−1 −Lq(y,θˆy ) ψy (y, t) − ϕy (ψy , ωy , t) −

∂ψy (y, t) ∂t

, ϕy : R × Rw × R+ → R

(15.32)

(15.33)

These functions transform the original equations (15.30), (15.31) into the following form

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I. Tyukin and C. van Leeuwen

ψ˙ x = −ϕx (ψx , ωx , t) + fx (x, θx , t) − fx (x, θˆx , t) + hy (x, y, t) ψ˙ y = −ϕy (ψx , ωy , t) + fy (y, θy , t) − fy (y, θˆy , t) + hx (x, y, t), where

(15.34)

hx (x, y, t) = Lγy (y,t) ψx (x, t), hy (x, y, t) = Lγx (x,t) ψy (y, t) fx (x, θx , t) = Lf (x,θx) ψx (x, t), fy (x, θy , t) = Lq(y,θy ) ψy (y, t)

Consider the following adaptation algorithms θˆx (x, t) = Γx (θˆP,x (x, t) + θˆI,x (t)); Γx ∈ Rd×d , Γx > 0 θˆP,x (x, t) = ψx (x, t)αx (x, t) − Ψx (x, t) ˆ˙ I,x = ϕx (ψx (x, t), ωx , t)αx (x, t) + Rx (x, θˆx , ux (x, θˆx , t), t), θ

(15.35)

θˆy (x, t) = Γy (θˆP,y (y, t) + θˆI,y (t)); Γy ∈ Rd×d , Γy > 0 θˆP,y (y, t) = ψy (y, t)αy (y, t) − Ψy (y, t) ˆ˙ I,y = ϕy (ψy (y, t), ωy , t)αy (y, t) + Ry (x, θˆy , uy (y, θˆy , t), t), θ

(15.36)

where Rx (·), Ry (·) are deﬁned as in (15.18), and the functions Ψx (·), Ψy (·) will be speciﬁed later. Now we are ready to formulate the following result Theorem 15.2 (Properties of the interconnected systems). Let systems (15.30), (15.31) be given. Furthermore, suppose that the following conditions hold: 1) The functions ψx (x, t), ψy (y, t) satisfy Assumption 15.1 for systems (15.30), (15.31) respectively; 2) The systems ψ˙ x = −ϕx (ψx , ωx , t) + ζx (t), ψ˙ y = −ϕy (ψy , ωy , t) + ζy (t)

(15.37)

satisfy Assumption 15.2 with corresponding mappings γx∞,2 (ψx0 , ωx , ζx (t)

2,[t0 ,T ] ),

γy∞,2 (ψy0 , ωy , ζy (t)

2,[t0 ,T ] ),

3) The systems (15.37) have L12 [t0 , ∞] → L12 [t0 , ∞] gains, that is ψx (x(t), t)

≤ Cγx + γx2,2 ( ζx (t)

2,[t0 ,T ] ),

ψy (y(t), t) 2,[t0 ,T ] ≤ Cγy + γy2,2 ( ζy (t) Cγx , Cγy ∈ R+ γx2,2 , γy2,2 ∈ K∞

2,[t0 ,T ] ),

2,[t0 ,T ]

(15.38)

4) The functions fx (x, θx , t), fy (y, θy , t) satisfy Assumptions 15.3, 15.4 with corresponding constants Dx , Dx1 , Dy , Dy1 and functions αx (x, t), αy (y, t); 5) The functions hx (x, y, t), hy (x, y, t) satisfy the following inequalities: hx (x, y, t) ≤ βx ψx (x, t) ,

hy (x, y, t) ≤ βy ψy (y, t) , βx , βy ∈ R+

(15.39)

Finally, let the functions Ψx (x, t), Ψy (y, t) in (15.35), (15.36) satisfy Assumption 15.5 for systems (15.30), (15.31) respectively, and there exist functions ρ1 (·), ρ2 (·), ρ3 (·) > Id(·) ∈ K∞ and constant ∆¯ ∈ R+ such the following inequality holds:

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

βy ◦ γy2,2 ◦ ρ1 ◦

Dy + 1 ◦ ρ3 ◦ βx ◦ γx2,2 ◦ ρ2 ◦ Dy,1

Dx + 1 (∆) < ∆ Dx,1

261

(15.40)

¯ Then for all ∆ ≥ ∆. C1) The interconnection (15.30), (15.31) with controls (15.32), (15.33) is forwardcomplete and trajectories x(t), y(t) are bounded Furthermore, C2) if properties H1, H3 hold for fx (x, θx , t), fy (y, θy , t), hx (x, y, t), hy (x, y, t), and also functions ϕx (ψx , ωx , t), ϕy (ψy , ωy , t), then lim ψx (x(t), t) = 0, lim ψy (y(t), t) = 0

t→∞

(15.41)

t→∞

Moreover, C3) if property H2 holds for fx (x, θx , t), fy (y, θy , t), and the functions αx (x, t), ∂ψx (x, t)/∂t, αy (y, t), ∂ψy (y, t)/∂t are locally bounded with respect to x, y uniformly in t, then lim fx (x(t), θx , t) − fx (x(t), θˆx (t), t) = 0,

t→∞

(15.42)

lim fy (y(t), θy , t) − fy (y(t), θˆy (t), t) = 0

t→∞

Let us brieﬂy comment on the conditions and assumptions of Theorem 15.2. Conditions 1), 2) specify restrictions on the goal functionals, similar to those of Theorem 15.1. Condition 3) is analogous to requirement to P3) in Theorem 15.1, condition 5) speciﬁes uncertainties in the coupling functions hx (·), hy (·) in terms of their growth rates w.r.t. ψx (·), ψy (·). We observe here that this property is needed in order to characterize the L2 norms of functions hx (x(t), y(t), t), hy (x(t), y(t), t) in terms of the L2 norms of functions ψx (x(t), t), ψy (y(t), t). Therefore, it is possible to replace requirement (15.39) with the following set of conditions: hx (x(t), y(t), t) hy (x(t), y(t), t)

2,[t0 ,T ] 2,[t0 ,T ]

≤ βx ψx (x(t), t) ≤ βy ψy (y(t), t)

2,[t0 ,T ] 2,[t0 ,T ]

+ Cx , + Cy

(15.43)

The replacement will allow us to extend results of Theorem 15.2 to interconnections of systems where the coupling functions do not depend explicitly on ψx (x(t), t), ψy (y(t), t). We illustrate this possibility later with an example. Condition (15.40) is the small-gain condition with respect to the L12 [t0 , T ] norms for interconnection (15.30), (15.31) with control (15.32), (15.33). In the case that mappings γx2,2 (·), γy2,2 (·) in (15.37) are majorated by linear functions γx2,2 (∆) ≤ gx2,2 ∆, γy2,2 (∆) ≤ gy2,2 ∆, ∆ ≥ 0, condition (15.40) reduces to the much simpler βy βx gx2,2 gy2,2

Dy +1 Dy,1

Dx +1 Dx,1

<1

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I. Tyukin and C. van Leeuwen

Notice also that the mappings γx2,2 (·), γy2,2 (·) are deﬁned by properties of the target dynamics (15.37), and, in principle, these can be made arbitrarily small. This eventually leads to the following conclusion: the smaller the L2 -gains of the target dynamics of systems S1 , S2 , the wider the class of nonlinearities (bounds for βx , βy , domains of Dx , D1,x , Dy , D1,y ) which admit a solution to Problem 15.1. Example Let us illustrate application of Theorem 15.2 to the problem of decentralized control of two coupled oscillators with nonlinear damping. Consider the following interconnected systems:        x˙ 1 = x2  y˙ 1 = y2 (15.44)     x˙ 2 = fx (x1 , θx ) + k1 y1 + ux ,   y˙ 22 = fy (y1 , θy ) + k2 x1 + uy , where k1 , k2 ∈ R are uncertain parameters of coupling, functions f (x1 , θx ), f (y1 , θy ) stand for the nonlinear damping terms, and θx , θy are unknown parameters. For illustrative purpose we assume the following mathematical model for functions fx (·), fy (·) in (15.44): fx (x1 , θx ) = θx (x1 − x0 ) + 0.5 sin(θx (x1 − x0 )), fy (y1 , θy ) = θy (y1 − y0 ) + 0.6 sin(θy (y1 − y0 ))

(15.45)

where x0 , y0 are known. Let the control goal be to steer states x and y to the origin. Consider the following goal functions ψx (x, t) = x1 + x2 , ψy (y, t) = y1 + y2

(15.46)

Taking into account equations (15.44) and (15.46) we can derive that x˙ 1 = −x1 + ψx (x(t), t), y˙ 1 = −y1 + ψy (y, t)

(15.47)

This automatically implies that x1 (t) y1 (t)

∞,[t0 ,T ]

≤ x1 (t0 ) + ψx (x(t), t)

∞,[t0 ,T ]

∞,[t0 ,T ]

≤ y1 (t0 ) + ψy (y(t), t)

∞,[t0 ,T ]

Hence, Assumption 15.1 is satisﬁed for chosen goal functions ψx (·) and ψy (·). Notice also that equalities (15.47) imply that x1 (t) y1 (t)

2,[t0 ,T ] 2,[t0 ,T ]

≤ 2−1/2 x1 (t0 ) + ψx (x, t) ≤2

−1/2

y1 (t0 ) + ψy (y, t)

2,[t0 ,T ]

(15.48)

2,[t0 ,T ]

Moreover, according to (15.47) limiting relations lim ψx (x(t), t) = lim x1 (t) + x2 (t) = 0,

t→∞

t→∞

lim ψy (y(t), t) = lim y1 (t) + y2 (t) = 0

t→∞

t→∞

(15.49)

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

263

guarantee that lim x1 (t) = 0, lim x2 (t) = 0, lim y1 (t) = 0, lim y2 (t) = 0

t→∞

t→∞

t→∞

t→∞

Hence, property (15.49) ensures asymptotic reaching of the control goal. According to equations (15.32), (15.33) control functions ux = −λx ψx − x2 − fx (x1 , θˆx ) uy = −λy ψy − y2 − fy (y1 , θˆy ), λx , λy > 0

(15.50)

transform system (15.44) into the following form ψ˙ x = −λx ψx + fx (x1 , θx ) − fx (x1 , θˆx ) + k1 y1 ψ˙ x = −λx ψx + fx (x1 , θx ) − fx (x1 , θˆx ) + k2 x1

(15.51)

Notice that systems ψ˙ x = −λx ψx + ξx (t), ψ˙ y = −λy ψt + ξy (t) satisfy Assumption 15.2 with γx2,2 =

1 ψx (x(t), t) λx

2,[t0 ,T ] ,

γy2,2 =

1 ψy (y(t), t) λy

2,[t0 ,T ]

respectively, and functions fx (·), fy (·) satisfy Assumptions 15.3, 15.4 with Dx = 1.5, Dx,1 = 0.5, αx (x, t) = x1 − x0 , Dy = 1.6, Dy,1 = 0.4, αy (y, t) = y1 − y0 Hence conditions 1)-4) of Theorem 15.2 are satisﬁed. Furthermore, according to the remarks regarding condition 5) of the theorem, requirements (15.39) can be replaced with implicit constraints (15.43). These, however, according to (15.48) also hold with βx = k1 , βy = k2 . Given that αx (x, t) = x1 −x0 , αy (y, t) = y1 −y0 , Assumption 15.5 will be satisﬁed for functions αx (x, t), αy (y, t) with Ψx (·) = 0, Ψy (·) = 0. Therefore, adaptation algorithms (15.35), (15.36) will have the following form: θˆx = Γx ((x1 + x2 )(x1 − x0 ) + θˆx,I ), ˙ θˆx,I = λx (x1 + x2 )(x1 − x0 ) − (x1 + x2 )x2 θˆy = Γy ((y1 + y2 )(y1 − y0 ) + θˆy,I ), ˙ θˆy,I = λy (y1 + y2 )(y1 − y0 ) − (y1 + y2 )y2

(15.52)

Hence, according to Theorem 15.2 boundedness of the solutions in the closed loop system (15.51), (15.52) is ensured upon the following condition k1 k2 λx λy

1+

Dx Dx,1

1+

Dy Dy,1

< 1 ⇒ k1 k2 <

λx λy 20

(15.53)

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I. Tyukin and C. van Leeuwen

Moreover, given that properties H1– H3 hold for the chosen functions ψx (x, t), ψy (y, t), condition (15.53) guarantees that limiting relations (15.41), (15.42) hold. Trajectories of the closed loop system (15.44), (15.50), (15.52) with the following values of parameters Γx = Γy = 1, λx = λy = 2, x0 = y0 = 1, θx = θy = 1 and initial conditions x1 (0) = −1, x2 (0) = 0, y1 (0) = 1, y2 (0) = 0, θˆx,I (0) = −1, θˆy,I (0) = −2 are provided in Fig. 15.2. a

b

0.5

0.5

0 0 −0.5

−1

0

5

10

15

20

−0.5

0

5

c

10

15

20

15

20

d

1

0.4 0.2 0

0.5

−0.2 −0.4 0

−0.6 −0.8

−0.5

0

5

10

15

−1

0

5

10

Fig. 15.2. Plots of trajectories x1 (t) (panel a), x2 (t) (panel b), y1 (t) (panel c), y2 (t) (panel d) as functions of t in closed loop system (15.44), (15.50), (15.52). Dotted lines correspond to the case when k1 = k2 = 0.4, and solid lines stand for solutions obtained with the following values of coupling k1 = 1, k2 = 0.1

15.6 Conclusion We provided new tools for the design and analysis of adaptive decentralized control schemes. Our method allows the desired dynamics to be Lyapunov unstable and the parametrization of the uncertainties to be nonlinear. The results are based on a formulation of the problem for adaptive control as a problem of regulation in functional spaces (in particular, L12 [t0 , T ] spaces) rather than of simply reaching of the control goal in Rn . This allows us to introduce adaptation algorithms with

265

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

new properties and apply a small-gain argument to establish applicability of these schemes to the problem of decentralized control. In order to avoid unnecessary complications, state feedback was assumed in the main-loop controllers which transform original equation into the error coupled model. Extension of the results to output-feedback main loop controllers is a topic for future study.

15.7 Proofs of the Theorems 15.7.1 Proof of Theorem 15.1 Let us ﬁrst show that property P1) holds. Consider solutions of system (15.4), (15.10), (15.17), (15.18) passing through the point x(t0 ), θˆI (t0 ) for t ∈ [t0 , T ∗ ] . ˆ˙ I ) = ˆ t): θ(x, ˆ˙ ˆ˙ P + θ Let us calculate the time-derivative of function θ(x, t) = Γ (θ ˙ ˆ˙ I ). Notice that ˙ Γ (ψα(x, t) + ψ α(x, t) − Ψ˙ (x, t) + θ ˆ˙ I = ψ(x, t) ∂α(x, t) x˙ 1 + ψ(x, t) ∂α(x) x˙ 2 + ˙ ψ α(x, t) − Ψ˙ (x, t) + θ ∂x1 ∂x2 ∂α(x, t) ∂Ψ (x, t) ∂Ψ (x, t) ∂Ψ (x, t) ˆ˙ ψ(x, t) x˙ 1 − x˙ 2 − − + θI ∂t ∂x1 ∂x2 ∂t According to Assumption 15.5, (15.54), we obtain

∂Ψ (x,t) ∂x2

ˆ˙ I = ˙ ψ α(x, t) − Ψ˙ (x, t) + θ

(15.54)

= ψ(x, t) ∂α(x,t) ∂x2 . Then taking into account ψ(x, t)

∂Ψ ∂α(x, t) − ∂x1 ∂x1

x˙ 1

∂α(x, t) Ψ (x, t) − + ψ(x, t) ∂t ∂t

(15.55)

Notice that according to the proposed notation we can rewrite the term ψ(x, t)

∂Ψ ∂α(x, t) − ∂x1 ∂x1

x˙ 1

in the following form: ˆ t). ψ(x, t)Lf1 α(x, t) − Lf1 Ψ (x, t) + (ψ(x, t)Lg1 α(x, t) − Lg1 Ψ (x, t)) u(x, θ, ˆ˙ I = ϕ(ψ)α(x, t). ˙ Hence, it follows from (15.17) and (15.55) that ψ α(x, t)− Ψ˙ (x, t)+ θ ˙ˆ Therefore, the derivative θ(x, t) can be written in the following way: ˆ˙ = Γ (ψ˙ + ϕ(ψ))α(x, t) θ

(15.56)

Asymptotic properties of nonlinear parameterized control systems with adaptation algorithm (15.56) under assumption of Lyapunov stability of the target dynamics

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were investigated in [20]. In the present contribution we aim to provide characterizations of the closed loop system in terms of functional mappings between functions ˆ ψ(x(t), t), ε(t), and f (x(t), θ, t) − f (x(t), θ(t), t) and without requiring Lyapunov stability of the target dynamics (15.11). For this purpose consider the following positive-deﬁnite function: ˆ θ, t) = 1 θˆ − θ Vθˆ(θ, 2

2 Γ −1

+

∞

D 4D12

t

ε2 (τ )dτ

(15.57)

Its time-derivative according to equations (15.56) can be obtained as follows: ˆ θ, t) = (ϕ(ψ) + ψ)( ˙ θˆ − θ)T α(x, t) − D ε2 (t) V˙ θˆ(θ, 4D12

(15.58)

Hence using Assumptions 15.3, 15.4 and equality (15.10) we can estimate the derivative V˙ ˆ as follows: θ

ˆ θ, t) ≤ −(f (x, θ, ˆ t) − f (x, θ, t) + ε(t))(θˆ − θ)T α(x, t) − D ε2 (t) V˙ θˆ(θ, 4D12 1 ˆ t) − f (x, θ, t))2 + 1 |ε(t)||f (x, θ, ˆ t) − f (x, θ, t)| ≤ − (f (x, θ, D D1 −

D 2 1 ε (t) ≤ − 4D12 D

ˆ t) − f (x, θ, t)| − |f (x, θ,

D ε(t) 2D1

2

≤0

(15.59)

It follows immediately from (15.59), (15.57) that ˆ −θ θ(t)

2 Γ −1

ˆ 0) − θ ≤ θ(t

2 Γ −1

+

D ε(t) 2D12

2 2,[t0 ,∞]

(15.60)

ˆ − θ 2 −1 ≤ In particular, for t ∈ [t0 , T ∗ ] we can derive from (15.57) that θ(t) Γ D 2 2 ∗ 2 ˆ ˆ θ(t0 ) − θ Γ −1 + 2D2 ε(t) 2,[t0 ,T ∗ ] . Therefore θ(t) ∈ L∞ [t0 , T ]. Furthermore 1 ˆ |f (x(t), θ(t), t) − f (x(t), θ, t)| − D ε(t) ∈ L1 [t0 , T ∗ ]. In particular 2

2D1

D ˆ |f (x(t), θ(t), t) − f (x(t), θ, t)| − ε(t) 2D1 D ˆ 0) θ − θ(t 2

2 Γ −1

+

D2 ε(t) 4D12

2 2,[t0 ,T ∗ ]

≤

2 2,[t0 ,T ∗ ]

(15.61)

ˆ Hence f (x(t), θ(t), t) − f (x(t), θ, t) ∈ L12 [t0 , T ∗ ] as a sum of two functions from ˆ L12 [t0 , T ∗ ]. In order to estimate the upper bound of the norm f (x(t), θ(t), t) − f (x(t), θ, t) 2,[t0 ,T ∗ ] from (15.61) we use the Minkowski inequality: D ˆ f (x(t), θ(t), t) − f (x(t), θ, t)| − ε(t) 2D1 D ˆ 0) θ − θ(t 2

2 Γ −1

0.5

+

D ε(t) 2D1

2,[t0 ,T ∗ ]

2,[t0 ,T ∗ ]

≤

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

267

and then apply the triangle inequality to the functions from L12 [t0 , T ∗ ]: ˆ f (x(t), θ(t), t) − f (x(t), θ, t)

2,[t0 ,T ∗ ]

≤

D ˆ f (x(t), θ(t), t) − f (x(t), θ, t) − ε(t) 2D1 D ε(t) 2D1

2,[t0 ,T ∗ ]

≤

D ˆ 0) θ − θ(t 2

2 Γ −1

2,[t0 ,T ∗ ] 0.5

+

+

D ε(t) D1

(15.62) 2,[t0 ,T ∗ ]

Therefore, property P1) is proven. Let us prove property P2). In order to do this we have to check ﬁrst if the solutions of the closed loop system are deﬁned for all t ∈ R+ , i.e. they do not go to inﬁnity in ﬁnite time. We prove this by a contradiction argument. Indeed, let there exists time instant ts such that x(ts ) = ∞. It follows from P1), however, ˆ that f (x(t), θ(t), t) − f (x(t), θ, t) ∈ L12 [t0 , ts ]. Furthermore, according to (15.62) ˆ the norm f (x(t), θ(t), t) − f (x(t), θ, t) 2,[t0 ,ts ] can be bounded from above by a ˆ 0 ), Γ , and ε(t) 2,[t ,∞] . Let us denote this bound continuous function of θ, θ(t 0 by symbol Df . Notice that Df does not depend on ts . Consider system (15.10) ˆ t) − ϕ(ψ, ω, t) + ε(t). Given that both for t ∈ [t0 , ts ]: ψ˙ = f (x, θ, t) − f (x, θ, ˆ f (x(t), θ, t) − f (x(t), θ(t), t), ε(t) ∈ L12 [t0 , ts ] and taking into account Assumption 15.2, we automatically obtain that ψ(x(t), t) ∈ L1∞ [t0 , ts ]. In particular, using the triangle inequality and the fact that the function γ∞,2 (ψ(x0 , t0 ), ω, M ) in Assumption 15.2 is non-decreasing in M , we can estimate the norm ψ(x(t), t) ∞,[t0 ,ts ] as follows: ψ(x(t), t)

∞,[t0 ,ts ]

≤ γ∞,2 ψ(x0 , t0 ), ω, Df + ε(t)

2 2,[t0 ,∞]

(15.63)

According to Assumption 15.1 the following inequality holds: x(t)

∞,[t0 ,ts ]

≤ γ˜ x0 , θ, γ∞,2 ψ(x0 , t0 ), ω, Df + ε(t)

2 2,[t0 ,∞]

(15.64)

Given that a superposition of locally bounded functions is locally bounded, we conclude that x(t) ∞[t0 ,ts ] is bounded. This, however, contradicts to the previous claim that x(ts ) = ∞. Taking into account inequality (15.60) we can derive that both ˆ θ(x(t), t) and θˆI (t) are bounded for every t ∈ R+ . Moreover, according to (15.63), (15.64), (15.60) these bounds are themselves locally bounded functions of initial ˆ conditions and parameters. Therefore, x(t) ∈ Ln∞ [t0 , ∞], θ(x(t), t) ∈ Ld∞ [t0 , ∞]. Inequality (15.21) follows immediately from (15.62), (15.12), and the triangle inequality. Property P2) is proven. Let us show that P3) holds. It is assumed that system (15.11) has L12 [t0 , ∞] → L1p [t0 , ∞], p > 1 gain. In addition, we have just shown that ˆ f (x(t), θ, t) − f (x(t), θ(t), t), ε(t) ∈ L2 [t0 , ∞]. Hence, taking into account equation (15.10) we conclude that ψ(x(t), t) ∈ L1p [t0 , ∞], ˆ t), ϕ(ψ, ω, t) are locally bounded with rep > 1. On the other hand, given that f (x, θ, spect to their ﬁrst two arguments uniformly in t, and that x(t) ∈ Ln∞ [t0 , ∞],ψ(x(t), t) ∈

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ˆ ∈ Ld [t0 , ∞], θ ∈ Ωθ , the signal ϕ(ψ(x(t), t), ω, t) + f (x(t), θ, t) − L1∞ [t0 , ∞], θ(t) ∞ ˆ f (x(t), θ(t), t) is bounded. Then ε(t) ∈ L1∞ [t0 , ∞] implies that ψ˙ is bounded, and P3) is guaranteed by Barbalat’s lemma. To complete the proof of the theorem (property P4) consider the time-derivative ˆ t): of function f (x, θ, d ˆ t) = L ˆ f (x, θ, ˆ f (x, θ, t)+ f (x,θ)+g(x)u(x,θ,t) dt ˆ t) ˆ t) ∂f (x, θ, ∂f (x, θ, ˙ Γ (ϕ(ψ, ω, t) + ψ)α(x, t) + ∂t ∂ θˆ Taking into account that the function f (x, θ, t) is continuously diﬀerentiable in x, θ; the derivative ∂f (x, θ, t)/∂t is locally bounded with respect to x, θ uniformly in t; functions α(x, t), ∂ψ(x, t)/∂t are locally bounded with respect to ˆ t)) is bounded. Then given that x uniformly in t, then d/dt(f (x, θ, t) − f (x, θ, 1 ˆ f (x(t), θ, t) − f (x(t), θ(t), t) ∈ L2 [t0 , ∞] by applying Barbalat’s lemma we conclude ˆ τ ) → 0 as t → ∞. The theorem is proven. that f (x, θ, τ ) − f (x, θ, 15.7.2 Proof of Theorem 15.2 Let us denote ∆fx [t0 , T ] = fx (x, θx , t)−fx (x, θˆx , t) 2,[t0 ,T ] , ∆fy [t0 , T ] = fx (y, θy , t)− fy (y, θˆy , t) 2,[t0 ,T ] . As follows from Theorem 15.1 the following inequalities hold ∆fx [t0 , T ] ≤ Cx +

Dx hy (x(t), y(t), t) D1,x

2,[t0 ,T ]

(15.65)

∆fy [t0 , T ] ≤ Cy +

Dy hx (x(t), y(t), t) D1,y

2,[t0 ,T ] ,

(15.66)

where Cx , Cy are some constants, independent of T . Taking estimates (15.65), (15.66) into account we obtain the following estimates: ∆fx [t0 , T ] + hy (x(t), y(t), t) Cx +

Dx +1 D1,x

hy (x(t), y(t), t)

∆fy [t0 , T ] + hx (x(t), y(t), t) Cy +

Dy +1 D1,y

2,[t0 ,T ]

2,[t0 ,T ]

2,[t0 ,T ]

hx (x(t), y(t), t)

≤ (15.67)

≤

2,[t0 ,T ] ,

(15.68)

The proof of the theorem would be complete if we show that the L12 [t0 , T ] norms of hx (x(t), y(t), t), hy (x(t), y(t), t) are globally bounded uniformly in T . Let us show that this is indeed the case. Using the widely known generalized triangular inequality [10] γ(a + b) ≤ γ((ρ + Id)(a)) + γ((ρ + Id) ◦ ρ−1 (b)), a, b ∈ R+ , γ, ρ ∈ K∞ , equations (15.67), (15.68) and also property (15.39), we conclude that

15 Decentralized Adaptation in Systems with Nonlinear Parameterization

hy (x(t), y(t), t) βy · γy2,2 ◦ ρ1 hx (x(t), y(t), t) βx · γx2,2 ◦ ρ2

2,[t0 ,T ]

≤

Dy +1 D1,y 2,[t0 ,T ]

269

hx (x(t), y(t), t)

2,[t0 ,T ]

+ Cy,1 (15.69)

≤

Dx +1 D1,x

hy (x(t), y(t), t)

2,[t0 ,T ]

+ Cx,1

where ρ1 (·), ρ2 (·) ∈ K∞ , ρ1 (·), ρ2 (·) > Id(·). Then, according to (15.69), the existence of ρ3 (·) ∈ K∞ ≥ Id(·), satisfying inequality βy ◦ γy2,2 ◦ ρ1 ◦

Dy + 1 ◦ ρ3 ◦ βx ◦ γx2,2 ◦ ρ2 ◦ Dy,1

Dx + 1 (∆) < ∆ ∀ ∆ ≥ ∆¯ Dx,1

for some ∆¯ ∈ R+ ensures that the norms hy (x(t), y(t), t)

2,[t0 ,T ] ,

hx (x(t), y(t), t)

2,[t0 ,T ]

are globally uniformly bounded in T . The rest of the proof follows from Theorem 15.1. The theorem is proven.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Arecchi FT (2004) Physica A, 338:218–237 Armstrong-Helouvry B (1993) IEEE Trans on Automatic Control, 38(10):1483–1496 Boskovic JD (1995) IEEE Trans on Automatic Control, 40(2):347–350 Canudas de Wit C, Tsiotras P (1999) Dynamic tire models for vehicle traction control. In Proceedings of the 38th IEEE Control and Decision Conference, Phoenix, Arizona, USA Dayan P, Abbott LF (2001) Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press Gavel DT, Siljak DD (1989) IEEE Trans on Automatic Control, 34(4):413–426 Ioannou PA (1986) IEEE Trans. on Automatic Control, AC-31(4):291–298 Jain S, Khorrami F (1997) IEEE Trans on Automatic Control, 42(2):136–154 Jiang ZP (2000) IEEE Trans on Automatic Control, 45(11):2122–2128 Jiang ZP, Teel AR, Praly L (1994) Mathematics of Control, Signals and Systems, (7):95–120 Kitching KJ, Cole DJ, Cebon D (2000) ASME Journal of Dynamic Systems Measurement and Control, 122(3):498–506 Krstic M, Kanellakopoulos I, Kokotovic P (1995) Nonlinear and Adaptive Control Design. Wiley and Sons Inc. Miroshnik I, Nikiforov V, Fradkov A (1999) Nonlinear and Adaptive Control of Complex Systems. Kluwer Narendra KS, Annaswamy AM (1989) Stable Adaptive systems. Prentice–Hall Raﬀone A, van Leeuwen C (2003) Chaos, 13(3):1090–1104 Sastry S, Bodson M (1989) Adaptive Control: Stability, Convergence, and Robustness. Prentice Hall Shi L, Singh SK (1992) IEEE Trans on Automatic Control, 37(2):1106–1118 Spooner JT, Passino KM (1996) IEEE Trans on Automatic Control, 41(2):280–284

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19. Tsuda I, Fujii H (2004) A Complex Systems Approach to an Interpretation of Dynamic Brain Activity I: Chaotic itinerancy can provide a mathematical basis for information processing in cortical transitory and nonstationary dynamics. In Lecture Notes in Computer Science, (3146): 109–128, Springer-Verlag 20. Tyukin IY, Prokhorov DV, Terekhov VA (2003) IEEE Trans on Automaitc Control, 48(4):554–567 21. Tyukin IY, Prokhorov DV, van Leeuwen C (2003) Finite form realizations of adaptive control algorithms. In Proceedings of European Control Conference, Cambridge, UK, September 1–4 22. Tyukin IY, Prokhorov DV, van Leeuwen C (2004) Adaptive algorithms in ﬁnite form for nonconvex parameterized systems with low-triangular structure. In Proceedings of the 8-th IFAC Workshop on Adaptation and Learning in Control and Signal Processing (ALCOSP 2004), Yokohama, Japan, August 29–31 23. Ye X, Huang J (2003) IEEE Trans on Automatic Control, 48(2):276–280

16 Controlled Synchronisation of Continuous PWA Systems N. van de Wouw1 , A. Pavlov2, and H. Nijmeijer1 1 2 3

Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected] NTNU, Department of Engineering Cybernetics, N7491, Trondheim, Norway, [email protected] Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected]

Summary. In this paper, the controlled synchronisation problem for identical continuous piecewise aﬃne (PWA) systems is addressed. Due to the switching nature of these systems, strategies common for controlled synchronisation can not be used. In this paper, an observer-based output-feedback control design solving the master-slave synchronisation problem for two PWA systems is proposed. The design of these dynamic controllers is based on the idea of, on the one hand, rendering the slave system convergent by means of feedback (which makes all its solutions converge to each other) and, on the other hand, guaranteeing that the closed-loop slave system has a bounded solution corresponding to zero synchronisation error. This implies that all solutions of the closed-loop slave system converge to this bounded solution with zero synchronisation error. The results are illustrated by application to a master-slave synchronisation problem of two mechanical systems with one-sided restoring characteristics.

16.1 Introduction Synchronisation of dynamical systems has received considerable interest because of the wide variety of systems in which synchronisation can occur or is desirable, e.g. in secure communication [1], biological systems [2], (electro-)mechanical systems [3], such as rotor dynamic systems or cooperating robots [4]. Many more illustrative examples can be found in [5]. Diﬀerent kinds of synchronisation [6] can be deﬁned; namely, natural synchronisation, which is established without interaction between the systems involved, self-synchronisation, which occurs due to a proper coupling between the systems while this coupling is inherently in the system and not enforced externally, and controlled synchronisation, which implies synchronisation enforced by active control. This paper deals with the controlled synchronisation problem for continuous PWA systems. Another distinction between diﬀerent types of synchronisation can be made in terms of the variables being synchronised. One can speak of phase synchronisation, see e.g. [7], when the responses of the systems only comply in terms of a certain phase variable. Here, we will discuss full-state synchronisation,

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 271–289, 2006. © Springer-Verlag Berlin Heidelberg 2006

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N. van de Wouw, A. Pavlov, and H. Nijmeijer

which implies the exact correspondence of all the states of the system. Other notions of synchronisation are partial synchronisation [8, 9, 10], in which only part of the state of the systems synchronise, and generalised synchronisation, in which correspondence of certain functionals of the state is established [6]. The controlled synchronisation problem can be divided into the master-slave synchronisation problem and the mutual synchronisation problem. In the master-slave variant, which is considered here, the slave system is unidirectionally coupled (by means of control) to the master system, whereas in mutual synchronisation a bilateral coupling ensures adaptation of the systems with respect to each other [11]. Many results on controlled synchronisation exist, e.g. [6, 12, 13, 14, 4], where both state-feedback and (observer-based) dynamic output-feedback control strategies are proposed. In [14, 15], the controlled synchronisation problem is considered in the scope of the regulator problem and in [16, 17] the strong link between the synchronisation problem and the observer design problem is illuminated. Robustness issues with respect to diﬀerences between the systems to be synchronised are treated in [13, 18, 19]. Furthermore, the synchronisation of chaotic oscillators has received a huge amount of attention, e.g. in [1, 13] and many other publications. Currently, PWA systems are receiving wide attention due to the fact that the PWA framework [20] provides a means to describe dynamic systems exhibiting switching between a multitude of linear dynamic regimes. Such switching can be due to piecewise-linear characteristics such as dead-zone, saturation, hysteresis or relays. A common strategy in achieving synchronisation is the stabilisation of the error dynamics between the systems to be synchronised. One could then think of translating the controlled synchronisation problem for PWA systems into some stabilisation problem for PWA systems and subsequently applying known results for the stabilization of PWA systems, see for example [21, 22, 23, 24, 25]. As we will illuminate in the next section, the switching nature of the vector-ﬁeld of PWA systems seriously complicates such an approach. Some PWA systems can be represented in the form of a Lur’e system (as is the case for the famous Chua circuit), for which the master-slave synchronisation problem is considered in [12, 26, 27, 13]. It is also worth mentioning the work in [28] on state-feedback tracking control of bimodal PWA systems, since master-slave synchronisation and tracking are closely related problems. Here, we propose a diﬀerent approach towards the controlled synchronisation problem for general continuous PWA systems. In this approach, the notion of convergence plays a central role. A system, which is excited by an input, is called convergent if it has a unique globally asymptotically stable solution that is bounded on the whole time axis. Obviously, if such a solution does exist, all other solutions, regardless of their initial conditions, converge to this solution, which can be considered as a steady-state solution [29, 30]. Similar notions describing the property of solutions converging to each other are studied in literature. The notion of contraction has been introduced in [31] (see also references therein). An operator-based approach towards studying the property that all solutions of a system converge to each other is pursued in [32, 33]. In [34], a Lyapunov approach has been developed to study both the global uniform asymptotic stability of all solutions of a system (in [34], this property is called incremental stability) and the so-called incremental

16 Controlled Synchronisation of Continuous PWA Systems

273

input-to-state stability property, which is compatible with the input-to-state stability approach (see e.g. [35]). In the scope of synchronisation we use the convergence property in the following way. The design of the synchronising controllers is based on the idea of, on the one hand, rendering the closed-loop slave system convergent by means of feedback (which means that all its solutions converge to each other) and, on the other hand, guaranteeing that the closed-loop slave system has a bounded solution corresponding to zero synchronisation error. This implies that all solutions of the closed-loop slave system converge to the synchronising solution. The paper is structured as follows. In Section 16.2, the problem of master-slave synchronisation of PWA systems is stated and it is illuminated that the common approach of synthesising synchronising controllers by providing asymptotically stable error dynamics does not lead to tractable solutions due to the switching nature of PWA systems. The notions of convergence and input-state convergence are introduced and suﬃcient conditions for these properties for PWA systems are proposed in Section 16.3. The latter properties are used in Section 16.4 to design state- and (observer-based) dynamic output feedback controllers achieving synchronisation. An illustrative example of the master-slave synchronisation of two mechanical systems with one-sided restoring characteristics is presented in Section 16.5 and Section 16.6 gives concluding remarks.

16.2 Problem Formulation Consider the state space Rn to be divided into polyhedral cells Λi , i = 1, . . . , l, by hyperplanes given by equations of the form H Tj x + hj = 0, for some H j ∈ Rn and hj ∈ R, j = 1, . . . , k. We will consider the master PWA system to be of the form: x˙ m = Ai xm + bi + Bum (t) for xm ∈ Λi , i = 1, . . . , l, y m = Cxm

(16.1)

and the slave system of the form x˙ s = Ai xs + bi + Bus for xs ∈ Λi , i = 1, . . . , l, y s = Cxs .

(16.2)

Here B ∈ Rn×m , C ∈ Rq×n , Ai ∈ Rn×n and bi ∈ Rn , i = 1, . . . , l, are constant matrices and vectors, respectively. The vectors xm ∈ Rn and xs ∈ Rn represent the state of the master and slave system, respectively, the vectors y m ∈ Rq and y s ∈ Rq are the corresponding measured outputs, um (t) is a time-dependent input of the master system and the vector us ∈ Rm is the control input for the slave system. The hyperplanes H Tj x + hj = 0, j = 1, . . . , k, are the switching surfaces for both systems. Now, we adopt the following assumptions: Assumption 16.1. The right-hand sides of (16.1) and (16.2) are continuous in the corresponding states. It is known (see e.g. [15]) that this assumption can be formalized in the (necessary and suﬃcient) requirement that for any two cells Λi and Λj having a common

274

N. van de Wouw, A. Pavlov, and H. Nijmeijer

boundary H Tij x + hij = 0 the corresponding matrices Ai and Aj and the vectors bi and bj satisfy the equalities Gij H Tij = Ai − Aj Gij hij = bi − bj ,

(16.3)

for some vector Gij ∈ Rn . Assumption 16.2. The input um (t) of the master system and the corresponding solutions xm (t) are bounded for t ≥ 0. The problem considered in this work is formulated as follows: Master-slave synchronisation Design a control law for us for the slave system that, based on information on the measured outputs y s and y m and the input um (t) of the master system, renders xs (t) → xm (t) as t → ∞ and the states of the closed-loop slave system are bounded. As mentioned in the introduction, a common strategy in achieving synchronisation is the stabilisation of the error dynamics between the master and slave systems. The topic of stabilisation of PWA systems is currently receiving wide attention. One could then think of translating the controlled synchronisation problem into some stabilisation problem and subsequently applying known results for the stabilization of PWA systems. Yet, this common way of solving the problem does not lead to tractable solutions. In order to illustrate this, let us consider the master-slave synchronisation problem for systems (16.1), (16.2) as formulated above for the simplest case in which the entire states of both the master and slave system are measured, i.e. y m = xm and y s = xs , and see how this problem would be approached in a conventional way. The ﬁrst step in this approach would be to decompose the control law into a feedforward part um (t) and a feedback part uf b (xs , xm (t)): us (xs , xm (t), um (t)) = um (t) + uf b (xs , xm (t)).

(16.4)

This results in the following closed-loop slave system: x˙ s = Ai xs + bi + Bum (t) + Buf b (xs , xm (t)) for xs ∈ Λi , i = 1, . . . , l. (16.5) The feedforward ensures that a solution xs (t), t > t0 , of the closed-loop slave system (16.5) will match the solution of the master system if it matches the solution of the master system at t = t0 , i.e. if xs (t0 ) = xm (t0 ) (and if uf b (xm (t), xm (t)) = 0). In other words, the feedforward generates the solution of the master system xm (t) in the slave system (16.2). Subsequently, asymptotic synchronisation is assured by designing the feedback part uf b (xs , xm (t)) such that uf b (xm , xm ) = 0 and the dynamics of the synchronisation error e = xs − xm are globally asymptotically stable. These error dynamics follow from (16.1) and (16.5): e˙ = Ai (e + xm (t)) − Aj xm (t) + (bi − bj ) + Buf b , for (e + xm (t)) ∈ Λi , i = 1, . . . , l, m

and x (t) ∈ Λj , j = 1, . . . , l.

(16.6)

16 Controlled Synchronisation of Continuous PWA Systems

275

Now, the problem in this approach for PWA systems lies in the fact that the error-dynamics in (16.6) not only switches when the state xs of the slave system switches from one polyhedral cell to another but also when the state xm (t) of the master system switches from one polyhedral cell to another. Consequently, the error dynamics is described by (potentially) l2 diﬀerent vector ﬁelds (which vector ﬁeld applies depends on e and xm (t), see (16.6)). Moreover, one should realise that these dynamics are time-varying. This combined switching and time-varying nature seriously complicates the stability analysis of the equilibrium point e = 0 of (16.6) and keeps one from applying standard stability analysis methods for PWA systems. This can be illustrated by considering a Lyapunov-based stability argument using for example a positive-deﬁnite quadratic Lyapunov function candidate of the form: V = eT P e with P = P T > 0.

(16.7)

The time-derivative of this function V obeys: V˙ =eT (Ai P + P Ai ) e + (bi − bj )T + uTfb B T + (xm )T (t)(Ai − Aj )T P e + eT P ((bi − bj ) + Buf b + (Ai − Aj )xm (t)) , for (e + xm (t)) ∈ Λi , i = 1, . . . , l and xm (t) ∈ Λj , j = 1, . . . , l.

(16.8)

Clearly, the design of the feedback law uf b guaranteeing the negative-deﬁniteness of V˙ , satisfying (16.8), becomes a rather cumbersome task. The latter exposition aims at clarifying that the master-slave synchronisation problem for PWA systems, on the one hand, can not be tackled by applying known techniques for master-slave synchronisation of smooth systems and, on the other hand, it is signiﬁcantly more complex than the stabilisation problem for PWA systems. This example motivates our study of the master-slave synchronisation problem for PWA systems. Here, we will propose a new approach to this problem based on the notion of convergent systems [29, 30], which is introduced in the next section.

16.3 Convergent Systems In this section, we will brieﬂy discuss the deﬁnition of convergence, certain properties of convergent systems and suﬃcient conditions for convergence of non-smooth, continuous piecewise aﬃne systems. The deﬁnitions presented here extend the deﬁnition given in [29]. Consider the system x˙ = f (x, t),

(16.9)

where x ∈ Rn , t ∈ R and f (x, t) is locally Lipschitz in x and piecewise continuous in t. Deﬁnition 16.1. System (16.9) is said to be ¯ (t) satisfying the following conditions: ! convergent if there exists a solution x

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¯ (t) is deﬁned and bounded for all t ∈ R, (i) x ¯ (t) is globally asymptotically stable. (ii) x ¯ (t) is globally uniformly asymptot! uniformly convergent if it is convergent and x ically stable. ¯ (t) is globally exponentially ! exponentially convergent if it is convergent and x stable. ¯ (t) is called a steady-state solution. As follows from the deﬁnition The solution x of convergence, any solution of a convergent system “forgets” its initial condition and converges to some steady-state solution. This, in turn, implies that any two solutions x1 (t) and x2 (t) converge to each other, i.e. |x1 (t) − x2 (t)| → 0 as t → +∞. In the scope of our problem setting of controlled synchronisation, the timedependency is due to some input determined by the time-dependent trajectory of the master system (e.g. xm (t) and um (t) in (16.5)). Below we will consider convergence properties for systems with inputs. So, instead of systems of the form (16.9), we consider systems of the form x˙ = f (x, w),

(16.10)

with state x ∈ Rn and input w ∈ Rd . The function f (x, w) is locally Lipschitz in x and continuous in w. In the sequel, we will consider the class PCd of piecewise continuous inputs w(t) : R → Rd which are bounded on R. Below we deﬁne the convergence property for systems with inputs. Deﬁnition 16.2. System (16.10) is said to be (uniformly, exponentially) convergent if it is (uniformly, exponentially) convergent for every input w ∈ PCd . In order to emphasize the dependency on the input w(t), the steady-state solution is denoted by ¯ w (t). x In the scope of synchronisation problems, inputs are usually deﬁned not on the whole time axis R, but only for t ≥ 0. For this case, we can formulate the following property. Property 16.1 ([15]). If a convergent system (16.10) is excited by an input w(t), that is deﬁned and bounded only for t ≥ 0 (rather than for t ∈ R), then any two solutions x1 (t) and x2 (t) of (16.10) satisfy | x1 (t) − x2 (t) |→ 0 as t → ∞. The next deﬁnition extends the uniform convergence property to the input-tostate stability framework. Deﬁnition 16.3. System (16.10) is said to be input-to-state convergent if it is uniformly convergent and for every input w ∈ PCd system (16.10) is input-to-state ¯ w (t), i.e. there exist a KLstable (ISS) with respect to the steady-state solution x function β(r, s) and a K∞ -function γ(r) such that any solution z(t) of system (16.10) ˆ corresponding to some input w(t) := w(t) + ∆w(t) satisﬁes ¯ w (t0 )|, t − t0 ) + γ( sup |∆w(τ )|). ¯ w (t)| ≤ β(|x(t0 ) − x |x(t) − x t0 ≤τ ≤t

(16.11)

In general, the functions β(r, s) and γ(r) may depend on the particular input w(t).

16 Controlled Synchronisation of Continuous PWA Systems

w

f1 x2

f2

x1

w

f1 x2

w

Fig. 16.1. Series connection of two systems with inputs.

f2

277

x1

w

Fig. 16.2. Bidirectionally interconnected systems with inputs.

Similar to the conventional ISS property, the property of input-to-state convergence is especially useful for studying convergence properties of interconnected systems. One can easily show that the parallel interconnection of (exponentially, uniformly, input-to-state) convergent systems is again an (exponentially, uniformly, input-tostate) convergent system. A series connection of two input-to-state convergent systems, see Figure 16.1, is an input-to-state convergent system, as stated in the next property. Property 16.2 ([36, 37]). Consider the system     x˙ 1 = f 1 (x1 , x2 , w), x1 ∈ Rn1    x˙ 2 = f 2 (x2 , w),

(16.12)

n2

x2 ∈ R .

Suppose the x1 -subsystem, with (x2 , w) as inputs, is input-to-state convergent and the x2 -subsystem, with w as an input, is input-to-state convergent. Then, system (16.12) is input-to-state convergent. The next property deals with bidirectionally interconnected input-to-state convergent systems, see Figure 16.2. Property 16.3 ([36, 37]). Consider the system     x˙ 1 = f 1 (x1 , x2 , w), x1 ∈ Rn1    x˙ 2 = f 2 (x1 , x2 , w),

(16.13)

n2

x2 ∈ R .

Suppose the x1 -subsystem with (x2 , w) as inputs is input-to-state convergent. Assume that there exists a class KL function βx2 (r, s) such that for any input (x1 , w) ∈ PCn1 +d any solution of the x2 -subsystem satisﬁes |x2 (t)| ≤ βx2 (|x2 (t0 )|, t − t0 ). Then the interconnected system (16.13) is input-to-state convergent. Remark 16.1. Property 16.3 can be used for establishing the separation principle for input-to-state convergent systems. This will be used in Section 16.4 to design

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synchronising output feedback controllers. In that context, system (16.13) represents a system in closed loop with a state-feedback controller and an observer generating state estimates for this controller. The x2 -subsystem corresponds to the observer error dynamics. Now, we present suﬃcient conditions for exponential convergence and input-tostate convergence for the class of continuous PWA systems considered here. Consider a PWA system of the form: ˜ ˜ ix + ˜ bi + Bw for x ∈ Λ˜i , i = 1, . . . , l, x˙ = A

(16.14)

˜i , i = 1, . . . , l, and B ˜ i, b ˜ some matrices of appropriate dimensions and Λ˜i , with A i = 1, . . . , l, are the polyhedral cells deﬁned in Section 16.2. Theorem 16.1 ([36],[38]). Consider system (16.14) and assume that the righthand side of (16.14) is continuous in x. If there exists a positive deﬁnite matrix P = P T > 0 such that ˜ T P < 0, ˜i +A PA i

i = 1, . . . , l,

(16.15)

then system (16.14), with w as input, is exponentially convergent and input-to-state convergent. In [38], the following technical lemma is proven, which will be used in the next section to construct output-feedback synchronising controllers for the master-slave system (16.1), (16.2). We denote the right-hand side of (16.14) by f (x, w), i.e. ˜i + Bw ˜ for x ∈ Λ˜i , i = 1, . . . , l. ˜ ix + b f (x, w) := A Lemma 16.1 ([36],[38]). Under the conditions of Theorem 16.1 it holds that (x1 − x2 )T P (f (x1 , w) − f (x2 , w)) ≤ −α(x1 − x2 )T P (x1 − x2 ).

(16.16)

for all x1 , x2 ∈ Rn , w ∈ Rd , for some α > 0 and for the matrix P satisfying (16.15).

16.4 State- and Output-Feedback Design Let us now propose a convergence-based design of synchronising controllers that avoid explicitly investigating the stability of the synchronisation error-dynamics and thus avoids a cumbersome stability analysis as illustrated in section 16.2. Since we take into account that, for both the master and the slave system, we do not have the entire state available for measurement, we will present an observer-based outputfeedback control design. The main idea of this convergence-based controller design is to ﬁnd a controller that guarantees two properties: a. the closed-loop slave system has a bounded solution along which the synchronisation error (xs − xm (t)) is identically zero, b. the closed-loop slave system is uniformly convergent.

16 Controlled Synchronisation of Continuous PWA Systems

279

Condition b guarantees that any two solutions of the closed-loop slave system converge to each other (see Property 16.1). Together with condition a, this guarantees that all solutions of the closed-loop slave system converge to the bounded solution along which the synchronisation error is identically zero, i.e. the synchronisation control goal is achieved. Let us ﬁrst adopt the perspective that the entire state vectors of both systems can be measured, i.e. C = I in (16.1) and (16.2), where I is an n × n-identity matrix. Then, we propose the following synchronising control law for the slave system, incorporating a linear synchronisation error feedback law: u(xs , xm (t), um (t)) = um (t) + K (xs − xm (t)) ,

(16.17)

with K ∈ Rm×n a constant feedback gain matrix to be designed. The following lemma poses conditions (in the form of LMIs) under which asymptotic synchronisation is achieved with controller (16.17). Lemma 16.2. Consider the master-slave system (16.1), (16.2), with C = I, satisfying Assumptions 16.1 and 16.2. If the LMI P c = P Tc > 0, Ai P c + P c ATi + BY + Y T B T < 0, i = 1, . . . , l,

(16.18)

is feasible, then system (16.2) with the controller (16.17), with K := YP −1 and c xm (t) and um (t) as inputs, is input-to-state convergent. Moreover, the synchronisation error (xs (t) − xm (t)) converges to zero as t → ∞. Proof. The closed-loop slave system has the form x˙ s = (Ai + BK) xs +bi +Bum (t)−BKxm (t) for xs ∈ Λi , i = 1, . . . , l. (16.19) Since the right-hand side of system (16.2) is continuous, the right-hand side of the closed-loop slave system (16.19) is also continuous. Since the LMI (16.18) is feasible, for the matrix K := YP −1 c it holds that −1 T P −1 c (Ai + BK) + (Ai + BK) P c < 0,

i = 1, . . . , l.

Therefore, the closed-loop slave system (16.19) satisﬁes the conditions of Theom m rem 16.1 with the matrix P := P −1 c > 0. Hence, system (16.19) with (u (t), x (t)) as inputs is input-to-state convergent and exponentially convergent. The fact that xs (t) ≡ xm (t) is a solution of the closed-loop slave system (16.19) implies, by Property 16.1, that xs (t) − xm (t) converges to zero as t → ∞. Remark 16.2. If system (16.19) would have an extra input BK∆x(t), then under the conditions of Lemma 16.2 the closed-loop slave system is input-to-state convergent with respect to the inputs xm (t), um (t) and ∆x(t). This fact will be used later on. Remark 16.3. The closed-loop slave system (16.19) satisﬁes the conditions of Theorem 16.1, which by Lemma 16.1 implies that the quadratic Lyapunov function V (x1 , x2 ) = 21 (x1 − x2 )T P (x1 − x2 ) satisﬁes the inequality V˙ ≤ −2αV , for any

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N. van de Wouw, A. Pavlov and H. Nijmeijer m

u (t)

Master PWA

Observer Master

ym

ˆm x -

Feedback

uf b +

+

us

+

Slave

ys

PWA

ˆs x Observer Slave

Fig. 16.3. Block diagram of the controlled master slave system.

two solutions x1 (t) and x2 (t) of the closed-loop slave system. By taking x1 = xs and x2 = xm , one can easily see that V is a quadratic Lyapunov function of the form (16.7) for the synchronisation error-dynamics (16.6). Here, we have constructed this quadratic Lyapunov function using the convergence property. Still, the proof of the existence of such a Lyapunov function and the corresponding feedback law from expression (16.8) directly is signiﬁcantly more diﬃcult. The linear static error-feedback design in (16.17) can easily be extended to more sophisticated linear dynamic error feedback laws, while LMIs similar to those in (16.18) can be formulated to guarantee input-to-state convergence and exponential convergence of the resulting closed-loop slave system. Let us now turn to the case in which only the respective outputs y m , y s of the master and slave systems are measured. We will propose an observer-based outputfeedback design for the slave system. Here, we adopt an approach also taken in [4]. See Figure 16.3 for a block diagram of the entire controlled master-slave system. The ﬁrst step is the design of observers for the (switching) PWA slave system (16.2). Note that the same observer design can be used for the master system, since the systems are identical. Hereto, we use an observer design proposed in the next lemma: Lemma 16.3. Consider slave system (16.2) satisfying Assumption 16.1. If the LMI P o = P To > 0, P o Ai + ATi P o + X C + C T X T < 0, i = 1, . . . l,

(16.20)

is feasible, then the system ˆ s + bi + Bus + L(ˆ ˆ s ∈ Λi , x ˆ˙ s = Ai x y s − y s ), x ˆs = Cx ˆ s, y i = 1, . . . , l,

(16.21)

with L := P −1 o X , is an observer for system (16.2) with globally exponentially stable error dynamics. The observer dynamics (16.21) is input-to-state convergent with ˆ s − xs (the observer error). respect to the inputs us and y s . Denote ∆xs := x Moreover, the observer error dynamics ∆x˙ s = g(xs + ∆xs , us ) − g(xs , us ),

(16.22)

16 Controlled Synchronisation of Continuous PWA Systems

281

where g(xs , us ) := Ai xs +bi +Bus +LCxs for xs ∈ Λi , i = 1, . . . , l, is such that for any bounded xs (t) and any feedback us = us (∆xs , t) all solutions of system (16.22) satisfy (16.23) |∆xs (t)| ≤ ce−a(t−t0 ) |∆xs (t0 )|, where the numbers c > 0 and a > 0 are independent of xs (t) and us = us (∆xs , t). Proof. Let us ﬁrst prove the second part of the lemma. Consider the function g(xs , us ). After unifying the terms containing xs , we obtain g(xs , us ) := (Ai + LC)xs + bi + Bus for xs ∈ Λi , i = 1, . . . , l. Since the right-hand side of system (16.2) is continuous, g(xs , us ) is also a continuous piecewise-aﬃne function. Moreover, since the LMI (16.20) is feasible, for P := P o and L := P −1 o X it holds that P (Ai + LC) + (Ai + LC)T P < 0, i = 1, . . . , l. Hence, by Theorem 16.1 system (16.21) is input-to-state convergent with respect to the inputs us and y s . Applying Lemma 16.1 to the function g(xs , us ), we obtain T

T

∆xs P (g(xs + ∆xs , us ) − g(xs , us )) ≤ −a∆xs P ∆xs

(16.24)

for all xs , ∆xs and us and some constant a > 0 independent of xs , ∆xs and us . T Consider the function V (∆xs ) := 1/2∆xs P ∆xs . The derivative of this function along solutions of system (16.22) satisﬁes T dV = ∆xs P (g(xs + ∆xs , us )− g(xs , us )) ≤ −2aV (∆xs ). dt

This inequality, in turn, implies that there exists c > 0 depending only on the matrix P such that if xs (t) is deﬁned for all t ≥ t0 then the solution ∆xs (t) is also deﬁned for all t ≥ t0 and satisﬁes (16.23). It remains to show that system (16.21) is an ˆ s − xs (t). Since xs (t) is a solution of observer for system (16.2). Denote ∆xs := x s system (16.2), ∆x (t) satisﬁes equation (16.22). By the previous analysis, we obtain that ∆xs (t) satisﬁes (16.23). Therefore, the observation error ∆xs exponentially tends to zero. It should be stressed once more that, since the master and the slave system are identical, the observer (16.21), with the output injection matrix L satisfying (16.20), is also an observer for the master system (of course using y m instead of y s ). Note that this observer guarantees exponentially stable observer error dynamics and does not require knowledge on the moment of switching of the system. We note that if system (16.2) can be represented as a Lur’e system, one can also use the circle criterion-based observer design from [39], see also [40]. These observer designs are more general than (16.21). For general PWA systems, one can also extend the observer design in (16.21) based on the ideas from [39] and [40]; however, we will not pursue such an extension in this paper. Lemma 16.2 shows how to design a state feedback controller which, based on xm and xs , achieves the synchronisation goal. Lemma 16.3 provides observer designs to asymptotically reconstruct xm and xs from the measured outputs y m and y s . In fact, a combination of these controller and observers gives us an output feedback synchronising controller as stated in the following theorem.

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Theorem 16.2. Consider the master-slave system (16.1), (16.2) satisfying Assumptions 16.1 and 16.2. Suppose the LMIs (16.18) and (16.20) are feasible. Denote and L := P −1 K := YP −1 c o X . Then, the slave system (16.2) in closed loop with the controller ˆ m + bi + Bum + L(ˆ ˆ m ∈ Λi , y m − y m ), x i = 1, . . . , l, x ˆ˙ m = Ai x m m ˆ = Cx ˆ , y ˆ s + bi + Bus + L(ˆ ˆ s ∈ Λi , y s − y s ), x i = 1, . . . , l, x ˆ˙ s = Ai x s s ˆ = Cx ˆ , y

(16.25)

ˆ m ) + um , us = K (ˆ xs − x is input-to-state convergent with respect to the inputs y m (t) and um (t). Moreover, the synchronisation error (xs (t) − xm (t)) converges to zero as t → ∞, i.e. synchronisation is achieved asymptotically. Proof. In the coordinates (ˆ xm , ∆xs , xs ), the equations of the closed-loop slave system are given by the ﬁrst equation in (16.25) and ˆ m , for xs ∈ Λi , x˙ s = (Ai + BK)xs + bi + BK∆xs + Bum − BK x s

s

s

s

s

(16.26)

s

(16.27)

ˆ )+u , u = K (x + ∆x − x

(16.28)

∆x˙ = g(x + ∆x , u ) − g(x , u ), s

s

s

m

m

with the function g as deﬁned before. By the choice of the control gain K satisfying ˆ m ) as inputs is input-to-state converLMIs (16.18), system (16.26) with (∆xs , um , x gent (see Lemma 16.2). By the choice of the observer gain L, for any bounded inputs ˆ m , um and for the feedback law (16.28), any solution of system (16.27), (16.28) xs , x satisﬁes |∆xs (t)| ≤ ce−a(t−t0 ) |∆xs (t0 )|, (16.29) ˆ m (t) and um (t) where the numbers c > 0 and a > 0 are independent of xs (t), x (see Lemma 16.3). Applying Property 16.3, we obtain that the system deﬁned by (16.26), (16.27) and (16.28) is input-to-state-convergent with respect to the inputs ˆ m (t). um (t) and x Next, we consider another interconnected system consisting of the series connection of the system deﬁned by (16.26), (16.27) and (16.28) and the observer for the master system. By choice of the observer gain L, the observer for the master system is input-to-state convergent with respect to the inputs um and y m . Therefore, by Property 16.2, this series connection is input-to-state convergent with respect to the ˆ s (t), xs (t)) = (xm (t), xm (t), xm (t)) is a soxm (t), x inputs um and y m . Notice that (ˆ lution of the closed-loop slave system. It is bounded due to Assumption 16.2. Thereˆ m (t), xs (t)) → (xm (t), xm (t), xm (t)) as t → ∞ and, fore, by Property 16.1, (ˆ xm (t), x s m as a consequence, x (t) − x (t) → 0 as t → ∞, i.e. asymptotic synchronisation is achieved. Remark 16.4. The results in Theorem 16.2 for the master-slave synchronisation of two PWA systems can be readily exploited to design synchronising dynamic outputfeedback controllers for an interconnected system of PWA systems, where the connectivity has a tree-like structure, see Figure 16.4. In that case, we address the problem

16 Controlled Synchronisation of Continuous PWA Systems

283

Master PWA

Slave PWA

Slave PWA

Slave PWA

Slave PWA

Slave PWA

Slave PWA

Fig. 16.4. Interconnected system of PWA systems with a tree-like connectivity structure.

of synchronising all systems by coupling (through active control) each system to its neighbour up in the tree. For the sake of brevity, we will omit a formal statement here, however, a sketch of the idea is given below. For every slave system in the tree, the control design can be decomposed into a feedforward part uff and a feedback part uf b . The feedback control design for every slave system in the tree involves a linˆ m ) for slave systems xsi − x ear synchronisation error feedback of the form usfib = K(ˆ sj ˆ si ) for any slave system xsj − x i directly coupled to the master system and uf b = K(ˆ j coupled to slave system i up in the tree (the gain matrices K and output injection matrices L may diﬀer as long as the LMIs (16.18) and (16.20) are satisﬁed). This control design renders each closed-loop slave system input-to-state convergent. Note that the interconnected chains of closed-loop slave systems are also input-to-state convergent due to the fact that a series connection of two input-to-state convergent system is also input-to-state convergent, see Property 16.2. Consequently, all solutions of the total interconnected system converge to each other as t → ∞. The feedforward designs for the slave systems should now ensure that this solution coincides with the synchronising solution. Suitable choices for the feedforward design s are for example uffj = um (t) for all slave systems or usffi = um (t) for slave systems s i directly coupled to the master system and uffj = usi for any slave system j cousi pled to slave system i up in the tree (where u is the total control input for slave system i). Then, the closed-loop interconnected system exhibits a solution for which all systems are synchronised. Consequently, all closed-loop systems in the tree-like interconnection structure asymptotically synchronise.

16.5 An Illustrative Example In this section, an example illustrating the theory presented in this paper is given. The example concerns the master-slave synchronisation of two identical PWA systems. Every system is a two-degree-of-freedom (2DOF) mechanical system with onesided restoring characteristics as depicted in Figure 16.5. An engineering motivation

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N. van de Wouw, A. Pavlov, and H. Nijmeijer

for studying such models can be recognised in synchronising control of wire bonders; in this context the mass-spring-damper combinations reﬂect the dynamics of wire bonder and the one-sided spring reﬂects the ﬂexibility of the workpiece. Each system consists of two masses m1 and m2 interconnected by a linear spring-damper combination with stiﬀness k and damping coeﬃcient c1 . Mass m1 also experiences a damping force due to a linear damper attached to the ﬁxed world with damping coeﬃcient c2 . Moreover, mass m2 experiences a gap d > 0 before hitting the onesided spring with stiﬀness coeﬃcient knl . Mass m1 of the master system is driven by a time-dependent forcing um (t) = A sin(ωt) while the slave system is actuated by a control force us . The displacements of masses m1 and m2 of the master system are denoted by z1m and z2m , respectively, and their respective velocities by z˙ 1m and z˙2m . The displacements of the masses m1 and m2 of the slave system are denoted by z1s and z2s , respectively, and their respective velocities by z˙1s and z˙2s . Moreover, for both systems only the position z1 of mass m1 is available for measurement. The master system can be written in the form (16.1) and the slave system can be written in the form (16.2), with n = 4, l = 2, m = k = q = 1 and the state vectors deﬁned by xi = z1i z˙1i z2i z˙2i , i ∈ {m, s}. Moreover, Λi1 = {xi | xi3 ≥ d}, Λ2 = {xi | xi3 < d}, T

i ∈ {m, s}, b1 = 0 0 0

knl d m2

T

, b2 = 0, B = 0 z1m

1 m1

00

,C= 1000 ,

z2m

um (t) m1

k

c2

m2

Master PWA

d

c1 z1s

knl

z2s

us m1 c2

k c1

m2 d

knl

Slave PWA

Fig. 16.5. Master-slave system consisting of two two-degree-of-freedom mass-springdamper systems with one-sided restoring characteristics.

16 Controlled Synchronisation of Continuous PWA Systems





1  0   − k − c1 +c2  m1 m1 A1 =    0 0    k m2

and

c1 m1

0 k m1

0 nl − k+k m2

0    c1  m1  ,  1     c1 − m2

285

(16.30)



 1  0   − k − c1 +c2  m1 m1 A2 =    0 0    k m2

c1 m1

0 k m1

0 − mk2

0    c1  m1  .  1     c1 − m2

(16.31)

Now we adopt the controller design as in (16.25), see Theorem 16.2. We adopt the following parameter setting: m1 = m2 = 1, c1 = c2 = 2, k = knl = 10, d = 0.01, A = 0.02 and ω = 1.5. A solution to the LMIs (16.18) is represented by   31.8  115.7.0 −376.5 1.8     −376.5 1395.9 −17.1 −226.8    , K = 111.5 30.1 19.4 7.4 Pc =     1.8 −17.1 51.7 −65.8        31.8 −226.8 −65.8 782.8 and a solution to the LMIs (16.20) is represented by   

(16.32)



 20.9676   1575.2 −158.6 −737.6 −24.2         110.1598 −158.6 69.1 −140.5 13.9      . , L =  Po =       18.6509  −737.6 −140.5 1639.5 −40.0             −4.4789 −24.2 13.9 −40.0 100.7

(16.33)

Herewith, all conditions of Theorem 16.2 are satisﬁed and synchronisation of the systems is achieved. A simulation with the initial conditions xm (0) = 0 0.01 0 0.01 ˆ m (0) = for the master system, x

T

T

0.01 0.01 0.01 0.01

for the observer of the

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master system, xs (0) =

T

0.005 0 0.01 0.01

ˆ s (0) = for the slave system and x

T

0 0 0 −0.01

for the observer of the slave system is performed. In Figures 16.6-

16.9, the resulting time series for the states x1 to x4 of both the master and the slave system are compared. Moreover, in Figures 16.10 and 16.11, the corresponding synchronisation errors are displayed. Clearly, asymptotic synchronisation is achieved.

16.6 Conclusions In this paper, the controlled synchronisation problem for identical continuous piecewise aﬃne (PWA) systems is addressed. It is shown that due to the switching nature of these systems conventional strategies for controlled synchronisation, which are commonly based on stabilising the synchronisation error dynamics, lead to highly complex stabilisation problems. This complexity is due to, ﬁrstly, the fact that the error dynamics switches not only on the state of the slave system but also on the state of the master system, and, secondly, the fact that the error dynamics is timedependent, where the time-dependency is due to the time-dependent state trajectories of the master system (which are a priori unknown). We consider the master-slave synchronisation problem for two PWA systems with an arbitrary number polyhedral cells. An observer-based output-feedback control design solving this problem is proposed. The design of these dynamic controllers is based on the idea of, on the one hand, rendering the closed-loop slave system convergent by means of feedback (which means that all its solutions converge to each other) and, on the other hand, guaranteeing that the closed-loop slave system has a bounded solution corresponding to zero synchronisation error. This implies that all solutions of the closed-loop slave system converge to this bounded solution with zero synchronisation error. This convergence-based approach avoids explicitly

Fig. 16.6. Position of mass m1 for master and slave systems: Synchronisation is achieved.

Fig. 16.7. Position of mass m2 for master system and slave system: Synchronisation is achieved.

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Fig. 16.8. Velocity of mass m1 for master and slave systems: Synchronisation is achieved.

Fig. 16.9. Velocity of mass m2 for master system and slave system: Synchronisation is achieved.

Fig. 16.10. Synchronisation errors in positions of masses m1 and m2 .

Fig. 16.11. Synchronisation errors in velocities of masses m1 and m2 .

dealing with building up a stabilisation argument for the time-dependent switching synchronisation error dynamics. This result can be used to address the controlled synchronisation problem for interconnected PWA systems with a tree-like structure. The results are illustrated by application to the master-slave synchronisation of two mechanical systems with one-sided restoring characteristics.

Acknowledgements This research was partially supported by the Dutch-Russian program on interdisciplinary mathematics “Dynamics and Control of Hybrid Mechanical Systems” (NWO grant 047.017.018)

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17 Design of Convergent Switched Systems R.A. van den Berg1 , A.Yu. Pogromsky1, G.A. Leonov2 and J.E. Rooda1 1

2

Eindhoven University of Technology, Department of Mechanical Engineering, P.O.Box 513, 5600 MB Eindhoven, The Netherlands. [r.a.v.d.berg, a.pogromsky, j.e.rooda]@tue.nl St. Petersburg State University, Mathematics and Mechanics Faculty, Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia. [email protected]

Summary. In this paper we deal with the problem of rendering hybrid/nonlinear systems into convergent closed-loop systems by means of a feedback law or switching rules. We illustrate our approach to this problem by means of two examples: the anti-windup design for a marginally stable system with input saturation, and the design of a switching rule for a piece-wise aﬃne system operating in diﬀerent modes.

17.1 Introduction It is well known that any solution of a stable linear time-invariant (LTI) system with a bounded input converges to a unique limit solution that depends only on the input. Nonlinear systems with such a property are referred to as convergent systems. Solutions of the convergent systems “forget” their initial conditions and after some transient time depend on the system input that can be a command or reference signal. One of the main objectives of feedback in controller design is to eliminate the dependency of the system steady-state solutions on initial conditions. This property should be preserved for an admissible class of the inputs that makes the problem of the design of convergent systems an important control problem. The property of convergency can play an important role in the studies related to the group coordination and cooperative control. Particularly, if each agent from the whole network is described by a convergent model, and if the input signal is identical for all agents, after some transient time all the agents will follow the same trajectory. In other words, the synchronization between the agents from a network will be achieved if each agent is controlled by a local (dependent only on that agent state) feedback aimed at making the agent convergent. An advantage of this synchronization scheme is that it can be achieved via decentralized (local) controllers, i.e. no exchange of information between the agents is required. The main disadvantage, however, stems from the same origin: if the agents operate in diﬀerent environment, they are perturbed by diﬀerent disturbances and thus eventually will follow diﬀerent paths, that is the group of the agents will stay in an asynchronous mode. To overcome this diﬃculty a communication between the agents can be introduced that

K.Y. Pettersen et al. (Eds.): Group Coord. and Cooperat. Cntrl., LNCIS 336, pp. 291–311, 2006. © Springer-Verlag Berlin Heidelberg 2006

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can be considered as a sort of feedback, usually in a form of the mismatch between the agents states or outputs. In [34, 35] it was demonstrated that for passivity-based design of synchronizing networks via output feedback the convergency of an agent subsystem consistent with some algebraic constraint plays almost the same role as minimumphaseness in the conventional output feedback stabilization problem. This motivates studies related to design of convergent systems via diﬀerent sorts of feedback. The property that all solutions of a system “forget” their initial conditions and converge to some steady-state solution has been addressed in a number of publications [8, 39, 32, 20, 41, 22, 10, 11, 13, 2]. In this paper we continue the study originated in [33, 31] on convergency of piece-wise aﬃne (PWA) systems. This class of systems attracted a lot of attention over the last years, see e.g. [3, 17, 18, 38] and references therein. In this paper two extensions are given to the convergency theory as presented in [33, 31]. Each of the extensions is discussed in the context of a suitable application area. First, we present a convergence based approach to the anti-windup controller design for marginally stable systems with input saturation. For such systems one cannot directly use the results on quadratic convergency of PWA continuous systems presented in [33], since there exists no common quadratic Lyapunov function for this kind of system. To tackle the problem we developed a method that allows to establish uniform, but not necessary quadratic convergence. Secondly, we address the problem of switching control for PWA systems operating in diﬀerent modes, that is to ﬁnd a switching rule as a function of the state and the input such that the closed loop system demonstrates convergent behavior. This approach is diﬀerent from the theory discussed in [31], in which the switching rule is assumed to be known. The paper is organized as follows. In Section 17.2 we recall some basic results on conditions for stability of solutions of nonlinear systems. Section 17.3 gives various deﬁnitions of convergent systems. In Section 17.4 we apply the convergency based approach to the anti-windup controller design for a marginally stable system with input saturation. Section 17.5 deals with the design of a switching rule that makes the closed loop system convergent.

17.2 Stability via First Approximation The study of convergent systems focusses on the stability of solutions of nonlinear systems. Two Lyapunov methods are available for analysis of the stability of solutions, i.e. Lyapunov’s indirect and direct method. In this section we present a brief overview of the problem of stability analysis of solutions of nonlinear time-varying systems via its ﬁrst order approximation (i.e., the indirect Lyapunov method) and at the end we will conclude that the direct Lyapunov method is more promising for analytical purposes. Consider a classical question of stability analysis of a particular (or all) solution(s) of the following nonlinear time-varying system x˙ = F (x, t), x ∈ Rn , t ∈ R,

(17.1)

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where F satisﬁes some regularity assumptions to guarantee the existence of local solutions x(t, t0 , x0 ). For the sake of brevity if there are no confusions and if the meaning is apparent we will omit the dependence of the solutions on some parameters, i.e. initial time and data. ¯0 ) of system (17.1), deﬁned for all t ∈ (t∗ , +∞), Deﬁnition 17.1. A solution x(t, t0 , x is said to be

! stable if for any t0 ∈ (t∗ , +∞) and ε > 0 there exists δ = δ(ε, t0 ) > 0 such that ¯0 || < δ implies ||x(t, t0 , x0 ) − x(t, t0 , x ¯0 )|| < ε for all t ≥ t0 ; ||x0 − x ! uniformly stable if it is stable and the number δ in the deﬁnition of stability can be chosen independently of t0 ; ! asymptotically stable if it is stable and for any t0 > t∗ there exists δ = δ(t0 ) > 0 such that ||x0 − x ¯0 || < δ for t0 > t∗ implies limt→∞ ||x(t, t0 , x0 )−x(t, t0 , x ¯0 )|| = 0; ! uniformly asymptotically stable if it is uniformly stable and there exists δ > 0 (independent of t0 ) such that for any ε > 0 there exists T = T (ε) > 0 such that ||x0 − x ¯0 || < δ implies ||x(t, t0 , x0 ) − x(t, t0 , x ¯0 )|| < ε for all t ≥ t0 + T ; ! exponentially stable if there exist positive δ, C, β such that ||x0 − x¯0 || < δ implies ||x(t, t0 , x0 ) − x(t, t0 , x ¯0 )|| ≤ Ce−β(t−t0 ) ||x0 − x ¯0 ||;

! uniformly globally asymptotically stable if it is uniformly asymptotically stable and attracts all solutions starting in (x0 , t0 ) ∈ Rn × (t∗ , +∞) uniformly over t0 , i.e. for any R > 0 and any ε > 0 there is a T = T (ε, R) > 0 such that if ¯0 )|| < ε for all t ≥ t0 + T , t0 > t∗ . ||x0 || < R, then ||x(t, t0 , x0 ) − x(t, t0 , x Suppose F is continuously diﬀerentiable in x and continuous in t. Let A(x, t) =

∂F (x, t) ∂x

be the Jacobi matrix for the function F (x, t) at the point x ∈ Rn . Let us consider the solutions x(t, t0 , x0 ) of the system (17.1) with initial conditions x(t0 , t0 , x0 ) = x0 under the assumption that they are well deﬁned for all t ≥ t∗ . Together with system (17.1) consider its ﬁrst order approximation governed by the following equation ξ˙ = A(x(t, t0 , x0 ), t)ξ, ξ ∈ Rn , t ≥ t∗ .

(17.2)

Let Φ(t, x0 ) be a fundamental matrix for system (17.2) with Φ(t0 , x0 ) = In . The following lemma is crucial for stability analysis of solutions of the nonlinear system (17.1) via ﬁrst order approximation (17.2). Lemma 17.1 ([21]). For any two solutions x(t, t0 , x0 ) and x(t, t0 , y0 ) of system (17.1) the following estimate ||x(t, t0 , x0 ) − x(t, t0 , y0 )|| ≤ sup ||Φ(t, η)|| ||x0 − y0 || η∈B

is true, where B = {η ∈ Rn | ||x0 − η|| ≤ ||x0 − y0 ||}.

(17.3)

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Proof. The system equations can be rewritten in the following form dx(t, t0 , x0 ) = F (x(t, t0 , x0 ), t). dt The earlier mentioned assumptions imposed on F imply the solution x(t, t0 , x0 ) is diﬀerentiable in x0 . From the last equation one gets d ∂x(t, t0 , x0 ) ∂x(t, t0 , x0 ) = A(x(t, t0 , x0 ), t) dt ∂x0 ∂x0 and thus

∂x(t, t0 , x0 ) = Φ(t, x0 ). ∂x0 The multidimensional variant of Lagrange’s mean value theorem yields ||x(t, t0 , x0 ) − x(t, t0 , y0 )|| ≤ sup

0<θ<1

∂x (t, t0 , x0 + θ(y0 − x0 )) ||x0 − y0 ||. ∂x0

The previous lemma is a simple tool that allows to analyze stability of a particular solution x(t, t0 , x ¯0 ) of nonlinear system (17.1) via its ﬁrst order approximation (17.2). Indeed, Lyapunov stability of this solution is granted if for all x0 from a neighborhood of x¯0 the corresponding fundamental matrix Φ(t, x0 ) is bounded as a function of time. If additionally limt→∞ ||Φ(t, x0 )|| = 0 then the solution x(t, t0 , x ¯0 ) is asymptotically stable. Although this approach can lead to veriﬁable stability conditions (see [21]) the ﬁrst attempts to tackle the problem of stability via ﬁrst order approximation were ¯0 ). The latter based on analysis of system (17.2) for one given solution x(t, t0 , x approach however is only applicable if for system (17.1) some shift transformation ¯0 ) to the z = x−x ¯ allows to reduce the problem of stability of the solution x(t, t0 , x problem of stability of the origin of the following nonlinear system z˙ = A(t)z + f (t, z),

(17.4)

where together with mild assumptions on A(t), the nonlinear term f (t, z) must satisfy the following assumption ||f (t, z)|| ≤ ψ(t)||z||m ,

m>1

for a continuous function ψ with zero Lyapunov exponent (in the original statement due to Lyapunov ψ(t) = const). Lyapunov proved that if the ﬁrst approximation system of (17.4) ξ˙ = A(t)ξ (17.5) is regular in the sense of Lyapunov (see e.g. [1]) and has negative Lyapunov exponents then the origin of (17.4) is asymptotically stable. The results of this kind stimulated development of numerical methods to analyze stability via the ﬁrst Lyapunov method, i.e. the method of Lyapunov exponents.

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Researches based on the Lyapunov exponents are particularly popular in physical studies of chaotic systems and numerical calculation of those exponents is now a standard tool, implemented in numerous software packages. The most diﬃcult assumption to verify in the Lyapunov theorem is the regularity of system (17.5), that implies that all Lyapunov exponents of (17.5) should be strict. Later on this condition was relaxed by Malkin, Chetaev and Massera (see e.g. [24, 6, 25, 26]) who showed that the conclusion of the Lyapunov theorem remains true if the largest Lyapunov exponent of (17.5) is suﬃciently negative and strictly bounded from above by α , − m−1 where α ≥ 0 is the so-called irregularity coeﬃcient, that is, in turn, quite diﬃcult to calculate for practical examples. Another diﬃculty that should be taken into account for the problem of asymptotic stability via ﬁrst order approximation is that Lyapunov exponents of system (17.5) are not stable in general (see e.g. [1, 7]). In other words, even an inﬁnitesimal perturbation of (17.5) can change the sign of its Lyapunov exponents. Particularly, this means that the asymptotic stability of the solution x(t, t0 , x¯0 ) is not a robust property. Though necessary and suﬃcient conditions for stability of the Lyapunov exponents are known (see e.g. [5, 27, 1]), they are quite diﬃcult to verify in practice. Another approach to tackle stability via ﬁrst order approximation is by means of the general exponent, as originated by Bohl [4]. Particularly, negativity of the general exponent of the norm of the fundamental matrix ||Φ(t)|| for (17.5) gives necessary and suﬃcient conditions for exponential stability of (17.5) [7]. At the same time exponential stability of the ﬁrst order approximation (17.5) together with some mild assumptions on the nonlinear term f (t, z) gives suﬃcient conditions for uniform asymptotic stability of the origin of nonlinear system (17.4), see e.g. [37]. Moreover, in contrast to Lyapunov exponents, the general exponents are stable, see [7]. However, as follows from the deﬁnition of the general exponents, its numerical computation is a challenging problem. The brief survey presented above illustrates the main diﬃculties that arise in the problem of stability via ﬁrst order approximation if one tries to tackle this problem via method of characteristic exponents (either Lyapunov or Bohl). It seems that from an analytical point of view the most promising method to investigate stability is the direct Lyapunov method. This method also allows to estimate Lyapunov (and Bohl) exponents and gives veriﬁable conditions for (uniform asymptotic) stability of a solution x(t, t0 , x0 ) of nonlinear system (17.1). In the subsequent sections we present recent developments in this ﬁeld using the concept of convergent systems that is closely related to the concept of stability of all solutions of nonlinear system (17.1).

17.3 Convergent Systems In this section we give deﬁnitions of convergent systems. Those systems are closely related to systems with uniformly globally asymptotically stable solutions and the deﬁnitions presented here extend those given by Demidovich [8].

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Deﬁnition 17.2. System (17.1) is said to be

! uniformly convergent if there is a solution x¯(t) = x(t, t0 , x¯0 ) satisfying the following conditions: (i) x ¯(t) is deﬁned and bounded for all t ∈ (−∞, +∞), (ii) x ¯(t) is uniformly globally asymptotically stable; ! exponentially convergent if it is uniformly convergent and x¯(t) is globally exponentially stable. The solution x ¯(t) is called a limit solution. As follows from the deﬁnition of convergence, any solution of a convergent system “forgets” its initial condition and converges to some limit solution which is independent of the initial conditions. In general, if there is a globally asymptotically stable limit solution x ¯(t) it may be non-unique, in the sense that there can exist another solution x˜(t) bounded for all t ∈ (−∞, +∞) that is also globally asymptotically stable. For any two such solutions it obviously follows that ||¯ x(t)− x ˜(t)|| → 0 as t → ∞. At the same time for uniformly convergent systems the limit solution is unique, as formulated below. Property 17.1 ([30, 29]). If system (17.1) is uniformly convergent, then the limit solution x ¯(t) is the only solution deﬁned and bounded for all t ∈ (−∞, +∞). In systems theory time dependency of the right-hand side of system (17.1) is usually due to some input. This input may represent, for example, a disturbance or a feedforward (reference) control signal. Below we will consider convergence properties for systems with inputs. So, instead of systems of the form (17.1), we consider systems x˙ = f (x, w),

(17.6)

with state x ∈ Rn and input w ∈ Rm . In the sequel we will consider the class PCm of piecewise continuous inputs w(t) : R → Rm which are bounded for all t ∈ R. We assume that the function f (x, w) is bounded on any compact set of (x, w) and the set of discontinuity points of the function f (x, w) has measure zero. Under these assumptions on f (x, w), for any input w(·) ∈ PCm the diﬀerential equation x˙ = f (x, w(t)) has well-deﬁned solutions in the sense of Filippov. Below we deﬁne the convergence property for systems with inputs. Deﬁnition 17.3. System (17.6) is said to be (uniformly, exponentially) convergent if it is (uniformly, exponentially) convergent for every input w(·) ∈ PCm . In this paper we are going to consider the systems of the form (17.1) and (17.6) with non-smooth right hand sides under quite general regularity assumptions that guarantee the existence of solutions in some reasonable sense, e.g. in a sense of Filippov, see e.g. [9, 40]. According to Filippov one can construct a set-valued function F(x, t) such that an absolutely continuous solution of the diﬀerential inclusion x˙ ∈ F(x, t) is called a solution for system (17.1). For system (17.1) consider a scalar continuously diﬀerentiable function V (x). Deﬁne a time derivative of this function along solutions of system (17.1) as follows ∂V (x) V˙ := x(t, ˙ t0 , x0 ). ∂x

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Since V is continuously diﬀerentiable and the solution x(t, t0 , x0 ) is an absolutely continuous function of time, the derivative V˙ (x(t, t0 , x0 )) exists almost everywhere in the maximal interval of existence [t0 , T¯) of the solution x(t, t0 , x0 ). For the function V we can also deﬁne its upper derivative as follows V˙ ∗ (x, t) =

sup

ξ∈F(x,t)

∂V (x) ξ . ∂x

Then for almost all t ∈ [t0 , T¯) it follows that V˙ (x(t, t0 , x0 )) ≤ V˙ ∗ (x(t, t0 , x0 ), t). Remark 17.1. Notice that in the domains of continuity of the function F (x, t) the derivative of V (x) along solutions of system (17.1) equals V˙ = ∂V∂x(x) F (x, t). According to [9] p.155, for a continuously diﬀerentiable function V (x) it holds that if the inequality ∂V (x) F (x, t) ≤ 0 ∂x is satisﬁed in the domains of continuity of the function F (x, t), then the inequality V˙ ∗ (x, t) ≤ 0 holds for all (x, t) ∈ Rn+1 . Deﬁnition 17.4. System (17.6) is called quadratically convergent if there exists a matrix P = P T > 0 and a number α > 0 such that for any input w ∈ PCm for the function V (x1 , x2 ) = (x1 − x2 )T P (x1 − x2 ) it holds that V˙ ∗ (x1 , x2 , t) ≤ −αV (x1 , x2 ),

(17.7)

where V˙ ∗ (x1 , x2 , t) is the upper derivative of the function V (x1 , x2 ) along any two solutions of the corresponding diﬀerential inclusion, i.e. V˙ ∗ (x1 , x2 , t) =

sup

ξ1 ∈F(x1 ,w(t))

∂V (x1 , x2 )ξ1 ∂x1

+

sup

ξ2 ∈F(x2 ,w(t))

∂V (x1 , x2 )ξ2 . ∂x2

Quadratic convergence is a useful tool for establishing the exponential convergence, as follows from the following lemma. Lemma 17.2 ([30]). If system (17.6) is quadratically convergent, then it is exponentially convergent. The proof of this lemma is based on the following result, which will be also used in the remainder of this paper. Lemma 17.3 ([39]). Consider system (17.6) with a given input w(t) deﬁned for all t ∈ R. Let D ⊂ Rn be a compact set which is positively invariant with respect to dynamics (17.6). Then there is at least one solution x ¯(t), such that x ¯(t) ∈ D for all t ∈ (−∞, +∞). Note that for convergent nonlinear systems performance can be evaluated in almost the same way as for linear systems. Due to the fact that the limit solution of a convergent system only depends on the input and is independent of the initial

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conditions, performance evaluation of one solution (i.e. one arbitrary initial state) for a certain input suﬃces, whereas for general nonlinear systems all initial states need to be evaluated to obtain a reliable analysis. This means that for convergent systems simulation becomes a reliable analysis tool and for example ‘Bode-like’ plots can be drawn to analyse the system performance. An example of simulation based performance analysis can be found in Sect. 17.5.1.

17.4 Application of Convergent Systems Analysis to the Anti-windup Problem The presence of actuator saturation in an otherwise linear closed-loop system can dramatically degrade the performance of that system. This performance degradation is caused by the so-called ‘controller windup’. In the past, several linear and nonlinear anti-windup techniques have been developed to compensate for this windup eﬀect (see e.g. [15, 14, 36, 19, 12]). However, not all approaches, e.g. based on ﬁnite (incremental) L2 gain, are able to guarantee global anti-windup for a marginally stable plant. Here we propose another approach, based on uniform convergency, which is close to that introduced by [12]. The main diﬀerence between this and most other approaches is that whereas the other approaches focus on guaranteeing L2 stability from input to output, this approach focusses on ensuring convergency, i.e. a unique limit solution for each input signal, independent of initial conditions, which is constant (resp. periodic) if the input signal is constant (resp. periodic). In this section we show that under some mild conditions, it is possible to guarantee uniform convergency for an anti-windup scheme with a plant that is an integrator (i.e. marginally stable).

r

w3

kP

+ -

+ -

+

yc

u

+

kI / s

+

+

+

1 s

y

-

kA Fig. 17.1. Anti-windup scheme with integrator plant

Consider the system in Fig. 17.1, consisting of an integrator plant with input saturation, a PI-controller and a static anti-windup block. Assuming input r is differentiable, this system can be described as follows     x˙ = Ax + Bu + Ew(t)     (17.8) z = Cx        u = sat(yc )

17 Design of Convergent Switched Systems

where x = [y yc ]T , w(t) = [r(t) r(t) ˙ w3 (t)]T and    



299



0  1  0   0 0 1  , B =  , E =   , C = [0 1] , A=       −kI −kI kA −kP + kI kA kI kP −kP with kI , kP , kA > 0 and sat(yc ) = sign(yc ) max(1, |yc |). Deﬁnition 17.5. A continuous function t → w(t), w(t) = [w1 (t) w2 (t) w3 (t)]T is said to belong to the class W if there exist nonnegative constants C1 , C2 , C3 , C4 , C5 , with C3 < 1 such that 1. ∀t ∈ R1 |w1 (t)| ≤ C1 ; 2. ∀t ∈ R1 |w2 (t)| ≤ C2 ; 3. w3 (t) = w3c (t) + w3i (t) with 3a. ∀t ∈ R1 |w3c (t)| ≤ C3 ; t 3b. ∀t, t0 ∈ R1 t0 w3i (τ )dτ ≤ C4 ; 3c. ∀t ∈ R1 |w3 (t)| ≤ C5 .

Theorem 17.1. If kA kP > 1 then system (17.8) is uniformly convergent for all w(·) ∈ W. First we prove the following lemmata. Lemma 17.4. If kA kP > 1 then system (17.8) is uniformly ultimately bounded for all w ∈ W, that is, given input w(·) from W, there is a number R > 0 such that for any solution x(t, t0 , x0 ) starting from a compact set Ω there is a number T (Ω) such that for all t ≥ t0 + T (Ω) it follows that ||x(t, t0 , x0 )|| < R. t

Proof. Let yi (t) = y(t) − t0 w3i (τ )dτ . Then the system equations can be rewritten as     y˙ i = y˙ − w3i (t) = sat(yc ) + w3c (t) (17.9)    y˙ c = −kI yi − kI kA yc + (kI kA − kP )sat(yc ) + ζ(t) with ζ(t) = kI r(t) + kP r(t) ˙ − kP w3 (t) − kI Consider the following Lyapunov function candidate  T  

t t0

w3i (τ )dτ. 



 yi   yi  c 1    .   W (yi , yc ) =    P   , with P =   yc yc 1 kA This function is positive deﬁnite and radially unbounded for kA c > 1.

(17.10)

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In the linear mode (sat(yc ) = yc ) the time derivative of (17.10) satisﬁes  T    T    T    yi  1 0  yi   yi   yi  ˙ =   Q   + 2   P   ζ(t) + 2   P   w3c (t) W             0 1 yc yc yc yc

(17.11)

and the matrix Q is negative deﬁnite provided kA kP > 1 and c = kP + kI kA (or lies in some interval around this value). In the saturated mode (sat(yc ) = sign(yc )) the time derivative of W equals ˙ W 2 2 = − kI yi2 − 2kI kA yi yc − kI kA yc + (c − kP + kI kA )yi sign(yc ) 2 2 + (1 − kA kP + kA kI )|yc | + (yi + kA yc )ζ(t) + (cyi + yc )w3c (t). Denote µ = yi + kA yc . Then ˙ W = − kI µ2 + (c − kP + kI kA )µsign(yc ) − (c − kP + kI kA )kA |yc | 2 2 + (1 − kA kP + kA kI )|yc | + µζ(t) + (cµ + (1 − kA c)yc ) w3c (t), so that with

˙ W ≤ −kI µ2 + |µ|C6 + (1 − kA c)(1 − C3 )|yc |, 2

(17.12)

C6 = |c − kP + kI kA | + kI C1 + kP (C2 + C5 ) + kI C4 + cC3 .

Combining (17.11) and (17.12) one can apply the Yoshizawa theorem on ultimate boundedness (see e.g. [16]) since kA c > 1, C3 < 1 and kA kP > 1. This completes the proof. Consider a system consisting of two copies of (17.8) with identical inputs:        A 0   B 0   sat(yc1 )   E  x +    +   w(t), x˙ =         0 A 0 B E sat(yc2 )

(17.13)

with x = [y1 yc1 y2 yc2 ]T . Deﬁne the function ξ(t) as follows    satyc1 (t)−satyc2 (t)  if yc1 (t) = yc2 (t);  yc1 (t)−yc2 (t)    ξ(t) = 1 if yc1 (t) = yc2 (t) & |yc1 (t)| < 1 & |yc2 (t)| < 1;       0 otherwise. Since the function sat(·) satisﬁes the incremental sector condition it follows that ∀t 0 ≤ ξ(t) ≤ 1. We need the following result.

17 Design of Convergent Switched Systems

301

Lemma 17.5. Given w(·) from W, for any solution x(t, t0 , x0 ) of (17.13) starting from some compact set Ω there exist δ = δ(Ω) > 0, T¯ = T¯(Ω) > 0, such that for all t ≥ t0 it follows that t+T¯

t

ξ(s)ds ≥ δ.

Proof. First consider yc1max := lim supt→∞ |yc1 (t)| < 1. From the previous lemma this means that after some ﬁnite T (Ω) both subsystems of (17.13) approach the linear mode and will stay in this mode for all t ≥ T (Ω). If both |yc2 | < 1 and |yc1 | < 1 then ξ = 1 and the result follows. Secondly, consider the opposite: yc1max ≥ 1. From the system equations it follows that for any t ≥ t0 , T > 0 t+T

y1 (t + T ) − y1 (t) =

t

t+T

satyc1 (s)ds +

t

w3 (s)ds

and therefore 1 T

t+T t

satyc1 (s)ds =

1 y1 (t + T ) − y1 (t) − T T

t+T t

w3 (s)ds.

(17.14)

From Lemma 17.4 and the assumption imposed on the signal w3 (t) it follows that by making T suﬃciently large one can make the ﬁrst term of the right hand side arbitrarily small and the second term strictly smaller than 1 by absolute value. In other words, for suﬃciently large T¯ = T¯(Ω) there is an α that can be chosen independently of t0 , 0 < α < 1 such that 1 T¯

t+T¯ t

satyc1 (s)ds ≤ α.

Due to the mean value theorem there is a η ∈ (t, t+T¯] such that |satyc1 (η)| ≤ α. From Lemma 17.4 we know that the time derivative of yc1 (t) is bounded and therefore the function satyc1 (t) is uniformly continuous on [t0 , ∞). Now choose some ε > 0 such that α + ε < 1. This is always possible since α < 1. Since saty1c (t) is uniformly continuous there is a number ∆t > 0 such that |saty1c (τ )| ≤ α + ε < 1, ∀τ ∈ [η − ∆t, η + ∆t]. This number ∆t can be chosen independently of t0 since the right hand side of (17.13) and hence y˙ 1c (τ ) is uniformly bounded. Among all possible ∆t choose the largest possible satisfying ∆t ≤ T¯. Now, integrating nonnegative ξ from t till t + T¯ yields t+T¯

t

ξ(s)ds ≥

min{t+T¯ ,η+∆t}

max{t,η−∆t}

ξ(s)ds.

Since t ≤ η ≤ t + T¯ it follows that min{t + T¯, η + ∆t} − max{t, η − ∆t} ≤ min{T¯, ∆t} = ∆t

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and

min{t+T¯ ,η+∆t} max{t,η−∆t}

ξ(s)ds ≥ ∆tξmin ,

where ξmin is the lowest bound for ξ under restriction that |saty1c | ≤ α + ε < 1. ξ approaches this bound if |y1c | = α + ε and |y2c | = yc1max ≥ 1. Therefore, ξmin = and hence

t+T¯ t

1−α−ε >0 yc1max − α − ε

ξ(s)ds ≥ ∆t

1−α−ε > 0. yc1max − α − ε

The last inequality implies the statement of Lemma (17.5) with δ = ∆t yc11−α−ε −α−ε . max

Proof of Theorem 17.1: For system (17.8) consider the Lyapunov function:    P −P   x ≥ 0, V (x) = xT    −P P

(17.15)

with x = [y1 yc1 y2 yc2 ]T , P as deﬁned in (17.10), c = kP + kI kA and therefore kA c > 1. Denote e1 = y1 − y2 , e2 = yc1 − yc2 and ϕ = sat(yc1 ) − sat(yc2 ). Then the derivative of V satisﬁes V˙ 2 2 2 = −kI e21 − 2kI kA e1 e2 − kI kA e2 + 2kI kA e1 ϕ + (1 + kI kA − kP kA )e2 ϕ 2 2 2 2 = −kI (e1 + kA e2 ) + 2kI kA ϕ(e1 + kA e2 ) − kI kA ϕ 2 2 2 −kI kA e2 ϕ − (kP kA − 1)e2 ϕ + kI kA ϕ 2 = −kI (e1 + kA (e2 − ϕ))2 − (kP kA − 1)e2 ϕ − kI kA (e2 ϕ − ϕ2 ).

Since sat(·) satisﬁes the [0, 1]-incremental sector condition e2 ϕ − ϕ2 ≥ 0, it follows that

V˙ ≤ 0

and uniform stability of all solutions y1 (t), y1c (t) of (17.8) is proven. The last inequality is not suﬃcient to prove quadratic stability of all solutions. However, the exponential convergence from a given compact set can be deduced from the previous lemma. Indeed, V˙ 2 2 2 = −kI e21 − 2kI kA e1 e2 − kI kA e2 + 2kI kA e1 ϕ + (1 + kI kA − kP kA )e2 ϕ 2 2 2 e2 − kI ξ(t)e21 − (kP kA − 1)ξ(t)e22 = kI (1 − ξ(t)) −e21 − 2kA e1 e2 − kA ≤ −kI ξ(t)e21 − (kP kA − 1)ξ(t)e22 .

17 Design of Convergent Switched Systems

303

It follows then that V satisﬁes the following inequality V˙ ≤ λmax ξ(t)V, in which λmax < 0 is the largest solution of the following generalized eigenvalue problem     0   −kI    − λP  = 0.  det  2    0 −(kP kA − 1) t +T¯

Hence t00 λmax ξ(t) ≤ λmax δ < 0 with δ from the statement of Lemma 17.5. Using the Gronwall-Bellman lemma (see e.g. [1]) one can see that V → 0 as t → ∞ uniformly in time and uniformly in the initial conditions from Ω. Since Ω is an arbitrary compact set and due to Lemma 17.4, all solutions are globally uniformly asymptotically stable. Due to Lemma 17.3 there is a bounded solution x ¯(t) deﬁned on the whole time interval (−∞, +∞) and thus system (17.8) is uniformly convergent for all w(·) ∈ W. Note that due to Property 17.1 this solution x ¯(t) is the unique solution bounded on (−∞, +∞). As one can see, our analysis is based on a PE-like (persistency of excitation) property that follows from Lemma 17.5. More advanced results in this direction can be found in [28, 23]. 17.4.1 Example: Inﬂuence of Parameter kA on System Dynamics

2 1.5 1 0.5 0 -0.5 -1 -1.5 -2

kA = 0 kA = 0.05 kA = 0.2 kA = 20 r(t) 0

2

4

6 time

8

10

(a) x0 = (−2, 0)

12

14

3 2 1 y

y

Theorem 17.1 states that system (17.8) is uniformly convergent for kA > 1/kP . In this example, we consider system (17.8) with kP = 10, kI = 20 and w3 (t) = 0, and evaluate the system behavior for several values of kA . Note that the values of kP and kI are chosen in such a way that the system without the saturation has a satisfactory performance.

0 -1 -2 -3

0

2

4

6 time

8

10

12

14

(b) x0 = (−3, 0)

Fig. 17.2. System output for input r(t) = sin(t) and diﬀerent values of kA

In Fig. 17.2 the system output y is plotted for four diﬀerent values of kA and the input signal r(t) = sin(t). Figures 2(a) and 2(b) display the results for two diﬀerent initial conditions of the system. For kA = 0 (i.e. no anti-windup) the two initial

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conditions result in two diﬀerent limit solutions. An other observation is that for the other values of kA (even for kA = 0.05 < 1/kP ) the system seems to have a unique limit solution for diﬀerent initial conditions. However, to verify the existence of a unique limit solution for kA = 0.05 we should be able to evaluate all initial conditions and input signals, since it is possible that for another initial condition or input signal the limit solution for kA = 0.05 is not unique, or we should be able to expand Theorem 17.1 such that it holds for kA < 1/kP as well.

17.5 Quadratic Convergence of Switched Systems Consider the switched dynamical system x˙ = Ai x + Bi w(t), i = 1, . . . , k,

(17.16)

where x(t) ∈ Rn is the state, w(·) ∈ PCm is the input. Suppose the collection of matrices {A1 , . . . , Ak } and {B1 , . . . , Bk } is given, and Ai is Hurwitz for all i = 1, . . . , k. The general problem is to ﬁnd a switching rule such that the closed loop system is uniformly (exponentially) convergent. In this section, we focus on a switching rule that is based on static state feedback, i.e. i = σ(x). Note that in a similar way dynamic state feedback and static/ dynamic output feedback can be considered. These approaches are subject for future research. Suppose a common Lyapunov matrix P = P T > 0 exists that satisﬁes the following inequalities ATi P + P Ai < 0, i = 1, . . . , k. (17.17) Consider the following switching rule σ(x, w) = arg min{xT Zix x + xT Ziw w},

(17.18)

i

in which Ziw = 4P Bi and Zix are matrices to be deﬁned. Theorem 17.2. If there exist a P = P T > 0, α > 0, and Z1x , . . . , Zkx and if Zix = Zjx and/or Ziw = Zjw such that 

∀i, j ≤ k, i = j, 

(17.19) 



−(Ai P + P Aj )   P Ai + Ai P − (Zix − Zjx )  In −In   ≤ −α        −In In P Aj + AjT P + (Zix − Zjx ) −(AjT P + P Ai ) T

T

(17.20) for all i, j ≤ k, i = j, then the switching rule (17.18) with matrices Z1x , . . . , Zkx makes system (17.16) quadratically convergent. Proof. First, note that condition (17.19) implies that the set of discontinuities of the right-hand side of the closed loop system has zero measure, which means that a

17 Design of Convergent Switched Systems

305

Filippov solution exists for the closed loop system. Let P be a common Lyapunov matrix for the collection {A1 , . . . , Ak } and consider the Lyapunov function candidate V (x1 , x2 ) = (x1 − x2 )T P (x1 − x2 ).

(17.21)

If σ(x1 , w) = σ(x2 , w) the inequality V˙ ≤ −εV, ε > 0

(17.22)

is obviously satisﬁed. Let σ(x1 , w) = p and σ(x2 , w) = q, such that the derivative of (17.21) can be written as V˙ = x1T (ATp P + P Ap )x1 + xT2 (ATq P + P Aq )x2 − x1T (ATp P + P Aq )x2 (17.23) T T T T T T −x2 (P Ap + Aq P )x1 + 2x1 P Bp w + 2x2 P Bq w − 2x1 P Bq w − 2x2 P Bp w. Using the switching rule (17.18) the following constraint functions can be deﬁned 1 T (x (Zpx − Zqx )x1 + xT1 (Zpw − Zqw )w) ≤ 0, 2 1 1 S2 (x, w) = (xT2 (Zqx − Zpx )x2 + xT2 (Zqw − Zpw )w) ≤ 0. 2 S1 (x, w) =

The system is quadratically stable (see Deﬁnition 17.4) if  T     x1   In −In   x1      V˙ ≤ −α       −In In x2 x2 for all (x, w) satisfying S1 (x, w) ≤ 0 and S2 (x, w) ≤ 0. Using the S-procedure, the previous condition is satisﬁed if the following inequality holds  T     x1   In −In   x1     . V˙ − S1 − S2 ≤ −α       x2 −In In x2

(17.24)

This inequality is equivalent to (17.20). Remark 17.2. Note that (17.20) is an LMI with design variables P , Z1x , . . . , Zkx and α, which can be solved eﬃciently using available LMI toolboxes. Remark 17.3. In case Bi = B for all modes, then the switching rule (17.18) is independent of the input. This implies that under the conditions stated in Theorem 17.2 the system can be made convergent without regarding the input, even if the input for example represents a disturbance signal.

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Although Theorem 17.2 gives suﬃcient conditions for quadratic convergence, it does not give insight for what collection of matrices {A1 , . . . , Ak } a switching law can be found. In the case that we deﬁne in advance both Zix and Ziw Zix = AiT P + P Ai ,

Ziw = 4P Bi ,

∀i = 1, . . . , k,

(17.25)

then Theorem 17.2 can be simpliﬁed as follows. Theorem 17.3. If there exist a P = P T > 0 satisfying (17.17) and if Zix = Zjx and/or Ziw = Zjw and

∀i, j ≤ k, i = j

P (Ai − Aj ) − (Ai − Aj )T P = 0

∀i, j ≤ k,

(17.26) (17.27)

then the switching rule (17.18) makes system (17.16) quadratically convergent. Proof. Consider the same notations as in the proof of Theorem 17.2. Note that due to (17.27), xT1 (ATp P + P Aq )x2 + xT2 (P Ap + AqT P )x1 = (17.28) 1 T T 1 x (A P + P Ap + ATq P + P Aq )x2 + x2T (ApT P + P Ap + ATq P + P Aq )x1 . 2 1 p 2 Combining (17.23) with (17.28) and using notations (17.25) gives 1 1 1 V˙ = x1T Zpx x1 + xT1 Zpx x1 + x1T Zpw w 2 2 2 1 1 1 + xT2 Zqx x2 + xT2 Zqx x2 + xT2 Zqw w 2 2 2 1 1 − xT1 (Zpx + Zqx )x2 − x2T (Zpx + Zqx )x1 2 2 1 1 − xT2 Zpw w − x1T Zqw w. 2 2 The switching rule (17.18) implies that 1 T 1 (x Zpx x1 + xT1 Zpw w) ≤ (xT1 Zqx x1 + xT1 Zqw w) 2 1 2 1 T 1 T T (x Zqx x2 + x2 Zqw w) ≤ (x2 Zpx x2 + xT2 Zpw w) 2 2 2 and therefore

1 V˙ ≤ (x1 − x2 )T (Zpx + Zqx )(x1 − x2 ) ≤ −αV 2

for some α > 0. Remark 17.4. Note that condition (17.27) is always satisﬁed for symmetric matrices Ai , i = 1, . . . , k.

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307

Linear controller 1

w(t)

+

Mass-springdamper system

SWITCH

-

y(t)

Linear controller 2

Fig. 17.3. Graphical representation of system (17.29)

17.5.1 Example: Performance of a Convergent Switched System In this example we show how a simulation-based performance analysis can be realized for a switched systems that is made convergent by the design of a switching rule. Consider the switched system x˙ = Ai x + Bi w(t), i = 1, 2

(17.29)

y = Cx

which represents for example a mass-spring-damper system with two linear controllers (see Fig. 17.3). Here, x(t) ∈ R3 is the state, w(·) ∈ PC1 is the input, C = [ 1 0 0 ], and          13   −7 2 −6             A1 =  −10 −3 −5  , B1 =  15  ,         −6 7 4 −1 For the common Lyapunov matrix 

 3   −6 3 −8             A2 =  −9 0 −8  , B2 =  9  .         −6 5 1 −9 

 0.5163 −0.1655 −0.0038       P =  −0.1655 0.2609 0.0321  > 0     −0.0038 0.0321 0.2669 and



Z1x − Z2x



 −0.3584 −0.0312 0.5055       =  −0.0312 −0.5208 0.7083       0.5055 0.7083 2.2239

conditions (17.19) and (17.20) are satisﬁed, with α = 1. Switching rule (17.18) thus makes the system (17.29) quadratically convergent,

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˙

V ≤ −(x1 − x2 )2 ≤ −1.6704(x1 − x2 )T P (x1 − x2 ) = −1.6704V.

Subsequently, the fact that λmin (P )(x1 − x2 )2 ≤ (x1 − x2 )T P (x1 − x2 ) ≤ λmax (P )(x1 − x2 )2 , with λmin (P ) and λmax (P ) the minimum and maximum eigenvalue of P respectively, leads to the following upper bound |x1 (t) − x2 (t)| ≤ λmax (P )/λmin (P ) |x1 (0) − x2 (0)| e ≤ 1.8671 |x1 (0) − x2 (0)| e−0.8352t .

−1.6704 t 2

(17.30)

In order to analyse the performance of this switched system, only one solution of the system needs to be evaluated, since the limit solution of this (convergent) system is independent of its initial conditions. In Fig. 17.4 the performance of the switched system is compared with the performance of the two corresponding linear systems, i.e., x˙ = A1 x + B1 w(t) and x˙ = A2 x + B2 w(t). The performance measure applied here is the integrated tracking error: tl +T tl

(w(t) − y(t))2 dt tl +T tl

w(t)2 dt

,

(17.31)

where T is a time period that is long enough to obtain a good average of the tracking error and tl is a moment in time for which all considered solutions are close enough to the limit solution. The time tl is in this example determined visually, but a bound can be calculated as well using (17.30). The performance is evaluated for the following input signals: w(t) = sin(bt), b ∈ [10−2 , 103 ]. From Figure 17.4 it can be concluded that for the considered performance mea-

integrated tracking error

1.2 1

Controller 1 Controller 2 Switched

0.8 0.6 0.4

0.2 0 10−2

10−1

100

b

101

Fig. 17.4. Performance of switched system

102

103

17 Design of Convergent Switched Systems

309

sure (17.31) the switched system performs better then the linear systems for the input range b ∈ [101 , 103 ], i.e. the switched system is less sensitive to high frequencies in the input signal. More important, however, is the fact that the performance of the switched system can be determined by means of simulation, which is practically impossible for most nonlinear/switched systems that are not convergent.

17.6 Conclusions In this paper we considered the following problem deﬁnition for piece-wise aﬃne systems: is it possible to design a feedback law and/or switching rule such that the resulting closed-loop system is convergent? We have investigated this problem for two areas of interest, i.e. the anti-windup design for a marginally stable plant with input saturation and the class of switched linear systems. For an integrator plant with input saturation, we proved that by a simple static anti-windup rule a uniformly convergent closed-loop system can be obtained. It is also noted that within the range of the anti-windup rule for which the system is convergent, performance of the closed-loop system can be optimized using simulation. Furthermore, we demonstrated the use of convergency in the class of switched linear systems. It is proved that by deﬁnition of the switching rule the switched system can made convergent if the linear subsystems satisfy certain conditions. For the convergent switched system, a performance evaluation has been shown feasible using a Bode-like plot.

Acknowledgement This work was partially supported by the Dutch-Russian program on interdisciplinary mathematics “Dynamics and Control of Hybrid Mechanical Systems” (NWO grant 047.017.018) and the HYCON Network of Excellence, contract number FP6IST-511368. The authors also thank Elena Panteley (Sup´elec, France) for very valuable discussions. Finally the authors thank the anonymous reviewers for their valuable comments.

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