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Control Methods for Electrical Machines
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Control Methods for Electrical Machines
Edited by René Husson
First published in France in 2003 by Hermes Science/Lavoisier entitled “Méthodes de commande des machines électriques” © LAVOISIER, 2003 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd, 2009
The rights of René Husson to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Méthodes de commande des machines électriques. English Control methods for electrical machines / edited by René Husson. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-093-6 1. Electric machinery--Automatic control. 2. Electric machinery--Mathematical models. I. Husson, René. II. Title. TK2391.M48 2009 621.31'042--dc22 2008043200 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-093-6 Printed and bound in Great Britain by CPI/Antony Rowe Ltd, Chippenham, Wiltshire.
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Chapter 1. Overview of Mechanical Transmission Problems . . . . . . . . . Pascal FONTAINE and Christian CUNAT
1
1.1. Technological aspects. . . . . . . . . . . 1.1.1. General structures of the machines 1.1.2. Mechanisms and movement. . . . . 1.1.3. Engine-machine . . . . . . . . . . . . 1.1.4. Particular movements . . . . . . . . 1.1.5. Friction – elements of tribology . . 1.2. Bibliography . . . . . . . . . . . . . . . .
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Chapter 2. Reminders of Solid Mechanics . . . . . . . . . . . . . . . . . . . . . Jean-François SCHMITT and Rachid RAHOUADJ
31
2.1. Reminders of dynamics. . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Kinematics of a rigid body . . . . . . . . . . . . . . . . . . 2.1.2. Kinetic elements for a rigid body – Koenig’s theorems . 2.1.3. Newtonian dynamics . . . . . . . . . . . . . . . . . . . . . . 2.2. Application example: dynamic balance of a rigid rotor . . . . 2.3. Analytical dynamics (Euler-Lagrange). . . . . . . . . . . . . . 2.4. Linear energies in the neighborhood of the balance for a non-damped discrete system . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Equilibrium configuration without group motion . . . . . 2.4.2. Equilibrium configuration with group motion . . . . . . . 2.5. Vibratory behavior of a discrete non-damped system around an equilibrium configuration . . . . . . . . . . . . . . . . . . . . . . 2.6. Analytical study of the vibratory behavior of a milling machine table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1. Positioning the problem . . . . . . . . . . . . . . . . . . . .
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2.6.2. Setting up the model . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3. Direct resolution of the eigenvalue problem . . . . . . . . . . 2.6.4. Cancellation of the stiff mode and reduction of the problem 2.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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53 58 59 61
Chapter 3. Towards a Global Formulation of the Problem of Mechanical Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian CUNAT, Mohamed HABOUSSI and Jean François GANGHOFFER
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3.1. Presentation of the mechanical drive modeling problem . . . 3.2. Brief review on continuum mechanics . . . . . . . . . . . . . . 3.2.1. Conservation laws . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Principle of virtual powers (PVP) . . . . . . . . . . . . . . 3.2.3. Thermomechanics of continuous mediums . . . . . . . . . 3.2.4. Notions on strain . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Some material behaviors: elementary analog models. . . 3.2.6. Variational formulations in mechanics of the structures . 3.3. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4. Continuous-time Linear Control . . . . . . . . . . . . . . . . . . . Frédéric KRATZ
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4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. PID controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Various structures. . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Selection criteria for the adjustments . . . . . . . . . . . 4.2.4. Control mode . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. PID controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Adjustment by trial and error . . . . . . . . . . . . . . . . 4.3.2. Ziegler-Nichols method . . . . . . . . . . . . . . . . . . . 4.3.3. Cohen-Coon method . . . . . . . . . . . . . . . . . . . . . 4.4. Methods based on previous knowledge of a system model . 4.4.1. Presentation of the Bode method. . . . . . . . . . . . . . 4.4.2. Presentation of the Phillips and Harbor method . . . . . 4.5. Linear state feedback control systems . . . . . . . . . . . . . 4.5.1. Formulation of the control problem . . . . . . . . . . . . 4.5.2. The structure of the control law . . . . . . . . . . . . . . 4.5.3. Reconstruction of the state . . . . . . . . . . . . . . . . . 4.5.4. The controller as a combination of state feedback and an observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5. Design of feedback gain matrix F . . . . . . . . . . . . . 4.6. Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Optimal regulator at continuous time . . . . . . . . . . . 4.6.2. Stochastic optimal regulator at continuous time . . . . .
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91 91 91 94 95 97 100 100 102 103 104 104 106 107 107 107 108
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4.6.3. Discrete time optimal regulator. . 4.6.4. Stochastic time optimal regulator 4.6.5. LQG/H2 control. . . . . . . . . . . 4.7. Choice of a control . . . . . . . . . . . 4.8. Bibliography . . . . . . . . . . . . . . .
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122 125 125 129 130
Chapter 5. Overview of Various Controls . . . . . . . . . . . . . . . . . . . . . Frédéric KRATZ
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5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Internal model controller . . . . . . . . . . . . . . . . . . . . . 5.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Passage of the structure of regulation to that of control by internal model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. Properties of the control by internal model . . . . . . . . 5.2.4. Implementation . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Predictive control . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. General principles of predictive control. . . . . . . . . . 5.4. Sliding control . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Structure of the control law . . . . . . . . . . . . . . . . . 5.4.3. Equivalent method control . . . . . . . . . . . . . . . . . 5.5. Bang-bang control . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Control-based fuzzy logic . . . . . . . . . . . . . . . . . . . . 5.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2. Structure of the controlled loop. . . . . . . . . . . . . . . 5.6.3. Representation of fuzzy controllers . . . . . . . . . . . . 5.6.4. Basic concepts of fuzzy logic . . . . . . . . . . . . . . . . 5.6.5. Fuzzification. . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.6. The inference mechanism . . . . . . . . . . . . . . . . . . 5.6.7. Defuzzification . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Neural network control . . . . . . . . . . . . . . . . . . . . . . 5.7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2. Formal neurons . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3. Neural networks. . . . . . . . . . . . . . . . . . . . . . . . 5.7.4. Parsimonious approximation . . . . . . . . . . . . . . . . 5.7.5. Implementation of neural networks . . . . . . . . . . . . 5.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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133 134 139 141 141 144 149 149 149 152 152 154 154 155 156 157 161 162 163 164 164 164 165 166 166 167
Chapter 6. Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . Rachid OUTBIB and Michel ZASADZINSKI
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6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169 170
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6.3. Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. General features of non-linear systems . . . . . . . . . 6.3.2. Existence of a sliding mode . . . . . . . . . . . . . . . . 6.3.3. Chattering phenomena . . . . . . . . . . . . . . . . . . . 6.3.4. Determination of sliding dynamics . . . . . . . . . . . 6.3.5. Case of more than one commutation surface . . . . . . 6.4. Direct Lyapunov method . . . . . . . . . . . . . . . . . . . . 6.4.1. Affine systems with regard to control . . . . . . . . . . 6.4.2. Linear systems . . . . . . . . . . . . . . . . . . . . . . . 6.5. Equivalent control method . . . . . . . . . . . . . . . . . . . 6.5.1. Invariance condition . . . . . . . . . . . . . . . . . . . . 6.5.2. Existence conditions . . . . . . . . . . . . . . . . . . . . 6.5.3. Sliding mode for a perturbed system . . . . . . . . . . 6.5.4. Canonical forms. . . . . . . . . . . . . . . . . . . . . . . 6.6. Imposing a surface dynamic . . . . . . . . . . . . . . . . . . 6.6.1. A classic surface dynamic . . . . . . . . . . . . . . . . . 6.6.2. A particular case: dynamic with pure discontinuities . 6.7. The choice of sliding surface . . . . . . . . . . . . . . . . . 6.7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2. A specific linear surface choice . . . . . . . . . . . . . 6.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .
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172 172 174 178 180 183 184 184 187 189 189 190 194 196 198 198 199 200 200 202 203 203 204
Chapter 7. Parameter Estimation for Knowledge and Diagnosis of Electrical Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Claude TRIGEASSOU, Thierry POINOT and Smaïl BACHIR
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7.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Identification using output-error algorithms. . . . . . . . . . . 7.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. Least-squares algorithm in output-error. . . . . . . . . . . 7.2.3. Principle of the output-error method in the general case . 7.2.4. Sensitivity functions . . . . . . . . . . . . . . . . . . . . . . 7.2.5. Convergence of the estimator. . . . . . . . . . . . . . . . . 7.2.6. Variance of the estimator . . . . . . . . . . . . . . . . . . . 7.2.7. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Parameter estimation with a priori information . . . . . . . . 7.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . 7.3.3. Minimization of the compound criterion . . . . . . . . . . 7.3.4. Deterministic interpretation . . . . . . . . . . . . . . . . . . 7.3.5. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Parameter estimation of the induced machine . . . . . . . . . 7.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.4.2. Modeling in the three-phase frame. . . . . . . . . . . . . . . . 7.4.3. Park’s transformation . . . . . . . . . . . . . . . . . . . . . . . 7.4.4. Continuous-time state-space model . . . . . . . . . . . . . . . 7.4.5. Output-error identification . . . . . . . . . . . . . . . . . . . . 7.4.6. Output-error identification and a priori information . . . . . 7.5. Fault detection and localization based on parameter estimation . 7.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2. Principle of the method . . . . . . . . . . . . . . . . . . . . . . 7.5.3. Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 7.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 8. Diagnosis of Induction Machines by Parameter Estimation . . 245 Smaïl BACHIR, Slim TNANI, Gérard CHAMPENOIS and Jean-Claude TRIGEASSOU 8.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . 8.2. Induction motor model for fault detection . . . . . 8.2.1. Stator faults modeling in the induction motor 8.2.2. Rotor fault modeling . . . . . . . . . . . . . . . 8.2.3. Global stator and rotor fault model . . . . . . 8.3. Diagnosis procedure. . . . . . . . . . . . . . . . . . 8.3.1. Parameter estimation . . . . . . . . . . . . . . . 8.3.2. Implementation . . . . . . . . . . . . . . . . . . 8.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 8.5. Bibliography . . . . . . . . . . . . . . . . . . . . . .
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245 246 247 255 259 261 262 265 267 268
Chapter 9. Time-based Coordination . . . . . . . . . . . . . . . . . . . . . . . . Michel DUFAUT and René HUSSON
271
9.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Brief description system . . . . . . . . . . . . . . . . . . . 9.2.1. Functional structure . . . . . . . . . . . . . . . . . . . 9.3. Some ideas on the manipulator system models . . . . . . 9.3.1. Various model types . . . . . . . . . . . . . . . . . . . 9.3.2. Geometric models . . . . . . . . . . . . . . . . . . . . 9.3.3. Kinematic models. . . . . . . . . . . . . . . . . . . . . 9.3.4. Dynamic models . . . . . . . . . . . . . . . . . . . . . 9.4. Coordination of motion . . . . . . . . . . . . . . . . . . . . 9.4.1. Why coordinate movements? . . . . . . . . . . . . . . 9.4.2. Step response of a controlled shaft. . . . . . . . . . . 9.4.3. Speed representation in a point-to-point movement. 9.4.4. Partially specified trajectories . . . . . . . . . . . . . 9.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 10. Multileaf Collimators . . . . . . . . . . . . . . . . . . . . . . . . . . Sabine ELLES and Bruno MAURY 10.1. Radiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1. The medical prescription. . . . . . . . . . . . . . . . 10.1.2. Linear accelerators . . . . . . . . . . . . . . . . . . . 10.2. Multileaf collimators . . . . . . . . . . . . . . . . . . . . 10.2.1. Geometric characteristics of multileaf collimators. 10.2.2. Technical characteristics . . . . . . . . . . . . . . . . 10.2.3. Readout systems for leaf position checking. . . . . 10.2.4. Leaf command system . . . . . . . . . . . . . . . . . 10.2.5. Accuracy of command and leaf positioning . . . . 10.3. Intensity modulated radiotherapy . . . . . . . . . . . . . 10.3.1. How to realize a modulated intensity beam with a multileaf collimator . . . . . . . . . . . . . . . . . . . . . . . 10.3.2. Discretization into static elementary beams. . . . . 10.3.3. Discretization into dynamic beams . . . . . . . . . . 10.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 11. Position and Velocity Coordination: Control of Machine-Tool Servomotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Patrick BOUCHER and Didier DUMUR
331
11.1. Open architecture systems . . . . . . . . . . . . . . . . 11.1.1. Historical overview . . . . . . . . . . . . . . . . . . 11.1.2. Principle and advantages. . . . . . . . . . . . . . . 11.1.3. Modular architecture example. . . . . . . . . . . . 11.2. Structure and implementation of control laws. . . . . 11.2.1. Cascaded structure . . . . . . . . . . . . . . . . . . 11.2.2. Polynomial structure of controllers. . . . . . . . . 11.2.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 11.3. Application to machines-tools axis drive control . . . 11.3.1. Classic control scheme . . . . . . . . . . . . . . . . 11.3.2. Cascaded velocity-position predictive control of synchronous motors . . . . . . . . . . . . . . . . . . . . . . 11.3.3. Multivariable flux-position predictive control of asynchronous motors . . . . . . . . . . . . . . . . . . . . . 11.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 11.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
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331 331 332 332 334 334 336 338 339 339
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339
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348 362 364
List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
369
Preface
The control of electrical machines very much depends on the context and environment in which motors are considered. How a machine is used has a strong influence on its control laws. These are especially related to loading characteristics, variables requiring control, operating conditions and, consequently, to the models chosen. Loading intrinsically governs the choice of control laws and the way they are implemented. Common load torques are inertia torques, whether they are constant or variable, or possibly even randomly variable. Constant inertia torques are induced by rotating masses, which may be either symmetric homogenous, working as inertia flywheels, or dissymmetric heterogenous, causing torque fluctuations or vibrations. Inertia may vary due to a load geometrical distortion, as is the case in Watt regulators or hinged jibs, or to a change in the rotating mass, like in a winder. Generally, non-inertial load torques need to be considered on top of these. Gravity causes load torques that adds up to the inertia torques induced by movement, especially in handling devices. The viscosity of the ambient environment causes resisting torques proportional to speed (viscous friction torques). This is the case in fans, boat propellers and more generally in all lubricated devices such as main bearings. Dry frictions cause stress, determined by the surface roughness of materials in contact and by the traverse speed. Being highly complex to model, stress is tricky to take into account in equations. The result of this is ill-controlled torques as well as inaccurate positioning. Load driving by electrical motors is almost always operated through a transmission, the role of which is either to modify the range of accessible speeds and torques or to change the nature of the movement (rotation/translation). These transmitters alter the properties of loads quite significantly: not only do they
xii
Control Methods for Electrical Machines
adversely affect the order of magnitude of torques, but they also input nonlinearities that may compromise the operation of the system. This is why this work starts with a presentation of the main problems encountered in mechanical transmissions, which are covered by Chapters 1, 2 and 3. Chapters 4, 5, 6, 7 and 8 deal with the means available to drive a “converter/motor/transmission/load” unit. It intends to provide the reader the most useful and recent information on the techniques that make it possible to design the most accurate control laws for the problem that needs to be solved. Generally, in usual applications, we strive to control three mechanical variables: velocity, position or torque (separately or simultaneously). Therefore, automatic techniques are called for. Even though usual linear controls (continuous or discrete, IPD, etc.) are still frequently used (rightly, considering their benefits), more recent and effective methods in difficult cases can be implemented. Optimal controls (linear-quadratic, linear-quadratic-Gaussian, etc.), adaptive controls (with or without the reference model), sliding mode controls or “bang-bang” controls help lead to more satisfying solutions, but often necessitate more comprehensive and more detailed models than linear control. Predictive controls, neural network and fuzzy logic controls help refine the control and improve the performance of dynamic system controls when the models are fairly unknown. Finally, since the models used in electrical machine control involve several disciplines (automatic control, electrical engineering, computer science and mechanics), a common representation mode appears to be appropriate. Bond graphs fulfill this role perfectly and can be very helpful as useful tools for control. This topic will not be addressed here, since a specialized piece of work of more than 300 pages1 fully covers this issue. Control and model building are only applicable with accurate numeric values of parameters. Therefore, identification, applied to electrical machines, aims at providing these data. Although numerous methods of measurement make it possible to reach some variables, the majority of the models used in control reveal constants, which gather several electrical variables and are not directly measurable as a consequence. Moreover, the necessary values are dynamic values, which are impossible to obtain using traditional measurements. That is why a chapter is dedicated to the identification methods that are best adapted for our scope of activity.
1 Karnopp D., Margolis D., Rosenberg R. Systems Dynamic Modeling and Simulation of Mechatronic Systems, John Wiley & Sons, 2000; and Geneviève Dauphin-Tanguy, Les Bond Graph, Hermes Science Publications, 2000.
Preface
xiii
The correct operation of the electrical machine is a key factor for operating safety. This work tackles the subject through the diagnosis of the defects of the machine. This technique is based on the continuous identification of the system and on the monitoring of the variation of its parameters. Diagnosis and adaptive control are sectors in which identification techniques are paramount. Control techniques call upon traditional concepts in automatic control, such as stability, robustness and observers. The latter are essential to make use of some variables which are not accessible to measurement. However, such basic concepts are beyond the scope of this work. Indeed, developing them would require several chapters, the material of which could be found elsewhere, resulting in a global loss of generality and an unbalanced overview, with no obvious added value. Chapters 9, 10 and 11 relate to the control of systems animated by a set of electrical motors, which requires a coordination of the actions of each component. Each motor obeys by the same control laws as those that would be applied to it if considered separately; however, a coordination (subsequently a coupling) between motors is introduced at the set point-level. Allowing complex operations such as deformations of the curvature of large areas, simultaneous machining, movements of hinged jibs or variable geometry openings requires the introduction of several coordination modes. The most obvious is synchronization, which is a coordination based on time. Synchronization requires specific speeds or motions for each motor at certain times. The individual control laws of each actuator are thus coupled, either in a discrete way by an assignation method or in a continuous way when the durations of actions are imposed by the operating constraints of the system. Position is also a coupling variable in multi-motor systems. The set point given to one of them will interfere with those of the other motors. Such is the case, for instance, with the variable geometry openings used in medicine for the irradiation of tumours. The VLT (Very Large Telescope) is also an example, with its huge mirrors deforming under their own weight. They are placed on multiple small electrical jacks, which correct the radius of curvature very precisely, requiring the coupling in position of 1,000 motors. Finally, speed is also a coordination means for the control of a set of motors. Thus, on the machine tools (and many other devices), the feed and rotation speeds of pins are not independent and must be coordinated, like positions. This type of coordination also exists in assembly lines and conveying attachments: in such contexts, the velocity variations of a conveying belt must be reflected on all the others.
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Control Methods for Electrical Machines
Working out a general theory at this level is a difficult task, that’s why the third part of the work rather presents examples of multi-motors systems illustrating the main coordination types. Coordination by time: the robot, coordination by the position: the multileaf collimator, coordination by speed: the machine tools. Throughout this work, our main goal has been to explain the concepts covered in a way understandable to most readers. This book is especially dedicated to people faced with issues at the border of their area of expertise. We hope that the material presented here will help them to solve these problems and gain a better view of global complex system design. Our purpose has therefore been twofold: providing a tool allowing better communication between experts from different domains, while allowing engineers from any background to gain a broader insight into general scientific culture. René HUSSON
Chapter 1
Overview of Mechanical Transmission Problems
1.1. Technological aspects The direct approach of a system of mechanical transmission of power may be delicate. In this chapter, we propose to assimilate a real mechanism to a discrete mechanical system. We will then identify its main components and propose a classification based on their efficiency (in section 1.1.1; see also [SPI 97]). The general theorems of mechanical application to these simple models after an easy mathematical treatment will highlight the relevant parameters governing machine performance and point to directions of thought in order to improve the command of those mechanisms (sections 1.1.2, 1.1.3 and 1.1.4; see also [SPI 97]). Finally, the main elements of tribology will be presented in section 1.1.5. Indeed, the study of friction and lubrication and their consequences constitutes an essential part in the conception and the functioning of machines.
1.1.1. General structures of the machines These engines are connected with machines through transmission power systems or mechanisms such as gearings, belts or chains, clutches or brakes, systems connecting rod-crank or the sawnut systems, cams or eccentric, elastic coupling. Any mechanism is put in motion by an entrance element called the leading or driving element, which supplies the driving energy. An exit element called the led or receiving element is the element by which the energy connected to loads goes out of Chapter written by Pascal FONTAINE and Christian CUNAT.
2
Control Methods for Electrical Machines
the mechanism. The power circulates from the engine towards loads. The exit of a component constitutes the entrance of the next component. 1.1.1.1. Engine In a power converter, the engine receives as an input an electrical power; as an output this power is always a mechanical power (couple and angular speed or strength and linear speed). The variation curve of the couple or the strength according to the speed is the curve of capacity of the engine, the characteristic of the engine. This conversion involves losses and therefore the notion of the engine’s efficiency is introduced. 1.1.1.2. Loads Loads are the linear efforts and torques applied to the parts of the machine situated downstream of the studied part. Loads represent the resistances induced by the environment of the machine and result from their functioning. They are the loads of friction, gravity and slowness. 1.1.1.3. Energy efficiency The classification of the numerous existing mechanisms can rely on the energy efficiency expression. By definition, the efficiency is the ratio of supplied energy to the necessary energy to supply.
Figure 1.1. Powers acting on a mechanism
The mechanism of Figure 1.1 receives the power P1 and supplies the mechanical power P2 to the receiver 2. The mechanism 1 being real, there is a Pp power being lost and transformed into heat in 1. The efficiency on the mechanism 1 is written:
Overview of Mechanical Transmission Problems
K
3
P2 P1
The nature of the movement being assumed to be unchanged in 1 (here a rotation), we can write:
K
C2 : 2 C1 :1
C1, C2 are the torques applied to 1, 2 respectively.
: 1, :2, are the angular speeds of driveshaft 1, 2 respectively. In the case of a perfect mechanism, there are no losses and then P1 = P2 = P20, with P20 = C20 :20. By making reference to a perfect mechanism, the efficiency of a real mechanism can be written as:
K
C2 : 2 C 20 : 20
By dividing every term by C1 :1 and putting
ic
i
C2 C1 :1 :2
i co
i0
C 20 C1 :1 : 20
moment rates
velocity rates
the real mechanism can then be written as:
K
K c u KZ
°° Kc with ® °K °¯ Z
ic i c0 i0 i
static efficiency 1 g kinematic efficiency
4
Control Methods for Electrical Machines
where
Kc
expresses losses by friction and
KZ characterizes the internal gliding in
the mechanism. We can then write :2
1 i
:1
(1 g ) : 20
1 g i0
:1
i o the geometric rate of transmission,
with ®
¯ g the gliding factor, a function of the transmitted couple.
This approach allows the classification of mechanisms into two big categories: – Positive mechanisms: the transmission of efforts is done by contact without sliding and the gliding factor remains 0. This concerns the gearings, the chains, the eccentric and the sawnut systems for which $ = 1 and = c . – Non-positive mechanisms: the effort transmission is done by friction. There are mechanisms with friction such as the flat or trapezoidal belts, strip carrier, speed variators with friction, clutches with friction and brakes. 1.1.2. Mechanisms and movement 1.1.2.1. Movement The position of a mechanism is described by 6 generalized coordinates for each part. Certain coordinates are not independent because they are linked by the equations expressing the mechanical connections. The degree of mobility of the mechanism is equal to the difference between the number of coordinates and the number of connecting equations. The movements of a mechanism are defined by the knowledge of the time evolution of every coordinate of the driving elements and the laws of space linking the position of components to the driving elements. The movement of a mechanism results from all the applied efforts, which are as follows: – Driving efforts others than those deriving from a potential Qm. – Useful efforts making the work of the machine Qu.
Overview of Mechanical Transmission Problems
5
– Efforts of dissipative friction Qf. – Efforts deriving from a potential Qpot. 1.1.2.2. Differential equation for one degree of mobility Let us consider a plane mechanism compound of n components in movement with regard to the frame. We note by q the free coordinate of the unique leading element and by Q the effort applied to it.
Figure 1.2. Parameterization of a mobile part and applied efforts
For any element i (Figure 1.2) of mass mi, and moment of inertia JGi , let us call:
– Gi( xGi , yGi ) the center of mass; – (Gi , xi , yi) the referential linked to part i; – Fi , Ci , moments and efforts applied to part i. Since the parts are rigid, their positions are known by their space laws:
x Gi
x Gi (q )
y Gi
y Gi (q )
Mi
M i (q )
The application of the general theorems of the mechanics of rigid bodies to this mechanism leads to the following equations:
I ( q) q
1 2
I ' ( q) q 2 U ' ( q)
* Qm ( q, q, t ) Q * ( q, q , t )
[1.1]
6
Control Methods for Electrical Machines n
i i
n
U
>
¦ mi x' 2 y ' 2 JM ' 2
I (q )
ª
¦ «mi g y Gi
i i¬
@
º ki fi 2 » 2 ¼
1
generalized inertia
total potential energy
where: mi g y Gi is the lifting work of the mass i and
1 k i f i2 is the deformation 2
* represents the driving efforts which are work of springs attached to part i. Qm reduced to the leading element and Q * represents the resistant efforts which are reduced to the leading element.
We can note by Q
ª
wx i
i i ¬«
wq
n
Q
¦ « Fxi
* Qm Q * the reduced effort of the leading element with
F yi
wy wq
Mi
wM i º » wq ¼»
Equation [1.1] allows either the determination of the movement when the driving effort is known (direct problem) or the determination of the action of the necessary driving effort when the movement is known (inverse problem). The reasoning is applicable to a spatial mechanism. 1.1.3. Engine-machine
This section aims to define the moments or couples applied to the input or output shafts for part of a transmission mechanical power. We thus define an upstream or driving subset and a downstream driven subset. 1.1.3.1. Leading effort This is the necessary effort to apply to the input shaft of the driven subset to ensure its functioning.
Overview of Mechanical Transmission Problems
7
. Figure 1.3. Modeling of a driven subset
The application of relation [1.1] to the input shaft (Figure 1.3) gives:
C
1 d J (M ) : 2 dU (M ) C * J (M ) : e dM dM 2
[1.2]
J (M ) : the moment of inertia for the mechanism reduces to the input shaft; U (M ) : the potential energy reduced to the shaft; C e* : the useful efforts and dissipators of energy reduced to the shaft.
The terms in the right-hand side of [1.2] are: : the acceleration couple of inertia; J (M ) :
1 d J (M )
: 2 : the necessary couple for changing the motion of elements which
2 dM the speed varies with regard to the input shaft;
d U (M ) dt : the leading couple of efforts deriving from a potential; C e* : the leading couple of efforts not deriving from a potential.
8
Control Methods for Electrical Machines
1.1.3.1.1. Calculation of the inertia reduced to the input shaft In the presence of a reducer
Figure 1.4. Sketch of a set with reducer
The inertia J of the mechanism (Figure 1.4) rotating with speed : s reduced to the input shaft of the engine rotating with speed : e is Je: Je
J Kr i 2r
Kr the reducer return ir
:e :s
the rate of the speed reducer
In the presence of mass in translation
Figure 1.5. Sketch of a set with reducer and defect of maneuver
In this sawnut system (Figure 1.5), the rotation of the set turning with speed : s entails the translation of the mass m at speed V.
Overview of Mechanical Transmission Problems
9
The inertia of this rotating set and its moving mass, which is reduced to the engine shaft, is:
Je
J
K r i r2
m
K r K v (i r i v ) 2
with
Șr the reducer return, °Ș the set sawnut return, ° v ° ȍe the speed rate of the reducer, ®i r = ȍs ° ° ȍ °i v = s the speed rate of the sawnut set V ¯
1.1.3.1.2. Characteristic behavior of driving machines This concerns the average couple, in a stationary system, that varies cyclically. Figure 1.6 gives the characteristic behavior of some families of machines.
Figure 1.6. Behavior of resisting couples
Curve 1: Ce const Pe Ce :e ; lifting machines, conveyors. Curve 2: C e
a: e
Pe
a: e2 ; generators.
10
Control Methods for Electrical Machines
Curve 3: C e
b: e2
b: 3e ; ventilators, propellers, fast vehicles.
Pe
Curve 4: Ce for any value for :e; constant synchronous alternators. Curve 5: Ce
a :e
Pe a const; winding machine, lathe.
Curve 6: Ce for any value; mixers, rolling mills, construction site machines. 1.1.3.2. Driving effort This is the torque available on the input shaft of the driving subset.
Figure 1.7. Modeling of an engine set
Application of relation [1.1] in the driving subset shaft (Figure 1.7) gives:
C
We Cm
C
* 1 d J (M ) : 2 dU (M ) J (M ) : Cm 2 dM dM
C ms
define
C ms
1 dJ (M ) 2
dt
:2
* Cm
dU (M ) dM
as
the
[1.3]
static
torque
and
as the permanent torque (mean torque), hence
. C m J (M ) :
1.1.3.2.1. Torque characteristic Figure 1.8 gives the behavior of the mean torque of some engines.
Overview of Mechanical Transmission Problems
11
Figure 1.8. Torque behavior
Curve 1: force of gravity, engine with direct current Curve 2: synchronous engine Curve 3: asynchronous engine Curve 4: engine shunt with strong power Curve 5: engine shunt with average power Curve 6: engine with permanent magnets, turbine Curve 7: engine series: P = const Cm
P :m
Curve 8: heat engine Curve 9: hydraulic engine 1.1.3.3. Coupling of an engine to a machine 1.1.3.3.1. Instantaneous speed We connect a machine with an engine to constitute a group (see Figure 1.9), which gives : : m : e .
Figure 1.9. Modeling a group
12
Control Methods for Electrical Machines
By equating [1.2] and [1.3] we obtain the equation enabling study of the group movement: *
* Cm Ce
1 ( J ' J ' ) : 2 (U ' U ' ) (J m J e ) : m e m e 2
[1.4]
The resolution of [1.4] is achieved by numerical methods. 1.1.3.3.2. Speed regime The variations of the couples and the moments of inertia make the speed fluctuate around a mean value. The last two terms of [1.4] cancel each other out on a cycle, most of the time in one revolution. In terms of mean values, we have: * Cm C e*
(J m J e ) :
Figure 1.10. Functioning point
If moments of inertia are constant and the variations of potential energy are nil, we then have:
Cm Ce
(J m J e ) :
In the permanent regime, we have C m (: 0 ) C e (: 0 ) C 0 , at the point of functioning N, which is the intersection of the characteristic curves (Figure 1.10). 1.1.4. Particular movements
The movement of a machine breaks down globally into three different phases: starting up, permanent motion, slowing down. Let us describe these transitory regimes which generate dynamic loads within machines.
Overview of Mechanical Transmission Problems
13
1.1.4.1. Starting up Starting a machine consists of supplying kinetic energy. Let us follow the starting up stages of a machine through a clutch (Figure 1.11). A clutch is intended to synchronize the speed of a led shaft with that of a driving shaft, and to pass on the training couple.
Figure 1.11. Modeling of a start up command
The driving shaft initially rotates with the speed : m0 , the pulled shaft rotates with the speed : e0 (see Figure 1.12, where: : m0 > : e0 t 0; Ce is the input torque; and CE is the transmitted torque by the clutch. Between A and B, the clutch does not act because CE < Ce; on B, the machine accelerates; on C, the clutch reaches its nominal skating torque : m decreases, but Cm increases according to the characteristic of the engine (asynchronous engine, in the neighborhood of a point of stable functioning); and on D, there is synchronization, : s : m : e , and the peak of CE results from the increase of the friction coefficient which takes its static value.
14
Control Methods for Electrical Machines
Figure 1.12. Starting up process
0 < t < t1
delayed appearance of CE
t1 < t < t2
ineffective clutch
t = t4
peak of couple (at the end of the gliding)
t > t4
shafts are connected to the speed :m
tec
establishment time of clutch couple
ts
period of synchronization
1.1.4.2. Stationary regime In the stationary regime, machines often present speed variations. This means that kinetic energy, potential energy and speed vary around their average values. We then define the factor of irregularity as:
G
: max i : min i :
We can obtain the angular acceleration from the equation of motion of the group [1.4]:
Overview of Mechanical Transmission Problems
:
1 Jm Je
15
1 ' ª * º * ' ' J m J e' : 2 » « Cm Ce U m U e 2 ¬ ¼
Hence, the speed fluctuations originate from: * (piston engine); – variations of the engine couple C m
– variations of the efforts deriving from a potential; and – variations of the reduced moments of inertia Jm and Je. , we define: Among the solutions used to decrease (even to cancel) :
– the static balance; – the dynamic balance; and – the increase of inertia (Jm + Je). 1.1.4.3. Braking Slowing down or braking a machine consists of reducing its kinetic energy, which is being dissipated after transformation into heat.
Figure 1.13. Brake modeling
In Figure 1.13 let J be the reduced moment of inertia, and C the reduced moment of all the efforts applied to the shaft group, JF is the moment of inertia of the brake, and CF the braking couple. Supposing constant moments of inertia, equation of motion [1.4] is: C CF
(J F J ) :
[1.5]
16
Control Methods for Electrical Machines
1.1.4.3.1. Stopped brakes These brakes generally work using friction. To maintain the machine’s stopped state, it is necessary to verify the condition: CF t s C
where s is a security coefficient.
1.1.4.3.2. Slowing down brakes When slowing down the braking couple must be greater than the torque. The deceleration is deduced from relation [1.5] :
CF C J JF
1.1.4.3.3. Permanent regime brakes A permanent regime brake balances the torque and permanently dissipates the power Pf, while leaving the machine in movement. We have in this case: CF
C
and P f
C Z0
These brakes are found: – in test bench power for engines; – regulating the vehicles or machines movement of lifting (brake winch); – regulating the tension or the speed of a thread, a cable or a ribbon (brakeunwinding). These are called Froude brakes, and are the eddy current brakes, or electrical generating brakes, which transform kinetic energy into electrical energy, towards a battery or an electrical network. 1.1.5. Friction – elements of tribology
1.1.5.1. Introduction The description of power transmission, outlined in the previous sections, appeals to elements of mechanical connection whose surfaces are practically always put into contact. The description of these contacts in functioning is the object of the tribology, which can be defined as “the science and the technology of the interactive surfaces in relative motion”.
Overview of Mechanical Transmission Problems
17
This complex discipline involves at the same time rubbing (friction), wearing out (material tearing) and lubrication. The first two phenomena affect the efficiency on the transmissions of power, while the third can be considered as a remedy for this.
N1
Contact area
T2
Figure 1.14. The tribology: a rubbing-wear out-lubrication triptych
The high complexity of the tribology lies in the following facts: i) that it is at the crossroads of such disciplines as mechanics, physics and chemistry; and ii) it involves the whole range of spatial scales, from subatomic and atomic phenomena up to macroscopic phenomena (scale of the mechanics for which the geometry contour of the outline and the boundary conditions are determining), by way of all the intermediate mesoscopic scales which implement either the collective or noncollective behavior of microstructure defects (dislocations, gaps, impurities, etc.) within open systems (reactions with the atmosphere, such as oxidation) and multiphases (composites, multilayers, inclusions, etc.). 1.1.5.2. From friction to wear Rubbing finds its scientific foundations in the works of Leonardo da Vinci in 1499 [VIN 99]. However, the precise formulation of the coefficient of friction dates back to Guillaume Amontons (1699) [AMO 99], although sometimes it is attributed to Charles Augustine de Coulomb in 1781 [COU 21]. Today, we often translate it into the form:
P12
c
c
T2adh T2def T2lab
adh W def W lab W12 12 12
N1
V11
c
[1.6]
18
Control Methods for Electrical Machines
where N1 indicates the pressure effort applied in the normal direction of the contact adh
def
lab
and T2 indicate the pressure effort applied in the normal area, T2 , T2 direction of the contact area, and the contact asperities are respectively the tangential critical effort owed to the adhesion of surfaces, the resistance in the critical shearing due to the elasto-visco-plastic deformation and the critical resistance in the plowing of bourrelets, provoked by the indentation – their sum corresponds to the total tangential critical effort
T2c , which has to be applied to overcome the friction.
Adhesion is an interaction phenomenon at the molecular scale. The elasto-viscoplastic deformation is the appearance of phenomena which are involved at the scale of grains (mesoscopic scale), while the plowing is essentially macroscopic. For a real contact area, we can define the normal constraint of pressure Ar , the normal c , which trigger the constraint of pressure and the critical shearing constraint W12
gliding. It is clear that in order to decrease the friction, any action will concern a decrease of each of the critical shearing components for a given normal load. Figure 1.15 presents some of the typical behaviors of the dry friction in the static regime.
Figure 1.15. Some typical behaviors in dry friction: a) dependence of the friction coefficient on the normal effort [BOW 64]; b) the role of the contact area [OUD 90]; c) some modelings of the friction [OUD 90], [RAO 01]; d) relation between load and gliding [TAB 64]
Overview of Mechanical Transmission Problems
19
During the calculation of structures by finite elements, the contact conditions due to friction are expressed by the absence of tangential movement (that is u2
0 ) as
long as the effort of gliding does not reach the critical friction value:
& T2 &b T c
with
T2c T2adh T2def T2lab 12 & N1 &
To reproduce the dependence of the shearing to the normal effort (Figure 1.15c), we generally use empirical laws to correct Coulomb’s law ( P12 constant).
P12
Bowden
a12 N1
n 1
and
Tabor
[BOW
64]
propose
the
0 is a P12
empirical
law
, justified by Figure 1.15a. Shaw’s law also allows a progressive c
transition from the Coulomb behavior to a Tresca behavior for which T2 is independent of the normal load N1 (by analogy with the notion of plasticity threshold). Shaw’s law for the evolution of the real surface of contact can be written as P12
(*r / *c ) (k / N1 ) , with the parameter k representing the threshold value of
the plastic drainage by shearing, and where the ratio between visible contact surface and the real contact area linked to the roughness of surfaces (this ratio) is updated by a modeling of the compression under N1 . The power relation resulting from experimental observations, schematized in Figure 1.15b, establishes the equivalence with the law of Bowden et al. [BOW 64]. We sometimes schematize this behavior using the Coulomb-Orowan model. Figure 1.15c gives a simplified view of these laws [OUD 90], [RAO 01]. In practice, the values of the friction coefficient cover a wide range according to the conditions governing the nature of the contact. Figure 1.16 illustrates this variety of behaviors.
20
Control Methods for Electrical Machines
Vacuum friction Dry friction Limit Elasto-hydrodynamic lubrication Hydrodynamic lubricaton μ12 10-2
10-3
10-1
1
10
Figure 1.16. A rough estimate of the friction coefficient
The various contributions in the field of friction are still the object of numerous studies; we can usefully refer to the recent report by Savkoor [SAV 01]. The determining role of adhesion was recognized in the 18th century by John Theophilus Desagulier [DES 34], is generally approached at the molecular scale, with the electrostatic potential of pairs interaction Waa , Wbb and Wab (Dupré’s energy of adhesion, which can sometimes be approached by (Waa Wbb)1/ 2 ), and by defining the surface energies J ab from the surface energies of the constitutive phases J aa and J bb , according to:
J ab
J aa J bb Wab
[1.7]
This bond dissociation energy originates from a spring-like force between both surfaces, and is a force which must be overcome to obtain a relative gliding, and representing the contribution of the friction adhesion. From this point of view, the friction force can be compared to a break criterion by shearing the adhesive joint:
GE
2 J ab GAr
Gc GAr
[1.8]
We can thus bring in the stress intensity factor K II in the opening mode of plane fissure and factor K III in the anti-plane mode, mode I being associated with the normal prompting of the contact.
Overview of Mechanical Transmission Problems
Mode I
Mode II
21
Mode III
Figure 1.17. Representation of the various break modes
Irwin’s formula makes it possible to connect the rate of the energy efficiency, G , and the critical value given by 2 J ab , to the stress intensity factors of forces.
This is described by: G
1 ª k 1 K I 2 K II 2 K III 2 º»¼ 2 Pel «¬ 4
[1.9]
in which P el is the shear modulus and k is a characteristic factor of the stress state, linked to the Poisson coefficient Q (having the value (34Q) in plane deformation and (3Q) /(1Q) in plane stress). In the theory of adhesive contact, called the JKR-S theory (Johnson-Kendal-Roberts [JOH 71] and Sperling [SPE 64]), the contact area is linked to the stress intensity factor of balance in mode I and by the surface energy, varying as (K I / J ab)4 /3 . This approach is well known to be satisfactory for friction coefficients greater than 1. On the other hand, for the coefficients lower than 0.5, use of Deryaguin, Muller and Toporov’s (DMT) [DER 75] friction model, which introduces the notion of radius of action for these contact forces, is recommended. In fracture mechanics, it is the difference G Gc which represents the thermodynamic force of non-balance. Onsager’s relation links force and flux dissipation and determines the speed of the crack progression and, thus, here the contact change. We sometimes write: G Gc
kG (T) V n
[1.10]
22
Control Methods for Electrical Machines
The viscoplastic deformation of asperities involves the hardness H , defined as the value of the limit contact pressure ( N1 / Ar ), which activates the complete plastification of the asperities in the neighborhood of the indentor. Studies inspired by the works of Prandtl [PRA 20] approach this hardness using 3 V Y (where V Y is the limit stress for plastic flow). As the hardness is assumed to be constant, any variation of the normal load will thus induce a change in the contact area. Such assumptions are the base of the Bowden and Tabor model [BOW 54], [BOW 64]. To refine the description, we can take into account the heterogenous nature of the stresses in the neighborhood of the indentor, and introduce a mutual “assistance” to plastification, using the normal loads of pressure and the tangential loads of gliding. The notion of Von Mises equivalent stress allows this coupling. 2
Bowden and Tabor write: V11 D W12
2
Vm
2
D Wm
2
[1.11]
The friction coefficient then takes the form: c W12
P12 |
2
(V m D
1 1/ 2 c 2) W12
D1/ 2 (
Wm
[1.12]
2
W12
2
1)
1/ 2
An interesting extension of the theory consists of decomposing the critical shear stress according to experimental observations, giving:
c W12
0 P0 V W12 12 11
[1.13]
0 when normal This relation allows us to find Coulomb’s friction coefficient P12
pressure prevails. The previous description considers a combined action of shear-compression. Besides, for a constant pressure, the shear conditions can induce effects on speed, and more generally on the load history. It is thus not surprising to observe evolutions of the tangential friction force, which are similar to those of the elastoplastic materials. The corresponding dissipation is then given by usual elasto(visco)plastic theories.
Overview of Mechanical Transmission Problems
23
To complete this analysis, it is important to emphasize that this model foresees an evolution of the contact area, proportional to the applied normal load in the absence of gliding. Indeed, relation [1.11] can be rewritten as:
Ar
[(
N1
Vm
2
) D (
T2
Vm
2
) ]
Ar0
1D (
T2
)
2
[1.14]
N1
where Ar0 N1 / m is the contact surface required to trigger the plasticity in pure compression under the load N1 , thus varying with the load. A profilometric approach allows us to understand the origin of the real contact area’s dependence on the normal effort, in the case of friction. This is based on the asperities of contact statistics. The most common formulation is probably that of Greenwood and Williamson [WIL 66], who postulate a Gaussian, even exponential distribution, for the height of asperities. The treatment is done within the framework of a Hertz elastic contact [HER 81], and it is the progressive destruction of the highest asperity that determines the “contact surface/normal stress” relation. The plowing component T2lab often remains negligible, according to Bowden and Tabor [TAB 64], considering both previous contributions. This plowing results from the action of the hard asperities, which penetrate into the soft surface creeping under the load, which they then tear away during the gliding by causing a severe furrow (macroscopic scale). Usually, the models used for the plowing are relatively simple. For a given shape of asperities which penetrates into the least hard surface, we estimate the contact area, then calculate its projection on the normal plane for the gliding Ar2 and on the gliding plane Ar1 Ar ; the normal and tangential effort components are estimated by admitting a contact pressure V that is homogenous on walls ( N1
V Ar1 and T2
V Ar2 ). Thus, the friction coefficient given by the
ratio of these two efforts only depends on the geometry of the contact during the plowing. For plowing with a conical asperity angle, at the vertex E , which is normally pointed to the surface, we arrive at ( 2/S cot E ). According to Bowden and Tabor [BOW 64], the slope of the asperity rarely exceeds 18°, thus the friction coefficient, due only to plowing, remains lower than 0.15. However, this component is usually at the origin of the wear phenomenon. Indeed, we can define the wear as the loss of material on the surface of solids in relative movement. We generally distinguish (i) the wear by adhesion, (ii) the wear
24
Control Methods for Electrical Machines
by abrasion and (iii) the wear by erosion; like (iv) the wear by fatigue or (v) the wear by vibrations (fretting). To find the wear by adhesion, as for abrasion, we often use:
Vu
ku N L H
[1.15]
where Vu is the volume of the wear fragments, ku the wear coefficient, N the normal effort to the surface, H the hardness of the softest materiel and L the glide distance. The wear by adhesion is linked to the physico-chemical nature of materials in contact. The wear fragments are obtained by: plastic deformation of the asperity in contact with penetration of the surface of the film; formation of adhesion joints on cleared oxide layer zones; or by breaking the asperities and formation of the wear fragments. This scenario is common in metals, ceramics and polymers. Wear by abrasion can be seen as the result of a gliding or plowing process, or could be due to plastic deformation; it can be realized in two or three bodies (with an intermediate body between both surfaces). We generally admit that it is necessary to have a difference of hardness of at least 15 to 20% between materials for wear by abrasion to occur. The erosion as well as the abrasion removes material by plastic deformation, by plowing and micro-manufacturing, but here the particle is moved by a fluid, which considerably complicates the quantification of the phenomenon. Wear by fatigue induces the formation and distribution of cracks, under the influence of alternated strain rate. The location of cracks cannot be on-surface because of the stress triaxiality. All the materials undergo this type of wear which recovers from the damage. The “fretting” or wear by vibrations of weak amplitude (microscopic oscillating movement) often occurs in machines. Only static friction has been considered so far, but since Leonard Euler in the 18th century [EUL 58], the difference between static friction and dynamic friction has been recognized. More recently the important experimental work of Rabinowicz [RAB 58] firmly establishes the dependence of the friction coefficient on the compulsory gliding speed. We have already emphasized how the rheology of the contact can be responsible for these observations. The progressive transition of the s to the dynamical friction P d is described by the empirical static friction P12 12 relation:
Overview of Mechanical Transmission Problems
P12 where Vcr
d (P s P d ) exp((V /V )G s ) P12 G cr 12 12
25
[1.16]
ucr is a critical speed (due to Stribeck), which depends on the material,
and G s is a parameter. 1.1.5.3. Lubrication as the remedy for losses by friction Relation [1.6] clearly shows that any decrease in the critical shear stress decreases the friction. To reach this end, we can interpose a suitable interface between both materials, which can be a thin or thick film of a different nature at the level of the contact. This is the object of the lubrication. Indeed, the essence of the friction results from this shear stress limit and depends on a large number of effective factors at all scales, including the required normal load, the temperature, the hardening of the material, the interphases or the present interfaces (oxide layers, pollution), the roughness of surfaces, etc. Lubrication can lead to the use of solid films; it will be, for example, the layers of oxide that guarantee weak values of the friction coefficient. These films usually have thicknesses in the order of 10 to 30 μm. With fluid films, the thickness plays a determining role. For thick films (3 to 10 standard deviations as a characteristic of the roughness of surfaces), the asperities are no longer in contact with each other, and the friction is linked to the rheology of the lubricant (in particular its viscosity K and its thickness e). In practice, the viscosity of the fluid films strongly depends on the temperature and on the pressure ( K K0 exp(E*/ RT) exp(DP) ); thus, these two variables play an important role in lubrication. For example, the pressure will condition the lubrication of the tooth-gears. The same applies for temperature which rises, decreasing the viscosity of oil. When the pressure is important, we obtain a lubrication limit with a lubricant layer of average thickness lower than three standard deviations in roughness, which corresponds to many contacts between the considered surfaces. We see here the advantage of using smooth lubricants, for which the chemistry maintains connections between the lubricant and surfaces by adsorption, even for high pressures. Finally, note that in lubrication limit, the lubricant rheology is not of high enough importance for the friction. It is the ability to react with surfaces and the asperity deformation which are established that make gliding contact the preferred field for viscoplastic models. Between these two extreme lubrications, we find a
26
Control Methods for Electrical Machines
mixed regime for which the thickness of the film is around three standard deviations in roughness. The difficulty to control the fluid lubrication often arises from the misunderstanding of the thickness of the lubricant layer.
Hydrodynamic regime v a2 N 1
fg
Limit regime
Turbulent regime
Mixed friction N1
vb2
Limit regime v a2 N1
N1
Hydrodynamic friction
e
v b2
Figure 1.18. Various regimes of fluid lubrication according to the thickness of the film
These lubrication regimes, which depend on the thickness of the fluid, are responsible for Stribeck’s observations [STR 02], for example on the bearing behavior. Stribeck’s curve is comparable to that of Figure 1.18, the thickness being replaced by the rotation speed of the bearing. Indeed, during the loading, the friction is of dry or boundary layer type. When the speed increases, the lubricant gradually flows between the mechanical pieces, and the friction becomes mixed. It then passes through a minimum and the shear stress of the fluid leads to an increase in the friction. Finally, at high speed, a turbulent regime can settle down when the fluid is viscous, which is expressed by a strong increase in friction. We often classify these types of lubrication with respect to the ratio of the lubricating film thickness and the total roughness, defined from the quadratic average roughness of the two contact surfaces:
O e/ Rq21 Rq22 When O 0.6 , it is a limit friction and surfaces wear out. When 0.6 O 3 , it is a mixed lubrication, which always involves considerable wear of the surfaces. When 3 O 4 , surfaces are in weak contact, and the wear is weak. Finally, when O !4 , it is a fluid lubrication regime without any contact between surfaces, and thus without
Overview of Mechanical Transmission Problems
27
wear: this is the domain of the hydrodynamic or hydrostatic lubrication of surfaces in contact and elastohydrodynamic lubrication of Hertzian contacts. Obviously, the pressure also plays a role in the film thickness of oil during functioning; and the Stribeck curve is sometimes represented by the ratio (speed/pressure) instead of the thickness. For technological assembling, numerous dynamic friction models exist whose objective is to simulate the complex behaviors which represent the memory load. Figure 1.19 gives the evolution of the friction force, according to the displacement during a cyclic load. The observed behavior is similar to the ratchet phenomena in viscoplasticity.
T2
x2
Figure 1.19. Example of gliding force behavior presenting a memory effect of its load history
Among the available models is the Dahl model [DAH 68], written as:
dT2 dt
dT2 du2
u2
e (1
T2 T2C
sgn(u2))
D
u2
[1.17]
where T2C is the Coulomb friction force. In this model T2 only depends on the position and not on the velocity. It thus cannot account for the Stribeck effect. It is also impossible to take into account the adhesion phenomena. Consequently, in spite of its ability to express the hysteresis according to the load, this model’s inadequacies led to the proposal of other empirical approaches. It seems that the introduction of the hysteretic behavior of visco-plasticity should take into account the phenomenon studied by Dahl, for
28
Control Methods for Electrical Machines
example, Ruina [RUI 83] who introduces explicitly internal variables to report the memory of glidings. After elimination of these internal variables, we obtain the following relation: wT2 wt
N1 A wu2 u 2 (T2 T2ss) w t u2 dc
[1.18]
with
T2ss
where
N1 (P0 (A B) ln(
u2 uc
))
[1.19]
dc represents a critical distance of the asperities order of magnitude and
uc and (A B) are sizing parameters.
1.2. Bibliography [AMO 99] AMONTONS G., De la résistance causée dans les machines tant par les frottements des parties qui les composent que par la roideur des cordes qu’on y employe et la manière de calculer l’un et l’autre, Histoire de l’Académie Royale des sciences, report of 19 December 1699, printed Paris in 1732, p. 206 [ARC 53] ARCHARD J.F., “Contact and rubbing of flat surfaces”, J. Appl. Phys., vol. 24, 1953. [BOW 54] BOWDEN E.P and TABOR D., The Friction and Lubrication of Solids, Part I, Clarendon Press, Oxford, 1954. [BOW 64] BOWDEN E.P and TABOR D., The Friction and Lubrication of Solids, Part II, Clarendon Press, Oxford, 1964. [COU 21] (de) COULOMB C.A., Théorie des machines simples en ayant eu égard au frottement de leurs parties et à la roideur des cordages, Prix de l’Académie des Sciences en 1781, Mémoires de Mathématiques et de Physique de l’Académie royale des sciences, t. 10, Paris, 1785, p. 254, new edition, Paris, 1821. [DAH 68] DAHL P., “A solid friction model”, Technical Report TOR-0158 (3107-18), The Aerospace Corporation, El Segundo, CA, 1968. [DER 75] DERYAGUIN D.V., MULLER V.M. and TOPOROV Y., “Effects of contact deformations on the adhesion of particles”, J. Colloid Interface Sci., vol. 53, p. 314, 1975. [DES 51] DESAGULIER T.J., Course of Experimental Philosophy, 1734, translated into English, Paris, 1751.
Overview of Mechanical Transmission Problems
29
[EUL 58] EULER L., Du mouvement de rotation de corps solides autour d’un axe variable, Histoire de l’Académie royale des sciences et belles-lettres, Berlin, 1758. [GRE 66] GREENWOOD J.A. and WILLIAMSON J.B.P., “The contact of nominally flat surfaces”, Proc. Roy. Soc. London, vol. A195, pp. 300-314, 1966. [HER 81] HERTZ H., “Uber die beruhung Fester elasticher Korper”, Journ. Math. (Jour. De Crelle), vol. 92, 1881. [JOH 71] JOHNSON K.L., KENDAL K., ROBERTS A.D., “Surface energy and contact of elastic solids”, Proc. Soc. London, vol. A324, p. 301, 1971. [OUD 90] OUDIN J., LOCHENIES D., RAVALARD Y., RIGAUT J.M., GÉLIN J.C., Pysique et Approches expérimentales et numériques des conditions de contact et de frottement, Mécanique de la Mise en Forme des Matériaux, Presses du CNRS, IRSID, ed. Moussy, Franciosi, Paris, 1990, 406-435. [RAB 58] RABINOWICZ E., Friction and Wear of Materials, 2nd ed., John Wiley, USA, 1958. [RAO 01] RAOUS M., Constitutive Models and Numerical Methods for Frictional Contact, Handbook of Materials Models, vol. 2, pp. 777-786, Academic Press, 2001. [RUI 83] RUINA A.L., “Slip instability and state variable friction laws”, J. Geophys. Res., vol. 88, no. 12, pp. 10359-10370, 1983. [SAV 01] SAVKOOR A.R., Models of Friction, Handbook of Materials Models, Vol. 2, Academic Press, 700-759, 2001. [SPE 64] SPERLING G., Eine Theorie de Haftung von Feststofteilschen an festen Koerpern, Doctoral dissertation, Fakultaet der Machinenwesen, T. H. Karlsruhe, Germany, 1964. [SPI 97] SPINNER G., Conception des machines: Principes et Applications, Vol. 1, Statique, Vol. 2 Dynamique, Presse Polytechniques et Universitaires Romandes, 1997. [STR 02] STRIBECK R., Die Wesentlichen Eigenschaften der Gleit und Rollenlager. Zeitschrift Vereines Deutsche Ingenieure, vol. 46, no. 38, pp. 1341-1348, pp. 1432-1438; vol. 46, no. 39, pp. 1463-1470; 1902. [VIN 99] (da) VINCI L., Il codice atlantico, folio 198, 1499.
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Chapter 2
Reminders of Solid Mechanics
2.1. Reminders of dynamics After providing an outline of the technological aspects of mechanical transmission, we briefly recall the basic elements of rigid body mechanics, in order to obtain the dynamic equations of motion. 2.1.1. Kinematics of a rigid body The rigid body hypothesis for a solid (S) with total mass m requires that the velocity field is a moment field, hence v(Q)
v(P) : PQ
[2.1]
whatever the points P and Q of the rigid body (S), and where : is the instantaneous rotation vector. This vector is such that d dt
PQ
: PQ
[2.2]
We denote the kinematic torsor, or velocity torsor \ v ^ , with reduction elements at point P : and v(P) respectively. The rigid body motion is thus completely determined by the position of one of its points, and by its instantaneous rotation vector. The velocity field of a rigid body thus depends on six parameters, Chapter written by Jean-François SCHMITT and Rachid RAHOUADJ.
32
Control Methods for Electrical Machines
once again noting the fact that a rigid body presents six degrees of freedom in a 3D space. The rigid body motion is, at any time, the composition of a translation at the same time parallel to the instantaneous axis of rotation (the line corresponding to the set points at which the velocity has the same direction as : ) and a rotation around this axis [CHE 96]. 2.1.1.1. Referential, trajectory, position vector A referential (O, i, j, k) , still qualified as an absolute referential, allows us to locate the position of the rigid body at any instant. The location of any point M of the rigid body, at instant t, is achieved, using the best adapted coordinates system for the geometry of the studied rigid body (Cartesian, cylindrical or spherical). The position of M is thus a function of three space variables, (x, y, z), (r, T, z) or (U, T, I) , each of them being a function of time, t. The 4D space (x, y, z, t) is called the R0 referential. The trajectory of the point M in the R0 referential corresponds to successive positions of this point when the time t varies. We call the position vector of M the vector OM . 2.1.1.2. Absolute velocity and acceleration of a point M The absolute velocity of the point M in R0 corresponds to the variation, between two instants of the position vector in R0: v(M / R0)
(
dOM dt
[2.3]
)R
0
This vector is independent of the choice of the origin of the referential R0. The absolute acceleration of M in R0 corresponds to the variation of the absolute velocity vector of the same point M, and is given by:
J (M / R0)
(
d v(M / R0) )R dt
0
[2.4]
Reminders of Solid Mechanics
33
2.1.1.3. Motion composition by change of referential, velocity and composed accelerations The velocity and the acceleration at point M can be also be evaluated in a moving referential R, with associated axes (O' , i', j' , k' ) , linked either to the moving rigid body (S) or to the intermediate referential between the absolute referential R0 and the rigid body (S). The following relation of velocity composition can be established: v(M / R) v(O' / R0) : (R / R0) O' M
v(M / R0)
[2.5]
Trajectory of M in the mobile referential R R 0
Trajectory of M in the fixed referential R 0
V(M/R): relative velocity V(M/Ro): absolute velocity
Figure 2.1. Composition of velocity
The term: v(M / R)
(
dO' M dt
)R
[2.6]
is the relative velocity of M, evaluated in the moving referential R. The second term in [2.5] v(O' / R0) : (R / R0) O' M
[2.7]
corresponds to the drive velocity of M, or the velocity in R0, of the moving referential R, which coincides with point M at the instant t. This is written as: v(M R / R0)
34
Control Methods for Electrical Machines
The vector : (R / R0) is the rotation vector of the moving referential R, with respect to the fixed referential R0. The rigid body motion is thus the superposition of the group motion with the relative motion:
va(M)
ve(M) v r (M)
[2.8]
Finally, we can establish the following relation for the composition of accelerations:
* (M / R0) * (M / R)
* (M / R) * (M R / R0) * Coriolis (M; R / R0) d v(M / R)
[2.9] [2.10]
dt
the last term being the relative acceleration of point M in the referential R * (M R / R0)
* (O' / R0) : (R / R0) : (R / R0) O' M )
d: (R / R0) dt
O' M
[2.11]
The moving acceleration corresponds to the acceleration of any point of the moving referential which coincides to point M at instant t. The Coriolis acceleration: * Coriolis ( M ; R / R0 )
2:( R / R0 ) v( M / R)
[2.12]
corresponds to the coupling between the absolute and the relative motions. We can finally keep in mind the following formula for the composition of accelerations:
* a(M)
* r (M) * e(M) * Coriolis(M)
[2.13]
Reminders of Solid Mechanics
35
2.1.1.4. Euler angles The rotation of a rigid body (S) with regard to a point C is expressed by three Euler angles linked to the rigid body (S), which allow for the change from the referential (C, i, j, k) to the new referential (C, i' , j' , k' ) . These angles are defined as follows: –\
(i, u) , the precession angle, according to the rotation around vector k;
–T
(k, k' ) , the nutation angle, according to the rotation around vector u; and
–M
(u, i' ) , the intrinsic angle, according to the rotation around vector k' .
The rotation vector : characterizing the motion of the referential (C, i' , j' , k' ) according to (C, i, j, k) is:
:
\ k Tu Mk'
[2.14]
The decomposition of : in (i' , j' , k' ) and (i, j, k) bases are: : (# sin sin # cos!)i ' (# sin cos!sin!)j ' (# cos !)k'
[2.15]
: (! sin sin # cos #)i (! sin cos # sin #)j (! cos # )k
[2.16]
The matrix A allowing the change from basis (i, j, k) to basis (i' , j' , k' ) is: cosM sin\ cosT cos\ sinM sinM sinT · § cosM cos\ cosT sin\ sinM A ¨ sinM cos\ cosT sin\ cosM sinM sin\ cosT cos\ cosM cosM sinT ¸ ¨ sinT sin\ cosT ¸¹ sinT cos\ © where A
1
At .
[2.17]
36
Control Methods for Electrical Machines
2.1.2. Kinetic elements for a rigid body – Koenig’s theorems
When a rigid body (S) of mass m is animated by any motion (translation and rotation), its mass not only contributes to its behavior but also to the way it is distributed. The inertia tensor allows us to account for the mass distribution of this rigid body [BEL 88], [BEL 89], [BAM 81]. 2.1.2.1. Mass, gravity center: Koenig referential The mass m of rigid body (S) is given by: m
³ Udv
[2.18]
V
where U is the local mass density of the rigid body at the point P, dv an elementary volume element, and V the three-dimensional domain occupied by the rigid body at any instant t. The center of gravity G, of the rigid body (S), is defined in the following equality:
³ OM Udv
mOG
[2.19]
V
where O is any point not necessarily coinciding with the origin of any referential. We can thus write that point G is such that:
³ GPUdv 0
[2.20]
V
We call Koenig referential RK the referential having a fixed orientation with respect to the absolute referential R0, but with an origin moving in time with G. The corresponding referential is (G, i, j, k) . 2.1.2.2. Inertia tensor of a rigid body 2.1.2.2.1. Moment of inertia of a rigid body with respect to an axis By definition, the moment of inertia of a rigid body (S) with respect to an axis is: J'
2 ³ r Udv V
[2.21]
Reminders of Solid Mechanics
37
with r the distance of a running point M of (S) to the axis ' . This quantity (either positive or zero) characterizes the distribution of the material around the axis ' . By introducing the referential (O, i, j , k ) , with O belonging to ' , we can establish the following relation: J'
D 2 J x E 2 J y J 2 J z 2DEJ xy 2EJJ yz 2DJJ xz
[2.22]
– D, E, J are components of n , the unit vector of ' in the base (i, j, k) ; – Jx, Jy, Jz are moments of inertia of a rigid body (S) with respect to the three axes Ox, Oy, Oz (for example: J x ³ (y 2 z 2)Udv ); V
– J xy
³ xyUdv , J yz V
³ yzUdv , J xz V
³ xzUdv
[2.23]
V
are all products of inertia for the rigid body, with regard to the three axes. The sign of the products of inertia can be positive or negative. They allow for the characterization of the asymmetric distribution of the mass around the axes for a rigid body referential. 2.1.2.2.2. Inertia tensor The inertia tensor J O(S) of a rigid body (S) with respect to the point O is symmetric and its representing matrix in the (i, j, k) basis is given by:
§ Jx ¨ ¨ J xy ¨J © xz
J xy Jy J yz
J xz · ¸ J yz ¸ J z ¸¹
We thus have: J'
n.J O ( S ).n
[2.24]
38
Control Methods for Electrical Machines
There exists three particular orthogonal axes OX, OY, OZ (with unit vectors I, J , K respectively), situated such that the representing matrices of J O(S) in (I, J , K ) are diagonal: the products of inertia are zero and the moments of inertia are called principal moments of inertia. These three axes are the principal axes of inertia. In many cases, they are found by symmetry studies of the rigid body, as is shown by the two following examples:
– If OXY is a symmetry plane for the rigid body (S), the products of inertia are Jxz and Jyz at 0, and the gravity center G belongs to this plane along with two of its principal axes. – If the axis OZ is an axis of revolution (case of a vehicle wheel for instance), any plane orthogonal to this axis is a principal plane of inertia, and any axis of the same plane is a principal axis. The two principal moments of inertia corresponding to the two principal axes included in the plane are equal. 2.1.2.2.3. Huygens’ theorem This theorem allows us to deduce the inertia tensor J O(S) from the knowledge of J (S ) , using the following relationship: G
J (S ) O
m(OG 2 1 OG
OG ) J ( S ) G
[2.25]
with the symbol
denoting the tensorial product. 2.1.3. Newtonian dynamics
2.1.3.1. Resultant and kinetic momentums: kinetic torsor of a rigid body – Koenig’s first theorem The moment or kinetic resultant of (S) is: P
³ UV a ( M )dv V
mV a (G )
[2.26]
Reminders of Solid Mechanics
39
P allows us to account for the total translation momentum of the rigid body in the absolute referential R0.
The kinetic momentum of the rigid body (S) with respect to O is: - V a (0)
³ OM UV a (M )dv
[2.27]
V
V a(0) accounts for the rotation momentum of the rigid body with respect to the point O. V a (0) is a momentum field with resultant P , thus: V a ( A)
V a (0) P OA , points O and A
[2.28]
The kinetic torsor:
^Wkinetic (O)`
P ½ ® ¾ ¯ Va (0) ¿ O
[2.29]
describes the momentum of the rigid body. The elements of reduction for this torsor at point O are P and a (0). The kinetic momentum about O of the rigid body (S) is expressed according to its inertia tensor J O(S) as follows:
V a (O )
OG mV O J O ( S ).:
[2.30]
where : is the instantaneous rotation vector of the rigid body. The kinetic momentum of the gravity center G for (S) can be obtained by the following relation, which defines the intrinsic momentum:
V a (G )
J G ( S ).:
[2.31]
Thus, the first Koenig theorem is expressed as:
V a (O )
OG mV G J G ( S ).:
[2.32]
40
Control Methods for Electrical Machines
The kinetic momentum of a rigid body about a point O is equal to the sum of the kinetic momentum of the center of mass associated with the total mass of the rigid body and the kinetic momentum relative to the center of mass. 2.1.3.2. Kinetic energy: Koenig’s second theorem The kinetic energy of a rigid body is given by: T
1
2 ³ V Udv .
[2.33]
2V
In the particular case of a rigid body rotating about a fixed axis 'we have: :
:.n
( n is the unit vector of ')
T
1 2
I ':2
[2.34]
where I ' is the rigid body moment of inertia about ' . In the case of a rigid body rotation about a fixed point O, we have: T
1 2
:.J (O).:
1 2
:.V (O)
[2.35]
and we can show that: T
1 2
mVG2 TK
[2.36]
where TK is the kinetic energy relative to the center of mass, namely the kinetic energy in the Koenig referential RK: TK
1 2
:.J (G ).:
[2.37]
The kinetic energy of a rigid body is equivalent to the sum of the kinetic energy of a mass point located at the center of mass (and having the mass of the body) and the kinetic energy relative to the center of mass: this constitutes Koenig’s second theorem.
Reminders of Solid Mechanics
41
2.1.3.3. Theorem of the resultant kinetic momentum The theorem of the resultant kinetic momentum states that: R ext
m(
dP dt
[2.38]
) R0
where R ext represents the result of the external forces applied (at distance or by contact and bonds) to the rigid body (S): and where P is the momentum or the kinetic resultant of (S), as previously defined. This vector-like equation is not sufficient to determine the rigid solid motion with 6 degrees of freedom: it only allows us to determine the motion of the center of mass G. The kinetic momentum is expressed by the following theorem: * ext (O)
(
d V a (O) dt
) RO
[2.39]
where *ext(O) represents the resultant moment about the fixed point O of all applied forces on S, including those giving rise to moments, the results of which sum up to zero. V a (O) is the kinetic moment of the solid (S) with respect to O; Rext and *ext(O) are respectively the resultant, and resulting moment of the torsor in O of the external forces, noted: \ external _ forces (O)^ . Thus, both previous fundamental theorems can be expressed in a more condensed way: the torsor at point O of the forces applied to the rigid body (S) is the derivative of the kinetic torsor in R0 with respect to time: d \ kinetic (O)^ \ external _ forces (O)^ dt
[2.40]
The kinetic moment theorem can be written in the Koenig referential: * ext (G )
(
d V R (G ) K
dt
) RK
allowing us to use the kinetic moment relative to the center of mass.
[2.41]
42
Control Methods for Electrical Machines
2.2. Application example: dynamic balance of a rigid rotor
Here we present the theoretical study of the dynamic balance of a rigid rotor. Such a structure is modeled by a rigid body (S), of mass m, with center of mass G in a rotation about an axis Ox0 . The latter is connected to a fixed frame (B) through a perfect pivot connection (see Figure 2.2). We designate by R0 the fixed referential (O, x 0 , y , z 0 ) tied to the frame, and by 0
R the referential (H, x0, y, z) tied to the
rigid body, where H is the orthogonal projection of G on the rotation axis. We denote by a, the distance from G to the rotation axis, viz HG a y .
z0
z0 y
z G
(S) O
T
G x0
y0
a
H
B
H x, x0
Trace of the vertical plane including G
Figure 2.2. Sketch of the rigid rotor
T
y0
Reminders of Solid Mechanics
43
The inertia matrix tensor of the rigid body (S) in the referential R is the § A F E · following matrix: ¨¨ F B D ¸¸ . In the permanent regime, the applied external © E D C ¹ forces on the rigid body (S) are: 0 ½ – the engine action, represented by the torsor ® ¾ ; ¯W x 0 ¿ H R½ – the link with the frame, represented by the torsor ® ¾ , with M .x 0 O ; ¯M ¿ H since the connection is assumed to be perfect it can generate any moment according to the rotation axis.
The relative weight of the rigid body is characterized by the torsor: mg z 0 ½ ® ¾ ¯ mga cos T x 0 ¿ H The momentum equations are obtained from the resultant theorem and kinetic moment:
d ^Wkinetic (H)` dt
^W
external _ forces
`
(H)
maT z . The kinetic moment of the rigid body (S) is obtained from the equality: V ( H ) J ( H ).: , with : T x 0 the
The total momentum is: P
mV (G )
instantaneous rotation of (S). Thus:
V (H )
§ A ¨ ¨ F ¨ E ©
F E ·§T · ¸¨ ¸ B D ¸¨ 0 ¸ ¨ ¸ D C ¸¹¨© 0 ¸¹
AT x 0 FT y ET z .
and we deduce the kinetic and dynamic torsors:
^Wkinetic (H)`
maT z ° °½ ® ¾ °¯ ATx 0 FT y ETz °¿ H
44
Control Methods for Electrical Machines
° ½° Tz T 2 y) ma( ® 2 2 ¾ °¯ AT x 0 (ET FT)y (FT ET)z °¿ H
d ^Wkinetic (H)` dt
The total momentum and kinetic moment provide the problem equations: ma (Tz T 2 y )
R mg z 0
ATx0 ( ET 2 FT) y ( FT 2 ET) z
W x0 M mga cos T x0 .
The projection of the second equation according to the x0 axis gives the equation for the motion of (S):
AT W mga cosT . The torsor expressing the forces applied by a moving rigid body on the support is
^
`
^
`
denoted W(S)o support (B) . We easily access the torsor Wsup port (B)o (S) , which being R ½ accounts for the connection action on the rigid body. equal to the torsor ® M ¾ ¯ ¿H Considering the equation of motion of (S), we can establish the identity:
^W
support (B) o (S)
`
° ½° ma (Tz T 2 y ) mg z 0 ½ ® ¾ ® 2 ¾ 2 ¯ mga cos T x0 ¿ H ¯°( ET FT ) y ( FT ET ) z ¿°H
The latter torsor is broken down as follows:
^W
support (B) o (S)
` ^W ` ^W ` C
t
with ^W C ` a constant torsor in R0, and ^W t ` a torsor which is a function of time in R0. In order to obtain the dynamic balance of the rotor, we look for the condition:
^W t ` ^0` . Thus, when the rotor rotates with a constant speed : , the paddles (here
modeled by a perfect pivot link) do not support the forces, due to the weight relative to the center of mass of the rotor and to the inertia effect. This happens when the rigid body (S) characteristics are such that: a = 0 and E = F = 0. The condition a = 0 corresponds to the static balance, when the center of mass G belongs to the rotation axis. At rest, the static balance is of no interest. The condition E = F = 0 corresponds to the dynamic balance condition. If the rotor does not fulfill the previous
Reminders of Solid Mechanics
45
conditions, it must achieve the balance by modifying the distribution of mass. In practice, we add punctual masses at mi in particular points at Mi so that the obtained rigid body (S’), verifies a’ = E’ = F’ = 0. In the permanent regime, : const. We shall now analyze the incidence of a supplementary punctual mass mi, located at point Mi of solid (S) and positioned in R by ai ,M i , xi : HM i xi x 0 ai u i . z
Figure 2.3. Punctual balancing mass mi
The torsor of external forces due to Mi is then: mi g z 0 ° ½° ®m gx y m ga cos(T M ) x ¾ . i i i 0° °¯ i i 0 ¿H
Thus, the total momentum and kinetic energy theorems give: ma: 2 y mi ai : 2 u R mg z 0 mi g z 0 i °° 2 2 2 ®E: y F: z mi xi ai : v i M mi gxi y 0 ° W mga cos T mi gai cos(T M i ) °¯
The third equation corresponds to the projection onto x0 . The torsor expresses the action of the rigid body (S), modified by the addition of n punctual masses (mi, xi, ai, Mi: value of the ith mass and corresponding position) as:
^W
frame ( B ) o ( S )
`
2 2 ½° °(mg ¦ mi g ) z 0 °½ ° ma: y ¦ mi ai : u i ® ¾ ® ¾ 2 2 2 °¯ ¦ mi gxi y 0 °¿ H °¯ E : y F : z ¦ mi xi ai : vi ¿° H
46
Control Methods for Electrical Machines
^
which can be reduced to: Wframe(B)o(S)
` ^W ` ^ W ` . ' C
' t
The number and position of the masses must be such that:
^W ` ^0`. ' t
Generally, a dynamic balance is obtained with two corrective masses m1 and m2 located at different positions. 2.3. Analytical dynamics (Euler-Lagrange) Let us consider a non-conservative system, for which kinematic relations are holonomic1 [GER 96]. The Euler-Lagrange equations governing motion have the following form:
d wT wT wV wD Q nc s 0 ) ( dt wq s wq s wq s wq s
[2.42]
s 1,..., n
The variables qs denote the generalized coordinate of the system. The terms T and D represent the kinetic energy as a function of the dissipation, which will be discussed further. The generalized non-conservative external forces are described by the term Q nc s . The potential energy V is given by: V
Vext Vint
The external conservative forces are derived from the potential Vext : Qext s
sVext sqs
the virtual work of which is nil during a closed cycle: ³ Q ext s Gq s
0 .
The potential is linked to the internal forces, with Vint , corresponding to the virtual work: N
GWint
wVint
¦ wu k 1
k
n
.Gu k
¦Q s 1
int s
Gq s
GVint where
Q int s
wVint wq s
1 Links defined by explicit relations of the form f (x;t) = 0, with x denoting the 3N components of the position vector expressed in the actual configuration. Scleronomic links are represented by time-independent relations f(x) = 0.
Reminders of Solid Mechanics
47
where u k indicates the motion field of each of the N material points with index k, and n the number of degrees of freedom (dof) of the system are described by the generalized coordinates q1 , q 2, ..., q n , t . The dissipation is considered by the Rayleigh dissipation function D, defined as the general form:
D
N
v(k )
(k ) (k ) ¦ ³ C f ( v) dv
k 1 0
where v ( k ) indicates the speed of the material point k, and f ( k ) ( v) a function of speed which includes the dissipative phenomenon; C ( k ) is also a phenomenological constant. The real dissipation cases can be empirically described by the expression of D, which can be homogenous of order m=1, 2 or 3: mD
wD
¦ q s s wq s
¦ Q s q s s
depending on whether we are faced with a dry friction, a viscous friction or with aerodynamic drag phenomena due to turbulence [GER 96]. We may notice that the field vector of motion is written as u (k) ( X ( k ) , t ) or U (k) (q 1 , q 2 , ...q n , t ) ; depending on whether we wish to use the physical coordinates or the generalized coordinates, we can write the velocity field as the material derivative of U (k) (q1 , q 2 ,...q n , t) : u (k)
n wU (k) wU (k) q s ¦ wt s 1 wq s
taking into account the fact that qs is a function of time. From the last relation, we can show that the kinetic energy 1 N ( k ) (k) (k) T ¦ m u .u , expressed with the generalized coordinates, splits into three 2k 1 homogenous contributions with the order of 0, 1 and 2 successively, giving: T
T0 (q, t ) T1 (q , q, t ) T2 (q , t )
[2.43]
48
Control Methods for Electrical Machines
and therein denoting the kinetic energy of motion T0, the mutual-kinetic energy T1 and the relative kinetic energy T2 respectively: T0
1 N ( k ) wU ( k ) 2 ) ¦m ( wt 2k 1
T1
¦ ¦
n
s 1 k
T2
wU (k) ( k ) wU (k) m . q s wqs 1 wt
N
1 n n N ( k ) wU (k) wU (k) . q s q r ¦ ¦ ¦m 2s 1 r 1 k 1 wq s wq r
The term T0 only remains when the speeds do not explicitly depend on generalized coordinates qs . Euler-Lagrange equation [2.42] takes a different form, explicitly bringing in the kinetic energies T1 and T2 : wT d wT2 ( ) 2 dt wq s wq s
Q nc s ( t )
wV * wD w wT Fs ( 1 ) wq s wq s wt wq s
[2.44]
where V* V T0 indicates the potential energy modified by the kinetic energy of motion ( T0 ), and Fs the generalized gyroscopic forces, which can be expressed as: n
Fs
¦ q r G rs r 1
with terms G rs indicating the different components of the gyroscopic coupled matrix [GER 96].
Reminders of Solid Mechanics
49
2.4. Linear energies in the neighborhood of the balance for a non-damped discrete system 2.4.1. Equilibrium configuration without group motion Consider the oscillation around the equilibrium of q a discrete non-damped system ( Q s { 0 , D { 0 ). The equilibrium configuration is defined by: q s (t )
q s (t
q s ( t )
0
0)
sV * s(V T0 ) 0, s=1,…,n sqs sqs
The generalized coordinates q s denote in this case variation quantities relative to equilibrium. The Taylor series expansion of the potential energy in the vicinity of equilibrium delivers the following relation: wV 1 n n w 2V ) q 0 qs ¦ ¦ ( ) q 0 q s q r O(q 3 ) q w 2 w q w q s r s s 1 s 1 r 1 n
V(q)
V(0) ¦ (
In the absence of group motion, the kinetic energy reduces to the only homogenous second degree term T2 (q ) [2.43], the equilibrium reducing to: wV wq s
[2.45]
0
adopting the value V(0) 0 as a matter of convention. The second order approximation of the potential energy is given by: V(q)
1 n ¦ K sr q s q r 2r 1
1 t q.K.q ! 0 2
[2.46]
50
Control Methods for Electrical Machines
w 2V ) q 0 represents the symmetric components of the wq s wq r rigidity matrix K. Still keeping the assumption of absence of motion, the development of the kinetic energy T gives the following approximation:
in which K sr
(
1 n n ¦ ¦ M sr q s q r 2s 1 r 1
T (q )
with M sr
K rs
1 t q .M.q ! 0 2
[2.47]
w 2T
) q 0 representing the symmetric components of the wq s wq r positive-definite mass matrix. The introduction of expressions [2.45] and [2.47] in Euler-Lagrange equation [2.45] renders it able to characterize the vibratory motion of the discrete system around its stable equilibrium position, thus obtaining: M rs
(
n
¦ (M sr qr K sr q r ) 0
K . q s 1,..., n or M . q
0
[2.48]
r 1
Recall that the assumptions underlying this last equation are: (i) the existence of an equilibrium configuration, and (ii) the consideration of small perturbations in the vicinity of this configuration.
2.4.2. Equilibrium configuration with group motion When the system undergoes a group motion, the equilibrium configuration results from the opposition of return forces and centrifugal forces (wT0 wq s ) for a rotation motion. The generalized velocities q (variations relative to equilibrium) are zero. Linearization of the potential and kinetic energies leads to the following approximations: – the effective potential energy V * V * (q)
V T0 takes the form
1 n n * ¦ ¦ K sr q s q r 2s 1 r 1
with K sr the coefficients of the effective rigidity matrix, such that K *sr
K sr (
w2K0 )0 wq s wq r
[2.49]
Reminders of Solid Mechanics
51
– the expression of the relative kinetic energy does not change: 1 n n ¦ ¦ M sr q s q r 2 s 1r 1
T2 (q )
[2.50]
– the mutual kinetic energy is linear in the velocities q and is linearized in the generalized coordinates q , giving: n
s 1
with
n
n
¦ C s q s ¦ ¦ Fsr q s q r
T1
Cs
(
[2.51]
s 1 r 1
wT1 )0 wq s
Fsr
(
w 2 T1 )0 wq s wq r
The equations of free motion result from Euler-Lagrange equation [2.45], in which approximations [2.49]-[2.51] are introduced, obtaining: sT d sT2 sV * s sT ( ) 2 Fs ( 1 ) 0, s=1,…,n dt sqs sqs sqs st sqs
with the complementary inertia forces ª w 2 T1 w 2 T1 º q « » ¦ r r wq r wq s » r 1 ¬« wq s wq ¼ n
Fs
n
¦ q r G rs or M.q G.q K * .q 0 [2.52] r 1
The gyroscopic coupling matrix G is antisymmetric ( G sr
G rs ).
2.5. Vibratory behavior of a discrete non-damped system around an equilibrium configuration We consider the case of a discrete or discretized system undergoing free oscillations around an equilibrium configuration, in the absence of group motion (see equation [2.47]). The eigenmodes of vibration, x (i) , and the associated eigenpulsations, Z (i ) , are the solutions of the two following equations: det(K Z (2i) M ) (K Z (2i) M ).x (i)
[2.53]
0
0
[2.54]
52
Control Methods for Electrical Machines
The eigenmodes are orthogonal and thus form a basis t
x (i) .M.x (j)
G ij
[2.55a]
t
x (i) .K.x (j)
Z (2i) G ij
[2.55b]
with the eigenpulsations ordered in the following manner: 0 d Z1 2 d Z 2 2 d ... d Z n 2 .
A problem in the dynamics of structures boils down to the resolution of equation [2.48] following different methods, the choice of which depends upon the number of dofs, of the frequency spectrum being explored, and the form of the mass and rigidity matrices. The simplest method applicable when the number of dofs is less than 10 consists of an analytical or numerical development of characteristic equation [2.53], which further exploits the Rayleigh quotient [GER 96]. When the number of dofs is between 10 and 250, we use the standard numerical methods: Jacobi diagonalization, Householder tridiagonalization and iterative power. In the case of complex systems, with a more important number of dofs, we frequently use the finite element method, whereby the structure is divided into simple elements. This method incorporates resolution methods involving the eigenvalues, some of which are the method of the iterate power, the subspaces method or the Lanczos method, to mention only the most widely used. Note that in certain cases, the solution of equation [2.48] may be disturbed by the presence of rigid modes, modes for which the potential energy of elastic deformation vanishes: V (q)
1 t q.K.q 2
0
The most well known technique used to remedy this problem involves the imposition of a spectral shift, and the development of the calculation thanks to the shifted rigidity matrix [GER 96]. As an illustration of this technique, we propose to treat the case of the milling machine table in order to simplify the problem, and only four degrees of freedom are kept.
Reminders of Solid Mechanics
53
2.6. Analytical study of the vibratory behavior of a milling machine table 2.6.1. Positioning the problem Here we shall address the case of a milling machine table inspired from the preliminary study presented by Del Pedro and Pahud [DEL 89]. The milling machine table with mass m t is driven by a direct current engine having inertia J m , thanks to a notched belt and ball screws with pitch t. The inertia of each of the two pulleys, assumed to be identical, supporting the belt is denoted by J p , and the inertia of the screw having mass m v is denoted by J v . We shall call k c the rigidity of the belt under traction, and k b and k e the rigidities of the thrust and screw respectively. The screw is submitted to traction-compression efforts we will assume to be decoupled. k vc and k vt are the corresponding rigidities. 2.6.2. Setting up the model Located in Figure 2.4 are the generalized coordinates q s : this follows on from section 2.3 and corresponds to the variable x, the linear displacement of the table, M v , the rotation of the screw, M p the rotation of the pulley at screw tip and M m , the rotation of the driving pulley.
stop Milling table
screw belt
nut
engine
Figure 2.4. Simplified sketch of a milling machine table
54
Control Methods for Electrical Machines
Furthermore, by defining the active length L of the screw, the ratio of the transmission a t 2S and the pulley radius R p , we can evaluate the kinetic and potential energies of the system and then establish the Lagrange differential equations. 2.6.2.1. Description of the kinetic energy components – Table kinetic energy: – Screw kinetic energy:
Tt
1 m t x 2 2 Tv Tvt Tvr
These two components of kinetic energy are translation kinetic energy:
Tvt
2 1 m v L§ u 1 mv · 2 ³ ¨ x aM v ¸ du = 2 3 x aM v 2 L 0© L ¹
and rotation kinetic energy: Tvr
1 Jv 2 L
L
2
¬ u 1 Jv 2 2 ¨ !p L !v !p ® du = 2 3 M p M v M p M r 0
We also have – kinetic energy of the pulley at the screw tip: Tp
1 J p M 2p ; 2
– kinetic energy of the engine and of the driving pulley: Tm
1 J m J p M 2m . 2
Assuming that the kinetic energy of the belt is negligible, we can write the total kinetic energy in the following form:
T
m J 1§ · ¨ m t x 2 v x aM v 2 v M 2p M 2v M p M v J p M 2p J m J p M 2m ¸ 2© 3 3 ¹
[2.56]
Reminders of Solid Mechanics
55
2.6.2.2. Description of the various kinetic energy components – Traction-compression potential energy of the screw accounting for the screw and thrust rigidities: Vvc
1 k x aM v 2 2
– Torsion potential energy in the screw: Vvt
1 k vt M v M p 2 2
– Traction potential energy in the belt: Vc
1 k c R 2p M p M m 2 2
Hence, the total potential energy is the sum of these three contributions:
V
1 k x aM v 2 k vt M v M p 2 k c R 2p M p M m 2 2
[2.57]
2.6.2.3. Euler-Lagrange system equations The Euler-Lagrange equations giving the dynamic description of the system are given by the expression:
d § wL ¨ dt ¨© wq s
· wL ¸ ¸ wq s ¹
0
with L = T – V designating the Lagrangian of the system and q s its generalized coordinates. The global kinetic and potential energies of the system are represented by the previously calculated quantities T and V respectively. The various EulerLagrange equations corresponding to the generalized coordinates are successively: – The Euler-Lagrange equation relative to x mv § ¨¨ m t 3 ©
mv · v kx akM v ¸¸x a 3 M ¹
0
[2.58a]
56
Control Methods for Electrical Machines
– The Euler-Lagrange equation relative to Mv a
§ a 2m v J v · mv J ¸M v v M p akx a 2 k k vt M v k vt M p x ¨ ¸ ¨ 3 3 3 6 ¹ ©
0
[2.58b]
– The Euler-Lagrange equation relative to Mp
Jv §J · v ¨ v J p ¸M p k vt M v k vt k c R 2p M p k c R 2p M m M 6 © 3 ¹
0
[2.58c]
– Euler-Lagrange equation relative to Mm
J m J p M m k c R 2p M p k c R 2p M m
[2.58d]
0
2.6.2.4. Matrix form of the motion equations The four previously obtained Euler-Lagrange equations [2.58a-d] may be written in matrix form (according to the general set of equations [2.48]): K . q 0 M .q
where M and K are the mass matrix and rigidity matrix of the system respectively. These matrices characterize the system as a whole, and are of the following form:
K
M
ak ª k « ak a 2 k k vt « « 0 k vt « 0 «¬ 0
0 k vt k vt k c R 2p
mv ª «m t 3 « m « a v « 3 « 0 « « 0 ¬
mv 3
a a2
0 º 0 »» k c R 2p » » k c R 2p »¼
k c R 2p
mv Jv 3 3 Jv 6 0
0 Jv 6 Jv Jp 3 0
º » » 0 » » » 0 » J p J m »¼ 0
q
ª x º «M » « v» « Mp » « » ¬M m ¼
Reminders of Solid Mechanics
57
2.6.2.5. Numerical values of the various parameters The mentioned numerical values can be found in [DEL 89]. Characteristics of the ball screw are: mv
Jv
L -3
4 4.34.10 Kg Kgm2
T
d -3
0.7 5.10 m m
2.9.10 m
E -2
G 11
2.11.10 N/m2
0.8.1011 N/m2
From this we can then calculate the values of parameters a, k vc and k vt , a = t/2 = 7.96.104 m k vc
E d 2 1.98.108 m L 4
k vt
G d 4 7.94.103 Nm/rad L 32
thrust rigidity: kb 2.8.109 N/m screw rigidity: ke 9.2.108 N/m equivalent rigidity:
k 1.54.108 N/m
belt rigidity: kc 106 N/m pulley characteristics: Rp 4.3.102 m, 3 2 kc Rp2 1.85.103 Nm/rad, J p 1.45.10 Kgm
mass of the table: m t
170 Kg
engine inertia: J m 1.4.103 Kgm 2 The mass and rigidity matrices have the following numerical matrices respectively: 171.3 0 1.061.103 ¡ ¡1.061.103 1.455.104 7.233.105 M ¡¡ 0 7.233.105 1.595.103 ¡ ¡ 0 0 0 ¢¡
¯ 0 ° ° 0 ° ° 0 ° ° 2.85.103 ±°
58
Control Methods for Electrical Machines
K
ª 1.541.10 8 « 5 « 1.226.10 « 0 « 0 «¬
1.226.10 5 8.033.10 3
0 7.936.10 3
7.936.10 3 0
9.785.10 3 1.849.10 3
º » » 3» 1.849.10 » 1.849.10 3 »¼ 0 0
2.6.3. Direct resolution of the eigenvalue problem As the system only has four degrees of freedom, the posited problem is rather simple, as far as the evaluation of the eigenvalues can be made by directly solving the characteristic equation det(K Z (2i) M )
0 . By taking into account the matrices
M and K, we obtain the following eigenvalues: Z1
0 rad / s , Z 2
934 rad / s , Z3
1302 rad / s and Z 4
8173 rad / s .
The first mode is thus a stiff mode (with zero pulsation $1 ). The eigenvectors (noted x (j) and obeying relation [2.53]) associated with these pulsations are respectively: 5.423.104 ¯ ¡ ° ¡ 0.6815 ° ¡ ° x1 ¡ ° 0.6815 ¡ ° ¡ ° ¡¢ 0.6815 °±
9.975.104 ¯ ¡ ° ¡3.731.102 ° ¡ °, x2 ¡ 2 ° 2.170.10 ¡ ° ¡ 2 ° ¡¢ 6.288.10 °±
6.778.104 ¯ ¡ ° ¡ 0.7656 ° ° x3 ¡¡ ° ¡ 0.7499 ° ¡ ° ¢¡ 0.4648 ±°
4.565.105 ¯ ¡ ° ¡ 0.9963 ° °. x 4 ¡¡ ° ¡ 0.1315 ° ¡ 3 ° ¢¡ 1.290.10 ±°
These vectors can be normalized according to the mass matrix M, which means satisfying relation [2.54a-b]: 1.144.102 ¯ ¡ ° ¡ 0.6815 ° ¡ ° x1n ¡ ° 0.6815 ¡ ° ¡ ° ¢¡ 0.6815 ±°
x2n
7.379.102 ¯ ¡ ° ¡ 2.759 ° ¡ °, ¡ ° 1.605 ¡ ° ¡ ° ¢¡ 4.652 ±°
Reminders of Solid Mechanics
x 3n
1.615.102 ¯ ¡ ° ¡ 18.25 ° ¡ ° ¡ ° 17.87 ¡ ° ¡ ° ¡¢ 11.08 °±
x4n
59
3.69.104 ¯ ¡ ° ¡ 80.52 ° ¡ ° ¡ ° 10.63 ¡ ° ¡ 1 ° ¡¢ 1.043.10 °±
The indices “n” indicate the normalized vectors. NOTE: the first mode represents a stiff mode corresponding to a simple motion (or gliding) of the table, without any variation in the energy of deformation.
2.6.4. Cancellation of the stiff mode and reduction of the problem When the gliding eigenvector is well known, it is always possible to cancel one of the generalized coordinates of the system, in order to obtain a new system with the number of dofs reduced to just one. In this case, the vector x 0
>a
becomes t x0
1 1 1@ , and the corresponding pulsation is zero.
Using the orthogonality condition of the modes also gives: t
x 0 .M .x r
0.
Still without the knowledge of the other eigenvectors x r , we write: t
xr
>x
Mv
Mp
Mm
@ and the orthogonality of the modes above leads to the
equation: a mtx
Jv J · § M v ¨ J p v ¸M p J p J m M m 2 2 ¹ ©
0
This relation allows us to obtain one of the initial coordinates according to the others. Hence, to discover M v , we have to use the following orthogonality relation: Mv
§ 2Jp 2a mt x ¨¨1 Jv Jv ©
2 Jp Jm · ¸M p Mm ¸ Jv ¹
60
Control Methods for Electrical Machines
We can then reduce the system order as follows: ª x º «M » « v» « Mp » « » ¬M m ¼
ª x º « » C « M p » , with C «¬M m »¼
ª x º ª x º M . C «« M p »» K . C «« M p »» «¬M m »¼ «¬M m »¼
ª 1 « 2a mt « Jv « « 0 « ¬ 0
0 § 2Jp ¨¨1 Jv © 1 0
0 2 Jp Jm · ¸ ¸ Jv ¹ 0 1
º
» » » » » ¼
0
Only three degrees of freedom remain, namely x , M p , M m . Accordingly, multiplying the previous equation by t C gives:
t
ª x º ª x º » « t C . M . C . « M p » C . K . C . «« M p »» «¬M m »¼ «¬M m »¼
0
We can then write M ' t C . M . C and K ' two square matrices of third order.
t
C K C . Note that M ' and K ' are
Solving the characteristic equation det K 'Z 2 M ' provides:
0 of the reduced problem
$1 932 rad/s, , $2 1,302 rad/s and $3 8,173 rad/s
for the three modes. It is now easy to find the associated eigenvectors: 0.348.10-3 ¯ 0.899.10-2 ¯ 0.151.10-2 ¯ ¡ ° ¡ ° ¡ ° x1 ¡¡ 0.3166 °° , x 2 ¡¡ -1.650 °° and x3 ¡¡ 9.963 °° . ¡ -0.4334 ° ¡ 1.057 ° ¡-0.928.10-1 ° °± °± ¢¡ ±° ¢¡ ¢¡
Reminders of Solid Mechanics
61
Normalizing these vectors according to the mass matrix M ' , we obtain: 0.074443 ¯ 0.016158 ¯ 0.000371 ¯ ¡ ° ¡ ° ¡ ° ¡ ° ¡ ° x1n 2.619632 , x 2 n -17.648436 x3n ¡10.635697° ¡ ° ¡ ° ¡ ° ¡ -3.586161° ¡ 11.311671 ° ¡ -0.099027 ° ¢ ± ¢ ± ¢ ±
In more complex cases involving many degrees of freedom, the reduced problem leads to a more efficient numerical treatment of the eigenvalues problem.
2.7. Bibliography [BAM 81] BAMBERGER Y., Mécanique de l’ingénieur I – Systèmes de corps rigides, Hermann, 1981. [BEL 88] BELLET D., Cours de Mécanique Générale, CEPADUES Editions, 1988. [BEL 89] BELLET D., Problèmes de Mécanique des Solides , CEPADUES Editions, 1989. [CHE 96] CHEVALIER L., Mécanique des systèmes et des solides déformables, Ellipses, 1996. [DEL 89] DEL PEDRO M. and PAHUD P., Systèmes discrets linéaires, 1989, Lausanne. [GER 96] GÉRADIN M., RIXEN D., Théorie des vibrations, application à la dynamique des structures, Masson, 1996 (2nd edition). [INM 01] INMAN D.J., Engineering Vibration, Prentice Hall, New Jersey, USA, 2001.
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Chapter 3
Towards a Global Formulation of the Problem of Mechanical Drive
3.1. Presentation of the mechanical drive modeling problem The preceding chapters presented the bases of power transmission, by covering in particular the technological and the mechanical aspects of indeformable solids. Significant attention was given to the presentation of the dynamic aspects, and the research into the characteristic modes, which is one of the principal concerns when studying moving systems. By selecting the most significant modes we can reduce the complexity of the problem. It is this step which should be systematized during the treatment of the transmissions in order to make the characterization of the complete structure accessible. The study of the characteristic modes and the frequency response obligatorily passes by this simplification. However, it is of course important to avoid introducing too rough approximations at certain stages. The objective of this chapter is to introduce some necessary tools for a fine description of each subset of a transmission. A brief review of the continuum mechanics theory is first proposed. Then, the resolution of the resulting mathematical models is presented from the viewpoint of the variational methods. In this short introduction to the basic methods of continuum mechanics and the calculation of structures, we point out that the global dynamic modeling for the dimensioning of a structure looks to optimize the triptych “forces – shapes – materials”, according to technical and functional specifications. This optimization Chapter written by Christian CUNAT, Mohamed HABOUSSI and Jean François GANGHOFFER.
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Control Methods for Electrical Machines
underlines the role of the boundary conditions or the contacts between the transmission subsets, the geometry chosen for the system components and the stiffnesses and limit resistances of the selected materials. For complex mechanisms, global optimization requires us to take into account the couplings between the various technological elements: the overall behavior can no longer be treated like the superposition of elementary problems associated with each component. Thus, it appears that global models are suitable. Such models pose the question of the scale choice (micron, centimeter and meter) of observation and problem treatment. Such a difficulty concerning the contact modeling has already been faced in the treatment of friction problems (see Chapter 1). Any transmission of power uses elements of structure (casing, shafts, etc.), connected to each other by linking elements (gears, belts, bearings, etc.); the correct functioning of the unit requires clearances which control the loading transfer between the basic technological units, thus introducing geometrical non-linearities (“shape”) combined with “material” and contact (“force”) non-linearities. Analysis of the overall behavior requires a discretization of the unit by using for example the finite element method (FEM). However, this unit cannot be treated like a juxtaposition of elementary problems due to the phenomena of couplings. Figure 3.1 illustrates the origin of non-linearities in: a) a “clearance” (geometric non-linearity); b) a viscoelastic behavior (material non-linearity); and c) the evolution of friction according to the displacement on a plane (contact nonlinearity). Transferred charge
(a) Displacement
Stress
Sliding force
(b) Strain
(c) Sliding
Figure 3.1. a) Example of “geometrical” non-linearity associated with a running clearance; b) example of “material” non-linearity, caused by viscoelasticity; and c) example of nonlinearity associated with the “forces” induced during frictional slipping contact
The problem of the machine tools highlights the incidence of the structure behavior on the control [DUM 01]. The precision of machining can only be obtained thanks to a closed loop control, which requires a fine knowledge of the oscillatory modes to be excited. It is thus advisable to model the kinematic chain to suitably give an account of the first (most significant) modes. We then seek to limit the
Towards a Global Formulation of the Problem of Mechanical Drive
65
number of parameters by gathering them in the form of equivalent parameters (stiffnesses, masses, and dampers). The modeling of the transmission set implies the dynamic behavior of the complete system in order to determine its transfer function. (t) D.q (t)K.q(t) f(t) , in which the gyroscopic coupling G The relation M.q (encountered in Chapter 2) is replaced by the dissipation D, gives the dynamic equation in the temporal domain of a system with damping. The passage to the frequency domain, when this system is linearized, is written M.s2 D.s K .Q( s ) F(s), which gives direct access to the transfer function H of the transmission chain by the relation Q(s) H (s) F (s). Figure 3.2 represents a simplified linear dynamic system.
Figure 3.2. Transfer function of a simplified linear dynamic system with mass M, spring K and damper G
For the sake of effectiveness, the sub-structuration method with condensation is generally adopted in order to reduce the complexity of the global model. This method is illustrated for the case of a milling table (see Chapter 2). The condensation (reduction) of the global model should be made on each substructure and not on the complete model [BOU 92]. This method [BAL 97] is based on the
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Control Methods for Electrical Machines
fact that there generally exists, for a restricted field of dynamic solicitations, subspaces of low dimension, which allow for a good description of the behavior of the complete model. The projection of the model on this subspace using the Ritz method makes it possible to estimate the response with a high degree of accuracy. The dynamic behavior of the structures is often described by the relation (K Z(2i) M).x(i) 0 . The method of reduction then consists of extracting the n first
solutions from among N solutions of the problem (n<
Figure 3.3. Model by J.N. Fawcett (according to Monternot [MON 98])
The application of the dynamics equations leads to only one resonance frequency
Z
1 2S
2 K (r22 J1 r12 J 2 ) /(J1 J 2 ) .
To refine the analysis, we can have recourse to Gasper and Hawker’s model [GAS 89]. Figure 3.4 illustrates this model which uses linear springs, without mass, coupled with a linear viscous damper in parallel (the Voigt rheological unit) to describe the bits of the belt; a viscous damper used for describing the torsion of the pulley.
Towards a Global Formulation of the Problem of Mechanical Drive
67
Figure 3.4. Model by R.S. Gasper and L.E. Hawker (according to Monternot [MON 98])
Log (amplitude)
However, such transmissions often use belt tighteners, modeled in 1989 by Yan and Barker, who introduce them into the serpentine transmission presented in Figure 3.5a. They used Fawcett’s linear model for the bits; as the equations were non-linear because of the tensioner, they linearized them by blocking the tensioner: [ M ]{T} [C ]{T } [ K ]{T } [ F ] . The tensioner is studied separately to determine the concerned bit lengths. Yan and Barker realized a frequency analysis [MON 98] (see Figure 3.5b).
b)
For different damping coefficients
Frequency
Figure 3.5. a) Yan and Barker’s model; b) frequency response of the model (according to Monternot [MON 98])
The study by Banaszek et al. [BAN 93] is interesting in the sense that it compares two modeling approaches. The first one is founded on a discretization of the toothed belt in spring-damper-based units, the other used finite elements. Figure 3.6 presents the studied transmission.
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Control Methods for Electrical Machines
Figure 3.6. a) Transmission model studied by Banaszek (according to Monternot [MON 98]); b) discretization using rheological units for the teeth and the heart of the belt; c) 8 node finite element discretization
The comparison of the two approaches shows that the main advantage of the finite element method is to allow a detailed description of the elastomeric behavior of the belt. To conclude with this example, Monternot shows how it is possible to obtain a reduced global model starting from an analysis, using discrete rheological elements. Figure 3.7 illustrates this operation. The general principle of the reduction methods for solving (finding the eigenvalues O Z(i2) and U x(i) ) the discretized problem (K Z(2i) M).x(i) 0 is thus to extract the m first characteristic solutions from the set of N characteristic solutions, with m<
§ 0 · ¨ ¸ ¨ 0 ¸ © ¹
Towards a Global Formulation of the Problem of Mechanical Drive
Step 4
Step 1
Step 2
Step 3
69
Kglob
Figure 3.7. Stages of model reduction [MON 98]
The relations of the second group (2nd line) make it possible to define the matrix ªt º «¬ sm »¼ presented after condensation, which links the complete solution
U
U U to the reduced solution U : m
s
U U m
m
ª º 1 T K O M « K ss O M ss ms ms » (U m) »¼ ¬«
ªt º U «¬ sm »¼ m
This relation constitutes the starting point of the various forms of dynamic condensation. The Guyan [GUY 65] and Irons [IRO 65] approximation, largely used in dynamic analysis, minimizes the deformation energy with regards to the slave dof, wW / wU s
0
with
T
W 1/ 2 U K U .
This
minimization
leads
to
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Control Methods for Electrical Machines
ªt º «¬ sm »¼
ª « K ss «¬
K º»» where the terms of inertia M 1
T
ms
ij
are neglected; the whole
¼
§ Um · ¸ dof are then expressed depending on the main sub-vectors ¨¨ ¨ U s ¸¸ ¹ ©
§ I mm · ¸ U . ¨ m ¨¨ tsm ¸¸ ¹ ©
The Guyan reduction makes it possible to considerably accelerate the calculations in dynamics. The obtained precision is strongly related to the choice of a priori main dof. The presentation above of the reduction principle ignores the dissipative phenomena of damping, at the cost of increasing complexity, as this can be reintroduced in the analysis. Now, we will present, first, the problem of the gear teeth engaging, which introduces local shocks leading to parasitic vibrations, and second the role of the coupling between various elements of the structure. The studying of the contacts between teeth is then impossible to circumvent for the global dynamic modeling of the gear boxes. Figure 3.8 illustrates a toothed gearing and its modeling [BOU 97]. Shaft 1 Motor
Receiver Shaft 2
a)
b)
Figure 3.8. a) Diagrammatic representation of a toothed gearing; and b) problems of teeth contact interruption [BOU 97]
Towards a Global Formulation of the Problem of Mechanical Drive
71
The contact is studied based on a rheological modeling described in Figure 3.9.
Figure 3.9. Schematization of the engaging interface: introduction of a decomposable rigidity variable in Fourier series (keng ((T)); and of a discharger H W which represents the effort at the teeth traducing the bringing together of the bodies [BOU 97]
Figure 3.10 represents the complete device with the casing studied by Bourdon.
App lied tor que
Bloc ke d seconda ry shaft
Amplitude at node 49
experimental
numerical
Figure 3.10. Study of a simplified driveline: comparison, at a node of the loaded shaft, between the transfer function measured experamentally and the one obtained numerically
Figure 3.11 shows the role of the casing on dynamic transmission in the centers of the exiting pinions.
Control Methods for Electrical Machines
without casing
,
,
Amplitude of the dynamic transmission error (rad)
Amplitude of the dynamic transmission error (rad)
72
with casing
,
,
Figure 3.11. Description of the couplings between various structural elements on the dynamic response
To conclude this introduction, we can point out the need for carrying out a fine mechanical analysis of the transmission mechanism so as to limit the heaviness of the treatments: each body can be studied independently to understand, as well as possible, schematizations that will not denature it. The finite element method will often make it possible to satisfactorily carry out this phase of the analysis. The obtained results will then direct the users towards the required total schematization. 3.2. Brief review on continuum mechanics 3.2.1. Conservation laws
In solid mechanics, conservation laws are the equations expressing the exchanges of matter and energy inside a domain and through its border. These equations thus express the balances of mass, momentum, kinetic moment and energy on the domain. The mathematical equations corresponding to these balances are presented here in their global form. The local form can be established with the help of mathematical transformations. This presentation is inspired by the works of Salençon [SAL 88] and Sidoroff [SID 92], which are quoted at the end of the chapter. 3.2.1.1. General global form Let X be the volume density of a property contained in a moving domain D, as in Figure 3.12.
Towards a Global Formulation of the Problem of Mechanical Drive
n
73
V
Figure 3.12. Moving domain D
The variation of the quantity X comes from exchanges with the outside through the border wD of the domain D (surface density D) on the one hand and from the internal production x of the quantity in the domain on the other hand: d X dv ¨ ds ¨ x dv sD D dt ¨D
[3.1]
The first term of [3.1] corresponds to the rate of variation of the volume density X. In Figure 3.12, n represents the normal vector of the domain and V its velocity. To obtain the three traditional laws of conservation in solid mechanics, we use [3.1] and the quantities of the following table: D 0
Momentum
X U voluminal density) UV
Kinetic moment
OM V (M D)
OM T
Mass
T (stress vector)
x 0 f (volume
force) OM f
3.2.1.2. General local form To establish the local form of the balance equations, we need to use the concept of a particular derivative, which expresses the derivative of an integral defined on a moving volume: d ³ X dv dt D
wX
ª º ³D « wt div( X V )» dv ¬
¼
By using the divergence theorem, we obtain for [3.2]: d sX X dv ¨ dv ¨ XV nds dt ¨D s D sD
[3.2]
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Control Methods for Electrical Machines
While admitting in [3.1] that (M , n) (M , n ) and that, at any point M of the field D, there is a flux a(M ) , such that (M , n) a . n , we can, thanks to [3.1] and the divergence theorem, write: dX
¨ ( dt
X Vi,i ai ,i x)dv 0 with
D
dX dt
wX Vi X ,i wt
[3.3]
Here the convention on the repeated index is adopted. The index after the comma indicates space derivation and i = 1, 2, 3 for all the Latin indices. The general local form of the conservation equation is then: dX X Vi ,i ai,i x 0 dt
[3.4]
This local form supposes the continuity of the various quantities in question. In the presence of a surface of discontinuity, 6 whose normal vector is N , moving at the velocity W (see Figure 3.13), we define the relative velocity U of the “shock” by U (W V ). N .
M
N
n
M– W
V 6
Figure 3.13. Surface discontinuity
The local equations must then be supplemented by an “equation with discontinuities” of the form: XU a i N i
while indicating by h
0
[3.5] h ( M ) h( M ) the jump of a size through 6
Applied to the already quoted physical properties, relation [3.5] gives: – for the conservation of mass UU
0 or
Towards a Global Formulation of the Problem of Mechanical Drive
75
(W V ). N (W V ). N ;
– for the conservation of the momentum U U Vi V ij N j
0;
– for the conservation of kinetic momentum, the corresponding equation is automatically verified if the preceding equation is also verified (symmetric stress tensor). 3.2.1.3. Fundamental laws While taking D Ti V ij n j , V ij representing the ij-component of the Cauchy stress tensor and T is the stress vector, the momentum conservation law then gives “the equation of motion”:
UJ i
U
dVi dt
V ij , j f i
In static, i.e. when the acceleration
V ij , j f i
[3.6]
Ji
0 , the equation of motion becomes
0 expressing the local equilibrium of the medium. The relation of
discontinuity in this case only affects the equation V ij N j T
(N )
0, which means
T (N ) .
NOTE. When examining the conservation law of the kinetic momentum, we verify that it is equivalent, in the case of a symmetric stress tensor, to the dynamic’s fundamental law [3.6]. 3.2.2. Principle of virtual powers (PVP)
For any system, a virtual motion is a possible motion of the system, expressed by kinematically admissible displacements or velocities, i.e. satisfying the essential or kinematics boundary conditions of the system. On the basis of motion equation ~ [3.6], which we multiply by the virtual velocity V , the following relation is obtained while integrating on the entire solid of volume D: ~
~
³D U J i Vi dv ³D V ij , j Vi dv ³D
~ f i Vi dv
By using the divergence theorem, we show that
[3.7]
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Control Methods for Electrical Machines
~
~
³D V ij , j Vi dv As
V
~
³wD V ijVi n j ds ³D V ij Vi, j dV
~
~
[3.8]
³wD Ti Vi ds ³D V ij Vi, j dv
~ ~ is symmetric, we can replace Vi , j with the symmetric part Di, j . The
velocity tensor is such that: ~ Vi , j
~ ~ Dij : ij
~ Dij
~ ~ 1 / 2 V i , j V j ,i
~ : ij
~ ~ 1 / 2 V i , j V j ,i
~ is the rotation rate associated with the where Dij is the strain rate tensor and : ij ~ virtual velocity Vi . Finally, we obtain the expression of the PVP: ~
~
~
~
~
³D U J i Vi dv ³D fi Vi dv ³wD Ti Vi ds ³D V ij Vi, j dv V
[3.9]
This means that for any virtual motion, the virtual power of the quantities of acceleration is the sum of the virtual powers of the external (voluminal and of contact) and the internal (due to the deformation) forces. NOTE. The virtual power of the internal efforts is zero for any rigidifying velocity of a rigid solid (non-deformable). The PVP is equivalent to the fundamental law of dynamics. In general, the continuum mechanics can be built indifferently starting from the conservation laws or from the virtual powers. 3.2.3. Thermomechanics of continuous mediums
3.2.3.1. First principle or energy conservation The first principle of thermodynamics is expressed by the balance equation of the system energy. It says that the variation of the system’s total energy (internal + kinetic energies) is, for any material domain D, equal to the sum of the work of the external forces acting on D and the quantity of heat exchanged with D: d ( E T ) P (ext ) Q dt
where T
1
³D 2 U ViVi dv
energy, P (ext )
represents the kinetic energy, E
[3.10]
³D U e dv the internal
³D f iVi dv ³wD TiVi ds the power of the external efforts, and:
Towards a Global Formulation of the Problem of Mechanical Drive
³D r dv ³wD h ds
Q
77
[3.11]
the rate of heat exchanged with D, resulting from a volume contribution r (radiation) in D, and a surface contribution h (conduction) through the border wD. By substituting the last expressions into [3.10], we obtain the following equation for the conservation of energy: d ³ U dt D
1 § · ¨ e ViVi ¸ dv 2 © ¹
³D f iVi r dv ³wD TiVi h ds
[ 3.12]
which can be obtained from the general conservation law [3.1] while taking:
X
D
1 § · U ¨ e ViVi ¸ 2 © ¹
TiVi h
x f iV i r
By introducing the flux vector q which describes the heat transfers through sD by
q. n ,
h
the
use
of
relation
[3.4]
then
gives
de UJ i V ij, j f i Vi V ij Vi , j q j , j r . Taking into account the equation of dt the motion and the symmetry of V ij , from the first principle of thermodynamics, the
U
following local form remains:
U
de dt
V ij Dij q j , j r
[3.13]
We can also write discontinuity equation [3.5] associated with this conservation law: § de · ViVi ¸U V ijVi q j N j © dt ¹
U¨
0
3.2.3.2. Clausius-Duhem inequality The first principle expresses the balance of energy during a thermodynamic transformation, but does not allow for a description of the evolution. This role returns to the second principle of thermodynamics which expresses the irreversibility of the physical phenomena. Thus, a mathematical modeling which does not satisfy the 2nd principle or the Clausius-Duhem inequality is physically
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Control Methods for Electrical Machines
inadmissible. The 2nd principle utilizes the entropy variable S. In thermostatics, for a monothermal process, subjected to an exchanged heat rate Q, the 2nd principle gives the variation of the entropy S: dS t
1 dQ T
[3.14]
This principle is applied to the continuum mechanics in the form S i
dS Se t 0 dt
dS t S e and dt
expressing that, for any material domain D, the rate of the
internal production of entropy S i is positive, the internal production being defined like the difference between the variation of the entropy of the domain D, expressed by:
³ UKdv
S
[3.15]
D
where K is the mass specific entropy, and the entropy exchanged with the outside, which is related to the heat transfers [3.11] by S e
r
h
³ dv ³ T ds ( T is the absolute temperature) DT wD
[3.16]
By taking into account relations [3.14], [3.15] and [3.16], the second principle of thermodynamics is written under the form: q i ni r d ³ UKdv t ³ T dv ³ T ds dt D D wD
By using the divergence theorem, we obtain:
U
dK 1 dK r § 1 · t ¨ qi ¸ or UT t r q i ,i q iT ,i dt T dt T © T ¹ ,i
By eliminating r between [3.13] and [3.17], we obtain: dK · 1 § de U¨ T ¸ q iT ,i V ij Dij t 0 dt dt ¹ T ©
[3.17]
Towards a Global Formulation of the Problem of Mechanical Drive
79
This is the Clausius-Duhem inequality which can also be written in the form dT · 1 § d< U¨ K ¸ qiT , i V ij Dij t 0 where < e KT is the free energy per dt ¹ T © dt unit mass. By neglecting the thermal variables, we can rewrite this relation in the de form M U V ij Dij t 0 where e is the internal or free energy (this has no dt importance since the material thermal variables are neglected). By substituting this expression of M into the kinetic energy theorem, we obtain:
5 (ext)
dT dt
5 (int)
dT dt
dE dt
) (irr) with ) (irr )
³ Mdv t 0 D
The power of the external efforts helps to increase the kinetic energy and the internal energy of the material, and its dissipative part is given by ) (irr ) . 3.2.4. Notions on strain
3.2.4.1. Description In large deformations, the motion is described by the function xi a1 , a 2 , a3 , t , giving the position Mt at the time t of a material particle which, in the reference configuration (t = 0), occupies the position a1 , a 2 , a3 . xi
The xi variables are Eulerian or space variables, while the ai variables are Lagrangian or material variables. A material vector da MM ' becomes dx M t M t ' after deformation. To describe this transformation, we must define the tangent linear application F , also called the deformation gradient tensor: Fij
wx i wa j
[3.18]
This tensor characterizes the local deformation in the vicinity of the point M. However, it is not a satisfactory measurement of the strain in the naive meaning of this term (problem of the rigid modes). By “strain” we refer to the variation of form, and therefore the length of the angle or the variation of the scalar products. By giving d a and G a two material vectors, d x and Gx their deformations, we have:
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Control Methods for Electrical Machines
d x.G x
dx i Gx i
Fij Fik da i Ga k
C jk da i Ga k
Thus, the variation of the scalar product of two vectors is characterized by the symmetric bilinear form C, defined by: C jk
Fij Fik i.e. C
F t F and d x.G x
d aCG a
where C is the dilatation tensor or the Cauchy-Green tensor. This tensor is the foundation of descriptions of large strains. In the absence of strain, i.e. in a rigid solid movement, we obtain C jk Qij Qik G jk , since the matrix Q is orthogonal. The dilatation tensor is the unit tensor I, and there is conservation of the scalar 1 product. While posing E (C I ) , we define the Green-Lagrange strain tensor. It 2 gives the variation of the scalar product of two vectors: d x.G x d a.G a
2d a.E.G a.
The diagonal components of E characterize the stretchings in the directions of the axes, while the non-diagonal components characterize the shearing in the directions axes. x a are introduced in order to wu describe the body deformation, thus obtaining Fij G ij i from [3.18]. The wa j
In solid mechanics, the displacements u
Green-Lagrange strain tensor is then given by:
Eij
1 §¨ wu i wu j wu k wu k ·¸ wai wa j ¸¹ 2 ¨© wa j wai
[3.19]
This strain tensor constitutes one of the multiple strain measurements in large deformations. The choice between these measurements and the corresponding stress tensors is one of the principal difficulties which is encountered in the treatment of large deformations. A detailed presentation of these points is proposed in the bibliographical references quoted at the end of this chapter. We will limit here the analysis to the current case of small strains. 3.2.4.2. Small strains Within the framework of the small perturbations assumption, the strain tensor defined by [3.19] as a function of the displacement u becomes:
Towards a Global Formulation of the Problem of Mechanical Drive
H ij
1 §¨ wu i wu j 2 ¨© wx j wxi
81
· ¸ ¸ ¹
where İ ij is the tensor of the linearized strains. All measurements of large strains are reduced to this tensor in small strains allowing us to calculate the stretching in direction n , and the slip in two directions m and n , by the formulae:
H n H ij ni n j and J m, n 2H ij ni m j
[3.20]
The trace of the strain tensor H ii corresponds to the relative variation of volume, 'V H ii . V 3.2.4.3. Compatibility of the deformations To determine the displacements starting from known strains, it is necessary that these strains verify the following relations of compatibility:
H il , jk H jk ,il H jl ,ik H ik , jl
0
i, j, k , l 1,2,3
[3.21]
If this is the case, the displacement can be calculated starting with the relation:
H ij
1 §¨ wu i wu j 2 ¨© wx j wxi
· ¸ ¸ ¹
up to a rigid solid displacement. As quantity [3.21] is antisymmetric with respect to i and j and k and l, and symmetric with respect to the couples i, j and k , l , we obtain the six equations of compatibility for (ijkl)=(1212), (1213), (1221), (2121), (1321), (2131). 3.2.5. Some material behaviors: elementary analog models The analog models presented here allow a simple presentation of some material behavior laws currently used in practice. Besides this, we will notice the same type of models are used to represent the physics (stiffness, damping) due to connections, for coupled systems. Among these models are:
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Control Methods for Electrical Machines
– the spring model of elastic behavior, given by the Hookian relation V EH , which represents a linear (E=constant) or non-linear (E depending on H) behavior; – the Coulomb friction element model of plastic behavior where the rate of plastic strain is given according to the yield stress V 0 by: £¦% p 0 if 0 ¦¦ ¦ p ¤% §0 if 0 ¦¦ ¦¦% p 0 if 0 ¦¥
– the damping model of viscous behavior, V coefficient).
KH (where K is the damping
Other rheological models are obtained by a series or parallel combination of the aforementioned models. In a series configuration, the stresses are added and the strains are the same, whereas, in the parallel case, the strains are added and the stresses are the same. For example, we have: – the Maxwell model V ® ¯H
EH 1
KH2
H1 H 2
H
V E
V K
– the Kelvin-Voigt model
V
EH KH
In general, choosing a model for representing the behavior of a material is difficult due to the complexity of the material. Even for usual materials such as steel, many aspects of its behavior remain unknown. In each problem, it is appropriate in practice to choose the simplest model leading to satisfactory results for the desired use. 3.2.6. Variational formulations in mechanics of the structures The analytical resolution of the continuum mechanics governing equations in their local form (partial differential equations) is generally impossible. The reasons are, on the one hand, the complexity of the domain geometry where we cannot distinguish any symmetry and, on the other hand, because of the non-linearities of the material behavior. Here an alternative method is used, which is based on the
Towards a Global Formulation of the Problem of Mechanical Drive
83
integral or the global formulation – known as weak or variational form – of these local equations. For certain behavior laws, particularly in the simplest case, which is that of linear elasticity, the integral form leads to a potential energy – and some other alternative forms – defined at the structure scale, which admits a minimum expressing an equilibrium. Extremum principles are then available which introduce an approximate resolution of the initial problem. The discretization of the integral forms indeed leads to the numerical resolution method called the finite element method. 3.2.6.1. Generalized variational principles In the following we present the variational principles, referred to as generalized, in the mechanics of structures. These principles are based on functionals which are dependent on several variables (displacement, strain, constraint) considered to be a priori independent. The minimum principles of a functional field (displacement or stress) are particularly obvious cases that we shall present first. Let us consider a structure subjected to an imposed displacement equal to u , on a portion Su of its border, with tractions T on the complementary portion St of the border. The minimum principle of total potential energy consists of making the functional stationary with regard to the displacement field:
V>u @ ³: W Hij f i u i dx ³S t i u i ds , t
with the subsidiary conditions S : Hij
1 u i, j u j,i in : and ui 2
ui in Su .
The obtained local equations are the Euler-Lagrange equations of V [u] . They describe the equilibrium of the solid for which V [u] is then minimal. An alternative consists of including these conditions in a more general formulation which amounts to considering these conditions as constraints of V[u]. The idea of the hybrid formulations – still called mixed formulations – thus consists of relaxing some of the subsidiary conditions by using additional variables and constraints depending on these variables. Thus, for example, the definition of the strain as a gradient of displacement (relation (S)) is a constraint relation between u and H that cannot be imposed, the two variables being considered to be independent. For this purpose we use the traditional method of the Lagrange multipliers, which is based on the theorem of factorization.
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Control Methods for Electrical Machines
Let us consider by E1, E2, E3 three vector spaces and two linear applications: L1: E1 oE2 L2: E1 oE3 There exists a linear application L: E2oE3, such that L2 L D L1 if and only if [L1 (x) = 0 L2 (x) = 0] or KerL1 KerL2 (the kernel of L1 is included in that of L2). Let J be a functional of an affine space E in : GE is the vector space associated with E, and it is also the space of the variations of the elements u of E, that is to say Gu GE. The problem of the stationarity of J[u ] – which expresses the fact that the real solution minimizes the length of the virtual paths corresponding to the trajectories described by J>v@ , where v is an acceptable virtual displacement – gives: Gu GE , J ' >u @.Gu
0
where J ' >u @.Gu indicates the derivative of J>u @ in the direction of the variation įu. Notions on the calculation of variations available in [IMB]. Thus, a constraint condition is introduced into the variational problem: U is sought E such that l(u) = 0, with L: EoF a differentiable application of E in an affine space F. This last condition is still written in differential form l’(u). Gu = 0, and the extremum problem, with constraint, becomes: l (u ) = 0 Find U E such that: 㹻įu 㺃įE verifying l ' (u ) įu = 0 J ' [u ] įu = 0
The factorization theorem allows us to state the equivalent problem of extremum without constraints. Find v = (u,O) E x F such that: v u, , J '< u >¸ u l ' u ¸ u ¸ l u 0,
that is to say ¢J < u > l u ¯± 0
which shows that the new functional G< u > J < u > l u is stationary.
Towards a Global Formulation of the Problem of Mechanical Drive
85
The parameter O is a Lagrangian multiplier which transforms the problem of a functional minimization with constraints into a functional minimization problem without constraints. Note that O is an element of an affine space which is in general a space of tensors in mechanics. Applying the method of the Lagrangian multipliers to the equivalent formulation of the principle of minimum total potential energy V[u], matched with subsidiary conditions (S), is equivalent to making the functional stationary:
>
J u i , Hij , Vij , O ij , Pij
with Vij
@
wW wHij
ª
1 ½º u i, j u j,i Hij¾» dx ³S t i u i ds ³S Pi u i u i ds t u ¯2 ¿¼
³: « W Hij f i u i O ij ®
¬
symmetric tensor (Cauchy tensor). The coefficients
Oij i, j and
P j j are Lagrangian multipliers, and the stationarity of J is written: GJ
ª wW º §1 · Gij f i Gui GOij ¨ u i, j u j,i ¸ Hij» ³: « ©2 ¹ »¼ ¬« wHij
leading to the stationarity conditions: i) Oij Vij , which identifies the Lagrangian multiplier to the Cauchy tensor; 0 in :, which represents the static equilibrium;
ii) Oij, j f i iii) O ij n j iv) İ ij =
t i on the surface S t , which expresses the static boundary conditions;
1 u + u j ,i , the strain due to the displacements; and 2 i, j
(
)
v) u i = u i on the surface Su , which are the kinematic boundary conditions. The Euler-Lagrange equations derived from the functional minimization correspond to all the static equations: equilibrium, behavior law, kinematics and force boundary conditions. If the relations i) and v) are incorporated in J, the three field functional is obtained by:
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Control Methods for Electrical Machines
1 § · J ui , Hij, Vij ³: W Hij f i ui dx ³:Vij¨Hij ui, j u j,i ¸dx ³S ti ui ds ³SuVijnj ui ui ds . t 2 © ¹
>
@
>
@
A generalized variational principle was thus built, according to which the real state u i , Hij , Vij of the solid is that which makes the functional J ¡¢ui , Fij , Tij ¯°±
stationary and where the variables u i , Hij , Vij are regarded as independent. According to the choice of variables and the nature of the local relations to be obtained, different functionals can be built. Thus, by eliminating the strain Hij , and replacing it with the stress Vij , using the Legendre transform, according to
Wc V ij
V ij H ij W H ij , we obtain:
J R ui , Vij
ª º Vij u i, j u j,i »dx ³s t t i u i ds ³Su Vijnj ui u i ds ³: « Wc Vij f i u i 2 «¬ »¼
which starts from the preceding functional, the Reissner principle associated with a two variable (displacement and stress) functional. The real variables (u, V) which make JR stationary define a saddle point, as JR is convex with regards to u and concave relative to V. Note that the generalized variational principle is the starting point towards other mixed variational principles: for example, the Fraejis de Veubeke, the Pian and the Herrmann formulations. From a numerical point of view, the multivariable-based functionals are written in an algebraic form, after a discretization of their geometric domain of definition. The interest of the mixed formulations is to allow a more precise evaluation of some unknowns when it is necessary. Thus, a weak variation of Poisson’s ratio involves considerable variation of the stresses for quasi-incompressible mediums (Herrmann formulation). They are employed in addition when two distinct formulations are coupled, each one being applied to a part of the structure: problems of composites where it is necessary to know with precision the shear and peeling stresses (interlaminar stresses) and problems of coupling fluid and structure. 3.2.6.2. Example: torsion of a cylindrical shaft Let us consider a full cylinder with unspecified section, subjected to torsion around the axis (0z) (see Figure 3.15).
Towards a Global Formulation of the Problem of Mechanical Drive
87
The section located in z 0 undergoes an infinitesimal rotation Tz around the axis 0z. Saint-Venant’s assumption leads to the following expression of the displacements: u x , y, z y Tz y z T ° ®v x, y, z x Tz xz T °w x , y, z w x , y T w ¯
where Tz
1 v, x u, y , and T is the rotation per unit of length. 2
y
y x 6L
60
n 61
M
ds x
O z s Figure 3.14. Shaft subject to torsion
The lateral surface 6 L is free from efforts, and surfaces 6 0 , 61 are subjected to efforts whose resultants are respectively two opposite torques Me3 and –Me3. The function w ( x , y) represents the warping of the section, and w(x,y) is called the warping function. The center of torsion (0,0) is the point of the plane ( x, y) which remains fixed ( u v 0 ). For a shaft made up of an isotropic material whose rigidity modulus is G, the behavior law is written: °V xz ® °¯V yz
GJ x z
G T w , x y
GJ yz
GT w,y x
The equilibrium equation according to z gives: V xz, x V yz, y
0 , that is to say w , xx w , yy
0
[3.22]
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Control Methods for Electrical Machines
The boundary condition on 6 L is written V.n=0, with n normal vector, from where V xz n x V yzn y 0 , that is to say: w , x n x w , y n y yn x xn y
(n x , n y ,0) the unit
0
which leads to: w ,n
1 d 2 2 x y 2 ds
n x because ® ¯n y
The
applied
[3.23]
0
y,s x ,s
torque
is
M
³H x Vyz y Vxz dxdy M
GJT ,
with
2 ³H x w , y y w , x x 2 y dxdy being the torsion moment of inertia. D = GJ is the rigidity modulus of the shaft in torsion.
J
In the case of a circular section (radius R ), relations [3.22] and [3.23] are written: £w,r 0 on r=R ¦ ¦ ¦ ¤1 ¦ rw, r ,r 0 ¦ ¦r ¥
thus, w is constant, S 4 J R , and M 2 in torsion.
and the section does not warp. Here the moment of inertia is 3 § 3 4· ¨ G R ¸T ; D G R 4 is the rigidity modulus of the shaft 2 © 2 ¹
Variational displacement formulation – the total potential energy is: 2 2 1 2 W w G ¨ ¡ w,x y w, y x ¯° dA A¢ ± 2
with w being kinematically admissible, i.e. w = 0 at the center of torsion, and W
³A
1 V xz J xz V yz J yz dxdy 2
Towards a Global Formulation of the Problem of Mechanical Drive
89
Variational stress formulation – the complementary potential energy is:
W C V xz , V yz
1 2 2 ³A §¨ V xz V yz ·¸dxdy MT © ¹ 2G
The variational formulations give a framing of the approximate solutions. If wh is a kinematically admissible approximate solution, the relations W w h ! W w and W C w h W C w h can provide limits (thus estimates) for the rigidity modulus in torsion.
3.3. Bibliography [BAL 97] BALMÈS E., Modèles expérimentaux complets et modèles analytiques réduits en dynamique des structures, HDR Paris VI, 1997 [BAN 93] BANASZEK G., WITTBRODT G., ESAT E., “Dynamic behaviour of belt driven gears”, Int. Conf. FEMCAD CRASH, pp. 105-119, Paris, June, 1993. [BER 96] BERTHIER Y., “Maurice Godet’s third body theory”, in D. DOWSON (ed.), Proc. of the 22nd Leeds-Lyon Symposium on Tribology, Lyon, 1996. [BOU 92] BOUHADDI N., Sous-Structuration par condensation dynamique linéarisée, Thesis Besançon, 1992 [BOU 97] BOURDON A., Modélisation dynamique globale des boites de vitesse automobile, Thesis, l’INSA, Lyon, September, 1997. [DOW 80] DOWNS B., “Accurate reduction of stiffness and mass matrices for vibration analysis and a rationale for selecting master degrees of freedom”, Trans of the ASME, J. of Mech. Design, vol. 102, pp. 412-416, 1980. [DUM 01] DUMETZ E., BARRE P.J., HAUTIER J.P et CHARLEY J., Caractérisation du comportement dynamique des axes d’une machine-outil à grande vitesse, Revue Internationale d’Ingénierie des Systèmes de Production Mécanique, no. 5, IV-47-57, June, 2001. [DUV 72] DUVANT G., LIONS J.L., Les inéquations en mécanique et en physique, Dunod, 1972. [DUV 76] DUVANT G., LIONS J.L., Inequalities in Mechanics and Physics, Springer, 1976. [FAW 89] FAWCETT J.N., BURDESS J.S., HEWIT J.R., International Power Transmission and Gearing Conference, pp. 25-28, Chicago, April, 1989. [GAS 89] GASPER R.G.S. and HAWKER L.E., “Resonance frequency prediction of automotive serpentine belt drive systems by computer modelling”, 12th ASME Biennal Conf. Mech. Vibr. Noise, pp. 13-16, Montreal, Canada, September, 1989. [GER 96] GÉRADIN M., RIXER D., Théorie des vibrations (2nd edition), Masson, 1996.
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Control Methods for Electrical Machines
[GUY 65] GUYAN R.J., “Reduction of stiffness and mass matrices”, American Instit. of Aero. and Astro. Journal, vol. 3, p. 380, 1965. [IPS 96] IPSI, Modélisation mécanique et numérique du contact frottement, 15-17 October, 1996. [IRO 65] IRONS B., “Structural eigenvalue: elimination of unwanted variables”, AIAA Journal, vol. 3, pp. 961-962, 1965. [KHA 95] KHAN A.S, HUANG S. J., Continuum Theory of Plasticity, Wiley & Sons, 1995. [KIK 88] KIKUCHI N., ODEN J.T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, Philadelphia, USA, 1988. [LAD 86] LADEVÈZE P., Mécanique des structures, Formation doctorale: C.A.O. et structures, June, 1986. [MON 98] MONTERNOT C., Comportement Dynamique des Transmissions de Puissance par Courroie Dentée, thesis, Lyon, 1998. [ODE 85] ODEN J.T. MARTINS J.A.C., “Models and comportational methods for dynamic friction phenomena”, Computer Methods in Applied Mechanics and Engineering, vol. 52, pp. 527-634, 1985. [QUO 00] NGUYEN Q.S., Stabilité et mécanique non linéaire, Hermes, Paris, 2000. [RAB 95] RABINOWICZ E., Friction and Wear of Materials (2nd edition), John Wiley, New York, 1995. [SAL 88] SALENÇON J., Mécanique des milieux continus, (I) Concepts généraux, Éditions Ellipses, 1988. [SID 92] SIDOROFF F., Mécanique des milieux continus, Tome 1, Cours du DEA de Mécanique, École Centrale de Lyon, 1992. [YAN 89] YAN Y.L. and BARKER C.R., “Dynamic analysis of an automotive belt drive system”, Proc. of the First Int; App. Mech. Syst. Design Conf. (IAMSDC-1), pp. 75.175.10, Nashville, Tennessee, June 1989.
Chapter 4
Continuous-time Linear Control
4.1. Introduction In this chapter, we quickly present PID controllers in a way similar to some current tuning methods, which are among the most frequently used for the control of industrial processes. After the omnipresent PID controller, we present techniques of control which are a little more complex and more multivariable directed, such as state feedback or optimal control. Certain basic concepts are assumed to be known: this is the case, for example, for the concept of transfer function, block diagram representation and the state space representation.
4.2. PID controllers 4.2.1. Overview Figure 4.1 shows the traditional diagram of a closed loop feedback system. The basic transfer function of PID controllers has the following simplified form: C s
u s H s
§ · 1 K p ¨¨1 Td s ¸¸ © Ti s ¹
Chapter written by Frédéric KRATZ.
[4.1]
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Control Methods for Electrical Machines
and a more exact form, which takes into account the real development possibilities of the derived action: u s H s
C s
§ T s · 1 K p ¨¨1 d ¸¸ © Ti s Ts 1 ¹
[4.2]
with T = Td /Kd where Kd is called the dynamic gain. The three adjustable parameters of a PID controller are: – Kp, proportional gain; – Ti, time constant of the integral action (integral time or repetition time); and – Td, time constant of the derived action (derivative time). Another way of writing PIDs which is used in an industrial environment is: C s
Kp
Kp
with K i
Ti Yc
Ki Kd s s
[4.3]
the integral gain and Kd = Kp Td the derivative gain.
u
H
+
Controller C(s)
Y Process G(s)
Ym +
Sensor H(s)
+ b
Figure 4.1. Closed loop feedback system. Symbols used: Yc, reference input; H, tracking error (H = Yc –Y); Y, output; u, controller output (control signal); Ym, output measurement; and b, measurement noise. The block “process” includes the actuators, the process itself and possibly the control loops already
Continuous-time Linear Control
93
4.2.1.1. Proportional action: C(s) = Kp The action u is proportioned as a function of the result reached and is reduced when the error H decreases. The proportional action is expressed by the gain Kp or by the proportional band Pb such that: Pb (%)
100 Kp
The concept of the proportional band very clearly has a physical direction, which is easily explained. In practice, the action u is limited in amplitude to a value umax (saturation of the actuators or voluntary limitation due to energy consumption). The range of variation in which the relation u = Kp.H is true is called the “proportional band”. u
umax H Pb
Figure 4.2. Proportional band
NOTE. If Kp is large action is strong and fast, but there is a risk of overshoot and oscillations. If Kp is small: there is a slow and gentle response.
4.2.1.2. Integral action: C s
1 Ti s
Ki s
The integral output of the block is proportional to the integral of the input and is expressed by: S t
1 t ³ E t dt Ti 0
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Control Methods for Electrical Machines
with Ti the integral time representing the time necessary for the variation of the output to be equal to the variation of the input. Ti is in minutes or seconds. Ki , the inverse of Ti , represents the number of times that the output repeats the input in unit time, so that if Ti is in minutes, Ki will be in revolutions per minute (RPM). The principal effects are, in static mode, the elimination of the error (H = 0 possible when u z 0) and, in dynamic mode, the deceleration of the transient and the increase in instability. 4.2.1.3. Derivative action: C s K d s The output of the derivative block is proportional to the derivative of the input. By calling Kd the proportionality factor: S t K d
dH (t ) dt
The principal effects are, in static mode, none and, in dynamic mode, the stabilizing effect and increasing in the speed of response (anticipative effect by taking into account the direction of variation of the error H)ҏҗ. It is important to note that the derived action can have harmful effects, hence the amplification of the noises in high frequencies (filtered PID solution (see [4.2])) and abrupt variations in the output of the controller during a modification of the reference signal (the solution being derivative action on the measurement).
4.2.2. Various structures In Table 4.1, some common structures of PID controllers are presented.
Continuous-time Linear Control
Structure (functional diagram)
No
Transfer function
Name
§ 1 · ¸ K c ¨¨1 Td s ¸ T is ¹ ©
Mixed (series/ parallel)
§ 1 · ¨ K c Td s ¸ ¨ Ti s ¸¹ ©
Parallel
§ 1 · ¸ k1 1 Td s k 2 ¨¨1 ¸ © Ti s ¹
Parallel
95
1 +
T is H
1
+
KC + T ds
1 T is
+ +
H
2
u
KC + T ds
k1(1+T ds) +
H
3 +
u
k2 (1+ 1 ) T is
Yc
4
+
H
k’p (1+ 1 ) Tis
u
u
1+Td s
Ym
§ 1 · ¸YC k 'p ¨¨1 ¸ T is ¹ © §
1 Td s k 'p ¨¨1 ©
1 · ¸Ym Ti s ¸¹
Derived on the measurement
Table 4.1. Some structures of PID controllers
4.2.3. Selection criteria for the adjustments The choice of adjustments is a daily problem for each operator and each control engineer. The need for simple methods, fast use and sufficient precision is enormous. This problem has always aroused the interest of theorists and experts in automatic control.
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Control Methods for Electrical Machines
It is obvious that the choice of adjustment depends on the desired performances of the system and will be different according to whether we want to have an aperiodic transitory mode or an oscillatory mode with a determined overshoot. The indices used to evaluate the quality of the control are generally: y max y f .100% ; yf
– overshoot, D
– rise time (tr), the time at which the response reaches the final value ( y f ) for the first time; and – settling time (ts), the time taken by the control system to reach the final specified value and thereafter remain within a specified tolerance (generally 'y = 5% yf ). These concepts are illustrated in Figure 4.3 below. 1.4
D%
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0 0
tr
ts
Figure 4.3. Second order step response
As for the selection criteria, among those most frequently used are: – aperiodic response (D=0%), with tr minimum; – oscillatory response at D=20%, with tr minimum;
Continuous-time Linear Control
97
– minimum integral of the square of the error: f
I
2 ³ H t dt
0
– minimum integral of absolute error (IAE): f
I
³ H t dt 0
– minimum integral of time multiplied absolute error (ITAE): f
I
³ H t t dt 0
which gives increasing weight to the absolute value of the error during its temporal evolution.
4.2.4. Control mode The selected control mode depends on the stability of the process in open loop. Indeed, the following rules make it possible to choose the recommended structure for the controller.
4.2.4.1. Stable process The system is modeled in the form of a delayed first order (known as Broïda form): G s
Ke Ws 1 Ts
[4.4]
To determine coefficients K, T and W, the tangent to the curve at the inflection point is plotted and then the delay W and the time-constant T are measured. The gain K is given directly by the ratio of the final amplitude of the output to the amplitude of the input.
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Control Methods for Electrical Machines
The value
T
(also called adjustability) makes it possible to determine the best W considered structure for the controller. This is represented in Table 4.2. 'Y K'YC
0.63*K'YC
0
0
2
W
4
6
8
10
12
14
16
18
20
t
T
Figure 4.4. Broïda modeling
T
2
5
10
20
W Type of controller
Smith predictor
PID
PI
P
Table 4.2. Adjustability of a stable process in open loop
On/off On/off control control
Continuous-time Linear Control
99
4.2.4.2. Unstable process The system is modeled in the shape of a delayed integrator (known as ZieglerNichols form): G s
Ke Ws s
[4.5]
10 9 8 7 6 5
K 4 3 2 1 0
0
2
4
6
8
10
12
14
16
18
20
t
W
Figure 4.5. Ziegler-Nichols modeling
Parameters W and K are determined in the following way: W is the X-coordinate of the intersection of the tangent with the time axis and K is the slope of the tangent. K has inverse time as a dimension, so KW is therefore dimensionless. The value of KW makes it possible to determine the suggested corrector.
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Control Methods for Electrical Machines
0.1
0.05
KW
Type of controller
On/off control
P
0.2
PI
0.5
PID
Smith predictor
Table 4.3. Adjustability of an unstable process in open loop
4.3. PID controllers In many cases, the engineer must manage an existing installation and make tuning on-site. A first category of methods is based on the knowledge of the step response of the system. The three main methods based on this principle are tuning by trial-error, the Ziegler-Nichols method and the Cohen-Coon method. The last two methods use the step response in an open loop. These methods make it possible to adjust the following type of PID: C s
§ · 1 K c ¨¨ 1 Td s ¸¸ Ti s © ¹
This can also be written as (see [4.3]): C s
K d s 2 K p s Ki s
4.3.1. Adjustment by trial and error This method is appropriate particularly when the system can be analyzed only in a closed loop. In many industrial situations, the opening of the loop is either impracticable or is inadvisable under normal operation. The tuning of PID controllers is done in the following way:
Continuous-time Linear Control
101
– Step 1: the controller is put into proportional mode; actions I and D are canceled while making Ti o f (in practice, it is set at its maximum value) and Td o 0 (it is set at its minimal value). – Step 2: a low value for the gain is chosen and the controller is put into automatic. A variation of the reference signal of the step type is carried out. – Step 3: the proportional gain Kc of action P is increased until the appearance of a maintained oscillation (system just oscillating). The gain thus obtained is noted as critical gain Kcr. – Step 4: the gain is reduced by a factor of two. – Step 5: Ti is decreased by small increments until it again has a maintained oscillation. Ti is increased to three times this value. – Step 6: Td is increased until a maintained oscillation is obtained. Td is set at a third of this value. During the tuning phase, saturation of the output controller should be prevented. If not, there can be oscillation with gain values lower than the critical gain Kcr. In theory, when the gain is lower than the critical gain, the answer y(t) in closed loop is under-damped or slightly oscillating. When the gain is higher than the critical gain, the system is unstable and there will most often be saturation. Caution is required, as this procedure can be dangerous and can destabilize the closed loop. Moreover, it does not apply to the unstable processes in open loop. In addition, the critical gain is not always obtained (as is the case with systems of the first or second order without delay). Nevertheless, the tuning steps indicated clearly show the influence of each action and provide a course to follow. NOTE. This method is an alternative to the method of oscillation supported by Ziegler and Nichols [ZIE 42]. This method consists of determining, as before, the critical gain and the period of oscillation Tosc. The tuning recommended by Ziegler and Nichols can then be used. Controller
Kp
Ti
Td
P PI PID
0.5 Kcr 0.45 Kcr 0.6 Kcr
Tosc/1.2 Tosc/2
Tosc/8
Table 4.4. Recommended tuning by Ziegler and Nichols
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Control Methods for Electrical Machines
This tuning corresponds to the oscillatory step response, with an overshoot D approximately equal to 35-40%. Here it is not a question of optimal tuning following a criterion, but of tuning allowing an exploitable method to be shown by the user. It may be that a type of tuning is appropriate when a variation of reference signal is made; however, this is not as good as when the system is subjected to a disturbance.
4.3.2. Ziegler-Nichols method This method makes it possible to adjust a PID when there is an open loop step response from the system. Ziegler-Nichols proposes to model the process as: G( s )
e Ls
R s
where the “apparent dead-time” L is the intersection of the tangent at the inflection point with the time axis; and R is the slope. For this model, the set of parameters minimizing the integral IAE were sought: f
I
³ H t dt
0
Figure 4.6. Open loop step response
Continuous-time Linear Control
103
The following tunings were obtained. Controller
Kp
Ti
P
1 R¸L 0.9 R¸L 1.2 R¸L
0.3 L 0.5 L
PI PID
Td
0.5 ¸ L
Table 4.5. Results from the Ziegler-Nichols method
4.3.3. Cohen-Coon method As in the preceding method, the Cohen-Coon method makes it possible to adjust a PID by knowing the step response of the system to be controlled in open loop. From the modeling of the system in the Broïda form (see equation [4.4] and Figure 4.4) we are given: G( s )
Ke Ws 1 Ts
However, coefficients K, T and W must be determined. The first solution consists of tracing the tangent to the curve at the point of inflection, then measuring the delay Wand the time-constant T. The gain K is given directly by the ratio of the final amplitude of the output and the amplitude of the input. Another approach, called the Broïda method, consists of measuring instants t1 and t2 at which the response respectively reaches 28% and 40% of its final value. The time and delay constants are obtained from the following relationships: T | 2.8t1 1.8t 2 and W | 5.5 t 2 t1
A similar approach consists of considering the times at which the response reaches 35.5% and 85.3% respectively, and has the advantage of being more clearly separated than in the preceding case. Then:
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Control Methods for Electrical Machines
T | 1.3t1 0.29t 2 and W | 0.67 t 2 t1
Controller
Kp
Ti
P
1 T ª 3T W º K W «¬ 3T »¼
PI
1 T ª10.8T W º K W «¬ 12T »¼
W
30 3 W / T 9 20 W / T
PID
1 T ª16T 3W º K W «¬ 12T »¼
W
32 6 W / T 13 8 W / T
Td
4W 11 2 W / T
Table 4.6. Coefficients of the PID obtained by the Cohen-Coon method
4.4. Methods based on previous knowledge of a system model This class of method is based on previous knowledge of a system model, and these methods are more or less sensitive to the uncertainties of this modeling. In this section, two techniques for PID design are described. The first method presented, known as the Bode method, uses existing links between frequency obtained parameters, and time parameters starting from the second-order approximation.1 The second method, the Phillips and Harbor method, uses a similar technique to the pole placement technique.
4.4.1. Presentation of the Bode method This method uses the following approximate relations: – Z ncl # Z co the natural frequency of the closed loop is given by the gain cutoff frequency of the open loop; and 'M q the damping factor of the closed loop is given by the phase margin 100 expressed in degrees divided by 100.
– ] cl
1 The second-order approximation consists of saying that the closed loop system is of the second order.
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105
Using the gain cut-off frequency Zco the open loop transfer function can be written in the following form:
CGH jZ co 1.e jT Z co where T represents the phase of the system. The equation of the open loop transfer function can be written in the following way: K p jZ co K d
K 1.e jT Zco j i GH jZ co Z co
From this last expression, the values of Kp and Kd can be directly deduced according to the pulsation Zco , and from the integral gain Ki. 4.4.1.1. Determination of the integral gain Ki If the non-corrected system is of type n, then the feedback system is of type n+1. In this case the steady-state error is looked at in response to the steady-state input of type n+1. The general expression of the error (difference between the reference signal and the measurement) is given by:
H s
1 .YC s with YC s 1 CGH s
1 s
n2
The steady-state error is directly obtained by the final value theorem:
H f
lim H t
t of
lim s .H s
s o0
taking into account type n of the non-corrected system, this can be written as: GH s
K sn
.
1 b1s ! bm s m 1 a1s ! a p s p
The steady-state error is equal to:
H f
1 K .K i
with m d p n
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Thus, by setting the value of the steady-state error (error of type n+1) in the specifications sheet of the regulation, the value of Ki is deduced. 4.4.1.2. Determination of the gain cut-off frequency Zco From the overshoot authorized by the specifications sheet of the regulation, the expression of the damping factor of the closed loop cl is obtained by second order approximation. The margin from the desired phase 'M in degrees is deduced from the damping factor and approximate relations (which were discussed at the beginning of this section), imposing the gain cut-off frequency Zco.
4.4.2. Presentation of the Phillips and Harbor method
From the regulation specification sheet, the natural frequency and damping factor desired of the closed loop are explained. This makes it possible to define the poles of the transfer function of the closed loop, in the estimation of the second order system [PHI 00], that is to say if s1 is a pole of the closed loop transfer function, then: CGH s s s 1
and if s1
1
s1 .e jE then:
GH s1
GH s1 .e j\
Phillips and Harbor proceeded to show that PID controllers have as parameters: 2 Ki cos E sin E \ GH s1 sin E s1
Kp
Kd
K sin\ i s1 GH s1 sin E s 2 1
The determination of the integral gain Ki is obtained in a way similar to the Bode method, i.e. by fixing the steady-state error value (error of order n+1) in the specification sheet of the regulation.
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4.5. Linear state feedback control systems
In these two last sections, synthesis of the controller was approached using temporal methods, i.e. by means of representing the process state. The advantage of the temporal approach compared to the frequency approach is that it proposes an automatic calculation of the synthesis parameters. We will begin our discussion with pole placement, a very popular method of synthesis.
4.5.1. Formulation of the control problem
Consider the linear time-invariant system using the state differential equation as a representation of the process: X t
AX t BU t
Y t CX t DU t
[4.6]
where X t IR n is the state vector, U(t) the input vector and Y(t) the output vector. A, B, C and D are matrices with appropriate dimensions. Without loss of generality information, we will suppose, for the continuation, that the matrix D is zero. The desired objective is to synthesize a linear control law, such that the poles of the controlled system coincide exactly with the zeros of a polynomial chosen by the user: P s s n a n 1 s n 1 ! a1 s a 0
[4.7]
The law of control, if it exists, which satisfies the preceding objective, is called: state feedback by pole placement. Such a name is justified by the fact that each pole of the resulting closed loop is assigned in the complex plan with a position chosen in advance by the user so as to guarantee the stability of the closed loop, as well as the other performances of the regulation.
4.5.2. The structure of the control law
To achieve the preceding goal, we propose using a control law of the form: U t FX t GYc t
[4.8]
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where the terms F and G indicate the parameters of the control law. Process [4.6] in closed loop with control law [4.8] is represented by the diagram block in Figure 4.7. This diagram highlights the two actions of the controller.
+
YC
U
Y Process
G
X F
Controller
Figure 4.7. State feedback
The first action, known as direct, produces term GYC(t). It is obvious that the tracking properties of the control device will especially depend on this action, i.e. the parameter G will have to be selected so that the tracking error H(t) = YC(t) –Y(t) is “small”. The second action generates the term – FX(t) and justifies the name “state feedback” allotted to the control. However, we notice that this control law suffers from a disadvantage which seriously limits its practical applicability. Indeed, this disadvantage is due to the assumption that the state vector is directly measured. However, in reality, the states are not all measured, for various reasons, e.g. economic, technological, physical, etc.
4.5.3. Reconstruction of the state
In this section, we present a solution for the practical implementation problem of a state feedback control law. It consists of building a system, whose inputs are U(t) and Y(t) and whose exit is a vector Xˆ(t) , estimated from the state vector X(t) of the process. Moreover, we desire that the variation: ~ X t
X t Xˆ t
which represents the estimation error tends towards zero exponentially.
[4.9]
Continuous-time Linear Control
U Y
109
^ X State Observer
Figure 4.8. State observer
Process 1
B
X(t)
s s
C
Y(t)
A U(t)
L 1 1
B
ss
^ X(t)
C
^ Y(t)
A
Observer
Figure 4.9. Block diagram of the process and of the observer. Notice that the observer contains a copy of the process
An observer is thus a system which provides an estimate Xˆ t of the state vector X(t), which is unknown, and based only on available information, i.e. the input and the output of the process. Within this framework, we present a particular observer called a Luenberger observer [LUE 67], which can be presented in the dual form of the controller using pole placement. In general, the observer can be written in the following form:
Xˆ t AXˆ t BU t L Y t Yˆ t with any X(0) Yˆ t CXˆ t
[4.10]
This form proves that the structure of the observer is in fact identical to that of the process (see [4.6]) with the only difference that the state equation of the observer
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is corrected by the term L Y t Yˆ t . The latter returns account for the difference between the output Y(t) of the process and what we can regard as the output of the observer, i.e. Yˆ t . The process-observer unit is further illustrated by Figure 4.9.
4.5.4. The controller as a combination of state feedback and an observer
4.5.4.1. Design of the controller Let us recall that the process that we want to control is a time-invariant linear system represented by a state model of the form: X t AX t BU t Y t CX t
[4.11]
We assume that the process is stabilizable and detectable. Let us recall that an invariant system is known as stabilizable if the uncontrollable part of the system is exponentially stable.2 In the same way, a system is known as detectable if the nonobservable part is exponentially stable. The detectability of the process guarantees to us the existence of the matrix gain L, such that the state Xˆ t , generated by the Luenberger observer, tends exponentially towards the state X(t) of the process. As for the assumption of stabilizability, it ensures the existence of the matrix gain F, such that the control law: U t FX t GYc t
stabilizes the process. The disadvantage of this control law is that it is not implementable, in the general case, since the state vector X(t) is not accessible to measurement. As Xˆ t is an estimate of X(t), the idea which comes naturally to mind is to substitute Xˆ t for X(t) in the control law. This gives a new control law: U t FXˆ t GYc t
[4.12]
X t
A t X t is exponentially stable if there are 2 positive
2 By definition, the system
real numbers D, E such that:
t t t0 X t d De E t t0 X t0
For an invariant-time system, exponential stability merges with asymptotic stability.
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Although this step seems natural, we can wonder whether it is legitimate, from the point of view, in particular, of the stability and performances of the closed loop system. We will thus interest ourselves in these equations. These equations will lead us to the separation theorem, which makes it possible to solve this question. The complete diagram of the system provided with its control is illustrated by Figure 4.10.
s
YC(t)
+ G -
1 s
Observer
F
Controller
Figure 4.10. Block diagram of a controller which combines state feedback with an observer
4.5.4.2. Closed loop equations Let us write the simultaneous differential equations of the process, the observer and of the controller: X t
AX t BU t
Xˆ t
AXˆ t BU t L Y t CXˆ t
[4.13a]
[4.13b]
U t FXˆ t GYc t
[4.13c]
Y t CX t
[4.13d]
~ introduce the estimation error: X t
~ X t Xˆ t y X t
X t Xˆ t .
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We obtain, starting from the first two equations above: ~ X t
A LC X~ t
[4.14]
The control law can be expressed in the form: ~ U t FX t FX t GYc t
[4.15]
Because of the presence of the observer, the dimension of the closed loop system is double that of the system without correction, that is to say 2n. While choosing the § X t · state vector ¨¨ ~ ¸¸ , we obtain the following equations of state and observation: © X t ¹ § X t · °¨¨ ~ ¸¸ °© X t ¹ ® ° §¨ Y t ·¸ ° ¨©U t ¸¹ ¯
BF º§ X t · § BG · ª A BF ¨ ~ ¸¨ ¸Yc t « 0 A LC »¼¨© X t ¸¹ ¨© 0 ¸¹ ¬ 0 º§ X t · § 0 · ªC « F F »¨¨ X~ t ¸¸ ¨¨ G ¸¸Yc t ¬ ¼© ¹ © ¹
[4.16]
4.5.4.3. Separation principle This arises directly from the diagonal form per blocks of the evolution matrix of the closed loop system (see [4.16]). By assuming the entire state to be measured we see appearing on the diagonal: matrix A-BF of the closed loop system and matrix ALC from the observer. Theorem. The set of eigenvalues of the closed loop system consists of the union of the eigenvalues of the observer and the eigenvalues which would be obtained if we directly established the control law starting with the state of the system. Proof. Indeed, the characteristic equation of the closed loop system is: § ª A BF det ¨¨ sI « ¬ 0 ©
BF º · ¸ A LC »¼ ¸¹
0
which develops in the form: det sI A BF . det sI A LC 0
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113
which is a remarkable result. The substitution of X(t) by Xˆ t does not modify the eigenvalues obtained during the calculation of the control law, but simply associates them with the eigenvalues of the observer. Moreover, there is a complete separation between the problem of control itself, i.e. the research of the matrix F, and the calculation of the observer. The independence of these two stages simplifies the problem resolution. We can deduce, in particular, that the stability of the closed loop system is not affected by this substitution; it is guaranteed if the control law starting from the state stabilizes the system (A – BF stable) and if the observer is without bias (A – LC stable). This result also gives us an indication of the choices of the eigenvalues of the observer, when this one is calculated by the modal approach (the control fixes the eigenvalues of A – BF). To avoid the behavior of the closed loop system being modified in a significant way, it is appropriate that the rebuilding of the state is fast in front of the dynamics of the closed loop system. However, an excessively fast rebuilding requires a very large gain L, which is likely to make the observer very sensitive to the noises of measurement (which are always present in practice even if we suppose them to be zero within the framework of the study). The dynamics of the observer thus result from a compromise between these two considerations.
4.5.4.4. Transfer function In order to supplement our analysis of the closed loop system, it is interesting to examine in terms of transfer function the behaviors obtained when the control law is established starting directly from the state or when an observer is used. In the first case, the system and the control law are written: X t
AX t BU t
U t FX t GYc t
[4.17a] [4.17b]
The closed loop system is thus given directly by: X t
A BF X t BGYc t
which is written in the Laplace field as: sX s x 0
A BF X s BGYc s
[4.18]
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Control Methods for Electrical Machines
i.e. X s
sI n A BF 1 BGYc s sI n A BF 1 x 0
[4.19]
In this relation, the first term highlights the matrix of transfer between the instruction (Yc(t)) and the state for zero initial conditions (X(0) = 0); the second term represents the evolution of the system in free mode (Yc(t) = 0). The poles intervening in each of the two terms are the eigenvalues of A – BF. When the control law is carried out using an observer, the closed loop system is written according to [4.16]: § X t · ¨ ~ ¸ ¨ X t ¸ © ¹
ª A BF « 0 ¬
BF º§ X t · § BG · ¨ ~ ¸¨ ¸Yc t A LC »¼¨© X t ¸¹ ¨© 0 ¸¹
and by taking the Laplace transform: sX s X 0 ~ ~ sX s X 0
A BF X s BFX~ s BGYc s A LC X~ s
~ from where, after elimination from X s we deduce: X s
sI n A BF 1 BGYc s sI n A BF 1 X 0 ~ sI n A BF 1 BF sI n A LC 1 X 0
[4.20]
The first two terms of this expression are identical to those in [4.19]. This means in particular that for initial null conditions ( X 0 0 and X 0 0 or in practice when the free mode is extinct), the closed loop system transfer function, obtained via the observer, is rigorously the same as the transfer function we obtain by directly applying the control to the real state. The third term of equation [4.20] corresponds to the free mode caused by an initial error of rebuilding. We find here for poles the eigenvalues of two matrices A – BF and A – LC, in accordance with the separation theorem.
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115
4.5.5. Design of feedback gain matrix F
The control which places poles with the state observer provides the user with the option of choosing the location of the closed loop poles. In order to make the system stable, it is of course necessary to choose eigenvalues with negative real part. However, this condition is not enough to guarantee the desired performances. Let us suppose that the desired performances are given in terms of speed (rise time for example) and of overshoot (damping ratio). We will be able to translate these performances in the complex plan. Thus, to obtain a correct transient mode, we will have to place the eigenvalues of A – BF in part of the complex plan, called the performance zone (see Figure 4.11) delimited by: – a vertical line corresponding to a sufficiently fast exponential attenuation; and – two oblique half-lines corresponding to a sufficient damping of the oscillations.
lm Minimal damping
Minimal speed Performance zone
Re
Figure 4.11. Performance zone delimited by the vertical line corresponding to a sufficiently fast attenuation of the exponential. The two oblique half-lines correspond to a sufficient damping of the oscillations
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Control Methods for Electrical Machines
Once the performance zone is determined, it is no longer necessary to choose the particular locations inside this zone, where the various poles of regulation of the closed loop will be placed. This choice can be guided by the following considerations: – The further the poles are located from the border of the performance zone, the more rapid the control device will be. Conversely, the faster the system, the more significant the variation of the control signal. However, a very variable control law tires the actuator, etc. – If the poles of the process are already in the performance zone, it is preferable to place the poles of regulation at the same places as those of the process. – Thus, certain poles of the process are located outside the performance zone, and we proceed as follows: - by retaining the place of the poles located inside the zone; - by substituting the poles located outside for new poles located on the border, as indicated in Figure 4.12. a)
b) Im
Im
x
x
x
x Re
x
Re
x x
x
x
x
x
x
a) Poles of the process
b) Poles of the closed loop
Figure 4.12. Choice of the location for the closed loop regulation poles
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117
4.6. Optimal control In the preceding section, we showed that under certain conditions, a linear system could be stabilized by a linear control and that the eigenvalues of the closed loop system could be arbitrarily selected. However, in practice, the desired dynamics of the closed loop system is necessarily limited by the maximum amplitudes that the actuators can transmit. The control must take account of this constraint, where the idea is to translate in the form of a criterion to optimize the compromise which must adapt the control to take account of the constraints.
4.6.1. Optimal regulator at continuous time 4.6.1.1. Formulation of the problem at the finished horizon Consider the continuous linear system: X t A t X t B t U t ;
X t IR n ; U t IR m
[4.21]
with the initial condition: X t0 X0 . The problem of the optimal regulator at the finished horizon consists of determining the control u opt t , which optimizes the criterion: t
J
>
@
1 1 T 1 X t Q t X t U T t R t U t dt X T t1 SX t1 ³ 2t 2
[4.22]
0
where: – Q(t) is a symmetric positive semi-definite matrix; – R(t) is a symmetric positive definite matrix; – S is a symmetric positive semi-definite matrix; – t0 and t1 are fixed and finished. We will suppose in what follows that the matrices A(t), B(t), Q(t) and R(t) are continuous functions of t, bounded on [t0, t1]. The objective of the regulator problem is to bring back as quickly as possible the state in the vicinity of the origin, the solution sought here corresponding well with a compromise between the previous goal (first term of the criterion) and the means implemented to reach that point (second term). In addition, the last term of the
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Control Methods for Electrical Machines
criterion translates the particular importance attached when the state is finally reached.
4.6.1.2. Solution The optimal control of the problem formulated above is given by: U opt t L t X t R 1 t B T t P t X t
[4.23]
where P(t) is the single positive semi-definite symmetric solution of the differential Riccati equation: P t P t A t A T t P t P t B t R 1 t B T t P t Q t
[4.24]
satisfying the terminal condition P t1 S. We note with interest that the solution obtained is a control in closed loop, for which the expression is independent of the initial state; moreover, the expression is linear according to the state. The closed loop system can be represented in the following way. 0
+
U (t)
+
1 s
B(t) -
X(t)
+
A(t)
R -1 (t)B T (t)P(t)
Figure 4.13. Structure of the optimal regulator at continuous time
However, the determination of P(t) is obtained by integrating the differential equation backwards, starting from the final condition, which forces us to calculate P(t) on the entire interval [t0, t1] before applying the control. 4.6.1.3. Optimal regulator at the infinite horizon Above, we provided the solution corresponding to one final moment t1 finished. In practice, it is often interesting to control a system with a large time interval [t0, t1],
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119
even freeing it from the existence of the upper limit. We will thus study the asymptotic properties of the solution obtained previously when t1 o f . If we restrict t1 towards infinite time, the presence of the final cost in the expression of the criterion does not increase the benefits, If the matrix Q is well chosen, convergence of the criterion ensures that the state converges towards 0. We will thus only be interested if S = 0. Let us recall that the optimal control obtained in the preceding section depends on the final moment t1, due to the terminal condition associated with the Riccati equation. We will thus note here P(t, t1), which is the matrix solution of this equation for the moment t and the state X(t). Let us consider the problem of the optimal regulator formulated above. Let us suppose moreover that S = 0 and that system [4.21] is: – completely commandable for all t [t0 , d[ ; – exponentially stable; and then that: – the solution P(t, t1) of the Riccati equation with the terminal condition P(t, t1) = 0 admits a positive semi-definite limit P t when t1 o f ; – P t checks Riccati equation [4.24]; – the expression of the optimal control for the criterion, t
J
>
@
1 1 lim ³ X T t Q t X t U T t R t U t dt 2 t1of t
[4.25]
0
is given by: U opt t R 1 t BT t P t X t
[4.26]
4.6.1.4. Optimal time-invariant regulator at the infinite horizon If the system to be controlled is time-invariant, it admits a representation of state with matrices A and B that are time-invariant: X t AX t BU t ; X t IR n ; U t IR m
with the initial condition: X t0 X0 .
[4.27]
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Control Methods for Electrical Machines
The problem of the optimal regulator invariant at infinite horizon is then to determine the control U opt t which optimizes the criterion: t
J
>
@
1 1 lim ³ X T t QX t U T t RU t dt 2 t1of t
[4.28]
0
where Q(t) is a symmetric positive semi-definite matrix and R(t) is a symmetric positive-definite matrix. If system [4.27] is stabilizable, then: – the solution P(t, t1) of Riccati equation [4.24] with the terminal condition P(t, t1) = 0 admits a positive semi-definite limit P t when t1 o f ; – P t is the solution of the algebraic Riccati equation: P A AT P P BR 1 B T P Q
[4.29]
0
– the expression of the optimal control is given by: U opt t R 1BT P X t LX t
[4.30]
If [4.27] is stabilizable and Q1 2 , A is a detectable pair, where Q 1 2 indicates
T Q1 2
any rectangular matrix such that Q 1 2
Q , then:
– P t is the single positive semi-definite solution of algebraic Riccati equation [4.29]; – the closed loop system is asymptotically stable.
Moreover, if Q1 2 , A is observable, P t is definitely positive. NOTE. In all the cases presented above, the solution obtained is a linear control in closed loop, a function of the complete state vector: U t L t X t .
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121
4.6.2. Stochastic optimal regulator at continuous time
When the process to be controlled is subjected to random disturbances, these can often be represented by a system of state equation excited by a white noise, so that the system formed by the process and the disturbance is in the form of: X t
A t X t B t U t v t
[4.31]
where v(t) is a white noise zero mean of covariance:
^
`
IE v t v T t W
V t G W
[4.32]
and X(t0) is a random variable, not correlated with v, of average quadratic value:
^
`
IE X t0 X T t0
X0
[4.33]
For a given input u(t) and a given realization of noise, the criterion employed in the deterministic case takes a certain value, which it is useless to seek to optimize because the evolution of the noise is unpredictable. We will on the other hand seek to optimize the average for all the possible achievements of the noise, by considering the criterion:
J
t ½ 1 °1 T ° IE ® ³ X t Q t X t U T t R t U t dt X T t1 SX t1 ¾ 2 °t °¿ ¯0
>
@
[4.34]
The optimal linear control in the presence of a white noise on the state is identical to that obtained in the deterministic case: U opt t R 1 t B T t P t X t
[4.35]
° P t P t A t AT t P t P t B t R 1 t B T t P t Q t ® °¯ P t1 S
[4.36]
with
in the same way, when we restrict t1 towards infinite time, we can show that the optimal solution obtained in the deterministic case also optimizes the criterion:
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Control Methods for Electrical Machines
t1 ½ 1 1 ° ° IE ® ³ X T t Q t X t U T t R t U t dt ¾ lim 2 t1of t1 t 0 °t °¿ ¯0
>
J
@
[4.37]
Generally, we will be able to extend, to the stochastic case, the methods implemented and the optimal controls obtained by treating the deterministic formulation of the various optimization problems considered in this chapter.
4.6.3. Discrete time optimal regulator
4.6.3.1. Formulation of the problem at the finished horizon Let the discrete linear system take the form: X k 1
A k X k B k U k ;
with initial condition: X k0
X k IR n ; U k IR m
[4.38]
X0 .
The problem of the optimal regulator at the finished horizon consists of determining the control U opt k which, in order to bring the state back into the vicinity of the origin, optimizes the criterion: J
1 k1 1 T 1 ¦ X k 1 Q k 1 X k 1 U T k R k U k X T k1 SX k1 2k k 2
>
@
[4.39]
0
where: – Q(k+1) is a symmetric positive-semi-definite matrix; – R(k) is a symmetric positive-definite matrix; – S is a symmetric positive-semi-definite matrix; and – k0 and k1 are fixed and finished. We will assume in the following that the matrices A(k), B(k), Q(k+1) and R(k) are limited for k = k0 … k1.
Continuous-time Linear Control
123
4.6.3.2. Solution A solution can be obtained rather quickly starting from Bellman’s principle of optimality [CUL 94]. The optimal control of the regulator in discrete time and finished horizon is given by the relations: U opt k L k X k ;
k0 b k b k1 1
[4.40]
1 £ ¦ L k ¡ BT k P k 1 Q k 1
B k R k °¯ ¡ BT k P k 1 Q k 1
A k °¯ ¦ ¦ ¢ ± ¢ ± ¦ T ¦ ¤ P k A k P k 1 Q k 1
¢A k B k L k ¯± ¦ ¦ ¦ P k1 S [4.41] ¦ ¦ ¥
The solution obtained is a linear control in closed loop, independent of the initial state. Note the analogy of the obtained solution with the problem in continuous time; be aware, however, that the differential Riccati equation is replaced here by a recurrence.
4.6.3.3. Optimal discrete time regulator at the infinite horizon We study here the asymptotic behavior of the solution above, when the final moment k1 tends towards the infinite moment. The results are identical to those of the continuous problem (by replacing the integrals with summations, and the differential Riccati equation with the recurrence above). Let us consider the problem of the optimal regulator formulated above. Let us assume moreover that S = 0 and that the system is either completely observable for k t k 0 or exponentially stable. Thus: – the solution, noted P k , k1 in [4.41] with the terminal condition P k1 , k1 0 admits a semi-definite positive limit P k when k1 l d; – P k checks [4.41]; and – the expression of the optimal control for the criterion: J
k1 1 1 lim ¦ X T k 1 Q k 1 X k 1 U T k R k U k 2 k1of k k
>
0
@
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Control Methods for Electrical Machines
is given by U k L k X k
where L k is obtained by replacing P(k) with P k in [4.41]. 4.6.3.4. Optimal invariant discrete time regulator at the infinite horizon For a time-invariant system: X k 1 AX k BU k ;
with the initial condition X k0
X k IR n ; U k IR m
[4.42]
X0 .
The optimal regulator problem at the infinite horizon consists of determining the control U opt k , which optimizes the criterion: J
k1 1 1 lim ¦ X T k 1 QX k 1 U T k RU k 2 k1of k k
>
@
[4.43]
0
where Q is a symmetric positive-semi-definite matrix and R is a symmetric positivedefinite matrix. If system [4.42] is stabilizable, then: P k , k1 in [4.41] with the terminal condition
– the solution, noted
P k1 , k1 0, admits a semi-definite positive limit P k when k1 o f ;
– the matrix P c Pc
P Q is the solution of the equation
AT P cA AT P cB B T P cB R
1 BT PcA Q
[4.44]
– the optimal control is given by
U k LX k B T P cB R
1 BT PcAX k
[4.45]
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125
4.6.4. Stochastic time optimal regulator
We suppose here that the system to be controlled is modeled by: X k 1
A k X k B k U k v k
[4.46]
where v(t) is a white noise zero mean with covariance:
^
`
IE v k v T k j
V k G kj
[4.47]
and X(t0) a random variable, not correlated with v, of average quadratic value:
^
`
IE X k 0 X T k 0
X0
[4.48]
The problem of the optimal regulator is then to optimize the criterion:
J
½° 1 ° k11 T 1 IE ® ¦ X k 1 Q k 1 X k 1 U T k R k U k X T k1 SX k1 ¾ 2 °k k 2 °¿ ¯ 0
>
@
[4.49]
where Q(k+1) is a symmetric positive-semi-definite matrix, R(k) is a symmetric positive definite matrix and S is a symmetric positive-semi-definite matrix. The optimal linear control in the presence of a white noise on the state is identical to that obtained in the deterministic case: U opt k L k X k
[4.50]
where L(k) is defined by [4.41].
4.6.5. LQG/H2 control
Again taking the optimal control problem, if the system is disturbed by noises and account holding, it is due to the fact that at the time of implementation of this control, we cannot have all the state measurements. It is possible to seek the observer gain which optimizes the variance of the reconstruction error. The observer thus determined is called the “Kalman filter”.
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4.6.5.1. The discrete time case Our process is modeled by the following equations: X k 1
A k X k B k U k v x k
[4.51a]
Y k C k X k v y k
[4.51b]
We will assume that our process is stabilizable and detectable. Moreover, LQG (linear quadratic Gaussian) control consists of making the probabilistic assumption that noises vx and vy are white, Gaussian respective covariance matrices Q(k) and R(k) and of seeking the regulator which minimizes the criterion:
J
ª k1 1 ½°º 1 ° lim « IE ® ¦ X T k 1 Qc k 1 X k 1 U T k Rc k U k ¾» [4.52] 2 t1of « °k k °¿»¼ ¬ ¯ 0
>
@
where Qc(k+1) and Rc(k) are positive weighting matrices. The solution of this problem respects a principle of separation (identical to the case of the state feedback control). The regulator consists of two functions: an optimal statistical filter (Kalman filter [LAB 78]) and an optimal linear return (LQ) on the reconstructed state. The Kalman filter makes it possible to reconstruct the state of system [4.51] starting from the observation of the controls and the outputs. Using this approach, two filters can be obtained simultaneously. Indeed, the reconstruction of X(k+1) fact of necessarily intervening U(k0), …, U(k) and outputs Y(k0), …, Y(k): we speak in this case of the predictor filter, another possibility being the use of outputs Y(k0), …, Y(k+1); in this last case, we speak about estimator filter. Note that Xˆ k 1,k is the state reconstructed at the instant k+1 starting from the controls and outputs until the moment k (predictor filter), by taking account of Y(k+1) (estimator filter). When the reconstructor allows the calculation of a control, the limited speed of the calculator necessarily introduces a shift. We prefer to use a filter predictor which makes it possible to calculate Xˆ k 1,k and thus U(k+1), the application of the preceding control U(k). Thus, the task of the calculator is simplified. The equations which make it possible to calculate a predictor filter are:
Continuous-time Linear Control
Xˆ k 1, k K( k )
>
@
A k Xˆ k , k 1 B k U ( k ) K ( k ) Y k C k Xˆ k , k 1
>
@
1 A( k )P k , k 1 C k T R k C k P k , k 1 C k T
P k 1, k
127
>A k K k C k @P k , k 1 A k T
[4.53]
Q k
To initialize the filter, we must assume that the initial state is known, along with the variance of the error. We then adopt: Xˆ k 0 IE^X k 0 ` X 0
P k 0 IE ® X k 0 X 0 X k 0 X 0 ¯
T ½¾¿
P0
[4.54]
the first relation ensures that the estimator is without skew, the zero error remaining on average at any moment k t k0. The optimal regulator in the reconstructed state is given by [4.50], in which we substitute X(k) with Xˆ k ,k 1 : U opt k L k Xˆ k , k 1
>
[4.55]
@ >
@
1 T T ° L k B k Pc k 1 Qc k 1 B k Rc k B k Pc k 1 Qc k 1 A k ° T ® Pc k A k Pc k 1 Qc k 1 >A k B k L k @ ° P k , k 0 ° c 1 1 ¯
4.6.5.2. The continuous time case Our process is modeled by the following equations: X t
A t X t B t U t v x t
Y t C t X t v y t
[4.56a] [4.56b]
We will suppose, as for the discrete time case, that our process is stabilizable and detectable. We will also place ourselves within the framework of the LQG control, which consists of making the probabilistic assumption that noises vx and vy are white, Gaussian respective covariance matrices Q(t) and R(t), and of seeking the regulator which minimizes the criterion:
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J
t1 ½ 1 1 ° ° IE ® ³ X T t Qc t X t U T t Rc t U t dt ¾ lim 2 t1of t1 t 0 °t °¿ ¯0
>
@
[4.57]
where Qc(t) and Rc(t) are positive weighting matrices. The solution of this problem respects a principle of separation (identical to the case of state feedback control). The regulator will consist of two functions: an optimal statistical filter (Kalman filter) and an optimal linear return (LQ) on the reconstructed state. In the continuous case, the Kalman filter is given by the following equations:
Xˆ t
A t Xˆ t B t U t K t Y t C t Xˆ t
K t
P t CT t R 1 t
P t
A t P t P t AT t P t C T t R 1 t C t P t Q t
[4.58]
with the initial values: Xˆ t 0 IE^X t 0 ` X 0 ° T½ ® °¯ P t 0 IE ®¯ X t 0 X 0 X t 0 X 0 ¾¿
P0
[4.59]
The optimal linear control is: U opt t Rc1 t BT t Pc t Xˆ t
[4.60]
with ° P t P t A t AT t P t P t B t R 1 t BT t P t Q t c c c c c c ® c °¯ P t1 0
4.6.5.3. Control by H2 optimization To solve an optimal control problem by optimization H2 we can use the algorithmic tools of LQG control [LAR 96]. The difference between the two methods is at the level of interpreting the matrices R, Q, Rc and Qc.
Continuous-time Linear Control
129
LQG control assumes a stochastic context, where the matrices R and Q are presumed to be known variance matrices, but from a practical point of view these are generally not known exactly, and Rc and Qc are the weightings given in a criterion (possibly with an economic interpretation). Optimization H2 poses a problem where the four matrices R, Q, Rc and Qc are synthesis parameters intended to regulate compromises of performance in the closed loop system.
4.7. Choice of a control
The choice of a control remains a delicate problem [MIN 99]. In this section we highlight the various degrees of freedom at the disposal of a user wishing to implement a control. The first choice to be made is between the “continuous” approach and the “discrete” approach. Indeed, the process to be ordered is in general analog, while the development of the control is completed using a microprocessor. Two approaches are possible. We can, on the one hand, calculate the control by assuming the whole of the system (regulating and process) is analog, therefore adopting the continuous approach to discretize the obtained regulator equations. The validity of this step presumes that the period of sampling is small compared to the dynamics of the buckled system, so that the correction where possible approaches the calculated analog correction. In this case, the discretization can be carried out in a simplified way by replacing the derivative with differences, integrations and equations with the recurrences, etc. This approach is often adopted for the PID synthesis (which is calculated analogically then transposed digitally). In addition, a discrete model of the process and analog-to-digital and digital-toanalog converters (samplers and blockers) can initially be calculated, and the problem solved, by adopting the discrete formulation leading directly to the equations of the regulator. This approach is more rigorous due to the fact that the period of sampling in front of the dynamics of the closed loop system is not necessarily small. If we retain a control law like pole placement, state feedback or optimal control, the choice of the eigenvalues of the closed loop system or the criterion weighting matrices must make it possible to regulate the dynamics of the closed loop system, taking into account the antagonistic constraints: obtaining a stable dynamic that is more rapid than in open loop, while avoiding overly strong actuator stresses. Moreover, if the system is disturbed and the state is not entirely accessible, the Kalman filter is essential, while in the case with no noise an observer is sufficient.
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4.8. Bibliography [CUL 94] CULIOLI J.C., Introduction à l’optimisation, Ellipses, Paris, 1994. [LAB 78] LABARERE M., KRIEF J.P., GIMONET B., Le filtrage et ses applications, Cépaduès, Toulouse, 1978. [LAR 96] DE LARMINAT P., Automatique, commande des systèmes linéaires, Hermes, Paris, 1996. [MIN 99] MINISTRY OF ECONOMICS, FINANCES commande avancée, Industry editions, 1999.
AND
INDUSTRY, Guide des solutions de
[PHI 00] PHILLIPS C.L., HARBOR R.D., Feedback Control Systems (4th edition), Prentice Hall, 2000. [ZIE 42] ZIEGLER, J.G., NICHOLS N.B., “Optimum settings for automatic controllers”, Trans. ASME, no. 64, p. 759-768, 1942.
Chapter 5
Overview of Various Controls
5.1. Introduction Any device of automated control requires a certain modeling of the process which is to be controlled. In the case of the PID, even reduced to its simplest version, this concept exists behind the principle of the “feedback” which supposes that we know at least a priori the way the action is to be carried out to reduce the variation observed. On the other hand, it is clear that a system whose dynamic behavior is perfectly suited to be modeled, and whose disturbances to which it is subjected are completely observable and measurable, can be ordered completely in “open loop”, i.e. without “feedback”. The controls required to transform this system from one state to another can be calculated exactly from all these elements. This concept of modeling of the process is at the core of the techniques of advanced control. To implement an application of advanced control must overlook the development of a model of the process to order so as to limit the role of the simple “feedback”, and to increase the share of the anticipation permitted by the model in the development of the controls. If this concept of model is omnipresent in advanced control, the ways of implementing it are numerous; there are various ways of working out, of representing and of using these models of the process which will guide us in the definition of the various types of advanced control. Chapter written by Frédéric KRATZ.
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The first cutting consists of distinguishing between algorithmic approaches and heuristic (or cognitive) approaches. The algorithmic approach rests on a modeling of the process based on mathematical equations representative of the physical behavior of the system (in its spontaneous evolution and its response to the external stresses). This model is exploited in the form of a programming algorithm. The cognitive approach is based on a “situations-actions” model of the process, which is based on an empirical description of the process’s response to the various stresses. Let us note that we should not oppose these two types of approach and that they appear more and more frequently complementary. Indeed, the “elaborate” – or advanced – approaches to process control are, with increasing frequency, using a combination of heuristic and algorithmic approaches: this type of union has proved reliable in a large number of cases. In this chapter, we will quickly present some techniques of control known as advanced controls.
5.2. Internal model controller 5.2.1. Introduction For the psychologist analyzing the work of a human operator, a “traditional” approach consists of saying that the operator builds an operational image of the system or process which it leads. Indeed, just as a pilot knows the reactions of his vehicle (car, plane, etc.), the operator of a process has a precise idea of the response times and gains of the various chains of his process. This operational image can, according to the human operator, take a more or less abstract form and can in extreme cases be a mathematical model; this is why it is more generally called an “internal model”, an abstract representation of the controlled process. If we replace the human operator with a calculator, then the approach described above consists of equipping the calculator with an internal model of the process. We will show how to pass from a traditional structure of regulation to an internal model control strategy, then we will develop the properties and we will conclude with the rules of implementation. The whole of this section is presented in a simple way, for didactic reasons using a monovariable system – as the extension to the multivariable systems does not present difficulties.
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5.2.2. Passage of the structure of regulation to that of control by internal model Let us consider the traditional loop of regulation (Figure 5.1) of a process represented by its transfer function G(s), a controller C(s), a reference signal YC and a stationary disturbance d(s) at the output of the process. d(s) YC (s) +
Y(s)
U(s) C(s)
G(s)
-
Figure 5.1. Principle of traditional regulation
Let us suppose that we have at our disposal a model of the process GM(s); we would like to use this knowledge in the loop. Figure 5.2 below uses this knowledge while preserving the performances. d(s) YC (s)
H(s)
U(s) C(s)
G(s)
GM(s)
GM(s)
Y(s)
-
-
-
W(s)
Figure 5.2. Introduction of the internal model interns in the closed loop structure
Indeed, if we compare control signals U(s) in both cases, we respectively obtain: Case 1:
U s C s >YC s Y s @
Case 2:
U s C s >H s G M s U s @
H s YC s W s from which after calculation we obtain: U s C s >YC s Y s @
W s Y s G M s U s
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There is thus equivalence between the two representations. In the traditional closed loop diagram, we assume that the model of the process is perfect and we make use of it to determine the controller or the control law. In Figure 5.2, the difference between the process and knowledge that we have is revealed. If we compare the loop ^C s , G M s ` with a controller D(s), the control structure given in Figure 5.3 is obtained. d(s) YC(s)
H(s)
U(s) D(s)
Y(s)
G(s)
GM(s)
W(s)
Figure 5.3. Block diagram of a closed loop with a controller based on the internal model principle
5.2.3. Properties of the control by internal model We showed above the passage of traditional regulation to internal model control by explicitly introducing the process model GM(s) into the control loop. We will examine the properties of the internal model control by studying more particularly the control signal U(s), the return signal W(s) and the output signal Y(s). For reasons of facility of presentation, we will not take into account the constraints which can intervene on the level of the control U(s) (saturations in amplitude and or speed). The return signal W(s) is written: W s G s U s d s G M s U s ,
however: U s D s >YC s W s @ , then:
W s
>G s G M s @D s YC s W s d s
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135
which highlights that W(s) looping directly uses the difference between the process and the internal model. From this we obtain: W s
>G s G M s @D s Y s d s C 1 >G s G M s @D s 1 >G s G M s @D s
[5.1]
5.2.3.1. Analysis of properties of control by the internal model We place ourselves in the ideal case where the internal model is adapted perfectly, i.e. GM (s) = G(s). Then: W(s) = d(s) Looping makes it possible to estimate the disturbance d(s) (see Figure 5.4). If the disturbance is zero, d(s) = 0 involves W(s) = 0; we find a structure in open loop; but as D(s) contains the process, we find the traditional control. YC (s)
U(s)
D(s)
Y(s) G(s)
Figure 5.4. Control by internal model in the ideal case GM(s) = G(s)
Y s G s D s YC s with D s
C s 1 G s C s
so: Y s
G s C s YC s 1 G s C s
In general, the control by internal model includes: – a classical control; – a perturbation estimation; – an error signal as a difference between the process and its internal model.
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The following question that we will study is the computation of the control signal U(s): U s D s YC s W s
By replacing W(s) with the expression given by equation [5.1], we obtain: U s
§ · >G s GM s @D s Y s d s ¸ D s ¨¨ Yréf s C 1 >G s GM s @D s 1 >G s GM s @D s ¸¹ ©
D s
YC s d s 1 >G s G M s @D s
[5.2]
YC s d s d s 1 >G s G M s @D s
[5.3]
and Y s G s D s
5.2.3.2. Analysis of stability We study at the same time the stability of the control law and that of the loop. The stability of a continuous linear system is given by the position of the roots of the equation characteristic with respect to the complex left half-plane, i.e. with a negative real part. U s
D s >YC s d s @ 1 >G s G M s @D s
Its characteristic equation is written: 1 >G s G M s @D s 0 or
1 G s G M s 0 D s
[5.4]
With respect to the output, the characteristic equation is written: G s G M s 1 G s D s G s
0,
In the ideal case: G s G M s .
[5.5]
Overview of Various Controls
137
1 0 which lays down a stable control structure and D s 1 0 which lays down that the process is stable. equation [5.5] involves G s D s
Equation [5.4] involves
To implement a internal model control, it is necessary that the G(s) process is stable and that the controller D(s) is also stable. If the process is naturally unstable or has a pure integrator, it will be necessary, as a preliminary step, to stabilize the process by a local loop. In general, the characteristic equations use the difference between the transfer functions of the process and of the internal model, and we see in a clear way the link between stability and robustness.
5.2.3.3. Choice of the control law D(s) In the ideal case, we have the structure in open loop of Figure 5.4. A simplistic approach consists of saying: in order for Y(s) to resemble YC(s) as much as possible, it is enough to choose the control law D(s) as the inverse of the process. Actually, this law is known only through its internal model; thus, we will choose: D s
1 G M s
The closed loop transfer function of the system controlled by an internal model (Figure 5.3) by again using equation [5.3] is written: Y s
G s G M s >Y s d s @ d s G s G M s C 1 G M s
where: Y s YC s
Theoretically, even if the internal model is not adapted perfectly, the output copies the reference signal on the condition of being able to invert GM(s). It is not, however, always possible to invert GM(s), and it must be stable. If GM(s) contains unstable zeros (zeros with positive real parts, thus a system with a non-
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minimum phase) or time delays, it is not possible to invert it. To solve this difficulty, we will separate GM(s) into two operators: one containing the invertible part of GM(s): G M s , the other G M s gathering the time delays and the unstable zeros and having a unit static gain. G M s G M s G M s
The internal model controller D(s) is obtained starting from the invertible part of the model to which a filter is added [MOR 88]: 1 D s F s G M s
[5.6]
In general, the filter F is chosen as a low-pass filter: F (s)
1 (1 c s)r
where Wc is the time-constant desired for the closed loop and where r is a natural integer ensuring the causality of D(s). Under these conditions, the closed loop becomes: Y s
1 G s F s G M s 1 1 >G s G M s @F s G M s
>YC s d s @ d s
then we obtain: Y s
G s F s >YC s d s @ d s G M s >G s G M s @F s
In the ideal case: G s G M s G M s G M s Y s
G s F s >YC s d s @ d s G M s
Y s G M s F s >YC s d s @ d s Y s G M s F s YC s 1 G M s F s d s
[5.7]
Overview of Various Controls
139
We see, according to equation [5.7], that it is impossible to compensate for time delays or non-minimum phases. 5.2.3.4. Study of steady state mode The final value theorem is:
Y f
lim sY s
s o0
Maybe if we assume a step disturbance as well as a step input: ª § Y0 d 0 · d 0 º G s D s ¨¨ ¸ lim s « », s ¸¹ s ¼ s o0 ¬1 >G s G M s @D s © s Y f
G 0 D 0 Y0 d 0 d 0 1 >G 0 G M 0 @D 0
The first factor must be equal to 1 so that there is no error in the steady state mode, i.e. G 0 D 0 1 >G 0 G M 0 @D 0
consequently: D 0
1 G M 0
In conclusion, there will be no steady state error if the gain of the controller is equal at the inverse to the gain of the internal model, regardless of the loss of adaptability of the internal model with respect to the process.
5.2.4. Implementation We will successively review the various stages of the construction of a internal model control: establishment of the internal model, the calculation of the control law and analysis of robustness.
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5.2.4.1. Establishment of the internal model Identification ensures that the establishment of the internal model is simplified (Strejc method) or advanced (technical numerical). It is, however, significant to recall that on the one hand the internal model will not be perfect whatever the method of identification used; and that on the other hand in the methods of PID tunings, for example, it is not necessary to have a very precise model. These remarks will be taken into account during the calculation of the control algorithm robustness.
5.2.4.2. Calculation of the control law We have shown that the inverse of the realizable part of the internal model provides a sufficient law of control, to which a filter (equation [5.6]) is added: 1 D s F s G M s
The determination of the control law is immediate if the internal model is represented in the form of a rational fraction. However, if the model is known in another form (impulse response for example), the inversion is not easily realizable and generally requires optimization ([RIV 87]).
5.2.4.3. Analysis of robustness Compared to the structure presented up to now, implementation introduces two additional operators: a trajectory generator T(s) to the reference signal and a controller R(s) (generally a lead-lag compensator) in the loop of return. Figure 5.5 has the structure of control for the implemented internal model. d(s) YC(s)
reference T(s)
Y(s) D(s)
+
+
U(s)
H(s) F(s)
G(s)
-
+ +
GM(s)
W(s) R(s)
Figure 5.5. Implementation of the internal model
Overview of Various Controls
141
If we examine robustness, we have: Y s d s
G s D s F s >YC s R s d s @ 1 >G s G M s @R s D s F s
[5.8]
The comparison of [5.8] and [5.3] shows that the controller R(s) acts on the variation G s G M s which is similar to its effect on the disturbance d(s). U s
D s F s >YC s R s d s @ 1 >G s G M s @R s D s F s
[5.9]
The analysis of the characteristic equation highlights the effect of R(s). Indeed, stability depends on the zeros of: 1 >G s G M s @R s 0 D s F s
The operators, controller R(s) and trajectory generator T(s), can be grouped together. If only the function of regulation is of interest, this approach is satisfactory. However, if we carry out tracking and regulation, the two operators should be dissociated.
5.3. Predictive control 5.3.1. Introduction Many alternative methods for the predictive control-based modeling (modelbased predictive control (MPC)) can be found in the literature [CAM 95], [BOU 96]. Nevertheless, they all fit on the diagram of control provided by Figure 5.6. Reference trajectory Reference
Reference model
Control signal Optimizer
Plant
Predictor
Figure 5.6. Diagram block of a predictive control
Output
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The control algorithm for every sampling time is as follows (see Figure 5.7):
>
@
– Prediction Yˆ k i , k , i 1, N p of the output of the process to be controlled, on the prediction horizon [k+1, k+Np] starting from the inputs applied to the process until the moment (k-1) and of the outputs measured until the moment k.
– Simulation, on the same prediction horizon, of the desired exit YC (k) obtained like the output of a reference model excited by the reference to be followed and representing the desired behavior in closed loop. – Optimization, compared to a sequence of future controls u(k+i) i >0 , N u 1@ of an index performance expressed in terms of the differences between the predicted outputs and the desired outputs, on the prediction horizon [k+N1, k+Np]. Nu ( d Np) called the control horizon; the posterior controls at the time (k + Nu - 1) are presumed to be constant, i.e. u(k + i) = u(k + Nu – 1) for i t N u . N1 t 1 depends on the delay of the process and the duration of the transient relating to the inversion of the direction of the instruction, in the case of a process with non-minimum phase. Only the first control signal u(k) of the optimized sequence is applied to the process; the prediction horizon is then shifted a step forwards, and all calculations are remade for the next moment (control at moving horizon).
reference consigne trajectory of reference : Y
C (k) YC(k)
predicted output
^(k+i,k) :Y
horizon of prediction Output of the process: Y(k)
k
k+N 1
k+N p
Overview of Various Controls
143
control horizon
k
k +N u - 1
Figure 5.7. Output and input signals for predictive control
The various methods of predictive control differ mainly by: – the type of model used to present the process and to predict its output; and – the performance index to be minimized. Among the methods most commonly used by industries are: – the MAC (model algorithmic control) method, which uses the impulse response model ([RIC 93]); – the DMC (dynamic matrix control) method, which uses the step response [MOR 85]; – the GPC (generalized predictive control) method, which uses a transfer function model or a state space model [ORD 93]. The advantages of a predictive control are in: – its applicability with various process classes (linear/non-linear, single/multivariable, stable/unstable, with/without time delay, with minimal/nonminimal phase, etc.); – its capacity to take into account constraints on the controls or their outputs, and to take into account disturbances as well as uncertainties on the parameters of the model;
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– its relative facility of tuning, related to the fact that the parameters to be adjusted have a physical sense and thus their variations have a foreseeable effect.
5.3.2. General principles of predictive control
5.3.2.1. Model and predictor The Y(k) output of a causal linear system is expressed like the convolution of its impulse response with the input U(k): Y k
f
¦ h jU k j
[5.10]
j 1
The coefficients hj are the coefficients of the impulse response, because they represent the values of the output when the system is excited by an impulse unit. Equation [5.10] implies that the system admits a delay unit (h0 = 0). An unspecified delay d can also be taken into account by canceling the coefficients hj for j = 0, …, d-1. In the case of a stable system, the sum intervening in equation [5.10] can be truncated with order n: Y k
n
¦ h j U k j e k H q U k e k
[5.11]
j 1
where e(k) represents the error of modeling due to truncation, and also if required includes a term of disturbance. H(q) is a polynomial of degree n, defined as: H q h1q 1 h2 q 2 ! hn q n
[5.12]
here q-1 represents the backward shift operator: ( q 1U k U k 1 ). From equation [5.11], it is easy to deduce the output prediction with i steps ( 1 d i d N p ): Yˆ k i , k
n
¦ h j U k i j eˆ k i , k
[5.13]
j 1
where eˆ k i , k represents the prediction of the modeling error. In the absence of information on this error, its predicted value can be selected as constant on the prediction horizon, and equalizes with e(k):
Overview of Various Controls
145
n
eˆ k i , k e k Y k ¦ h j U k j
[5.14]
j 1
i.e. the variation, at the time k, between the output of the process and the output of the model. In the PFC (predictive functional control) method, Richalet et al. [RIC 87] propose predicting this error using a polynomial model: eˆ k i , k e k
M
¦ D m k i m
m 1
The coefficients Dm(k), m = 1, …, M are given as the solution (in the least squares sense) of a system of N a t M algebraic equations. These equations are obtained by considering the last variations e(j), j = k – Na +1, …, k filtered using a low-pass filter of degree M to reduce the effect of the high frequency disturbances of the measures. This procedure thus utilizes three tuning parameters: degree M, horizon Na and pole z0 of the filter (of multiplicity order M). It is important to note that the choice of these parameters has a strong influence on the precision of the predictor and the stability and robustness of the closed loop system. The equation of output prediction [5.13] can be broken up in the following way: Yˆ k i , k
i
n
j 1
j i 1
¦ h j U k i j ¦ h j U k i j eˆ k i , k
Yˆ k i , k Yˆ f k i , k Yˆl k i , k
where Yˆl k i , k
[5.15]
n
¦ h jU k i j eˆ k i , k represents the free response of the
j i 1
system, i.e. the value of the output at the time (k+i) predicted using the outputs measured until the time k and the control signals applied to the process until the moment (k-1); and where Yˆ f k i , k
i
¦ h j U k i j corresponds to the forced j 1
response, i.e. with the component of the prediction resulting from the action of the future controls which are to be optimized. By introducing the following vectors:
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ª Yˆ k 1, k « ˆ « Y k 2 , k « # « ˆ Y k N p ,k ¬«
Yˆ N p k
º » » ; Yˆ f k » Np » ¼»
ª Yˆ f k 1, k « ˆ « Y f k 2 , k « # « «¬Yˆ f k N p , k
º » » ˆl » ; Y N p k » »¼
ª Yˆl k 1, k « ˆ « Yl k 2 , k « # « ˆ Y k N p ,k ¬« l
º » » » » ¼»
equation [5.13] can be rewritten in a matrix form as: f Yˆ N p k Yˆ
Np
k Yˆ Nl k p
HU N p k Yˆ l
Np
k
[5.16]
where U N p k represents the vector of the future controls to optimize: U k ª º « U k 1 » « » « » # « » ¬«U k N p 1 ¼»
U N p k
and where Yˆ l
Np
k
is calculated using equations [5.15] and [5.14], and matrix H,
from dimensions (Np, Np) easily results from Yˆ f k i , k in [5.15]:
H
ª h1 « h « 2 « # « «¬h N p
"
0º " 0 »» # % #» » h N p 1 " h1 » ¼ 0 h1
5.3.2.2. Algorithm of the control law The performance index mainly used in MPC algorithms is the following quadratic criterion:
J U N p k
N N p 1 º 1 ª« p ˆ 2 ¦ Y k i , k YC k i ¦ O i U 2 k 1 » 2 «i N » i 0 ¬ l ¼
>
@
[5.17]
O i ! 0 is a weighting, generally considered constant and equal to O which makes it possible to carry out a compromise between the quality of the tracking and
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the energy cost, measured respectively using the first and second sum of the criterion. The prediction horizon [k+Nl, k+Np] represents the interval of time over which we minimize the standard deviations between the predicted and the desired outputs. In what follows, we will take Nl = 1. Let
us
introduce
>YC k 1
Np YC k
the
extended
vector
of
the
reference
trajectory
@
! YC k N p T on the prediction horizon. The criterion
can be rewritten in the following matrix form:
J U N p k
2º ª 1 2 » Np 1« ˆ « Y N p k YC k / 2 U N p k » 2« » ¬« ¼»
[5.18]
where / is a positive-definite diagonal weighting matrix, whose diagonal elements are the coefficients O(i). By using decomposition equation [5.16] of the predictor, and by omitting the temporal index to simplify the presentation, the criterion becomes:
J U Np
T
Np · § Np · 1 T 1§ l l ¨ HU N p Yˆ N YC ¸ ¨ HU N p Yˆ N YC ¸ U N /U N p p p p 2© ¹ © ¹ 2
which is developed as:
J U Np
1§ T · T ¨U N AU N p 2b U N p c ¸ p 2© ¹
[5.19]
with
A
H T H / ;b
N º ª H T «Yˆ l YC p » ; c N ¼ ¬ p
T
Np º ªˆ l Np º ªˆ l «Y N p YC » «Y N p YC » ¼ ¼ ¬ ¬
The minimization of criterion [5.19] is obtained by canceling the gradient using U Np :
wJ U N p UNp
AU N p b
0
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Matrix A is positive-definite, leading to the following solution: U Np
A 1b
>H
@
T H / 1 H T ªY N p Yˆ l º « C N p »¼ ¬
[5.20]
The first optimized control U(k) being applied to the process, is enough to
>
@
1
calculate the first line of the matrix H T H / of dimensions (Np, Np). In the case of DMC and GPC algorithms, which are expressed according to the increments of control 'U k i defined by: 'U k i U k i U k i 1
it is possible to decrease the computational load by introducing a control horizon Nu < Np. The control signals are assumed constant on the horizon k N u 1, k N p 1¯ , i.e. 'U k i 0 for i N u , N p 1¯ optimization is ¢ ± ¢ ± carried out compared to the following 'U N p k vector, of dimension Nu: 'U N p k
>'U k
'U k 1 ! 'U k N u 1 @T
[5.21]
The matrix to be inversed is then of reduced size (Nu, Nu). 5.3.2.3. Properties and extensions The law of control given by equation [5.20] is easy to implement from a numerical point of view. However, this implementation must be preceded by a tuning phase of the corrector’s parameters of synthesis, which characterize the predictor of the modeling error, the reference model (in general a first order or even a second order model) and the function cost (prediction Np, control Nu and weighting O horizons). In spite of the predictive control’s great success in industrial applications, in particular in the oil refineries, the chemical, petrochemical, metallurgical or agroalimentary industries (an investigation of the Ministry for Economy, Finances and Industry, launched in 1996, listed more than 10,000 in the world), many research tasks are still devoted to predictive control, and in particular to the following points: – choice of the parameters of synthesis and influence of these parameters on the performances and the stability of the closed loop system;
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– taking into account clarified constraints on the output and input signals in the optimization phase of the performance index; and – studying the robustness related to the influence of uncertainties of the process model on the performances in closed loop, and taking account of these uncertainties during the synthesis of the corrector.
5.4. Sliding control 5.4.1. Introduction
In this section, we are interested in the controls with fast commutation. The objective of this type of control is to control the state trajectory of the non-linear system towards a particular surface, chosen by the user, and to maintain the state trajectory on this surface. This surface is called the “switching surface”; however, the plant state trajectory is “above” the surface, a feedback path has one gain that changes if the trajectory drops “below” the surface. Thus, this surface defines the commutation law because the gain switches from one value to another. The commutation surface is also called the sliding surface (sliding manifold) because once the trajectory is brought to the surface, the commutated control maintains this trajectory on this surface. It is obvious that the significant point of the synthesis of this type of control law is the definition of the sliding surface which must ensure dynamic stability and performances of tracking and regulation of the system.
5.4.2. Structure of the control law
5.4.2.1. Modeling of the system We will study a particular class of non-linear system compared to the state x(.) and linear system compared to the control u(.) form: x t F x , t , u
f x , t B x , t u x , t
[5.22]
where
x t IR n , u t IR m , and B x, t IR nqm . Moreover, each component of f(x,t) and B(x,t) is assumed to be continuous and the derivative compared with x is continuous and limited. In the case of timeinvariant linear systems, equation [5.22] becomes:
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x t
Ax t Bu t
with A (n u n) and B (n u m) constant matrices. We can define the sliding surface of dimension (n – m) associated with the system: S
^ x,t IR n1 V x,t 0`
[5.23]
where
V x , t
>V 1 x , t
! V m x , t @T
0
If there is no temporal dependence, the sliding surface of dimension (n – m) in the state space IRn is determined by the intersection of m surfaces of dimension (n – 1) V i x , t 0 . These surfaces are conceived in such a way that the restriction of the state trajectory on the sliding surface has the desired behavior (from the point of view of stability, tracking, etc.).
5.4.2.2. The control law After having defined the sliding surface, the control u x , t >u1 x , t ! u m x , t @T is built according to the following structure: °u x , t if u i x , t ® i °¯u i x , t if
V i x , t ! 0 V i x , t 0
law
[5.24]
According to equation [5.24], we see that the control u(x, t) is not defined on the sliding surface. Outside the sliding surface, the values of controls u ir are selected such that the tangent vectors with the trajectory are directed towards the surface, so the states are directed towards and maintained on x, t 0. The control law u(x, t) is conceived so that the trajectory of the state vector is attracted by the sliding surface, and once the trajectory meets this surface it remains on this surface for every following moment. The state vector trajectory can thus be seen as slipping along the sliding surface and consequently the system is in slipping mode. An ideal sliding mode exists only when the state trajectory x(t) of the controlled system checks the equation: V x t , t 0 at every t p t1 . This requires infinitely fast switching. For a real system, because of the imperfections of the commutated
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regulator, delay, hysteresis, etc., we limit commutation to a finished frequency. The point representative of the trajectory of the state vector can oscillate in the neighborhood of the sliding surface. This phenomenon, known as “chattering”, is to be taken into account and must be eliminated within the possible limit [BAR 89]. To summarize, the sliding mode takes place at three times (see Figure 5.8): – the surface reaching phase; – the “chattering” around the sliding surface; – the sliding on the surface. reaching phase
x2
sliding mode
x1
Figure 5.8. Sliding mode
5.4.2.3. Existence condition of the sliding mode The existence condition of the sliding mode is to guarantee the motion of the state trajectory to the surface. The use of Lyapunov’s second method makes it possible to find a condition of this motion [UTK 78]. Taking as a Lyapunov function: V t , x ,V
Like
1 2 V x 2
[5.25]
dV t , x ,V dV x this gives us a sufficient sliding condition: V dt dt
VV 0
[5.26]
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5.4.3. Equivalent method control
The equivalent control is a control which, applied to the system, produces the evolution of the system on the sliding surface for all the times that the initial state is on this surface. Let us suppose that at the instant t1, the state trajectory meets the sliding surface, and that a sliding mode exists. The existence of a sliding mode implies that, at every t t t1 , V x t , t 0 and also x t , t 0 . We define the equivalent control ueq for systems given by equation [5.22] like a vector that satisfies:
V
wV wV x wt wx
wV wV wV f x , t B x , t u eq wt wx wx
0
[5.27]
wV If we suppose that the matrix product B x , t is regular for any t and any x, wx the equivalent control is given by:
u eq
ª wV º « B x , t » ¼ ¬ wx
1
§ wV wV · f x , t ¸ ¨ wx © wt ¹
[5.28]
and the sliding vector is: x
f x , t B x , t u eq
[5.29]
When the slide takes place, everything occurs as if the system were controlled by the equivalent control, also known as average control, on the set of switching which takes place. The control by sliding mode is a robust control in the sense where it is enough to ensure the existence of the mode of sliding, i.e. the attractivity of the sliding surface, so that the system evolves with the equation of the sliding mode.
5.5. Bang-bang control
In Chapter 4, we presented in section 4.6 optimal control whose objective is to find a control ready to lead the system towards the desired state, while minimizing a quadratic criterion including the final error, the trajectory follow-up and the control energy. In the present section, we are interested in a different strategy problem from control. Let the continuous linear system be:
Overview of Various Controls
X t
AX t BU t ;
X t IR n ,U t IR m
153
[5.30]
The objective of this control is to reach the final state of the state vector in a minimal time on the basis of a given initial state. This objective can be reformulated in the following way, seeking a control which minimizes the criterion: J t 0
T
³ 1dt
[5.31]
t0
The initial instant, the initial state and the final state are imposed. The final instant T is to be determined. The solution of this problem is to use a control with infinite energy, which does not make any sense and requires the problem to be reformulated: the obtained control must satisfy a constraint of saturation, such that: u t d 1
[5.32]
for all t >t 0 ,T @ . This constraint means that each component of the vector u(t) should not be larger than 1. In our example, this means that the control is forced in an acceptable field of IRm (for which our case is hyper-cubic). If the constraints on the components of u(t) have a value different from 1, it is enough to put on the scale the B columns in order to obtain a constraint on u(t) identical to that of equation [5.32]. The problem of optimal control posed here is thus to find a control u(t) which makes it possible to pass from an initial state given x(t0) to an imposed final state while the criterion [5.31] is minimized. The solution of this problem is given by the maximum principle ([PON 74]). A characteristic of the solutions obtained in the case of the linear dynamic systems is that the optimal control is necessarily on the border of the subset of the acceptable controls. When this set is a closed paving stone of IRm (as in our case), the control “jumps” from one border to the other, at moments of precise commutation. We use the term of bang-bang control to insist on the fact that only the maximum and minimal accepted values of the control are used.
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5.6. Control-based fuzzy logic1 5.6.1. Introduction Tuning and process control are an application of fuzzy logic. At the time of the process control, the operators responsible for this operation are able to choose actions to undertake, starting from fuzzy implications. Generally, they can interpret measurements in the form of vague qualifiers “small”, “large”, “almost no one”, “slow”, “fast” and infer the actions of control corresponding to these states. However, if measurements are too numerous, too vague, or if major information is missed, interpretation can be made difficult. To a certain extent, the vague set theory makes it possible to take these difficulties into account. This theory, relating to vagarities and dubiousness, handles concepts badly defined in badly definite situations [KIC 75]. This summary definition can be reactualized, because fuzzy logic is now based on well established scientific foundations and a more rational analysis of human expertise. To compare the operation of a conventional controller with that of a fuzzy controller, we can say that the first models the behavior of the process to deduce from them, by calculation, the laws of control, while the second seeks to model the behavior of an operator, i.e. its reactions compared with the evolution of the process. The process can be modeled in an analytical way by a set of algebraic-differential equations, which allows us to choose the structure and calculate the coefficients of the controller. Alternatively, this adjustment can be carried out by rules resulting from a logical model of the process behavior. A fuzzy controller generally uses, at the time of its phase of design, the same logic monitoring as that used by traditional controllers (PID in particular); this logic makes it possible to enumerate a certain number of rules of operation of the type: “if the error of control is Ei then the control is Uj”, where Ei and Uj are qualifiers, of the error and control respectively. This rule is to be compared with U(t) = K.E(t) used for a traditional proportional control. For certain situations, there can be equivalence between the two modes of control. This is the case for the modal values Ei and Uj which correspond to the particular situation where the fuzzy translations of Ei and Uj are worth 1 in these points [GAL 92]. In the current state, it seems that the tuning by fuzzy logic lends itself particularly well to two design categories: design of controllers for approximately modeled processes and design of non-linear controllers.
1 This section uses the PhD of E. Tan as a starting point [TAN 97].
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155
In the first case, we must apply a heuristic process by also employing operator experiments. To obtain suitable results, a long and difficult approach is often necessary. In the second case, we benefit from the non-linear characteristics improving the performances of the conventional tunings.
5.6.2. Structure of the controlled loop In this section, we only consider the control of a single-input single-output system. The diagram of the control is the same classic design used for several types of controllers, for example: PI, PID. It concerns a simple loop of regulation represented in Figure 5.9 where the fuzzy controller uses as entry information the E(k) error and its variation 'E(k) (for example, we could use a more complex structure taking account of the increased degree of the variation error). GE G 'U
E(k) Y C (k) +
-
G' E
Fuzzy controller
'U(k)
6
Y(k) Plant U(k)
' E(k)
Figure 5.9. Synoptic diagram of the fuzzy control
The following notations are used: E k YC k Y k
[5.33]
'E k E k E k 1
[5.34]
U k U k 1 G 'U .'U k
[5.35]
YC(k), U(k), 'U(k) and Y(k) respectively represent the reference signal, the control, the increment of the control and the output of the system at the instant k. E(k) and 'E(k) are the error and its variation at the instant k. GE, G'E and G'U are the tuning gains of the error, its variation and the increment of the control. Such a diagram is frequently encountered in the literature describing fuzzy control. As indicated in Figure 5.9, in general the fuzzy controller does not directly provide the value of the control, but rather the increment of the control. This choice seems logical, as during the construction of the controller it is easier to know if it is
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necessary, in a given context, to increase or decrease the control. The value U is obtained by integrating the output of the fuzzy controller. These two variables (E and 'E) are directly related to the expertise on the system. When a human operator controls a system, they use observations of the type of “error” and “variation of the error”. In other words, they observe if the objective to be reached is distant or close to the present situation, and whether the approach is quick or slow. To achieve this goal, several trajectories are possible because the control of a system is not inevitably single. If several trajectories would have been followed, it is possible to estimate if one of them is better than the others according to a performance criterion. This stage is equivalent to the phase of training, where several tests are useful in order to control a system suitably.
5.6.3. Representation of fuzzy controllers The fuzzy controller included in the regulation loop of Figure 5.8 can be schematized by: ~ E k F >G E .E k @
[5.36]
~ 'E k F >G 'E .'E k @
[5.37]
~ ~ ~ where F[ ] represents the action of fuzzification. E k , 'E k and 'U k characterize the fuzzy sets associated with the standardized error G E .E k , the variation of the standardized error G 'E .'E k and the increment of control signal.
Rule base
E(k)
GE
˜ (k) E Fuzzification
'E(k)
G
'E
~ 'E(k)
G Inference mechanism
~ ' U(k)
Defuzzification
'U
'U(k)
Figure 5.10. Diagram of fuzzy control
This diagram accounts for the three essential steps in the operation of such a controller: fuzzification, the inference mechanism and defuzzification. First of all, fuzzification makes it possible to transform a measured variable into a fuzzy set. In
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157
this diagram, it transforms the numerical values E (error) and 'E (variation of the error) into symbolic values. The inference mechanism makes it possible to calculate the fuzzy set associated with the control. This calculation depends on the representation mode of the rules; this reproduces human behavior in general. Lastly, the defuzzification makes it possible to transform the fuzzy set, obtained by the preceding calculation, in a manipulated variable (the increment of the control) to apply to the system.
5.6.4. Basic concepts of fuzzy logic 5.6.4.1. Linguistic terms The description of a certain situation, a phenomenon or a process contains fuzzy expressions in general such as: “some”, “much”, “often”, “heat”, “cold”, “rapid”, “slow”, “large”, “small”, etc. Expressions of this kind form the values of the linguistic terms of fuzzy logic. In order to enable a digital processing, it is essential to subject them to a definition using membership functions.
5.6.4.2. Membership functions Figure 5.11 shows three simple functions of membership. In Figure 5.11a, the value x0 receives qualifier A with a maximum membership degree (for value x0 called a modal value of A). Set A is a fuzzy set which qualifies variable x. For Figure 5.11b, we speak about fuzzy singleton because qualifier B corresponds to the only element x0: B is worth 1 in x0 and 0 everywhere else. Lastly, Figure 5.11c shows a Gaussian membership function which has the advantage of being continuous, like its derivative, on all the selected support.
P(x)
P(x)
A
B 1
1
0
P(x)
x0
x
0
1
x0
x
0
x0
support (a)
(b)
Figure 5.11. Membership functions
(c)
x
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Clearly, many other choices for the shape of the membership function are possible (e.g. triangular and trapezoidal shapes), and these will each provide a different meaning for the linguistic values that they quantify.
5.6.4.3. Fuzzy deductions (inferences) In general, several linguistic terms, suitably defined by membership functions, are dependent on each other by rules, making it possible to draw some conclusions. These rules are called fuzzy deductions or interfaces. Inference in the fuzzy rules implies the deduction of new conclusions starting from information. It follows the rule of the generalized modus ponens rule. This consists of deducing a conclusion starting from a rule, as illustrated below in Figure 5.12. The closer condition is to the condition of the rule, the closer the obtained conclusion is to that of the rule. rule:
if E is P then 'U is N
condition:
E is P'
____________________________________ conclusion: 'U is N'
P(E)
1
P('U)
P
N
1
N'
P'
0
E
E0
E is P'
E
'U
0
then
'U is N'
Figure 5.12. Inference of a fuzzy rule
Figure 5.13 has the inference of a fuzzy rule where E (error) and 'U (increment of the control) are numerical variables. The linguistic terms are P and N (P: positive and N: negative). Thus (see Figure 5.12 (left)), if the variable E takes the particular value E0, its membership degree of the qualifier P is P(E = E0) = P'. The activated rule imposes itself whereas the qualifier N of control to be applied is such that 'U, given by the hatched area represented in Figure 5.12.
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The condition is contained in the premise of the rule, and the conclusion indicates the action to be applied. These rules can easily be generalized in the case of several entries, for example: “if E is P and 'E are P then 'U is N”, with 'E the variation of the error.
5.6.4.4. Fuzzy logic operators The linguistic terms are related between them to the level of the inferences by operators of fuzzy logic. Just as for traditional logic, in fuzzy logic, there are negation, intersection and union operators. – Fuzzy complement: the complement (“not”) of a fuzzy set A with a membership function ȝA(x) has a membership function given by 1 í ȝA(x):
P C x 1 P A x
[5.38]
– Fuzzy intersection (AND): in the case of fuzzy logic, the AND operator is carried out in the majority of the cases, like the formation of the minimum between the functions of membership PA(x) and PB(x) of the two sets A and B, thus leading to the minimum operator:
PC(x) = MIN [PA(x), PB(x)]
[5.39]
This operation is represented in Figure 5.13. As shown, it is possible for the function of the resulting membership PC(x) not to reach value 1.
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Control Methods for Electrical Machines P A (x) 1
x 0 P B (x) 1
x 0 P C (x) 1
x 0
Figure 5.13. AND operator, carried out by the formation of the minimum
– Fuzzy union (OR): the development of the OR operator on the level of fuzzy logic is generally achieved by forming the maximum, applied to the membership functions PA(x) and PB(x) of the two sets A and B, thus obtaining the maximum operator:
PC(x) = MAX [PA(x), PB(x)]
[5.40]
Figure 5.14 shows this operation. Note that it is possible for the function of resulting membership PC(x) to reach value 1 twice, as seen in Figure 5.14.
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P A (x) 1
x 0 P B (x) 1
x 0 P C (x) 1
x 0
Figure 5.14. OR operator, carried out by the formation of the maximum
5.6.5. Fuzzification Fuzzification consists of transforming the data of a numerical space towards a symbolic space using a qualifier or a symbol, in general, of the linguistic terms. Each qualifier is a whole fuzzy set of the universe of discourse and is described by a characteristic function which we call the membership function, where to each point of the support a degree of membership corresponds to the qualifier considered. The universe of discourse or the reference frame is the field of definition of a variable. Let us note that all of the qualifiers must cover the entire discourse universe of discourse of the variable, in other words each real variable must belong to at least a qualifier with a non-zero degree. In Figure 5.15, variable x takes its values in the interval [0, 10], which is the interval corresponding to the universe of discourse of this variable. It is represented by linguistic terms such as very low (VL), low (L), medium (M), high (H) and very high (VH). If variable x has a very precise value, x0 = 7 for example, the fuzzification can qualify the specified value x0 with vector {0/VL, 0/L, 0.3/M, 0.7/H, 0/VH}. This vector indicates that the value x0 is medium with a 0.3 membership
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degree and high with a 0.7 membership degree. The notation normally used for a membership degree is:
PVL(x0) = 0, PL(x0) = 0, PM(x0) = 0.3, PH(x0) = 0.7, PVH(x0) = 0 P(x)
1 0.7
VL
L
0.3 0
M
H
x0
7
VH
x 10
Figure 5.15. Example of fuzzification
5.6.6. The inference mechanism The inference mechanism makes it possible to calculate the fuzzy set associated with the control. In this phase, we find all of the rules which are generally presented in the form of a table called the table of inference. These rules are used to represent knowledge of a human being on a particular system, and are of the type: “if ... then ...” A rule indicates that if a condition is present in the operation of the system, then an action is necessary in order to put the system in the desired operating condition. The maximum number of rules used depends on the number of definite linguistic terms for each entry variable of the fuzzy controller. Thus, for example, if the variables E and 'E (error and its variation) are fuzzified according to five linguistic terms or five fuzzy subsets: Negative High (NH), Negative (N), Zero (Z), Positive (P), Positive High (PH), then the maximum number of rules is 25 rules. Among the most frequently used tables of inferences (or the tables of rules), we find Buckley et al.’s [BUK 89] proposal. Two variables are used at input (the error and variation of the error) and one at output (the increment of control). Each of them moves in a universe of discourse on which several fuzzy subsets are defined. More particularly, the universe discourses of the inputs variables are cut out in five noted fuzzy subsets: Negative High (NH), Negative (N), Zero (Z), Positive (P), Positive High (PH). That of the output variable is cut out in nine fuzzy subsets: Negative Very High (NVH), Negative High (NH), Negative Medium (NM), Negative Low (NL), Zero (Z), Positive Low (PL), Positive Medium (PM), Positive High (PH), Positive Very High (PVH).
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The rules are of the form: “if (error is negative high) and (variation error is negative high) then (increment of the control is positive very high)”. The inference table is given below, the values for which represent the increment of the control. NH
N
Z
P
PH
NH
PVH
PH
PM
PL
Z
N
PH
PM
PL
Z
NL
Z
PM
PL
Z
NL
NM
P
PL
Z
NL
NM
NH
PH
Z
NL
NM
NH
NVH
Table 5.1. Inference table: with E error in lines and 'E variation error in columns
5.6.7. Defuzzification Defuzzification makes it possible to transform a fuzzy size into a numerical value. From a practical point of view, this is applied to transform the fuzzy part of the control into a manipulated variable, to be applied to the system (conversion of a space symbolic system to a numerical space). To carry out this transformation, the discourse universe of the variable must be defined, as well as the membership functions associated with the various linguistic terms characterizing the variable. We propose the calculation of the value of the output of the controller by the center of gravity method. The equation most often used for this method in the case of a discrete discourse universe is: j
¦ P n 'U .u n 'U
n 1 j
[5.41]
¦ P n 'U
n 1
where j is the number of zones of the discourse universe qualifier of the control.
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5.7. Neural network control2 5.7.1. Introduction Neural networks are an extension of the traditional statistical techniques. They are able to approach any function (linear or non-linear) that is sufficiently regular. They can thus be used in any application requiring the development of a non-linear function, where the analytical form is unknown, but where a certain number of values are known.
5.7.2. Formal neurons A formal neuron is a non-linear limited algebraic function, whose value depends on weighting parameters called coefficients (or weight). The variables of this function are the inputs of the neuron, and the value of the function is the output. Figure 5.16 graphically represents a neuron fulfilling a limited non-linear function y f x1 , x 2 ,! , x n ,Z 1 ,Z 2 ,! ,Z p , where ^xi ` are to them the variables and
^Z j ` are the adjustable parameters. In the majority of cases, the function f is a non-
linear function (generally a hyperbolic tangent) from a combination of the inputs: y
· § n tanh¨ ¦ Z i x i ¸ ¸ ¨ ¹ ©i 1 x1
[5.42]
Input s
x2
Output
f
y
xn
Figure 5.16. Graphic diagram of a neuron
2 This section, drawn from various articles, is written by G Dreyfus. Readers interested in the question will find further information on http://www.neurones.espci.fr.
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A neuron is nothing other than a balanced sum followed by a non-linearity. The association of such simple elements in the form of networks makes it possible to fulfill useful functions for industrial applications.
5.7.3. Neural networks There are two great types of neural network architecture: non-buckled neural networks and buckled neural networks.
5.7.3.1. Feedforward networks A non-closed loop neural network carries out one or more algebraic functions of its inputs, by composition of the functions fulfilled by each of its neurons. A nonclosed neural network is schematized by all the neurons being connected to each other, the information of the inputs circulating towards the outputs without backward return. Figure 5.17 represents a non-closed loop neural network: it includes inputs, a layer of hidden neurons and the output neurons. The neurons of the hidden layer are not online between them. Non-closed loop neural networks are static objects: if the inputs are independent of time, so are the outputs. They are used mainly to carry out approximations of non-linear function, classification or modeling of non-linear static processes. input layer
hidden layer
output layer
x0
y
x1
x2
Figure 5.17. Simplified view of a neural network with hidden layer
5.7.3.2. Feedback networks Contrary to non-closed loop neural networks, closed loop neural networks have an unspecified architecture of connections, in particular including loop circuits
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which return the numerical values of one or more outputs to the inputs. It is obvious that to ensure such a system is causal, it is necessary to associate a delay with each loop. A net of looped neurons is thus a dynamic system, governed by differential equations or by equations at the differences if we place ourselves within the framework of discrete-time systems. The general form of the equations governing a looped neural network is: x k 1 M >x k , u k @
[5.43a]
y k \ >x k , u k @
[5.43b]
where M and \ are non-linear functions fulfilled by a non-looped neural network, and where k indicates the temporal moment. This equation of state is called a user canonical form. Looped neural networks are used to carry out dynamic system modeling tasks, process control or filtering.
5.7.4. Parsimonious approximation The specificity of neural networks lies in the parsimonious nature of the approximation: with equal precision, neural networks require fewer adjustable general options (weights) than universal approximators normally use (functions splines, wavelet, etc.). Indeed, the number of general options varies linearly with the number of variables of the function to approximate, whereas it varies exponentially for the majority of the other approximators [HOR 94]. This parsimony makes it possible to explain the industrial interest in neural networks. In general, a neural network makes it possible to approximate, with precision comparable to the approved methods, with less data. In the case of interest to us, process control, the advantage of neural networks is due to the fact that the desire to order a process amounts to imposing a behavior defined in advance according to the actuating signals. The unit orders and process can thus be regarded as a system which fulfills a (non-linear) function that a neural network can approximate.
5.7.5. Implementation of neural networks To carry out the approximation of the function starting from generally disturbed samples, three stages are necessary.
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5.7.5.1. Architecture of neural networks It is necessary to choose the external inputs, the number of hidden neurons and the layout of the neurons between them in such a way that the network is able to reproduce what is deterministic in the data. The number of adjustable weights is one of the fundamental factors of the success of applications. There are various methods to determine an optimal architecture, based on statistical evaluations [URB 94].
5.7.5.2. Calculation of neural network weights This corresponds to the calculations of the general parameters of the non-linear regression. These calculations are carried out starting from a different set of data, called user examples, by minimizing the error of approximation on the entire training points in such a way that the outputs of the network are as close as possible to the desired outputs. These calculations of the coefficients form the supervised training for the neural network. The algorithms used are non-linear algorithms of optimization such as the algorithms of the gradient (modified to take account of the specificity of the neural networks, and known as the backpropagation method), etc.
5.7.5.3. Validation of the quality of the regression The estimation of quality of the neural network obtained is carried out by presenting examples (games of data) which do not form part of the entire training.
5.8. Bibliography [BAR 89] BARTOLINI G., “Chattering phenomena in discontinuous control systems”, Int. J. of Systems, vol. 20, no. 12, p. 2,471-2,481, 1989. [BOU 96] BOUCHER P., DUMUR D., La Commande Prédictive, editions Technip, 1995. [BUC 89] BUCKLEY J.J., YING H., “Fuzzy controller theory: limit theorems for linear fuzzy control rules”, Automatica, vol. 25, no. 3, p. 469-472, 1989. [CAM 95] CAMACHO E., BORDONS C., Model Predictive Control in the Process Industry, Springer, 1995. [GAL 92] GALICHET S., DUSSUD M., FOULLOY L., “Contrôleurs flous : équivalences et études comparatives”, Neuro-Nimes, p. 229-236, 1992.
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[HOR 94] HORNIK K., STINCHCOMBE M., WHITE H., AUER P., “Degree of approximation results for feedforward networks approximating unknown mappings and their derivatives”, Neural Computation, vol. 6, p. 1 262-1 275, 1994. [KIC 75] KICKERT W.J.M., “Further analysis and application of fuzzy logic control”, Internal Rep. F/WK2/75, Queen Mary College, London, 1975. [MOR 85] MORSHEDI A., CUTLER C., SKROVANEK T., “Optimal solution of dynamic matrix control with linear programming techniques”, Proc. ACC, p. 199-208, Boston, 1985. [MOR 88] MORARI M., ZAFIRIOU E., Robust Process Control, Prentice Hall, Englewood Cliffs, 1988. [ORD 93] ORDYS A., CLARKE D., “A state space description for GPC controllers”, Int. Journal Systems Sci., vol. 24, no. 9, p. 1 727-1 744, 1993. [PON 74] PONTRIAGUINE L., BOLTIANSKI V., GAMKRÉLIDZÉ R., MICHTCHENJO E., Théorie Mathématique des Processus Optimaux, Mir, 1974. [RIC 87] RICHALET J., ABU EL ATA S., ARBER D., KUNTZE H., YACUBASCH A., SCHILL W., “Predictive functional control: application to fast and accurate robust”, Proc. of 10th IFAC World Congress, Munich, 1987. [RIC 93] RICHALET J., Pratique de la commande prédictive, Hermes, Paris, 1993. [RIV 87] RIVERA D.E., MORARI M., “Control relevant model reduction problems for SISO H2, H f , and ȝ-controller synthesis”, Int. J. Control, vol. 46, no. 2, p. 505-527, 1987. [TAN 97] TAN E., Etude comparative de différentes structures de contrôleurs flous, Thesis, National Polytechnic Institute, Lorraine, 1997. [URB 94] URBANI D., ROUSSEL-RAGOT P., PERSONNAZ L., DREYFUS G., “The selection of neural models of nonlinear dynamical systems by statistical tests”, Neural Networks for Signal Processing I, p. 229-237, IEEE Press, 1994. [UTK 78] UTKIN V.I., Sliding Modes and their Applications in Variable Structure Control, Mir, 1978.
Chapter 6
Sliding Mode Control
6.1. Introduction Control is one of the most important aspects of modern automation. Over the past decades, numerous techniques have been proposed to deal with process control. Representing a system’s instantaneous state remains one of the most traditional methods for modeling the evolution of a real process. It involves considering a system of differential equations, linear and non-linear, that are in general founded on the laws of physics and obtained by considering the nature of the process, then making certain implicit assumptions about it. In the literature, many works have been devoted to this subject. However, many of these relate to an ideal model, do not take inaccuracies and uncertainties into account, and are therefore not applicable to real processes. For this reason, taking into account the imprecision of modeling and all possible uncertainties became a research necessity, and many interesting findings on the construction of robust and adaptive correctors have been established. One of the approaches used to control complex systems is called the “sliding mode”. This method, which is applicable to linear and non-linear models alike, is both simple and robust. It has been applied to various processes, including aerospace and vehicle dynamics, robotics, chemical processes and fluids under pressure. A Chapter written by Rachid OUTBIB and Michel ZASADZINSKI.
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typical area of application is in the control of electrical power systems. Indeed, electronic circuits themselves control machines and consist of components (transistors or thyristors) that impose dynamics and are therefore prone to discontinuous behavior. Thus, they cannot be modeled using the classical theories used for linear systems, and the sliding mode is very well suited to them (see for example [BUH 86]). Historically, the technique of sliding mode control emerged in the 1950s and was developed by the Russian school. Since then, the approach has aroused the interest of many researchers, and a great deal of work has been devoted to it. A modest list of articles on the topic is given in the bibliography. The results are primarily of two types: on the one hand, those that are theoretical in nature, in which the aim is to improve the methodology, and on the other hand, those which use theoretical results that have already been established and apply them to practical situations (see for example [ACK 98, BAR 98, BOU 97, FLO 00, SIR 87, SIR 92a, SLO 83, SLO 84, UTK 93]). For reference works, we cite [UTK 92], which is dedicated to the basic concepts of sliding mode control, and [FIL 88], which is devoted to the study of discontinuous systems. There is also [SLO 91], which deals with this subject (see Chapter 7). This chapter is organized as follows: section 6.2 presents an objective example to illustrate use on systems with variable structures. Section 6.3 comprises a brief description of the basic concepts. In section 6.4, an approach using the Lyapunov method is presented. Section 6.5 is dedicated to equivalent control and section 6.6 discusses an approach by the imposition of surface dynamics. In section 6.7, we discuss the selection of a sliding surface. Finally, remarks and conclusions are given in section 6.8, and a summary of the notation used is presented in section 6.9.
6.2. Illustrative example In this section, we present a simple example, inspired by the article [UTK 77], to illustrate sliding mode control and, in general terms, its use in variable structure systems. Let us consider a system, described by the following equation: x u
The aim is to propose a control law of the form u
x (t) l 0 and x(t) l 0 when t l d
[6.1]
u (x) so that: [6.2]
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A simple argument shows that with a control law that simultaneously depends on x and x of the form u = u(x, x ), it is easy to solve the problem of stabilization by linear state feedback. In effect, it is sufficient to take the example: u = -x – D x with D > 0
Case II (0 < a < 1)
Case I (a > 1)
Case III
Figure 6.1. Illustrative example
However, use of a linear control law is not only dependent on x (i.e. of the form u =ax; D̓ R) requires a < 0 to avoid the instability of closed loop system. More precisely, in this case, the origin is stabilized, but not asymptotically (see Figure 6.1 (Case 1) and (Case II)). Now, if we consider a state feedback that commutates between two linear commands on x, of the form:
u
° a1 x if xx 0 ® °¯a2 x if xx 0
[6.3]
with a1
a1. Put another way, it is possible to build a system with variable structure that can be verified for all x (0) Rn:
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x ( t ) o 0 when t o + f which is obtained by commutation between two unstable systems. For proofs of the various statements in this section, the reader is invited to consult [UTK 77].
6.3. Basic concepts This section is devoted to introducing the basic concepts of sliding mode control. First of all, general facts about non-linear systems are presented. Then, a general theorem governing the existence of the sliding regime is given. Note that the theorem will not be used in its general form, but will be adapted to simple models. In section 6.3.4, we present traditional methods used for determining sliding dynamics.
6.3.1. General features of non-linear systems We consider the non-linear system below:
x =F(t, x, u)
[6.4]
where: – x Rn is the state of the system; – u Rm represents the control on the system; – f: R × Rn × Rm ĺ Rn. It is well known (see, for example, [COD 55]) that in a domain D of R × Rn, the field of vectors F meets the Lipschitz condition, for k > 0:
__F(t ,x1, u(t,x1)) - F(t, x2, u(t,x2))__ < k __ x1 - x2__
[6.5]
then for all points x Rn, there exists one solution only. If condition [6.5] is not satisfied, modeling the system given in [6.4] is generally difficult. Typically, in such cases, the systems have discontinuities. Consider the particular case where F is given by:
Sliding Mode Control
£¦F (t, x, u) if (t, x) 0 x F (t, x, u) ¦¤ ¦¦F (t, x, u) if (t, x) 0 ¥
173
[6.6]
where: – F+ and F- are the field of complete1 vectors in Rn; – ı is a surface of dimension n – 1. It divides the space into two disjoint subspaces, namely {ı > 0} and {ı < 0}. In the set, these two subsets will be termed respectively E ı>0 and E ı<0. E0 is also used to designate the surface {ı = 0}. The system described by [6.6] is said to have a variable structure because it has discontinuities. In fact, it acts as a system defined by the commutation between two structures, which is a function of time. It should be emphasized that certain models representing real processes are naturally discontinuous, for example, electrical circuits with switches. Taking into account discontinuous models, or use of an irregular (discontinuous) control law of the form: £¦u (t, x) if (t, x) 0 u ¦¤ ¦¦u (t, x) if (t, x) 0 ¥
[6.7]
in general leads to the study of discontinuous systems of the general form: £ ¦F (t, x) if (t, x) 0 x F (t, x) ¦ ¤ ¦ ¦ ¥F (t, x) if (t, x) 0
[6.8]
Unlike regular systems, in which the field of vectors F defining the system is at least differentiable, there is no complete theory for modeling systems such as [6.8]. Several different options can arise from of the discontinuity surface E0. Amongst the possible situations, a case that is particularly interesting is when two vectors F+ and F- are directed towards the surface E0. This results in the discontinuous surface shown in Figure 6.2. Having joined the surface, the state of the system remains there. This is a sliding mode, the surface of discontinuity, E0 is attractive and is called the sliding surface.
1. A field of vectors X is said to be complete if all solutions ij of the system x =X(x) are
limited for all time finite t.
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Figure 6.2. Attractive surface of commutation
6.3.2. Existence of a sliding mode One condition that must be met for a sliding mode to exist is that the surface of commutation E0 must be attractive. However, depending on whether the attractivity is local or global, different definitions have been proposed.
6.3.2.1. The sliding domain A sliding domain is described in the following manner: DEFINITION 6.1 – Say DG is a domain on the surface E0. DG is said to be a sliding domain for all values of H > 0, if there exists G > 0, for all initial conditions x0 in the G -vicinity2 of DG one solution of the system that passes from x0 and does not leave the H-vicinity of DG but crosses the H-vicinity of the external border of DG (see Figure 6.3).
Figure 6.3. Sliding domain 2. Where K > 0 andU R , a set in the n-vicinity of U is made of the points x R where n
d( x, U ) < H usually indicates the distance.
n
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175
That is to say, A V> 0 and A V< 0 are the respective domains of attraction for V > 0 and V < 0. So, the domain of sliding DG is defined by: DG = { x/V= 0 and x A V >0 A V<0} 6.3.2.2. Conditions necessary for a sliding mode In this section, we present results that determine whether a sliding mode exists. It concerns systems that are affine to controls, that is:
F(t, x, u) = f(t, x) + g(t, x)
[6.9]
THEOREM 6.1 – Let DG be a domain on V and :9a vicinity of DG. Let us suppose that a Lyapunov function V (t, xV exists that is continuously differentiable with respect of all argument and for which:
1) V (t, x, V is positive-definite for V(i.e., V ( t, x, V ) > 0 for Vz 0 and V (t, x, 0) { 0) 2) for all U> 0, the following condition is true: ¬ inf V v 0 and lim sup V 0 l0 ®
3) the derivative of V (t, x, V) on the trajectories of the system is negative and outside the switching surface with: § · sup ¨ sup V ¸ 0 ¨ ¸ a d V db © V U ¹
for all constants a, b, for which 0 < a
V
V2
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Then, in order to ensure the surface V = 0 is attractive over the entire operational range V , it is sufficient that the derivative of V with respect to time is negative, that is:
VV 0 for all x of V
[6.10]
The domains of attraction A V < 0 and A V> 0 are defined by: A V<0 = {x V /V d 0 and V > ̓ 0} and:
̓ 0} A V>0 = {x V V t 0 and V < To illustrate the sliding domain, we consider the example of a double integrator controlled by a relay switching on a linear surface.
x2
k Sliding domain
$V!
x1 $V
-k
Figure 6.4. Sliding boundary for example 6.1
EXAMPLE 6.1 – Consider the example: ° x1 ® °¯ x2
x2 ksign V
[6.11]
with k > 0, and where x = (x1 ,x2)t R2 and V = x1 + x2. The sliding condition defined by [6.10] gives: (Vҏ> 0 x2 < k) and (Vҏ< 0 x2 > –k)
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177
from this we can deduce that: A V >0 = {xV / x1 + x2 t 0 and x2 < k} and: A V<0 = {xV / x1 + x2 d 0 and x2 > –k} Finally, the sliding domain is given by: DG = {x V / x1 + x2 = 0 and –k < x2 < k} 6.3.2.3. Stability of the sliding mode Consider the following planar system: x ° 1 ® ° x ¯ 2
x1 x1 x2
x1 x1 x2
2
2
ksign V x1 x1 x2
3
[6.12]
where V = x1 + x2 and k > 0. In this example, we are interested in the attractivity of the point of operation x0 = (0, 0). Attractivity condition [6.10] of V becomes:
V4 (k sign(V) + x1) > 0 A simple argument shows that AV >0 and AV <0 are definite as: AҏV >0 = {x R2 /x1 t –x2 and x1 > –k} and: AV <0 = {x R2 / k < x1 d –x2} Therefore, we have: DG = {x R2 / x1 = –x2 > k} It should be noted that the sliding domain DG does not contain the operating point x0. Thus, the fact that a sliding mode exists does not ensure the stability of x0, even if the sliding mode is in accordance with Definition 6.1. In effect, the
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trajectories leave the range sliding domain only through the edge of the boundary x1 = k. For x AV >0, the trajectory reaches x0 without crossing the surface of commutation. x2 AV> 0
-k
+k x1
Exit of sliding domain
A V< 0 Sliding domain
Figure 6.5. Sliding mode existence and instability
NOTE 6.1 – The fact that an initial condition lies within the attractive domain does not ensure that the system’s trajectories will return to the sliding domain. Indeed, attractivity means only that the field F is directed towards the surface of commutation. Also, it should be noted that it is possible for a trajectory to join the sliding domain where the initial condition lies outside the attractive domain.
6.3.3. Chattering phenomena “Chattering” phenomena are one of the most important aspects of sliding mode control methods. Consider an ideal model, one that does not take possible perturbations and inaccuracies into account. Suppose the intention of the command is to make operating point xd attractive, on a surface given by the equation V = 0. One classic order control strategy would be to make this surface attractive, then to
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179
choose a control law to make this surface invariant and the point xd attractive for the model, restricted to {V = 0}. For example, suppose that the surface V obeys a dynamic such that: dV 2 d W V dt
with W > 0. In this case, it is not difficult to prove that the surface {V = 0} is attractive and will be reached, starting from any given initial condition, in a given time. Then, a wise choice for the command {V= 0 } will allow convergence towards xd (see Figure 6.6a). Now, if we take into account the imprecision of modeling and the perturbations to the system, the steps described above are connected to the phenomenon of chattering (see Figure 6.6b). In practice, but with certain exceptions, this phenomenon is not desired. Therefore a significant part of the literature on sliding mode control (see for example [BAR 89], [BAR 96], [BAR 98]) is concerned with chattering and its possible minimization. V V
Sliding mode convergence
x0
Phase of switching Surface obtainability a) Ideal behavior
.x
Chattering
.x
d
d
x0 b) Chattering as result of imperfect control switching
Figure 6.6. Chattering phenomena
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6.3.4. Determination of sliding dynamics 6.3.4.1. Method of equivalent control Equivalent control, usually written as ueq, functions like a control that makes the switching surface3 time-invariant. It involves ensuring that V = 0. It is a way of ensuring a trajectory can be maintained on a switching surface. The condition V = 0 is met when: wV ¢V, F (t, x, ueq)² + wt = 0
[6.13]
Generally, solving [6.13] is a very difficult task. To illustrate the technique, consider the case of an uncontrolled system (a case where the vector field F is given by [6.9]), and assume that (V, g)-1 z 0. In such a case, equation [6.13] provides the following explicit solution: ueq t,x
V ,g 1 §¨ V , f ©
wV · wt ¸¹
[6.14]
The closed loop control system defined when it leaves [6.4] with [6.9] and where the control is given by [6.14] is: x
wV · 1 § f t,x g t,x V ,g ¨ V , f wt ¸¹ ©
[6.15]
This method is widely used, and in effect is a constructive approach. However, for certain systems it is possible to give an explicit solution. To illustrate this, we consider the planar system given by: ° x1 ® °¯ x2
ax2 ux1
1 a x1 4u3 x1
[6.16]
with a ]0, ½[. Suppose that V = x1 + x2, and consider the “bang-bang” control given below:
3. A subset D V is said to be invariant under the action of a system x = X(x)(S) if, for all x0D, a solution \ of the system (S) (according to \ (0, x0) = x0) is that \ (t, x0) D for all values of t t 0.
Sliding Mode Control
£¦1 if x1 0 u ¦¤ ¦¦¥1 if x1 0
181
[6.17]
We have:
VV
V x1 x2
V aV 1 u 4u 3 x1
aV 2 1 u 4u 3 V x1
According to the values of u given by [6.17], it is easy to prove that sliding condition [6.10] is satisfied on x R2. Moreover, the equivalent control solution for [6.13] leads to:
V
1 u
eq
3 4ueq x1
[6.18]
It is clear that ueq = –½ is the solution of [6.18]. Now, if we substitute this value into [6.16], we obtain, on the switching surface, the following equations for the sliding mode: x1
1· § ¨ a 2 ¸ x1 © ¹
x2
1· § ¨ a 2 ¸ x2 © ¹
The sliding dynamics are therefore stable.
6.3.4.2. Regularization method This method is based on the introduction of ZI(G),thickness imperfection zone 2G around the surface of commutation E0 defined for |V(x)| d̓Gand G < 0. Indeed, the idea involves replacing one control u with another, u , which takes into account all the possible imperfections attributable to measurement errors and the effects of time delays, etc. It effectively substitutes, at the interior of this zone, a field of vectors F for FG and the sliding mode, and does not operate on the surface E0, but inside ZI(G (see Figure 6.7). It then produces a real sliding mode.
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Figure 6.7. Imperfection zone
6.3.4.3. Filippov method Consider the system given by [6.8] and let us assume it possesses only one discontinuity. Our aim is therefore to find an expression for the vector F0 that will ensure a sliding mode. At first, let us assume the system is autonomous. The idea proposed by Filippov [FIL 88] involves proposing that for all values of xV=0 on E0:
F0 = μF + V =0 + (1 – μ) F - V = 0 (0d μd1)
[6.19]
where F + Vҏ=0 and F - V = 0 are the respective limits of the vector fields F + (x) and F (x) when x approaches xV =0. The parameter P is given by virtue of the condition: V ,F0
0
[6.20]
By substituting the expression for F0 as a function of FV
0
and FV
0
given by
[6.19], into [6.20] we obtain:
P
V ,FV
V , FV
0
0
FV
0
[6.21]
The equation for the sliding mode according to Fillipov is therefore given by:
x
V ,FV
V , FV
0
0
FV
0
FV 0
V ,FV
V , FV
0
0
FV
0
FV
0
[6.22]
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183
Now, if system [6.8] is not autonomous4, similar reasoning shows that the sliding mode is governed by:
x
V ,FV
V , FV
0
0
FV
0
wV 1 wt V , F F V 0 V
FV 0
FV
0
0
V ,FV
V , FV FV
0
0
0
FV
0
FV
0
[6.23]
6.3.5. Case of more than one commutation surface In this chapter, to make things simple, the sliding mode method is explained for cases involving one surface of commutation. However, it should be noted that a general theory encompassing cases with multiple surfaces does exist. Indeed, most of the results stated here also apply to the case V = VVVp)t: Rn o Rp, where E0 is given by:
E0 = {x Rn /V(x) = V 2(x) = ... = Vn (x) = 0}
Figure 6.8. Filippov construction
4. A system is said to be non-autonomous if it is explicitly dependent on time.
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Figure 6.9. Example of a dynamic on two commutation surfaces
6.4. Direct Lyapunov method For the last century, the Lyapunov approach has played a major role in studies on the stability and stabilizing methods using the state feedback of complex systems. This approach is based on the use of functions called “Lyapunov functions”.5 The rest of this section shows how this approach was adopted as a means of imposing control by the sliding method.
6.4.1. Affine systems with regard to control Consider systems [6.4] to [6.9] and assume they are subject to the following perturbation: x
f t, x ug t, x [ t, x
[6.24]
where [ is the perturbation affecting the system. Suppose that a scalar function exists, D(t, x) R, for which:
[(t, x) = D(t, x)g(t, x)
[6.25]
5. A function V is said to be a Lyapunov function if V(x) t 0 and V (x) = 0 is true only when
x = 0. In the case of a global study, for example, of V = Rn, it is also assumed that lim V x f . x of
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185
Suppose also that a Lyapunov function V exists, for which: ¢V(x),f(t, x)² < \(__x__) for all x z 0
[6.26]
where \ is a regular function with \ (z) > 0 if z z 0. We have the following proposition: PROPOSITION 6.1 – For all scalar functions U for which it is true that:
U(t, x) > __\(t, x)__
[6.27]
the control law defined by: u
V x ,g t ,x
U t,x
V x ,g t ,x
[6.28]
is a state feedback that is stable and robust. Proof: the derivative of V along the trajectories of systems given in [6.24]–[6.28] is given by: V
V, f t, x ug t, x [ t, x § V x , g t, x · V x , f t, x ¨ [ t, x ¸ ¨ V x , g t, x ¸ © ¹
Because of inequalities [6.26] and [6.27], the preceding equation leads to: V x \ x
for all x z 0
hence the proposition.
The control law given in [6.28] is clearly discontinuous on the set E0 defined by: E0 = {(t, x) R×Rn / ¢V(x), g(t, x)² = 0}
[6.29]
Condition [6.27] for Uthat is sufficiently large guarantees that the feedback loop controlled system is asymptotically stable and the sliding mode exists on V= 0, which is the same on E0.
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NOTE 6.2 – In the case of an autonomous system, condition [6.27] becomes: ¢V(x), f(x)² < 0 for all x z 0
[6.30]
EXAMPLE 6.2 (DISPLACEMENT OF A PLATFORM BY AN ECCENTRIC model describes the evolution of a process which is given by: x1 ° ° ® ° x2 °¯
MASS)
– The
x2
G cos x1 d t G sin x1 x22 bx2 kx1 u
1 G
2
cos 2 x1
[6.31]
where:
– G(< 1) is the eccentricity of the rotational inertia; – d(t) is the displacement of the platform; and – x1 and x2 represent the angle of displacement and the speed of rotation of the mass. The uncertainty arises from the translational motion of the platform. The unperturbed model without control (i.e. where d = 0 and u = 0) has a point like the origin of equilibrium with global asymptotic stability. To prove this, we take the energy of the system:
E x1 ,x2
1 2 1 kx 1 G 2 cos 2 x1 x22 2 1 2
[6.32]
It is easy to see that the energy E given by [6.32] is a positive definite function and that its derivative relative to [6.31] with d = 0 and u = 0 is given by: E
wE wE x1 x wx1 wx2 2
kx x G 1
2 2
2
cos x1 sin x1 x2
x2 1 G 2 cos 2 x1 G 2 cos x1 sin x1 x22 bx2 kx1 2
2
1 G cos x1
bx22 d 0
Thus, the origin is a stable point of equilibrium. To prove its attractivity, we can use the LaSalle invariance principle (see for example [LAS 61]). Finally, we can conclude that in the absence of a perturbation (that is, if we are considering an ideal
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187
model), the system is asymptotically stable. Now, consideration of a more realistic model must take the perturbations into account. Since the energy of the system does not obey [6.30], then it will not be used to generate a control law using the methods developed in this section. To achieve this, we consider V1, the Lyapunov function given by:
V1 x1 ,x2
k1 2 1 x D 1 G 2 cos 2 x1 x1 x2 1 G 2 cos 2 x1 x2 2 1 2
[6.33]
Simple reasoning shows that V1 is a positive definite function. Calculation of its derivative along system [6.31] trajectories gives: V1
D x22 G 2 x1 cos x1 sin x1 D 1 G 2 cos 2 x1 b
D b k1 k
x1 x2 D kx12
[6.34]
Therefore, for the best choice of k1 and for x placed into a domain given by: {x R2 / __x__d r(a ,k, b, k )} it is possible to prove that condition [6.30] is satisfied. Finally while using the V1 Lyapunov function, we deduce that the control law is defined by:
u = -U(t, x1, x2)sign(Dx1 + x2) with:
U0(t, x1, x2) > _Gcosx1d(t)_ the model can be locally stabilized by taking into account the disturbance close to the origin.
6.4.2. Linear systems Consider the linear system subject to perturbation, defined by: x
Ax bu d x,t
[6.35]
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Control Methods for Electrical Machines
with x, d, b Rn, u R and A Rnun. If we suppose that the spectrum of A is strictly a negative real part, then P Rnun exists, a matrix positive and symmetric for which
Q = At P + PA
[6.36]
is negative-definite. It should be stated that if the pair (A, b) can be controlled it is also possible perhaps after preliminary feedback, to impose [6.36]. Let V be the Lyapunov function defined by: V (x) = xt Px The derivative of V along the system trajectories [6.35] is given by: V x
xt Qx 2bt Pxu
(x)
To ensure that V is negative, we use u, a rule for which:
btPxu d 0
[6.37]
If, for practical reasons, we assume that u is for umax >0:
u [- umax,+ umax]
[6.38]
then, the expression of u verifying [6.37] and [6.38] is given by:
u = - umaxsign(V(x))
[6.39]
In this case, the surface given by:
V(x) = btPx
[6.40]
represents an attractive switching surface. Sliding mode control is widely used for linear systems. Classical theory was adapted not only for guaranteeing asymptotic stability, but also to ensure a particular performance level or control criterion. For example, if we seek to ensure a system’s hyper stability, or to optimize a quadratic criterion, we can prove that the control type given by [6.39] is acceptable (see for example [BUH 86]). It is important to note that the matrix P used in [6.40], which characterizes the surface of commutation, can be obtained in a different way, depending on the level of performance required.
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189
6.5. Equivalent control method In this section, we confine our interest to a system which only upon input is autonomous and affine with regard to control, represented by the expression: x
f x g x u
[6.41]
where x V an opening of Rn, u: Rn o R represents the control law (which is eventually discontinuous) and f and g are the fields of regular vectors on V with g(x) = 0, for all x V . Throughout this section, the control law used outside of sliding surface will be: £¦u (x) u ¦¤ ¦¦u (x) ¥
if (x) 0
[6.42]
if (x) 0
The control law u+ and u- are assumed to be regular with regard to x and are of the form in which u+(x) > u-(x) locally on V. Let us assume that on applying [6.42]6 we have:
lim V , f gu 0 and lim V , f gu ! 0
V o 0
[6.43]
V o0
6.5.1. Invariance condition Under this approach, an ideal sliding mode is described by using the invariance condition:
V = 0 and ¢V, f + gueq² = 0
[6.44]
The control law ueq is called the equivalent control. According to [6.43], it is given explicitly by: ueq x
V x , f x V x ,g x
[6.45]
6. Throughout this chapter, lim Q v defined the limit of the quantity Q when v approaches v o0
0 with v > 0, and in a similar fashion, lim Q v . v o0
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Control Methods for Electrical Machines
6.5.2. Existence conditions This is best explained by defining it in terms of the unique conditions of invariance. LEMMA 6.1 – A necessary and sufficient condition by which equivalent control can be well defined using the following property, known as the transversal condition: ¢V (x), g(x)² z0
[6.46]
When it is checked locally, in the vicinity of E0. Proof: if [6.46] is true on E0, then ueq given by [6.45] is clearly defined. We can demonstrate this by arguing an absurdity. Suppose two solutions ueq and ueq exist for the invariance, then: ¢V,f + ueq g² = ¢V,f + u~eq g² = 0 and ( ueq - u~eq )¢V, g² = 0 Finally, due to [6.46], we have ueq { ueq . Suppose that the equivalent control is serviceable, and that condition [6.46] is not satisfied. We obtain the result in which ¢V, f² cancels itself. In this case, the uniqueness of the control is not assured because the invariance would be checked for all u. This completes the proof of the lemma. LEMMA 6.2 – If a mode of sliding exists locally on E0, then: ¢V(x), g(x)² < 0
locally on E0. Proof: by virtue of the definition of the sliding mode [6.43]:
¢V, f + gu+(x)² < 0 and ¢V, f + gu-(x)² > 0 then ¢V, g(u+(x) – u-(x))² = (u+(x) - u-(x))¢V, g² < 0
[6.47]
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191
This concludes the proof of the two preceding lemmas, since u+ > u-.
As a direct consequence of these two lemmas, we obtain Lemma 6.3. LEMMA 6.3 – A necessary condition for the existence of a local mode of sliding on E0 is that the equivalent control is well defined. THEOREM 6.2 – A condition necessary and sufficient for the local existence of a sliding mode on E0 is that locally on V, for x E0:
u-(x)
[6.48]
Proof: we assume that a sliding mode exists locally on E0. According to [6.43], we have: ¢V,f + gu+² = ¢V, f² + u+¢V, g² < 0 Under the terms of [6.45], according to Lemma 6.2, condition [6.47] is satisfied: V , f V ,g
u
ueq x u ! 0
On E0, we have ¢V,f + u-(x)g² > 0. Using a similar reasoning, we arrive at: – ueq(x) + u- < 0 To demonstrate the wider implications, let us consider ueq(x), a regular function verifying [6.47] and [6.48]. We have: 0 < ueq(x) – u-(x) < u+(x) – u-(x) it then follows that: 0 < veq < 1 where: veq x
ueq x u x u x u x
from which we can deduce that:
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Control Methods for Electrical Machines
0 = ¢V,f + ueq(x)g² = veq ¢V,f + u+(x)g² + (1 - veq)¢V,f + u-(x)g² The two quantities: ¢V,f + u+(x)g² and ¢V,f + u-(x)g²
are of opposite sign.
CORROLORY 6.1 – If a sliding mode exists on Eo, then a commutation to assure it is given by: u = k°ueq(x)°Sign(V (x)) with k > 1 EXAMPLE 6.3 (AN ELECTRICAL CIRCUIT) – Consider the electric circuit represented in Figure 6.10. Its evolution can be described by the system given below:
with x1
x1
0
x2
Z1
0
0 x2 u Z1
x3
0
0
0 x3
Z1 0 x1
I i Li i 1,3 ,x2
0 0 V C2 ,Z1
Z1
0
x1
Z2 x2
0
Z2 1/
0
x3
L1C2 and Z2
Figure 6.10. An electrical circuit
[6.49] 1/
L3C2 .
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193
The aim is to show that we can transfer the energy stored in L1 to L3 by a simple commutation on the capacitor. A simple argument establishes that: E
1 x2 2
The total energy of the system is constant (that is, E(t) = E(0) > 0 or all t t 0) and for u = 0, x3 is constant, and for u = 1, x1 is constant. Thus, two commutations are sufficient to achieve the energy transfer. The transfer can be assured by creating a sliding mode on a sphere of constant energy (E { E(0)) on the line x2 = K (where K is a constant number, 0 < K < 1). The first control strategy is to take u = 0, until x2 = K is reached. After this, a sliding movement is created along x2 = K, until reaching x1 = 0, then to make x1 = 1 until obtaining a maximum value for x3. It should be noted that during our analysis, the system evolves in an area of a sphere defined by: C+ = {x R3 / xi > 0 for i = 1, 2, 3} We consider a line defined for K ]0, 1[, by: E0K
^x
3
/ x
E0
and V x
x2 K
`
0
[6.50]
A simple calculation gives, on C+: ¢V, g² = - (Z1x1 + Z2x3) z 0 Thus, condition [6.46] is satisfied. The equivalent control defined by [6.45] is explicitly given by:
ueq
Z1 x1 Z1 x1 Z2 x3
In this example, we have:
u+ = 1 and u- = 0 And from the analysis on C+, it is clear that: 0 < ueq < 1 Thus, condition [6.48], necessary and sufficient for the existence of a sliding mode, is verified. Finally, the control:
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Control Methods for Electrical Machines
u
1 ° Z1 x1 ° ®Z x Z x 2 3 ° 1 1 °0 ¯
if if if
x2 ! K i x2
K
x2 K
allows the transfer of energy between the inductances L1 and L3.
Figure 6.11. Transfer of energy for controlling a sliding mode
6.5.3. Sliding mode for a perturbed system Consider the case where the system described by [6.41] is perturbed: x = f (x) + g (x) u + [
[6.51]
where [ represents the perturbation of the system. DEFINITION 6.2 – It can be said that the ideal sliding mode shows strong invariance property with respect to the perturbation [̓if the ideal sliding dynamic is independent of [. THEOREM 6.3 – A sliding regime on E0 in system [6.51] has a strong invariance property with respect to [ if a regularity function D exists for which:
[(x) = D(x)g(x)
[6.52]
Proof: showing that [6.52] holds is sufficient. Effectively system [6.51] becomes:
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195
x = f (x) + g(x)(u + D(x))
It is clear that for:
u = ueq (x) - D(x) the sliding dynamic is not affected by the perturbation. Now, to prove that [6.52] is a necessary condition, suppose that, on E0:
V x , f x g x u [ x
V
0
[6.53]
Thus, it follows that: x
f x
V x , f x V x ,g x
g x
V x ,[ x V x ,g x
g x [ x
[6.54]
Finally, to show the sliding dynamics are not affected by the perturbation, it is necessary that:
[ x
V x [ x V x g x
g x
0
[6.55]
and thus that condition [6.52] is satisfied for:
D x
V x [ x V x g x
thus proving the theorem.
THEOREM 6.4 – Suppose that [(x) { a(x)g(x) and that ueq represents the equivalent control corresponding to a sliding mode on V for the non-perturbed system [6.41]. If a sliding mode exists for the perturbed system [6.51], then:
u-(x) < D(x) – ueq (x) < u+(x) Proof: if a sliding mode exists for [6.51], then:
[6.56]
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Control Methods for Electrical Machines
lim V , f gu [
V o0
0
!0
lim V , f g u D
V o0
and lim V , f gu [
V o0
lim V , f g u D
V o 0
This is the same as saying that for the system described by [6.41], a sliding mode exists with:
u
°u x D x ® °¯u x D x
if
V x ! 0
if
V x 0
[6.57]
This results in, using Theorem 6.1: u-(x) + D(x) < ueq (x) < u+(x) + D(x)
thus proving the theorem.
6.5.4. Canonical forms By virtue of the regularity of V, the equation: V(x) = 0
[6.58]
demonstrates the interdependence between variables xi (i = 1 ... n). Thus, by using the implicit functions theorem (see [BOO 75]), we can express, without losing general information xn as a function of the other variables: xn = V 1(x1)
[6.59]
where x1 = (x1, x2, ..., xn-1)t. The system defined by [6.41] is said to be in its canonical form if it is expressed as: x ° 1 ® °¯ xn
f1 x1 ,xn
f n x1 ,xn g n x1 ,xn u
[6.60]
Sliding Mode Control
197
with f1 = (f1, f2, ..., fn-1)t and gn(x1, xn) z 0 for all x V . The sliding surface is defined by: E0 = {(x1, xn) / xn = Vң (x1)}
[6.61]
Under the conditions of an ideal sliding mode, the second equation for system [6.60] is used to give an expression for the equivalent control, whilst the first describes the resulting sliding dynamics. According to [6.60] and [6.61]:
ueq x1
x f x ,V x x ,V x V ,x f x ,V x
g n1 x1 ,V 1 x1 g n1
1
1
1
n
n
1
1
1
1
1
n
1
1
1
[6.62]
which shows that xn is considered to be a control variable for the reduced system: x 1
f1 x1 ,xn
On applying the existence condition expressed by [6.48] to [6.62], we obtain xn
u u
xn
u ueq
xn
u u
[6.63]
[SIR 87] demonstrates Theorem 6.5. THEOREM 6.5 – A condition necessary and sufficient to transform system [6.41] to its regular canonical form [6.60] is given by: g (x) z0 for all x V
[6.64]
EXAMPLE 6.4 – Consider a linear system with a single-input defined on Rn by:
x
Ax bu
[6.65]
where A Rnun, b Rn with b z0. There exists bi0, an element of b in which bi0z0. Therefore, condition [6.64] is met. Without losing general information, we can presume that io = n and that b = (b1,bn)t, with b1 = (b1, ...,bn-1)t. Simple argument shows that by using the transformation z = Tx, where:
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Control Methods for Electrical Machines
T
b1 º ª « I n 1 b » n » « « 0 » 1 ¬ ¼
[6.66]
and [6.60] can be put into its canonical form.
6.6. Imposing a surface dynamic Sliding mode control by imposing a dynamic on the surface involves combining two techniques. First of all, we impress a discontinuous dynamic on the surface V, which ensures the sliding surface E0 is attainable, and we can then deduce from the laws of discontinuous control whether E0 is attractive for the system. The second step involves selecting a manifold surface that ensures the asymptotic stability of the point of equilibrium.
Figure 6.12. Classic dynamic for the surface
6.6.1. A classic surface dynamic We consider the dynamic for the surface below (see Figure 6.12):
V
G1Sign V G2V for all V R*
[6.67]
where G1 and G2 are two strictly positive constants. We also have: VV
G1 V G2V 2 0
[6.68]
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199
The dynamic given by [6.67] guarantees that E0 is attractive. A simple argument shows that:
V
G1 exp G2 t 1 Sign V exp G2 t V t G2
tr
G2 1 §¨ ln 1 G2 ¨ G V 1 t ©
0
for all t[0, tr]
[6.69]
with · ¸ ¸ ¹
0
[6.70]
designating the time at which the surface E0 is rejoined. Expanding the expression for V given by [6.67] leads to: V , f gu
wV wt
G1 Sign V G2V 2
which then gives:
u
V ,g
wV 2· ¨ V , f wt G1Sign V G2V ¸ © ¹
1 §
[6.71]
6.6.2. A particular case: dynamic with pure discontinuities This concerns cases, in which, for [6.67], we have G2 = 0, that is, the proportionality constant is zero:
V
GSign V
[6.72]
where G is a strictly positive constant. In this case, the dynamics of V given by [6.71] become:
V and
GSign V t V t
0
for all t [0, tr]
[6.73]
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Control Methods for Electrical Machines
tr
Vt
0
G
[6.74]
Note that the sliding surface E0 is regained more quickly as the gain G increases. However, selecting a large gain can have practical disadvantages. For example, the control can be saturated.
Figure 6.13. Attainability of E0 using the dynamic given in [6.67]
6.7. The choice of sliding surface 6.7.1. Introduction In this section, we consider the particular case of a system of the form: n x
f x bu
[6.75]
where u represents the control, and the state vector is x
t
x,x, " x , where x n 1
(i)
is the ith-order derivative of x with respect to time; f is a non-linear function. After being given *d(t), we want to find a means of controlling the trajectory such that:
x t *
d
t
o 0 for all x(0) V
which is our control objective.
[6.76]
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201
More precisely, we want to find a sliding surface E0, for which [6.76] holds. It should be noted that there is no one general method for selecting a sliding surface. A simple case involves taking a linear function of state error x , as sliding surface V:
V
c , x *
c ,x
n
¦c
x
i 1 i
i 1
[6.77]
with c
c0 ,c1 ," ,cn1
t
and *
t
* ,* ,"* n 1
d
d
d
We have: x x 2 ° 1 ° x x3 ° 2 # ° ° ® xn 2 xn 1 ° ° xn 1 xn ° 1 ° c0 x1 c1 x2 " cn 2 xn 1 V ° cn 1 ¯
[6.78]
The sliding dynamics on E0 are given by: x x 2 ° 1 ° x x 3 ° 2 # ° ° ® xn 2 xn 1 ° ° xn 1 xn ° 1 ° c0 x1 c1 x2 " cn 2 xn 1 ° c n 1 ¯
[6.79]
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Control Methods for Electrical Machines
To ensure that this sliding mode will maintain the origin as a point of asymptotic equilibrium, it is crucial that c generates a Hurwitz polynomial7.
6.7.2. A specific linear surface choice This interesting choice results in a characteristic polynomial with real roots. More precisely, the surface Vis given by:
V
§d · ¨ dt O ¸ © ¹
n 1
x1
[6.80]
For example, for n =3,the expression for Vbecomes: 2
V
§d · ¨ dt O ¸ x1 © ¹ O 2 e 2e e
O 2 x1 2O x2 x3
where e indicates the deviation x – *. The surface defined by [6.80] has the advantage, on the one hand, of a dynamic with only real poles, and on the other, of an analysis with only one real parameter. The variable e can be considered as an output of first order systems cascade in which V is the input variable of the system (see Figure 6.14).
n-1 blocks
Figure 6.14. Error dynamics
7. Hurwitz polynomials are those in which all roots are real and negative.
Sliding Mode Control
203
6.8. Conclusion This chapter sought to present the techniques used for controlling complex systems using the sliding mode method. Clearly, the contents are not exhaustive, but are sufficient to familiarize the reader with our opinions on this method, with particular emphasis on its use in the area of power electronics. In this chapter, the sliding mode technique was presented for solving system control problems, and for cases where the models under consideration had continuous time. We would like to emphasize that this approach can also be adapted for discrete time systems and can be extended to deal with aims other than control, for example, the parameter identification or the observer development ([BEN 99], [FUR 90], [SAR 87], [SLO 87], [XU 93]).
6.9. Notations .
Scalar product
.
Euclidean norm
.
Absolute value
.
Gradient t
w wx Rnxn
Transpose Partial derivative for x
In
Set of n u n-matrices Identity matrix on Rnxn with real entries
V, D
Sub-space of Rn defining the study area
:
Sub-domain of V
x u
State vector Control variable
VV1
Surfaces
Vt=0
Value of V at the time t = 0
V=0
x ueq
Limit of x when Vo 0 Equivalent control
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Control Methods for Electrical Machines
F ,F0 ,F ,F ,FV 0 ½° ¾ FV 0 ,FG , f ,g °¿
[ u+ u
-
Field of vectors Perturbation affecting the system Control applied on the subspace V̓> 0 Control applied on the subspace V̓< 0
EV<0
Subspace of Rn when V< 0
EV>0
Subspace of Rn when V> 0
E0
Subspace of Rn when V = 0
DG
Sliding domain
As>0
Attractivity domain for V> 0
As<0
Attractivity domain for V< 0
x0
Initial condition
M
Finded trajectory
ZI(G)
Area of imperfection
*d G, G1, G2 tr E I1, I2 L1, L3 C2 V, V1 Sign
Desired state Gains Time to return to the sliding surface Energy of the system Electrical currents Inductances Capacity of the capacitor Lyapunov functions Sign functions
6.10. Bibliography [ACK 98] ACKERMANN J., UTKIN V.I., “Sliding mode control design based on Ackermann’s formula”, IEEE Transactions on Automatic Control, vol. 43, no. 2, p. 234-237, 1998. [BAR 89] BARTOLONI G., “Chattering phenomena in discontinuous control systems”, International Journal on Systems Science, vol. 20, no. 12, 1989. [BAR 96] BARTOLONI G., PYDYNOWSKI P., “An improved, chattering free, V.S.C., scheme for uncertain dynamical systems”, IEEE Transactions on Automatic Control, vol. 41, no. 8, p. 1220-1226, 1996.
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205
[BAR 98] BARTOLONI G., FERRARA A., USAI E., “Chattering avoidance by second-order sliding mode control”, IEEE Transactions on Automatic Control, vol. 43, no. 2, p. 241246, 1998. [BEN 99] BENCHAIB A., RACHID A., AUDREZET E., “Sliding mode input-output linearization and field orientation for real-time control of induction motors”, IEEE Transactions on Power Electronics, vol. 14, no. 1, p. 3-13, 1999. [BOO 75] BOOTHBY W.M., An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975. [BOU 97] BOURI M., Contribution à la commande non linéaire par mode de glissement: application aux actionneurs électropneumatiques, PhD Thesis, INSA de Lyon, Lyon, 1997. [BUH 86] BUHLER H., Réglage par mode de glissement: complément au traité d’électricité, Presses polytechniques romandes, Lausanne, Switzerland, 1986. [BUP 94] BUPP R.T., BERNSTEIN D.S., COPPOLA V.T., “Vibration suppression of multimodal translational motion using a rotational actuator”, in Proceedings of the Thirty-third Conference on Decision and Control (Lake Buena Vista, Florida), p. 4030-4034, December, 1994. [COD 55] CODDINGTON E., LEVINSON N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. [DRAK 90] DRAKUNOV S.V., UTKIN V.I., “Sliding mode in dynamic systems”, International Journal of Control, vol. 55, p. 1029-1037, 1990. [DRAZ 69] DRAZENOVIC B., “The invariance conditions in variable structure systems”, Automatica, vol. 5, no. 3, p. 287-295, 1969. [FIL 88] FILIPPOV V., Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Amsterdam, 1988. [FLI 91] FLIESS M., MESSAGER F., “Sur la commande en régime glissant”, Comptes Rendus de l’Académie des Sciences de Paris, Series 1, vol. 313, p. 951-956, 1991. [FLO 00] FLOQUET T., PERRUQUETTI W., BARBOT J.P., “Angular velocity stabilization of body via VSS control”, ASME, vol. 122, p. 669-673, 2000. [FLU 53] FLUGGE-LUTZ I., Discontinous Automatic Control, Princeton University Press, New Jersey, USA, 1953. [FUR 90] FURATA K., “Sliding mode control of a discrete system”, Systems and Control Letters, vol. 14, p. 145-152, 1990. [GUT 79] GUTMAN S., “Uncertain dynamic systems – a Lyapunov min-max approach”, IEEE Transactions on Automatic Control, vol. AC-24, p. 437-449, 1979. [HUN 93] HUNG J.Y., GAO W.B., HUNG J.C., “Variable structure control: a survey”, IEEE Transactions on Industrial Electronics, vol. 40, no. 1, p. 2-22, 1993. [ITK 76] ITKIS U., Control Systems of Variable Structure, John Wiley & Sons, New York, 1976.
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[LAS 61] LASALLE J., LEFSCHETZ S., Stability by Liapunov’s Direct Method with Applications, Academic Press, New York, 1961. [SAR 87] SARPURK S.Z., ISTEFANOPULOS Y., KAYNAK O., “On the stability of discretetime sliding mode control systems”, IEEE Transactions on Automatic, vol. 32, no. 10, p. 930932, 1987. [SIR 87] SIRA-RAMIREZ H., “Differential geometric methods in variable-structure control”,International Journal of Control, vol. 48, no. 2, p. 1359-1390, 1987. [SIR 92a] SIRA-RAMIREZ H., “Dynamical sliding mode control in the regulation of nonlinear chemical processes”, International Journal of Control, vol. 56, no. 1, p. 1-21, 1992. [SIR 92b] SIRA-RAMIREZ H., “On the sliding mode control of nonlinear systems”, Systems and Control Letters, vol. 19, p. 303-312, 1992. [SLO 83] SLOTINE J.E., SASTRY S.S., “Tracking control of non-linear systems using surfaces with applications to robot manipulator”, International Journal of Control, vol. 38, no. 2, p. 465-492, 1983. [SLO 84] SLOTINE J.E., “Sliding controller design for non-linear systems”, International Journal of Control, vol. 40, no. 2, p. 421-434, 1984. [SLO 87] SLOTINE J.E., HEDRICKS J.K., MISAWA E.A., “On sliding observers for nonlinear systems”, ASME Journal on Dynamical Systems Measurement and Control, vol. 109, p. 245-252, 1987. [SLO 91] SLOTINE J.E., LI W., Applied Nonlinear Control, Prentice Hall, New York, 1991. [SLO 93] SLOTINE J.E., “Sliding order and sliding accuracy in sliding mode control”, International Journal of Control, vol. 58, no. 6, p. 1247-1263, 1993. [UTK 77] UTKIN V.I., “Variable structure system with sliding modes”, IEEE Transactions on Automatic control, vol. AC-22, no. 2, p. 23-36, 1977. [UTK 92] UTKIN V.I., Sliding Modes in Control Optimization, Springer-Verlag, Berlin, 1992. [UTK 93] UTKIN V.I., “Sliding mode control design principles and applications to electric drives”, IEEE Transactions on Industrial Electronics, vol. 40, no. 1, p. 23-36, 1993. [XU 93] XU J.X., HASHIMOTO H., “Parameter identification methodologies based on variable structure control”, International Journal of Control, vol. 57, no. 5, p. 1207-1220, 1993. [YOU 99] YOUNG K.D., UTKIN V.I., OZGÜNER U., “A control engineer’s guide to sliding mode control”, IEEE Transactions on Control Systems Technology, vol. 7, no. 3, p. 328342, 1999.
Chapter 7
Parameter Estimation for Knowledge and Diagnosis of Electrical Machines
7.1. Introduction In automatic control, modeling and identification are fundamental and essential disciplines, which precede all operations of simulation, observation, synthesis of a control law or system monitoring. This stage of modeling and identification depends highly on the system and on the considered application. Thus, concerning synthesis of a control law, a discrete model associated with an algorithm derived from the least squares technique can be suitable. With an objective of simulation or state-observation, the same methodology can still be satisfactory. However, if the user wants to simulate the dynamic behavior of the system and simulate the influence of some characteristic parameters at the same time, the approach using discrete model is insufficient, and it will be necessary to use a continuous-time representation with differential equations nearer to the physical nature of the system. Finally, with a monitoring objective, a continuous-time model is preferable, especially when the user wishes to carry out a diagnosis of the system state starting from a parameter estimation representative of its safe or fault functioning. In the case of discrete models, where the parameters and the structure may have lost links with physics, it is common to speak about black-box models. However, when the model is of continuous type governed by differential equations and when the parameters and the structure can approach physical equations, we refer in this case to knowledge models or more reasonably gray-box models. Indeed, it is unrealistic
Chapter written by Jean-Claude T RIGEASSOU, Thierry P OINOT and Smaïl BACHIR .
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to describe the physical behaviors exhaustively; and in fact only approximate models (or reduced order models) are used, hence the term gray-box model, which better represents the modesty of the ambitions of the researcher. In electrical engineering, even for a control application, the user prefers to use a model close to reality, whose parameters (resistances, inductances, etc.) have a tangible significance. The same attitude appears in process engineering or in chemical engineering (for example, for the constant kinetics of chemical reaction) like in robotics (for parameters such as masses and stiffness). In this chapter, we are interested in the parameter estimation of electrical machines starting from continuous-time models, close to physics. The considered applications concern the estimation of electrical parameters, and particularly the monitoring of electrical machines based on parameter estimation. However, the same identification algorithms could be used for the state observation or the synthesis of control laws. In this context, special attention will be given to system identification using continuous-time models. Depending on the nature of the residuals, two types of algorithms are commonly used: equation-error and output-error. In practice, equation-error algorithms are suitable only for differential equation models having constant coefficients. For such models, many techniques [MEN 99] were imagined in order to express linearly the system output with respect to its parameters (LP model). This property of linearity enables the use of least squares methods, whose essential interest is to provide an analytical expression of the estimated parameters [LJU 87]. Unfortunately, it is shown that for any LP model whose regressor depends directly (or indirectly by filtering) on the output values, the residuals are of the equation-error type and consequently the estimator is biased. A solution to eliminate this bias consists of using an instrumental variable technique [YOU 70, SÖD 83]. This procedure complicates the identification algorithm and convergence can, in some cases, present some problems. Finally, because in general electrical machines are not governed by differential equations with constant coefficients, but rather by non-linear differential systems, it can be deduced that this identification methodology is not really adapted for the considered problem. However, these methods should not be rejected. Indeed, although their estimates are open to criticism, they can be used to initialize the output-error methods. The reader interested by this subject will be able to refer to some synthetic presentations [JEM 97, TRI 01] and in particular to a comparative study of these identification methods [MEN 99] where selection criteria are presented. Thus, this chapter is dedicated to the presentation of the second category of algorithms, which are of output-error type; in France they are also called “méthode du modèle” (model method), according to a term imposed by their promoter J. Richalet [RIC 71]. These algorithms present a wide range of applications; since
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they do not stay on restrictive linearity assumptions, they can be used for non-linear differential systems. They are not only used in electrical engineering but also in process and chemical engineering. Unfortunately, it is important to specify that they require a computational load much higher than the algorithms derived from least squares. This certainly explains the interest of many researchers considering all the least squares alternatives previously mentioned. It also explains why, although the theory of these algorithms was proposed in the 1960s, it could only truly start to be imposed recently thanks to the rise in computing power of digital calculators. The computational load of output-error algorithms is mainly due to the iterative nature of the minimization of a quadratic criterion by a non-linear programming (NLP) algorithm [HIM 72]. Indeed, at each iteration, it is necessary to simulate the model of the system together with its sensitivity model, according to the considered variant. Moreover, the convergence to a global optimum is not guaranteed, because of the non-convexity of the quadratic criterion. Beyond these difficulties, which can fortunately be overcome, the principal interest of output-error algorithms is to provide an unbiased estimator, and (as opposed to equation-error techniques) with a certain immunity with respect to the modeling errors. However, it is necessary to examine the influence of this modeling error with respect to a deterministic parametric bias. In the same way, these output-error algorithms must be modified, if necessary, when the system is in closed-loop, inherent for example with the functioning of modern electrical machines. An important problem is the problem of a priori knowledge on the parameters. This knowledge is necessary for the initialization of the NLP algorithm, but in some cases, we can arrive at the paradox that the obtained estimator proves to be nonsensical when compared to the initial physical knowledge. Thus, we propose a methodology that allows the introduction of a priori knowledge into the estimator, according to the general principle of the Bayesian estimation [PET 81]. Although this last methodology must be used cautiously, as will be specified, it can be useful to improve convergence of the algorithm each time that an initial and reliable knowledge is available, particularly within the framework of machine monitoring, using an extended model composed of the safe functioning model of the system (thus with a good a priori knowledge) and of a model dedicated to a type of fault. This chapter is composed of four sections. Section 7.2 gives the general methodology of output-error identification. Section 7.3 presents an approach that allows introduction of a priori knowledge in practice within the framework of the Bayesian estimation. An application to the parameter estimation of an induced machine in the Park’s reference is given in section 7.4. Finally, a monitoring methodology of an electrical system with fault model and a priori knowledge of the safe functioning are the subject of section 7.5. This monitoring methodology based
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on parameter estimation is used and developed in Chapter 8, which is dedicated to the monitoring of electrical machines. 7.2. Identification using output-error algorithms 7.2.1. Introduction These algorithms are “model method” algorithms [RIC 71, RIC 91] and are fundamentally characterized by the simulation of the model output based on the only knowledge of the input (error-equation algorithms are based on a prediction of the output using the knowledge of the input and past values of the output). Using this procedure, the simulated output is independent of the perturbation affecting the system (if the system is in open-loop); then, residuals correspond to this perturbation, hence the term of output-error, and some interesting properties of convergence. However, this simulation complicates the criterion minimization problem which requires the use of non-linear optimization techniques. Many approaches can be used with output-error algorithms. After a brief reminder on the properties of the least squares algorithm used in output-error, we present the case where the gradient is computed by using the sensitivity functions. 7.2.2. Least squares algorithm in output-error In some cases, the system output y(t) is linear with respect to its parameters. Let this output model be y = f (θ, u), where u(t) is the input and θ the parameter vector. The model is linear with respect to its parameters (LP) [WAL 97] if y(t) can be written as follows: y = ϕT (u) θ
[7.1]
where ϕ (u) is the regressor vector. Let θ be an estimation of θ. Then, using u(t) (or uk known at each sampling time tk ), the estimated output yk is obtained: yk = ϕTk (u) θ
[7.2]
Consider that we have K experimental data pairs {uk , yk∗ }, obtained with sampling period Te , such that t = kTe . A quadratic criterion can thus be defined by: J=
K k=1
ε2k =
K
2 yk∗ − ϕTk (u) θˆ
[7.3]
k=1
where yk∗ = yk + bk
[7.4]
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and u εk = yk∗ − yk θ,
[7.5]
with – yk∗ : output measurement; – yk : true value of the output; – bk : stochastic perturbation; – εk : residual. Since the model is linear with respect to its parameters, the value θM C of θ which minimizes J is obtained analytically [LJU 87]: θM C =
K
−1 ϕk ϕTk
k=1
K
ϕk yk∗ .
[7.6]
k=1
Using [7.4], we obtain: θM C = θ + Δθ M C
[7.7]
where Δθ M C is the estimation error such that ΔθM C =
K
−1 ϕk ϕTk
k=1
K
ϕk bk .
[7.8]
k=1
Consider that the system is in open-loop, i.e. the sequence {uk } is independent of the noise sequence {bk }. Moreover, let us suppose that the perturbation is zero-mean. Then:
[7.9] E ϕk bk = 0 and
E ΔθM C = lim Δθ M C = 0 K→∞
i.e. the estimate θ M C is asymptotically unbiased [LJU 87, WAL 97]. If we suppose that the perturbation {bk } is independent and stationary, then
Var bk = σ 2 .
[7.10]
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The variance of the estimate Var{θ M C } is given by [LJU 87]: K −1 ϕk ϕTk Var {θM C } = σ 2
[7.11]
k=1
Since the term σ 2 is unknown, it is necessary to estimate it: σ 2 =
J (θM C ) K −N
[7.12]
where: – J(θ M C ) is the value of the criterion with θ = θ M C ; – K is the number of measurements; – N is the number of parameters. N OTE. Expression [7.11] is attractive due to its simplicity; unfortunately, this expression is rarely used because it is unrealistic. In fact, it is rare that the random perturbation verifies the preceding hypotheses. Generally, the perturbation can be correlated and non-stationary. Moreover, a deterministic part corresponding to modeling error adds to the perturbation. This part is directly correlated to the input {uk }. Then E{ϕk mk } = 0 and the estimator is asymptotically biased. Hence, [7.11] cannot be used to define realistic uncertainty domains of estimated parameters. Nevertheless, this relation gives valued information on the sensitivity of the quadratic criterion in comparison with the variation of the parameters (in the neighborhood of the optimum), which is a relative uncertainty (and not an absolute one). 7.2.3. Principle of the output-error method in the general case Consider a system given by the general N th order state-space model depending on parameter vector θ: x˙ = g (x, θ, u) [7.13] y = f (x, θ, u) where y(t) and u(t) are considered as single output and inputs to simplify the presentation. Note that no hypothesis on the linearity is necessary: g and f are based on the physical laws, which are generally non-linear. The only hypothesis is that the system is identifiable [WAL 97]. Let θ be an estimation of θ. Then, using the input u(t) and a numerical integration technique, a simulation of the system output is obtained: u ˙ = g x , θ, x [7.14] u y = f x , θ,
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The optimum θ opt is the value minimizing the quadratic criterion: J=
K k=1
ε2k =
K
2 yk∗ − fˆk u, θˆ
[7.15]
k=1
where yk∗ is the measure of the noised output, and bk is the noise signal. a non-linear optimization technique is used Since y(t) is not linear in parameters θ, [RIC 71]. Many techniques can be used; our choice is based on techniques based on the gradient and more precisely on the Levenberg-Marquardt [MAR 63] algorithm:
−1 [7.16] θ j+1 = θj − [J θθ + λI] J θ ˆ θ=θ j
where: K – J θ = −2 k=1 εk σ k,θi is the gradient; K – J θθ ≈ 2 k=1 σ k,θi σ Tk,θi is Gauss-Newton’s approximation of the Hessian; – λ is the control parameter; and – θk,θi = ∂∂θyˆk is the output sensitivity function. i
This algorithm, using the control parameter λ, tends towards the gradient algorithm when the estimation is far from the optimum (then λ >> 1). Near the optimum, it tends towards Newton’s technique (when λ → 0) which allows acceleration of the convergence near the optimum. This procedure ensures robust convergence, even with a bad initialization. Fundamentally, this technique is based on the calculation of the gradient and Hessian, which are themselves dependent on the numerical integration of the sensitivity functions [TRI 88, KNU 94, WAL 97]. These sensitivity functions σ are equivalent to the regressor ϕ in the LP case. Thus, let us consider the simulation of y(t) obtained with the estimated parameters θ: ˆu yˆk = fk x ˆ, θ, Then and let dθ be a variation of θ. yˆk θˆ + dθˆ = yˆk θˆ + σ Tk,θˆdθˆ + · · ·
[7.17]
[7.18]
or, with a development limited to the first order: dˆ yk ≈ σ Tk,θˆdθˆ
[7.19]
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In the LP case, the prediction yk is given by: yˆk = ϕTk,θˆθˆ
[7.20]
dˆ yk = ϕTk,θˆdθˆ
[7.21]
or
Moreover, if θopt is the parameter vector which minimizes J, and if we consider a variation dθ around θ opt , it is easy to show that: K T + dθˆ σ σ T dθˆ [7.22] J θ + dθˆ ≈ J θ opt
k k
opt
k=1
In the LP case, if θM C is the parameter vector which minimizes J, we obtain: K T T ˆ ˆ + dθ + dθ = J θ ϕ ϕ [7.23] dθˆ J θ MC
MC
k
k
k=1
These two examples show the analogy between regressors and sensitivity functions and the interest of the sensitivity functions to analyze the output-error estimators, particularly their accuracy. 7.2.4. Sensitivity functions In practice, it is necessary to differentiate two kinds of sensitivity functions [KNU 94, MOR 99]: y (t) is the output sensitivity function, used for the calculation of the – σy,θi = ∂∂θ i gradient and the Hessian; – σxn ,θi =
∂xn (t) ∂θi
Let us recall that
is the state sensitivity function.
dim(x) = N
dim(θ) = I Then, σy,θ is a vector of dimension I and σxn ,θ is a matrix of dimension [N × I] such that: [7.24] σ x,θ = σ x,θ1 · · · σ x,θi · · · σ x,θI
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For each parameter θi , σxn ,θi is obtained by partial derivation of the equation [7.13]. Thus: ∂g (x, θ, u) ∂x ∂g (x, θ, u) ∂ x˙ = σ˙ x,θi = + ∂θi ∂x ∂θi ∂θi
[7.25]
and σxn ,θi is the solution of a non-linear differential system: σ˙ x,θi =
∂g (x, θ, u) ∂g (x, θ, u) σ x,θi + . ∂x ∂θi
[7.26]
Finally, ∂y/∂θi is obtained by partial derivation of equation [7.13]: ∂y = ∂θi
∂f (x, θ, u) ∂x
T σ x,θi +
∂f (x, θ, u) . ∂θi
[7.27]
7.2.5. Convergence of the estimator u) is non-linear in θ, the quadratic criterion Because the simulation y = f ( x, θ, J(θ) is not parabolic as in the LP case, and the uniqueness of the optimum θ opt is not guaranteed [WAL 97]. Secondary optima can exist and the optimization algorithm can converge to one of these optima. A solution to converge to the global optimum uses a global approach, like genetic algorithms [FON 95]. A “map” of the criterion and of its optima is obtained. The global optimum θopt is then obtained using the Marquardt algorithm. In the presence of noise affecting the output, the estimator converges to: θopt = θ + Δθ
[7.28]
where θ is the true parameter vector and Δθ the estimation error. Using the sensitivity functions, Δθ can be approached by: ⎧ −1 K ⎫ K ⎨ ⎬ σ k σ Tk σ k bk [7.29] Δθ ≈ ⎩ ⎭ k=1
k=1
θopt
which is equivalent to: Δθ =
obtained in the LP case.
⎧ K ⎨ ⎩
k=1
−1 ϕk ϕTk
K k=1
⎫ ⎬ ϕk bk
⎭ θopt
[7.30]
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Then, if E{bk } = 0 and if the system is in open-loop, uk is independent of bk and E{Δθ} = 0 (since σ k depends only on uk ). The output-error estimator is asymptotically unbiased, whatever the nature of the zero mean output noise [WAL 97]. N OTE. When the system is in closed-loop (which is the case with AC machines with vector control), the perturbation {bk } is necessarily correlated with the control input through the corrector (or the control algorithm). Then σk , which depends on the input {uk }, is correlated with the noise {bk }; i.e. the estimation θopt is asymptotically biased. Fortunately, this bias is only really very significant when the signal-to-noise ratio is weak. Thus, as a first approximation, it can be neglected. Moreover, output-error algorithms functioning in closed-loop have been proposed; they are able to reject this bias using a more complicated identification procedure [GRO 00, LAN 97]. 7.2.6. Variance of the estimator Using the analogy between regressor and sensitivity functions, it is possible to define an approximated expression of the variance of θopt . Thus, by replacing ϕk with σ k in expression [7.11], we obtain:
ˆ2 Var θopt ≈ σ
K
−1 σ k σ Tk
[7.31]
k=1
This expression is approximated for the following reasons: – since the output model is non-linear in parameters, σ k is a local approximation of ϕk , i.e. this expression is an approximation of the quadratic criterion by a paraboloid; – as with the LP case, the perturbation is not characterized by the only term σ 2 , i.e. this expression only gives an information on the relative accuracy. 7.2.7. Implementation Output-error algorithms are more complex than equation-error algorithms because of the non-linear optimization. However, this is not the only difficulty with implementing these algorithms. It is also necessary to study the numerical simulation, the problem of initial conditions and the normalization of the sensitivity functions. 7.2.7.1. Simulation of the differential system In the least squares case, the implementation of the predictor is trivial for discrete-time; with continuous-time, it is more difficult [TRI 01]. In the output-error case, we are confronted with a real problem of numerical simulation of differential systems for the simulation of the model output and the
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217
sensitivity functions. Simulation error or approximate calculations will give a systematic error or deterministic bias. When the system is linear, the differential equations can be integrated using the exponential matrix technique [ISE 92], which enables conciliation of accuracy and rapidity of computations. When the system is non-linear, which is almost always the case with physical systems, it is necessary to give importance to accuracy and numerical stability. Accuracy is guaranteed by an algorithm like Runge-Kutta of order 4. However, the discretization of the differential equations can make their integration unstable. Then, implicit techniques can be used, for instance, the Adams’ algorithms [NOU 89]. Nevertheless, some errors can subsist, even if all the preceding precautions are taken. In fact, it is very important to consider the input type in the numerical integration algorithm. The simplest case is when the input is derived from a numerical command. The applied input is then continuous and it is necessary to specify how the input varies between two sample instants: a linear change is sufficient in the major cases, otherwise a parabolic (or a higher order) extrapolation should be used [ISE 92]. 7.2.7.2. Initial conditions in OE algorithms Initial conditions of the state are supplementary unknown parameters which need identification [WAL 97]. When the data number is relatively small, particularly with regard to the transient of the system output, identification of initial conditions is the necessary solution. Then, an extended parameter vector is considered which contains the system parameters and the initial conditions. The identification algorithm remains unchanged, but the computation time and the convergence difficulties increase. When the acquisition time is higher than the transient tr [RIC 91], the kr first data (kr = tr /Te ) are not used in the criterion, avoiding the estimation of initial conditions. Then, the criterion becomes: J=
K
ε2k
[7.32]
k=kr
and the differential system is simulated from t = 0. Let us recall that in the two situations, if the initial conditions are not taken into account, an important bias appears. 7.2.7.3. Normalization In an academic situation, the user chooses numerical values which are close. In a real situation, numerical values can be very different, causing some difficulties for the convergence of the identification algorithm to appear. A solution consists of
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normalizing the parameters, which in practice is achieved by the normalization of the sensitivity functions [MOR 99, RIC 71]. Consider the parameter θn , with an initial estimation θn0 , such that θn = θn0 + Δθn . The estimation of θn is equivalent to that of Δθn . Let us define θn = (1 + μn )θn0 where Δθn = μn θn0 . The sensitivity function linked to θn is then: 1 ∂ yˆ ∂ yˆ = ∂θn θn0 ∂μn
[7.33]
where the sensitivity functions ∂ y /∂μn are now normalized and closed. In practice, μ and thus θ are estimated using the non-linear programming algorithm and we obtain: θn = (1 + μn ) θn0
[7.34]
7.3. Parameter estimation with a priori information 7.3.1. Introduction Despite all the numerical precautions stated in the preceding section, output-error algorithms can, in certain situations, provide incoherent estimates such as negative resistances or inductances (see [ALM 95, JEM 97] in the electrical engineering case). It is necessary to seek the cause of these anomalies in the optimization mechanism: indeed, the latter determines the set of parameters which allows best fit of the data by the selected model, without any physical constraint. Moreover, when that occurs, only some parameters are affected by the anomaly. It is fundamentally a problem of parameter sensitization: although theoretically identifiable, the concerned parameters are almost unidentifiable and balancing phenomena can be observed. The reflex response to such a problem is to propose to improve the excitation [LJU 87, KAB 97]: however, in many situations, this optimal excitation can turn out to be unrealistic in practice (even dangerous for the process) or can transgress the validity conditions of the model! In addition, these problems of excitation must be connected to the selected model and its modeling error. Within the framework of control using black-box models, the engineer increases the complexity of these models until the residuals become independent and uncorrelated to the input [LJU 87]. Although the “physicist” uses models of voluntarily reduced complexity, adapted to the description of a specific phenomenon, this does not prevent him from estimating the corresponding parameters without systematic error, but with specific and dedicated approaches. Then, the
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question is can these same models, used with the identification techniques developed in automatic control, provide estimates which are coherent with physical laws? A solution suggested in this chapter consists of explicitly introducing the physical knowledge in order to replace the lack of excitation, or to improve it. In addition, even with a good excitation, it is interesting to improve the convergence of the estimator (and its accuracy) by introducing an a priori knowledge (given that this knowledge is not erroneous!). 7.3.2. Bayesian approach A possible misunderstanding should be immediately cleared up: effectively, it is recommended to initialize the optimization algorithm with parameters close to the optimum in order to accelerate the convergence and to reduce the computation time. For this, an initial knowledge is used but the optimization algorithm quickly forgets this initialization! In the best situation, convergence to a secondary optimum can be avoided. It is therefore necessary to explicitly introduce this prior knowledge: thus, a modified or compound criterion is defined. The major justification of this new criterion can be found in the Bayesian approach [EYK 74, GOO 77, PET 81, TUL 93]. This approach consists of considering the estimation problem in a probabilistic context. Consider a set of experimental data uk , yk∗ (or U , Y ∗ ); we propose to estimate θ by ∗ (or maximizing the probability density of θ conditionally to the data Y ∗ , i.e. P θ/Y a posteriori density). Otherwise, we have a priori information on θ characterized by Then, the Bayes’s relation is: P θ. ˆ Y ∗ θˆ P θP [7.35] P θˆ Y ∗ = PY ∗ the maximization of P θ/Y ∗ is Because P Y ∗ does not explicitly depend on θ, ∗ equivalent to the maximization of P θP Y /θ. This technique is known as a posteriori maximum. In order to tackle this problem, some hypothesis are necessary: usually, P θ and P Y ∗ /θ are considered as Gaussian densities, allowing us to write: T 1 ˆ P θˆ Y ∗ =A exp − θ − θ0 M0−1 θˆ − θ0 2 [7.36] 1 ∗ ˆ ˆ T −1 ∗ ˆ ˆ − Y − Y θ, U Rb Y − Y θ, U 2
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Control Methods for Electrical Machines
where: – A is a constant; – θ0 is the initial knowledge of θ; – M0 is the covariance matrix of θ 0 ; – Rb is the covariance matrix of the stochastic perturbation. Finally, by considering the natural logarithm of this expression, it is easy to show ∗ is equivalent to the minimization of the compound that the maximization of P θ/Y criterion: T JC = θˆ − θ0 M0−1 θˆ − θ 0 J0
ˆU + Y ∗ − Yˆ θ,
T
ˆU Rb−1 Y ∗ − Yˆ θ,
[7.37]
J∗
JC = J0 + J ∗ In practice, the covariance matrix of the perturbation is unknown and Rb is replaced by σb2 I (where σb2 is the variance of the noise and I the identity matrix). Then, except σb2 (representing the variance of output noise), J ∗ represents the usual quadratic criterion, containing experimental information. However, J0 is a second quadratic criterion which introduces an “elastic” constraint in the minimization of the global criterion JC : in fact, it prevents θ from moving away from θ 0 , with a “return force” dependent on θ − θ0 . 7.3.3. Minimization of the compound criterion Let K T 2 1 ∗ ˆu yk − yˆk θ, JC = θˆ − θ0 M0−1 θˆ − θ0 + 2 σb
[7.38]
k=1
This new criterion is minimized using the Marquardt’s algorithm with: ! " K 1 −1 ˆ εk σ k J Cθ = 2 M0 θ − θ0 − 2 σ ˆb
[7.39]
k=1
and
!
JCθθ ≈ 2
M0−1
K 1 + 2 σ k σ Tk σ ˆb k=1
" [7.40]
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221
Let θC be the value of θ minimizing JC and obtained using the optimization algorithm. In order to demonstrate the interest of a priori information, we propose to show the links between θ C , θopt and θ0 . Thus, let us consider that θopt and θ0 are near to θ C , i.e. the sensitivity functions θ are approximately equal. Then, we can write: T −1 JC θˆ ≈ JC min + θˆ − θC Jθθ [7.41] θˆ − θC where
Jθ = 2
M0−1
and
Jθθ ≈ 2 with
ST ε ˆ θ − θ0 − 2 σb
M0−1
ST S + 2 σb
[7.42]
[7.43]
⎡
⎤ σ T1 ⎢ ⎥ S = ⎣ ... ⎦ σ TK
When the Newton’s algorithm is used to minimize JC from the initial value θ0 , this Then: optimum θC is obtained in one iteration because JC is a quadratic form of θ.
−1 Jθ [7.44] θ C = θ 0 − Jθθ θ0
with
Jθ
θ0
=
−2S T ∗ ˆ Y − Y θ0 σb2
[7.45]
which gives: θ C = θ0 +
M0−1
ST S + 2 σb
−1
S T ∗ ˆ Y − Y θ0 σb2
[7.46]
−1 Let Popt = Sσ2S where Popt is the covariance matrix linked to the conventional b criterion minimized by θopt . Then, let us define: T
−1 P −1 = M0−1 + Popt
[7.47]
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Control Methods for Electrical Machines
i.e. θC = θ0 + which can be written as:
P S T ∗ ˆ Y − Y θ0 σb2
θC = θ 0 + K Y ∗ − Yˆ θ0
[7.48]
[7.49]
where K is the gain of a Kalman filter applied to a particular system [RAD 84]: ⎧ ⎨θi+1 = θ i = θ [7.50] ⎩Y ∗ = Y (θ, u) + b i i i This interpretation confirms the preceding probabilistic and stochastic approach: from θ0 , the optimal value to obtain θC is determined, taking into account the gain K and the information given by the dataset Y ∗ (corresponding to θopt ). Moreover, taking into account [7.47], the estimate θC is necessarily more precise than θ 0 or θ opt . 7.3.4. Deterministic interpretation The Bayesian approach can be interpreted like a Kalman filter and can also be interpreted like a regularization technique [TIK 77, JOH 97, SAY 98]. Unfortunately, this interpretation failed in the case of physical parameter estimation using a model of reduced complexity. Note that the output perturbation {bk } is not really a random perturbation but a deterministic one: the major part of the output perturbation is due to the modeling error. The perturbation is deterministic because it is generated by the input u. Thus, a new interpretation is proposed, essentially based on the geometry of the quadratic criterion (see [MOR 99] for more information). Let σ b2 be the pseudo-variance of the output perturbation which is obtained using: σ b2 =
J(θ opt ) K
[7.51]
where J is the conventional criterion and K the number of samples. ∗ 2 We can notice that J θopt = K takes into account the k=1 yk − fk uk , θ opt b2 is a pseudo-variance. noise bk and the modeling error, i.e. σ Using σ b2 , a pseudo-covariance matrix can be defined by: −1 Popt = σ ˆb2 S T S
[7.52]
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223
In the neighborhood of θopt , it is possible to write: T Jopt J θˆ ˆ − θ T S S θˆ − θ ≈ θ opt opt + 2 2 σ ˆb σ ˆb σ ˆb2
[7.53]
Finally: T T −1 JC ≈ θˆ − θ0 M0−1 θˆ − θ0 + θˆ − θopt Popt θˆ − θ opt + K
[7.54]
This expression shows that JC is the result of two paraboloids, centered on θ0 and θ opt , characterized by the spreads M0 and Popt . JC is a unique paraboloid, centered on θC , such that: −1 −1 −1 −1 M0 θ 0 + Popt θopt θC ≈ θ 0 + M0−1 + Popt
[7.55]
(θ C is the barycenter of θ0 and θopt ) characterized by the spread MC , such that: −1 MC−1 = M0−1 + Popt
[7.56]
T JC ≈ θˆ − θC MC−1 θˆ − θC + JC θC .
[7.57]
Let
Let us consider a monovariable example, in order to illustrate this result (see Figure 7.1). When θ0 is close to θopt , the criterion JC is more convex than the conventional criterion, and secondary optima tend to disappear. Consequently, the introduction of {θ 0 , M0 } accelerates the convergence of the optimization algorithm. The estimation θC is generally more precise than θ0 or θopt because the paraboloid JC is less “open” than the preceding criteria. Moreover, if the estimation θ opt is ill-sensitized, i.e. the optimum is straight, then the global optimum θ C will be attracted by θ0 which is equivalent to an “elastic” return force (especially if the paraboloid J0 is “closed”). N OTE. The conventional probabilistic interpretation has certainly been an obstacle for use of the Bayesian approach. However, the determinist interpretation shows the advantages of this technique on the convergence of the optimization algorithm and on the uniqueness of the optimum. Moreover, it is important to notice that the covariance matrices of θ0 and of the noise can be significantly simplified, with no influence on their properties. Thus,
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J 0 , J*
ˆ opt
0
JC
C
ˆ
Figure 7.1. Deterministic interpretation of the compound criterion – monovariable case
matrix M0 can be favorably reduced to a diagonal form, with an overevaluation of each variance if necessary, as will be shown in sections 7.4, 7.5 and Chapter 8. Moreover, the covariance matrix of the noise can be reduced to a single coefficient constituted by the variance of the perturbation, which is particularly justified when a modeling error occurs. 7.3.5. Implementation The implementation of this methodology requires two types of information to be handled: – the a priori information {θ0 , M0 } obtained from a preceding global estimation or from specific and partial estimations. In this more realistic case, the matrix M0 is at best diagonal (some examples are proposed in the next section);
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225
– the variance σ b2 of the perturbation: this is an essential parameter in the weighting b2 is linked to the between J0 and J. The deterministic interpretation has shown that σ 2 spread of the paraboloid J/ σb : a lower value gives more weight to experimental data, while a high value gives more importance to the initial information θ0 . In practice, it is necessary to define a practical procedure for its determination because it principally depends on the modeling error, unknown before optimization. Concretely, a prior value is used, which is approximately the variance of the stochastic perturbation; when θ C is estimated, a new value σ b2 is obtained using J(θC ) (where J 2 is the conventional criterion). If σ b is very far from its initial value, a new estimation of θ C is performed. This last point is considered in the next section. The value of σ b2 obtained using J (θ C ) can also be used as a guideline of the coherence between initial information and prior knowledge. Thus, when σ b2 , θ0 and 2 M0 are reliable information, σ b allows us to test if the experimental information is compatible with the knowledge on the system: a value of σ b2 significantly greater than 2 σb can correspond to a non-stationarity of the system which involves a modification of its model. Then, the modeling error increases as does σ b2 . NOTE. M0 and σ b2 play an essential part in the weighting between a priori information and experimental data. The variance of the a priori information M0 should result from a decisive choice: a low variance increases confidence in θ0 . Thus, in practice, this value can be increased in order to allow the variation of θC , if this variation is predictable (see section 7.5, which is dedicated to fault diagnosis). The estimated value σ b2 will need to be adjusted thanks to a new identification, as was previously described. In practice, the example treated in the next section shows that this convergence is very fast. Finally, let us recall that the use of the Bayesian approach must be justified by true a priori information {θ0 , M0 }: it is necessary to be persuaded that erroneous initial information will irremediably bias the estimator. However, when this use is justified, this technique significantly improves the convergence of the optimization algorithm (and thus computing time) and guarantees the existence of only one optimum. In addition, it allows reconsideration of some techniques based on parameter estimation, like fault detection. 7.4. Parameter estimation of the induced machine 7.4.1. Introduction We propose to illustrate the application of output-error techniques (with and without a priori knowledge) to the induced machine in the dynamic case.
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At first, it is necessary to specify the model of this machine, firstly in the three-phase context and secondly using Park’s transformation, which is well adapted for parameter estimation of that kind of machine. Then, we present the model identification starting from experimental data with the two types of algorithms previously defined. 7.4.2. Modeling in the three-phase frame The general model of the induced machine is obtained by considering a uniform air-gap and a sinusoidal distribution of the magnetic field. The machine is assumed to work in an unsaturated magnetic state and the iron losses are neglected. In these conditions, the dynamic model of the machine with the leaks added up at the stator can be described by: ⎧ d ⎪ ⎪us = [Rs ] is + dt φs ⎪ ⎪ ⎪ ⎪ ⎨0 = [Rr ] ir + d φ dt r [7.58] ⎪ ⎪ φs = [Ls ] is + [Msr ] ir ⎪ ⎪ ⎪ ⎪ ⎩ φr = [Msr ]T is + [Lr ] ir with:
Rs = Rs · I and [Rr ] = Rr · I ⎛ ⎞ Lp Lp L − + L − f sa ⎜ p 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ L L p p ⎟ ⎜ Ls = ⎜ − Lp + Lf − 2 2 ⎟ ⎜ ⎟ ⎝ ⎠ Lp Lp − Lp + Lf − 2 2 ⎞ ⎛ Lp Lp − − L ⎜ p 2 2 ⎟ ⎟ ⎜ ⎜ Lp Lp ⎟ ⎜ [Lr ] = ⎜− − ⎟ Lp 2 ⎟ ⎟ ⎜ 2 ⎠ ⎝ L Lp p − Lp − 2 2 ⎞ ⎛ 2π 2π Lp cos θ − Lp cos(θ) Lp cos θ + ⎜ 3 3 ⎟ ⎜ ⎟ ⎜ ⎜ 2π 2π ⎟ ⎟ Msr = ⎜Lp cos θ − Lp cos(θ) Lp cos θ + ⎟ ⎜ 3 3 ⎟ ⎜ ⎟ ⎝ ⎠ 2π 2π Lp cos θ − Lp cos(θ) Lp cos θ + 3 3
Parameter Estimation of Electrical Machines
227
where: – us , is and ir respectively represent the voltage vector, the stator current vector and the rotor current vector; – φs and φr : vectors of stator and rotor fluxes; – Rs (resp. Rr ): resistance of a stator phase (resp. rotor); – Lp and Lf : principal inductance and leakage inductance added up at the stator; – θ: electrical angle of the rotor position. The equations of voltages obtained above are relatively simple, even if the number of state-variables is high. Nevertheless, the matrix of mutual inductances [Msr ] depends on the electrical angle θ. Then, the writing of the state-space representation remains complex. These equations are simplified using Park’s transformation [CHA 87, CAR 95, GRE 97] which is described below. 7.4.3. Park’s transformation Park’s transformation, largely used for the modeling of AC machines, corresponds to a projection of the three-phase variables on a turning diphasic frame, the goal being to eliminate the position of mutual inductances in the matrices. To achieve this, it is sufficient to carry out a transformation of the three-phase system abc to the diphasic system αβ using the transformation T23 (Concordia’s transformation), followed by a rotation of reference frame P (−θ) in order to link the reference frame to the rotation Park’s axes dq (see Figure 7.2). as
T cr
d ar
cs
bs
q
br
Figure 7.2. Park’s transformation linked to the rotor
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Control Methods for Electrical Machines
In a matrix form, the essential variables of the machine become [CAR 95]: – in the stator: xdqs = P (−θ) T23 xs – in the rotor: xdqr = T23 xr with
⎡ 0 T23 =
cos(0) 2 ⎢ ⎢ ⎢ 3 ⎣ sin(0)
⎡ ⎢cos(θ) ⎢ P (θ) = ⎢ ⎣ sin(θ)
cos
2π 3
⎤ 4π 3 ⎥ ⎥ ⎥ 4π ⎦ sin 3
cos
2π sin 3 ⎤ π cos θ + 2 ⎥ ⎥ ⎥: rotation matrix of angle θ. π ⎦ sin θ + 2
Equations of the induced machine [7.58] in the Park’s frame linked to the rotor with leaks added up at the stator are: ⎧ π d ⎪ ⎪ = R i + + ω P U φ φdq s dqs ⎪ dqs ⎪ r dt dqs 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎨ 0 = Rr idqr + φdq dt r [7.59] ⎪ ⎪ ⎪ ⎪ φ = Lm + Lf idqs + Lm idqr ⎪ ⎪ ⎪ dqs ⎪ ⎪ ⎪ ⎩φ =L i +i dqr
m
dqs
dqr
where: represents the electrical pulsation (where θ = p θmechanical and p is the – ω = dθ dt number of pairs of poles per phase); – Lm = 32 Lp is the magnetizing inductance. The obtained model of the induced machine is essentially characterized by four physical parameters Rs , Rr , Lm and Lf . These parameters are the parameters that need estimation. 7.4.4. Continuous-time state-space model For the majority of the industrial applications of the induced machine, the inertia of the rotating parts is significant. Consequently, the rotor speed is generally slowly variable when compared with other electrical parameters of the machine [MOR 99].
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229
Thus, a 4th non-linear state-space representation of the induced machine is obtained (because of the dependency on the speed), by associating the state-vector which contains the stator currents and rotor fluxes as well as the input and the output of the corresponding system respectively with the voltages and stator currents of axis d and q [CAR 95, MOR 99]: = A(ω) x(t) + B u(t) x(t) ˙ [7.60] y(t) = C x(t) with
T x = ids iqs φdr φqr : state-vector " ! ! " Uds ids , y= : respectively machine input and output u= Uqs iqs ⎡ R +R s r − ⎢ Lf ⎢ ⎢ ⎢ −ω ⎢ ⎢ A(ω) = ⎢ ⎢ ⎢ Rr ⎢ ⎢ ⎣ 0 ⎡
1 ⎢ Lf B=⎢ ⎣ 0
0 1 Lf
−
Rs + Rr Lf 0 Rr
⎤T 0 0 ⎥ ⎥ , ⎦ 0 0
⎤ ω Lf ⎥ ⎥ Rr ⎥ ⎥ Lm · L f ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ Rr ⎦ − Lm
Rr Lm · Lf ω − Lf Rr − Lm
ω
0
C=
1 0 0 0 0 1 0 0
7.4.5. Output-error identification The considered system is multivariable, with two inputs (Uds and Uqs ) and two outputs (ids and iqs ). Thus, a criterion J composed of two quadratic terms is considered: J=
K k=1
i∗ds k
− ˆidsk
2
+
K
i∗qs − ˆiqsk
k=1
k
2
[7.61]
where i∗ds and i∗qs are sampled measurements with sampling period Te = 0.7 ms k k (t = kTe , k varying from 1 to K = 4,500). ˆidsk and ˆiqsk represent the simulation of model [7.60] based on the estimation θ where θT = Rs Rr Lm Lf .
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Control Methods for Electrical Machines
At each iteration, it is also necessary to simulate the sensitivity functions ∂iqs ∂ θ
∂ids ∂ θ
and
, as pointed out in the preceding section.
Experimental data are obtained with an induced machine of 1.1 kW supplied by a generator with a vector control. The machine is regulated to its nominal speed and coupled with a continuous generator which acts as a load. The machine input is a pseudo-random binary sequence (PRBS) of ±90 tr/mn added to the speed reference of 750 tr/mn. Rotor currents and voltages are measured, as well as the mechanical position of the rotor (which allows the computation of the pulsation ω). The input vector {Uds , Uqs } and the output vector {i∗ds , i∗qs } are obtained by using Park’s transformation. Minimizing quadratic criterion [7.61] using the Marquardt algorithm, the optimum is given by: ⎤ ⎡ ⎤ ⎡ 9.507 Ω Rs ⎢ R ⎥ ⎢ 4.010 Ω ⎥ ⎥ ⎢ r⎥ ⎢ θ opt = ⎢ ⎥ = ⎢ ⎥ ⎣Lm ⎦ ⎣0.4364 H⎦ 0.0751 H Lf Figure 7.3 shows the estimation θ according to the iterations of the identification algorithm. 10
(:)
4.1
Rs
9.8
4
9.6
Rr
3.9
9.4 0 0.437
(:)
2
4
6
8
(H)
3.8 0 0.077
2
4
0.435 0
Lm 2
4
8
6
8
Lf
0.076 0.436
6
(H)
0.075
6
8
0.074 0
2
4
Figure 7.3. Estimation of θ according to algorithm iterations
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231
At the optimum, an estimation of the noise variance is obtained using: σ b2 =
Jopt = 0.0462 2 (K − N )
7.4.6. Output-error identification and a priori information Here we estimate the same parameters, but with a priori information, constituted by the average of ten preliminary estimations (corresponding to the knowledge of the “healthy” functioning of the machine). For this, the composite criterion is minimized: 2 2 K ∗ ids − ˆidsk + i∗qs − ˆiqsk T −1 k k Jc = θˆ − θ 0 M0 θˆ − θ0 + [7.62] δˆ2 with ⎤ ⎡ ⎤ ⎡ 9.81 Ω Rs 0 ⎢ R ⎥ ⎢ 3.83 Ω ⎥ ⎥ ⎢ r0 ⎥ ⎢ θ0 = ⎢ ⎥=⎢ ⎥ ⎣Lm0 ⎦ ⎣ 0.436 H ⎦ Lf0 and
⎡
2 σR S
0 2 σR r 0
0.0762 H
0 0
⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎦ 2 σL f
⎢ 0 ⎢ M0 = ⎢ 2 σL ⎣ 0 m 0 0 0 ⎡ 0 0 2 10−3 ⎢ 0 −4 2 10 0 ⎢ =⎢ ⎣ 0 0 6 10−7 0 0 0
0 0 0
⎤ ⎥ ⎥ ⎥ ⎦
10−7
The previous data is used, and we choose: b2 = δ2 = σ
Jopt = 0.0462 2 (K − N )
Minimizing JC using the Marquardt algorithm, we obtain θ C : ⎤ ⎡ ⎤ ⎡ 9.667 Ω Rs ⎢ R ⎥ ⎢ 3.920 Ω ⎥ ⎥ ⎢ r⎥ ⎢ θC = ⎢ ⎥ = ⎢ ⎥ ⎣Lm ⎦ ⎣0.4366 H⎦ 0.0762 H Lf
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Control Methods for Electrical Machines
Figure 7.4 shows the evolution of θC according to algorithm iterations: it is obvious that the addition of a priori information has significantly accelerated the convergence of the algorithm. 10
(:)
4.1
R
9.8
R
4
s
9.6
r
3.9
9.4 1 0.437
(:)
2
3
4
(H)
3.8 1 0.077
L
2
4
3
4
Lf
0.075 0.435 1
3
0.076
m
0.436
2
(H)
3
4
0.074 1
2
Figure 7.4. Evolution of θC according to algorithm iterations
Considering the variance of the a priori information (better precision on inductances than on resistances), only resistance estimations are slightly different from those corresponding to θ0 . NOTE. The second part of the criterion enables us to estimate the variance of the noise for θ = θC . Thus, we obtain σ b2 = 0.0483, which is close to the value initially chosen 2 for δ : it is thus useless in this case to reiterate the algorithm to determine the optimal value of δ 2 . Nevertheless, the question of the choice of δ 2 must be evoked if initial information on the residuals variance is erroneous. Again using the same data, JC was initialized with an erroneous value of δ 2 and the estimation of θopt was iterated. The results are shown in Figures 7.5 and 7.6. It can be seen that this iterative process converges almost entirely in only one iteration to δ 2 = 0.0483, regardless of the initial value of δ 2 . 7.5. Fault detection and localization based on parameter estimation 7.5.1. Introduction The fundamental assumption made when monitoring (or supervising) a system by parameter estimation is that a fault results in the variation of one (or several)
Parameter Estimation of Electrical Machines
233
0.05
0.04
0.03
0.02
0.01
0
0
1
2 Iterations
3
4
Figure 7.5. Evolution of δ 2 according to algorithm iterations: case, initial value JC < 0.0483
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0
1
2 Iterations
3
4
Figure 7.6. Evolution of δ 2 according to algorithm iterations: case, initial value JC > 0.0483
characteristic parameter(s) of the system, thus constituting the signature of this fault. According to this assumption, supervising a system involves monitoring its parameters using an identification algorithm, either off-line (by parts of samples) or in a recursive way.
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Control Methods for Electrical Machines
In fact, this assumption can easily be invalidated by the fact that this methodology is not able to distinguish a normal parametric variation (possibly foreseeable) from that corresponding to a fault occurring randomly. This is due to the fact that in order to estimate parameters, a model should initially be defined. The reflex is indeed to use the model of normal operation of the system. However, a fault tends to modify this model and also modifies, in some cases, its structure, and in most cases a modeling error is introduced. Thus, we will propose a methodology again based on parameter estimation, but combining two characteristics: – the general model will include a model of safe functioning (nominal model) and a fault model (specific to each considered fault); – parameter estimation will be used with a priori information, which corresponds to the expertise (or knowledge) of the user on the safe functioning of the system. 7.5.2. Principle of the method The principle of the method is shown here in the case of linear systems governed by differential equations with constant parameters, although this methodology is general. Let Hn (s) be a system of nominal transfer function, characterized by a vector θn . When a fault occurs, a modeling error δHi (s) signing the fault also appears (δHi (s) is characterized by a vector θ i ). Thus, the input/output transfer function becomes: H(s) = Hn (s) + ΔHi (s)
[7.63]
The general model of the system, in a fault situation, is shown in Figure 7.7, where b(t) is a random perturbation, u(t) is the input and y ∗ (t) is the measured output. 'H i s b t +
u t
H n s
+
y* t +
+
Figure 7.7. General model of the system corresponding to the fault di
The nominal model Hn (s), or safe functioning model, summarizes the user expertise on the functioning of the system, i.e. the knowledge on the nominal parameters θn and on their variance Var{θn }, as well as noises affecting the output,
Parameter Estimation of Electrical Machines
235
i.e. their variance σb2 . In addition, the modeling error δHi (s) must constitute a true signature of the fault, not only by its structure but also in its parameters θ i . The general model of the system H(s) is then composed of a “common mode” term (the nominal model Hn (s)) and of “differential mode” term (the fault model δHi (s)) only sensitized when a fault di appears. In addition, the nominal model must take into account foreseeable variations of the parameters, whereas the fault model must for its part remain insensitive to these same variations. Finally, the nominal model must include the expertise of the user, summarized by {θn , var{θn }}. Thus, this methodology is naturally linked to identification with a priori information. An extended parameter vector is thus defined: ! " θn [7.64] θe = θi Moreover, an extended covariance matrix is defined: ! " 0 Var {θn } Var {θe } = 0 Var {θi }
[7.65]
The a priori knowledge can be essentially defined on the nominal model. We thus obtain: ! " θˆ [7.66] θ e0 = n 0 and ⎤
⎡
σθ21n
⎢ ⎢ ⎢
⎢ Var θe0 = ⎢ ⎢ ⎢ ⎢ ⎣
..
σθ2N n 0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0
. ∞
..
.
[7.67]
∞ Notice that Var{θn0 } takes into account only the diagonal terms resulting from Var{θn }. In addition, the terms σθ2jn , resulting from a safe functioning, must be overestimated in order to tolerate foreseeable parameter variations (for example, according to the temperature or to the magnetic state).
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Control Methods for Electrical Machines
On the other hand, as we do not know if the fault will occur, its a priori value θi is zero while its initial variance is infinite (or very large). Thus, the optimization algorithm responsible for the minimization of the criterion: K T 2
−1 1 ∗ JC = θˆe − θe0 Var θe0 yk − yˆk θˆe θˆe − θe0 + 2 σb
[7.68]
k=1
will affect the expertise of the user in the nominal model, by tolerating foreseeable variations (included in σθ2jn ) and will be very sensitive to the variations θ i of the differential model, characteristic of the fault di . 7.5.3. Simulations In order to specify the methodology which has just been presented, let us consider an academic electrical example. A reel of N turns wound up on a magnetic circuit of section S, of average length l (see Figure 7.8) is considered.
i
v
l N S Figure 7.8. Magnetic circuit
The iron of the magnetic circuit is assumed to be characterized by the relation B = μH (where μ = cst). Neither the iron saturation nor the magnetic hysteresis is taken into account. In addition, it is assumed that the iron losses are negligible (at first approximation). Then, according to Ampere’s theorem: H = Nl i , the total flux φ is given by: φ = N BS = μ
N 2S i l
[7.69]
The reel inductance L can be defined according to φ = Li: L=μ
N 2S l
[7.70]
Parameter Estimation of Electrical Machines
237
Moreover, the reel resistance R is proportional to the length of the electric wire, i.e. to the number of turns. We can thus write that R = k1 N and L = k2 N 2 . Let us define a problem where the fault is constituted by a variation δN of the number of turns of the reel (compared to nominal number N ). In practice this could be due to a winding with commutation of the number of turns. Then, if N varies by ΔN : – R varies by ΔR, with ΔR = k1 ΔN ; dL ΔN , i.e. ΔL = 2k2 N ΔN ; – L varies by ΔL, with ΔL = dN that is, the variations of R and L are linked. In the nominal state of the reel: L k2 N =τ = R k1
[7.71]
where τ is the time constant of the reel while: 2k2 N ΔL = = 2τ ΔR k1
[7.72]
Then, a nominal model (nominal impedance) of the reel can be defined by: Zn (s) = Rn + Ln s
[7.73]
and a fault model can be defined by: ΔZ (s) = ΔR + ΔL (s) = ΔR (1 + 2τ s)
[7.74]
leading to the extended model of the reel: Z (s) = Zn (s) + ΔZ (s) = Rn + Ln s + ΔR (1 + 2τ s)
[7.75]
In addition, R and L can vary without the appearance of a fault, for example with changes in the temperature or in the magnetic state of iron. Since μ was assumed to be constant, we cannot consider variation of L without modification of this assumption. However, we can consider a variation of the resistance (alone) under the effect of heating, therefore of an increase in temperature T . Then R (T ) = Rn + ΔR (T )
[7.76]
L (T ) = Ln
[7.77]
and
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Control Methods for Electrical Machines
NOTE. An important problem concerns the identifiability of the parameters of the fault (s) model. For this, let us consider the sensitivity functions. Knowing that I (s) = U Z(s) , it can easily be shown that: ⎧ −I (s) ⎪ ⎪ σRn (s) = L {σRn (t)} = ⎪ ⎪ Z (s) ⎪ ⎪ ⎪ ⎪ ⎨ −s I (s) σLn (s) = L {σLn (t)} = [7.78] ⎪ Z (s) ⎪ ⎪ ⎪ ⎪ ⎪ − (1 + 2τ s) I (s) ⎪ ⎪ ⎩ σΔR (s) = L {σΔR (t)} = Z (s) where L{·} is the Laplace’s transform. Then: σΔR (t) = σRn (t) + 2τ σLn (t)
[7.79]
Since σΔR (t) is a linear combination of σRn (t) and σLn (t), the fault parameter T ΔR is unidentifiable as the pseudo-Hessian J θθ ≈ 2 K k=1 σ k,θ i σ k,θ i of the direct method is non invertible. However, when the composite criterion JC is used, which incorporates a priori information {θ0 , M0 }, we obtain the corresponding Hessian [7.43]: JCθθ ≈ 2M0−1 +
J θθ σb2
which is now invertible thanks to M0 and the fault parameter ΔR becomes identifiable. This example clearly shows the interest in associating a model dedicated to a type of fault with the knowledge obtained on the safe functioning in the framework of a strategy of fault detection. 7.5.4. Numerical simulations 7.5.4.1. Study of the safe functioning Let us consider a reel characterized by R = 4 Ω and L = 0.1 H. The functioning of this reel was simulated numerically with Te = 1 ms and a PRBS input voltage. The current output was disturbed by a white noise in such a way that the signal-to-noise ratio S/B = 10. An input/output data file was then formed (see Figure 7.9).
Parameter Estimation of Electrical Machines
239
Voltage (V) 1
0.5
0
Ŧ0.5
Ŧ1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
Current (A) 0.4 0.3 0.2 0.1 0 Ŧ0.1 Ŧ0.2 Ŧ0.3
0
0.1
0.2
0.3
0.4
0.5 Time (s)
Figure 7.9. Input/output data
Using the output-error identification algorithm (without a priori knowledge), the values of R and L have been estimated. These values will be the basis of our expertise on the safe functioning. Thus, for the model Zn (s) = Rn + Ln s, we have: ⎧ ˆ n = 4.012 Ω σR = 3.85 10−2 Ω R ⎪ n ⎪ ⎪ ⎨ ˆ n = 0.0989 H σL = 1.81 10−3 H L n ⎪ ⎪ ⎪ ⎩ σ ˆb2 = 1.64 10−3 τˆn = 0.0247 s The prior information is then defined by: ⎧ R0 = 4.012 Ω, L0 = 0.0989 H, τ = 0.0247 s ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σR0 = 1 Ω > σRn (which authorizes the variations of R with the temperature) ⎪ σL0 = σLn ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 σb = σ ˆb2
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Control Methods for Electrical Machines
7.5.4.2. Study of the functioning with fault conditions Let us consider a variation of ΔN of the total number of turns, which corresponds to the fault model: with θ Te = Rn
Ln
Z (s) = Rn + Ln s + ΔR (1 + 2τ s) ΔR .
[7.80]
Let us consider in addition variations of Rn due to the temperature, that is to say δR (T ). For each considered situation, the previous input is used, but the realization of the white noise was different (although the same signal-to-noise ratio was the same). All the results are presented in Table 7.1. Test
1
2
3
4
5
R (Ω)
4
5
5
5
5
ΔR (Ω)
1
0
0.2
1
−0.2
(Ω) R (H) L
3.966
4.965
4.962
5.066
4.979
0.0990
0.0988
0.0989
0.0989
0.0989
(Ω) ΔR
1.066
0.047
0.172
0.886
−0.148
Table 7.1. Results of parameter estimation
Test 1 corresponds to an increase of N (corresponding to ΔR = 1 Ω) without temperature variation. The increase of R in the fault model is perfectly detected (taking into account of the noise level). Reciprocally, test 2 corresponds to a temperature variation (increase of R of the common mode), without variation of δR of the differential mode. Only the resistance of the common mode varied. Tests 3, 4 and 5 correspond to simultaneous variations of the temperature and of the number of turns (increase or decrease in δR). The results show the corresponding resistance changes of the common and differential modes, in close connection with their cause (always taking into account of the noise level) and independently of their amplitude. In conclusion, the association of a fault model (with common and differential modes) and an algorithm of parameter estimation with a priori knowledge constitutes a tool for fault detection, making it possible to effectively distinguish them from the parameter variations of common mode.
Parameter Estimation of Electrical Machines
241
7.6. Conclusion This chapter was devoted to output-error identification and particularly to the estimation of physical parameters within the framework of electrical engineering. Two approaches held our attention: – The traditional approach is the extension of the least squares method to non-linear systems. Despite its computational load, its essential attribute lies in its natural immunity to random disturbances, thus guaranteeing an unbiased estimator. – The Bayesian approach makes it possible to include a priori knowledge available on the system, mainly when the user has to deal with physical problems; however, this initial knowledge must be completed by the variance information so as to avoid the risk of biasing the estimator. The error-equation approaches, afflicted with a bias inherent to the construction of the regressor, should not be systematically rejected. They enable, despite this bias, initialization of the research of the optimization algorithm (thus avoiding possible secondary optima) and, if required, they can take part in the development of the a priori information within the framework of the Bayesian approach. The two output-error techniques of parameter estimation have been tested and compared in the case of the induced machine. In addition, we have proposed a new methodology of fault detection, based on the Bayesian approach (the a priori knowledge corresponds to the expertise of the user on the safe functioning of its system), and on the use of a fault model, a true signature of this fault. This methodology, validated by a numerical simulation, will be taken again and generalized with the case of the asynchronous machine in the next chapter, which is devoted to the detection of stator and rotor faults. 7.7. Bibliography [ALM 95] A L M IAH H., Modélisation et identification en ligne des paramètres d’une machine asynchrone saturée en régime statique, PhD Thesis, University of Poitiers, 1995. [CAR 95] C ARON J.P. and H AUTIER J.P., Modélisation et commande de la machine asynchrone, Editions Technip, 1995. [CHA 87] C HATELAIN J., Machine électriques, Dunod, 1987. [EYK 74] E YKHOFF P., System Identification, Parameter and State Estimation, Wiley, London, 1974. [FON 95] F ONTEIX C., B ICKING F., P ERRIN E. and M ARC I., “Haploïd and diploïd algorithms, a new approach for global optimization: compound performances”, Int. J. Systems Sci., vol. 26, no. 10, p. 1919–1933, 1995. [GOO 77] G OODWIN G. and PAYNE R., “Dynamic system identification: experiment design and data analysis”, Math. in Sc and Engineering, vol. 136, Academic Press, 1977.
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[GRE 97] G RELLET G. and C LERC G., Actionneurs électriques. Principes, modèles et commande, Eyrolles, Paris, 1997. [GRO 00] G ROSPEAUD O., Contribution à l’identification en boucle fermée par erreur de sortie, PhD Thesis, University of Poitiers, 2000. [HIM 72] H IMMELBLAU D.M., Applied Non-linear Programming, McGraw-Hill, 1972. [ISE 92] I SERMAN R., L ACHMANN K. and M ATKO D., Adaptative Control Systems, Prentice Hall, 1992. [JEM 97] J EMNI A., Estimation paramétrique des systèmes à représentation continue – Application au génie électrique, PhD Thesis, University of Poitiers, 1997. [JOH 97] J OHANSEN T., “On Tikhonov reguralization, bias and variance in non-linear system identification”, Automatica, vol. 33, no. 3, p. 441–446, 1997. [KAB 97] K ABBAJ H., Identification d’un modèle type circuit prenant en compte les effets de fréquence dans une machine asynchrone à cage d’écureuil, PhD Thesis, INP de Toulouse, 1997. [KNU 94] K NUDSEN M., “A sensitivity approach for estimation of physical parameters”, 10th IFAC Symposium on System Identification, SYSID’94, vol. 2, p. 231–237, 1994. [LAN 97] L ANDAU I.D. and K ARIMI A., “Recursive algorithms for identification in closed loop: a unified approach and evaluation”, Automatica, vol. 33, p. 1499–1523, 1997. [LJU 87] L JUNG L., System Identification: Theory for the User, Prentice Hall, USA, 1987. [MAR 63] M ARQUARDT D.W., “An algorithm for least-squares estimation of non-linear parameters”, Soc. Indust. Appl. Math, vol. 11, no. 2, p. 431–441, 1963. [MEN 99] M ENSLER M., Analyse et étude comparative de méthodes d’identification des systèmes à représentation continue. Développement d’une boîte à outil logicielle, PhD Thesis, University of Nancy I, 1999. [MOR 99] M OREAU S., Contribution à la modélisation et à l’estimation paramétrique des machines électriques à courant alternatif: Application au diagnostic, PhD Thesis, University of Poitiers, 1999. [NOU 89] N OUGIER J., Méthodes de calcul numérique, Masson, Paris, 1989. [PET 81] P ETERKA V., “Bayesian approach to system identification”, in E YKHOFF P. (Ed.), Trends and Progress in System Identification, Pergamon, Oxford, p. 239–304, 1981. [RAD 84] R ADIX J., Filtrage et lissage statistiques optimaux linéaires, Cepadues, Toulouse, 1984. [RIC 71] R ICHALET J., R AULT A. and P OULIQUEN R., Identification des processus par la méthode du modèle, Gordon and Breach, 1971. [RIC 91] R ICHALET J., Pratique de l’identification, Hermes, Paris, 1991. [SAY 98] S AYED A. and K AILATH T., “Recursive Least-Squares adaptive filters”, in M ADISETTI V.K and W ILLIAM D.B. (Eds.), Digital Signal Processing Handbook, CRC Press, 1998.
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[SÖD 83] S ÖDERSTRÖM T. and S TOICA P., Instrumental Variable Methods for System Identification, Springer Verlag, Berlin, 1983. [TIK 77] T IKHONOV A. and A RSENIN U., Solutions of Ill-posed Problems, Winston, Washington, 1977. [TRI 88] T RIGEASSOU J.C., Recherche de modèles expérimentaux assistée par ordinateur, Technique et Documentation Lavoisier, Paris, 1988. [TRI 01] T RIGEASSOU J.C. and P OINOT T., “Identification des systèmes à représentation continue – Application à l’estimation de paramètres physiques”, in Identification des systèmes, p. 177–211, Hermes, Paris, 2001. [TUL 93] T ULLEKEN H., “Grey-box modeling and identification using physical knowledge and Bayesian techniques”, Automatica, vol. 29, no. 2, p. 285–308, 1993. [WAL 97] WALTER E. and P RONZATO L., Identification of Parametric Models from Experimental Data. Communications and Control Engineering Series, Springer, 1997. [YOU 70] YOUNG P., “An instrumental variable method for real-time identification of noisy process”, Automatica, vol. 6, p. 271–287, 1970.
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Chapter 8
Diagnosis of Induction Machines by Parameter Estimation
8.1. Introduction In simple technology, the asynchronous machine or induction motor is intensively used in most electrical drives, especially for constant speed applications such as ventilation and pumping. The progress of power electronics associated with modern control made it possible to consider efficient variable speed applications that were previously reserved for DC engines and more recently in synchronous drives. An illustration is the three generations of high-speed trains used in France (TGV): the first one (south-east) commercialized in 1981 is equipped with a DC motor, the second (south-west in 1989) with synchronous motors and the latest (Eurostar in 1994) with asynchronous motors. Thus, in view of all these economic issues, a general reflection has been initiated for safety operating oriented to the diagnosis of induction machines. The aim of this monitoring is to detect the electrical and mechanical faults in the stator and the rotor of induction motors. The diagnosis of induction machines under fixed speed has been intensively studied in the literature, unlike applications under variable speed. Indeed, the signals being highly non-stationary, approaches based on conventional Fourier analysis of current lines [ABE 99, INN 94, FIL 94], stator voltages and electromagnetic torque [MAK 97, MAL 99] prove inadequate. Considerable efforts have been undertaken in the last decade in parameter identification of continuous models
Chapter written by Smaïl BACHIR, Slim T NANI , Gérard C HAMPENOIS and Jean-Claude T RIGEASSOU.
246
Control Methods for Electrical Machines
[TRI 88, MEN 99, MOR 99]. Requiring rich excitation of machine modes, parameter estimation is well suited for diagnosis in variable speed drives. Thus, algorithm development dedicated to a realistic physical parameter estimation [MOR 99, TRI 99], taking into account a prior knowledge of the machine, has enabled promising advances in the diagnosis of induction machines. This approach, based on parameter identification of a model, which is one of the most important goals, is the development of mathematical models that are really representative of default operations. In the faulty case, the induction machine present in addition to a conventional dynamic behavior a default behavior [BAC 01a, BAC 01b, BAC 06]. In modeling for diagnosis, it is essential to consider two modes: a “common” and a “differential” mode. The common mode describes the dynamic model of the induction machine and translates the healthy model of the machine. The differential mode gives information on a defect. The parameters of this mode should be essentially sensitive to the faults. This situation is useful to the effective detection and localization. Indeed, a change in temperature or magnetic state is reflected by a change in the state of the common-mode parametric model, but there is no change in the differential mode [SCH 99, BAC 01a]. This diagnosis method made it necessary to carry out a global parameter estimation of the two model modes. Thus, the electrical parameters of the common-mode indicate the dynamic state of the machine (constant rotor time, magnetizing inductance, etc.). Differential mode parameters explain the default information and the monitoring of these parameters enable the detection and localization of the imbalance. In this chapter, we first study two fault models which take into account the effects of inter-turn faults resulting in the shorting of one or more circuits of stator phase winding and broken rotor bars. To take into account simultaneous stator and rotor faults, a global fault model of the machine will be presented. The corresponding diagnosis procedure based on parameter estimation of the stator and rotor fault model and more experimental results are presented later in the chapter. 8.2. Induction motor model for fault detection For a diagnosis of induction motors, it is useless to consider an unbalanced two axis Park’s model [MOR 99, SCH 99]. The deviation of their electrical parameters is certainly an indication of a new situation in the machine, but this evolution can be due to heating or an eventual change in the magnetic state of the motor [MOR 99]. On the other hand, it is very difficult to distinguish stator faults from rotor faults. The use of fast Fourier analysis of identification residuals is an original method to localize a fault, but the estimation of electrical parameters is unable to produce the fault level. A good solution is the introduction of an additional model to explain the faults [BAC 02, BAC 01b]. The parameters of this differential model enable the detection and localization of the faulty windings.
Diagnosis of Induction Machines by Parameter Estimation
247
8.2.1. Stator fault modeling in the induction motor In order to take into account the presence of inter-turn short circuit windings in the stator of an induction motor, an original solution is to consider a new winding dedicated to the stator fault [SCH 99]. The new model is composed of an additional shorted winding in three phases axis. Figure 8.1 shows a three phase, 2-pole, induction machine in the case of short circuit winding. This fault induces in the stator a new winding Bcc short circuited and localized by the angle θcc . as T
T cc Bcc
cr
ar
cs
bs
br Figure 8.1. Motor windings with a short circuit
Two parameters are introduced to define the stator faults: – The localization parameter θcc which is a real angle between the short circuit inter-turn stator winding and the first stator phase axis (phase a). This parameter allows 4π the localization of the faulty winding and can take only three values 0, 2π 3 or 3 , corresponding respectively to a short circuit on the stator phases a, b or c. – The detection parameter ηcc is equal to the ratio between the number of inter-turn short circuit windings and the total number of inter-turns in one healthy phase. This parameter makes it possible to quantify the unbalance and to obtain the number of inter-turns in short circuit. 8.2.1.1. Short circuit model On three-phase windings, we define the vector of stator voltages and currents, respectively us and is , and the vector of rotor currents ir : ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ua isa i ra ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ u s = ⎝ ub ⎠ is = ⎝ isb ⎠ ir = ⎝ irb ⎠ . uc isc i rc
248
Control Methods for Electrical Machines
In general, as a result of a short circuit, vibrations and torque oscillations synonymous with the presence of new components in the electromagnetic torque and therefore in the stator currents result from this [MOR 99, BAC 06]. Indeed, a short-circuiting of turns is at the origin of a new stator winding with a strong current and, consequently, an additional magnetic field in the machine. For example, consider the case of a healthy machine with p pole-pairs. When the three phase currents system with a pulsation of ωs = 2πfs flows through the stator windings, three stationary magnetic excitations directed along the axis of each phase will be created. It is the sum of these excitations which creates a rotating field in the airgap at the pulsation of Ωs = ωps according to the original winding. When a stator fault occurs, an additional shorted circuit winding Bcc appears in the stator. This winding creates a stationary magnetic field Hcc at the pulsation Ωs oriented according to the faulty winding. In this case, a strong current, noted icc , flows through the short circuit winding Bcc . It is the interaction of Hcc with a rotating motor field which introduces a torque ripple and new electromagnetic forces. With assumption of system linearity, this situation is equivalent to a superposition of a “common” operating mode producing a rotating field and a “differential” mode producing a fault field. Voltage and flux equations for fault model of induction machine can be written as: d [8.1] us = [Rs ] is + φs dt d 0 = [Rr ] ir + φr [8.2] dt d 0 = Rcc icc + φcc [8.3] dt
where
⎛
φs = [Ls ] is + [Msr ] ir + [Mscc ] icc
[8.4]
φr = [Mrs ] is + [Lr ] ir + [Mrcc ] icc
[8.5]
φcc = [Mccs ] is + [Mccr ] ir + Lcc icc
[8.6]
⎛ ⎞ ⎞ Rsa Rra 0 0 0 0 ⎜ ⎜ ⎟ ⎟ 0 ⎠ 0 ⎠ Rsb Rrb [Rr ] = ⎝ 0 [Rs ] = ⎝ 0 0 0 Rsc 0 0 Rrc ⎛ Lsac ⎞ Lsab − − Lpsa + Lf sa 2 2 ⎟ ⎜ ⎟ ⎜ ⎜ Lsab Lsbc ⎟ ⎟ Ls = ⎜ − Lpsb + Lf sb ⎟ ⎜ − 2 2 ⎟ ⎜ ⎠ ⎝ Lsbc Lsac − Lpsc + Lf sc − 2 2
Diagnosis of Induction Machines by Parameter Estimation
⎛ ⎜ ⎜ ⎜ Lr = ⎜ ⎜ ⎜ ⎝ ⎛
Lpra + Lf ra Lrab − 2 Lrac − 2
−
Lrab 2
Lrac 2 Lrbc − 2
−
Lprb + Lf rb −
Lrbc 2
Msa ra cos(θ) ⎜ ⎜ ⎜ 2π ⎜ Msr = ⎜Msb ra cos θ − ⎜ 3 ⎜ ⎝ 2π Msc ra cos θ + 3
249
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Lprc + Lf rc 2π Msa rb cos θ + 3 Msb rb cos(θ) 2π Msc rb cos θ − 3
⎞ 2π Msa rc cos θ − 3 ⎟ ⎟ 2π ⎟ ⎟ Msb rc cos θ + ⎟ 3 ⎟ ⎟ ⎠ Msc rc cos(θ)
[Mrs ] = [Msr ]T Rsx (resp. Rry ): proper resistance of stator phase (resp. rotor phase). Lpsx and Lf sx : inductance and leakage stator inductance. Lpsx + Lf sx : proper inductance of stator phase. Lsxy (resp. Lrxy ): mutual inductance between two stator phases (resp. rotor phases). Msx ry : mutual inductance between stator phase x and rotor phase y. Mscc (resp. Mrcc ): mutual inductance between of stator phase (resp. rotor phase) and short circuit winding. θ = p · θmechanical : electrical rotor angle. p: number of pole-pairs. H YPOTHESIS 8.1. The previous electrical equations can be simplified with these usual hypotheses: – symmetry and linearity of the electrical machine; – magnetomotive force in both the airgap and the flux is sinusoidal; – the magnetic circuit is not saturated and has a constant permeability; – skin effect and core losses are neglected. With these assumptions, we can write: Rsx = Rs Rry = Rr Lpsx = Lpry = Lsxy = Lrxy = Msx ry = Lp .
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Control Methods for Electrical Machines
In the previous electrical equations, the leakage is divided between the stator and the rotor phases. This method generates two coupled parameters Lf sx and Lf sy . One solution to simplify these equations is to globalize the leakage in the stator phase according to the relations: Lf ry = 0 [8.7] Lf sx = Lf .
The winding resistances are proportional to the number of inter-turns, then the resistance Rcc of faulty winding Bcc can be written as: Rcc = ηcc Rs with ηcc =
ncc Number of inter-turn short circuit windings = ns Total number of inter-turns in healthy phase
[8.8]
According to the previous hypothesis, the expressions of inductance and mutual inductances can be simplified: 2 Lp + Lf Lcc = ηcc Mccs = ηcc Lp cos θcc
2π 2π cos θcc + cos θcc − 3 3 2π 2π Mccr = ηcc Lp cos θcc − θ cos θcc − θ − cos θcc − θ + 3 3
T Mrcc = Mccr ,
T Mscc = Mccs .
8.2.1.2. Two-phase stator fault induction model To minimize the number of model variables, we use Concordia transformation which gives αβ values of the same amplitude as abc value. Thus, we define three to two axis transformation T23 as: xαβs = T23 xs : stator variables xαβr = P (θ) T23 xr : rotor variables
[8.9]
Diagnosis of Induction Machines by Parameter Estimation
251
where xαβ is a projection of x following the α and β axes. Matrix transformations are defined as: ⎡ 0 ⎢cos(0) 2⎢ [T23 ] = ⎢ 3⎣ sin(0)
cos sin
2π 3 2π 3
cos sin
⎤ 4π 3 ⎥ ⎥ ⎥ 4π ⎦ 3
⎤ π cos(θ) cos θ + ⎢ 2 ⎥ ⎥ ⎢ P (θ) = ⎢ ⎥: rotational matrix ⎣ π ⎦ sin(θ) sin θ + 2 ⎡
The short circuit variables are localized on one axis; these projections on the two Concordia axes α and β is defined as: ! iαβcc =
cos(θcc )
!
"
sin(θcc )
· icc ,
φαβ
cc
=
" cos(θcc ) sin(θcc )
· φcc
[8.10]
Thus, equations [8.1]-[8.6] become: U αβs = Rs iαβs + 0 = Rr iαβr
d φ dt αβs
[8.11]
d π φ φαβ + −ωP r dt αβr 2
0 = ηcc Rs iαβcc +
d φ dt αβcc
[8.13] 0
φαβ = (Lm + Lf ) iαβs + Lm iαβr + s
0 φαβ = Lm (iαβs + iαβr ) + r
0 φαβ
cc
=
[8.12]
2 ηcc Lm iαβcc 3
2 ηcc Lm iαβcc 3
2 ηcc Lm Q(θcc ) (iαβs + iαβr ) 3 2 2 Lm + Lf ηcc Q(θcc ) iαβcc + 3
[8.14]
[8.15]
[8.16]
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Control Methods for Electrical Machines
where ω = inductance
dθ dt
is the rotor electrical pulsation and Lm = ! Q(θcc ) =
3 2
Lp is the magnetizing
" cos(θcc ) sin(θcc )
cos(θcc )2
sin(θcc )2
cos(θcc ) sin(θcc )
.
If we neglect the leakage inductance Lf according to magnetizing inductance Lm in short circuit flux expressions [8.14]-[8.16], we can write new flux equations as: ⎧ φαβ = φαβ + φαβ ⎪ ⎪ s f m ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = Lf iαβs + Lm (iαβs + iαβr − ˜iαβcc ) [8.17] ⎪ φαβ = φαβ = Lm (iαβs + iαβr − ˜iαβcc ) ⎪ ⎪ r m ⎪ ⎪ ⎪ ⎪ ⎩ φ˜ = ηcc Q(θcc ) φαβ αβ cc
m
where
0 ˜iαβ = − cc
0
2 ηcc iαβcc , 3
˜ φ αβ
cc
=
3 φ 2 αβcc
[8.18]
φαβ , φαβ are the magnetizing and leakage fluxes respectively. m
f
Then, short circuit current equation [8.13] becomes: ˜iαβ = 2 ηcc Q(θcc ) d φ . cc 3 Rs dt αβm i DE s
i'DE s
Rs
Lf
[8.19]
Z .P ( S / 2 ).I DE
i DE
~ i DE cc U DE s
Q cc
m
Lm i DE
r
Rr
m
Figure 8.2. A short circuit induction machine model
According to this equation, the faulty winding Bcc becomes a simple unbalanced resistance element in parallel with magnetizing inductance. The existence of localization matrix Q(θcc ) in equation [8.19] makes the state space representation in Concordia’s axis complex. In a large range of industrial applications, voltage drop in
Diagnosis of Induction Machines by Parameter Estimation
253
Rs and Lf is neglected according to stator voltage U αβs then, we can put a short circuit element Qcc into the input voltage border (see Figure 8.2). Line currents iαβs become the sum of short circuit current ˜iαβcc and usual current iαβs in the classic Concordia model. It is much simpler to work in the rotor reference frame because we only have two stator variables to transform. Therefore, in state operation, all the variables have their pulsations equal to s ωs (where s is the slip and ωs is stator pulsation). We define Park’s transformation as: xdq = P (−θ) xαβ
[8.20]
Afterwards, the fault model will be expressed under Park’s reference frame. So, short circuit current [8.19] becomes: ˜idq = 2 ηcc P (−θ) Q(θcc ) P (θ) U dq . cc s 3 Rs
[8.21]
8.2.1.3. Example of stator fault model validation by spectral analysis It is interesting to study the properties of current ˜idqcc in the short circuit winding. In the literature, it is shown that a spectral analysis of stator current allows us to specify the nature of the defect [MOR 99]. Indeed, a failure in the stator is reflected in the power spectral density by the appearance of spectrum lines around frequencies of 2 ω, whose origins can be explained in the following manner. The three phase stator currents create in the machine airgap, a magnetic field turning at synchronous speed ω . This magnetic field sweeps the rotor windings, which causes rotation ωs = 1−g of the motor. When the stator defect appears, it creates with the direct stator field an opposite field running at the speed − ωs . The stator currents are now direct and inverse following the imbalance of windings. The interaction of this field with that from the stator windings induce an electromagnetic forces at the frequency equal to 2 ωs . Therefore, with Park’s transformation, we can find in the stator currents measurement a harmonic frequency at 2 ω. For example, to validate a previous fault model, the stator current in Park’s frame idqs is required to present a sinusoidal component around this frequency. We will use the additional short circuit current term of ˜idqcc [8.21] to justify the default model. For example, consider the case of a short circuit occurring on the first stator phase, localized by the angle θcc = 0. In this case, we can write: 1 0 Q(θcc ) = 0 0 The short circuit current becomes: ˜idq = R(θ) U dq cc s
[8.22]
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Control Methods for Electrical Machines
with:
! ηcc cos(2θ) + 1 2 ηcc P (−θ) Q(θcc ) P (θ) = R(θ) = 3 Rs 3 Rs − sin(2θ)
− sin(2θ)
"
1 − cos(2θ)
.
The input voltages U dqs are almost continuous in the Park’s frame, so they vary slowly compared to the terms of the matrix R(θ) (except during the transient corresponding to a change of torque). The short circuit current ˜idcc and ˜iqcc are then a linear combinations of terms whose instantaneous pulsation is at 2θ˙ = 2 ω. These sinusoidal components at 2 ω can be found in the measurement of stator currents idqs , explaining a possible stator imbalance. (dB)
(dB)
60
60
20
20
a. Healthy case
−20
−20
−60
−60
−100 0
50
100
Harmonics
150
200
(Hz)
−100 0
b. Stator fault case
50
100
150
200
(Hz)
Figure 8.3. Stator current power spectral density (fs = 25 Hz)
For illustration, we present in Figure 8.3 a comparison between the power spectral density (Fourier transform) of direct current Park ids in both healthy and stator fault cases (with short-circuiting of 58 turns on phase a). For a 1.1 kW induction machine, with 4-poles, whose rotational speed is around 750 rpm (25 Hz), we measured the stator currents vector is and performed the Park’s transformation. Thus, in Figure 8.3 we can observe the appearance of additional spectrum lines in the stator fault case around 2 · fs = 2 · 25 Hz. 8.2.1.4. Global stator fault model Fundamentally, we show that in the faulty case, an induction machine can be characterized by two equivalents modes. The common mode model corresponds to the healthy dynamics of the machine (Park’s model), whereas the differential mode model explains the faults. This model, which is very simple to implement because expressed in Park’s frame, offers the advantage of explaining the defect through a short circuit element dedicated to the faulty winding. However, it is unsuitable in the case of simultaneous defects on several phases. Indeed, this representation is only adapted in the case of a single phase defect. In the presence of short circuits on several phases, this model translates the defect by aberrant parameters values, because it takes into account only a single winding.
Diagnosis of Induction Machines by Parameter Estimation
255
To remedy this, we generalize this model by dedicating to each phase of the stator a short circuit element Qcck to explain a possible faulty winding [BAC 01a, BAC 01b]. So, in the presence of several short circuits, each faulty element allows for the diagnosis of a phase, by watching the value of the parameter. This simple deviation indicates the presence of unbalance in the stator. The short circuit current, noted ˜idqcc , k in the k th differential model can be expressed as: ˜idq = 2 ηcck P (−θ) Q(θcck ) P (θ) U dq cck s 3 Rs
[8.23]
where Q(θcck ) is the localization matrix. If the faults occurs on phase a (resp. b and and 4π ). c) then the angle θcck is equal to 0 rad (resp. 2π 3 3 i dq
i'dqs
s
~ i dqcc
U dq s
Qcc1
1
~ i dqcc
Qcc2
2
~ i dqcc
Qcc3
Rs
Lf
Z .P ( S / 2 ).I dq
s
i dq
r
3
Rr
Lm
i dq
m
Figure 8.4. A global stator fault model in the dq frame
Figure 8.4 shows the global stator fault model in the dq Park’s axis with global leakage referred to the stator. 8.2.2. Rotor fault modeling As with the stator fault, the rotor fault is modeled by a new axis B0 which is related to the first rotor axis ar by the angle θ0 [BAC 02]. This additional short circuited winding is at the origin of a stationary rotor field H0 (t) steered according to rotor fault axis (see Figure 8.5). Recently, rotor faults occurring in induction motors have been investigated. Various methods have been used, including measurement of rotor speed which indicates speed ripple, in the same way as spectral analysis of the line current [FIL 94, INN 94, MAN 96]. The main problem concerning these monitoring methods is that they are essentially invasive, requiring obvious interruption of operation. Moreover, they are inappropriate under varying speeds. For these reasons, parameter estimation is preferred for fault detection and diagnosis of induction motors [MOR 99].
256
Control Methods for Electrical Machines
ar B ro k en ro to r b a r a x is B0
T0 bar 2
bar 1
bar nb
bar nb - 1
E n d -rin g
br
cr
Figure 8.5. Broken rotor bar representation
Parameter estimation is based on the simulation of a continuous state-space induction motor model. This model assumes sinusoidal magnetomotive forces, non-saturation of the magnetic circuit and negligible skin effect. Under these assumptions, the stator in dq Park’s axis and squirrel cage rotor made of nb bars can be modeled by an equivalent circuit. So, two additional parameters are introduced in “differential” mode to explain rotor faults: – the angle θ0 between the fault axis (broken rotor bar axis) and the first rotor phase, allowing the localization of the broken rotor bar; and – a parameter η0 equal to the ratio between the number of equivalent inter-turns in defect and the total number of inter-turns in one healthy phase, introduced to quantify the rotor fault:
η0 =
Number of inter-turns in defect Total number of inter-turns in one phase
[8.24]
The number of turns in one rotor phase is indeed fictitious. For nb rotor bars, if we assume that the rotor cage can be replaced by a set of nb mutually coupled loops, each
Diagnosis of Induction Machines by Parameter Estimation
257
loop is composed by two rotor bars and end ring portions [ABE 99, BAC 01a]; then the total number of rotor turns in one phase for a three-phase representation is equal to n3b . For nbb broken rotor bars, fault parameter η0 becomes: η0 =
3 nbb nb
[8.25]
8.2.2.1. Model of broken rotor bars During stator fault modeling, we can write voltage and flux equations of new faulty winding B0 in dq Park’s frame [BAC 02]: 0 = η0 Rr io + φ0 =
dφ0 dt 0
2 2 η Lm i 0 + 3 0
[8.26] 2 η0 Lm cos θ0 3
sin θ0 idqs + idqr
[8.27]
The current i0 in the faulty winding B0 creates a stationary magnetic field H0 being directed related to the broken rotor bar axis. This additional magnetic field is at the origin of flux φ0 . By throwing i0 and φ0 on dq Park axis, we associate the stationary vectors: ! ! " " cos(θ0 ) cos(θ0 ) i0 , φdq = φ0 idq0 = 0 sin(θ0 ) sin(θ0 ) Equations [8.26] and [8.27] become relations between stationary vectors according to the rotor frame. Thus, the voltage and flux equations of stator, rotor and faulty winding of the induction motor are given by: d π φdq [8.28] U dqs = Rs idqs + φdq + ω P s s dt 2 0 2 [8.29] η0 idq0 φdq = Lf idqs + Lm idqs + idqr + s 3 0 = Rr idqr +
d φ dt dqr
[8.30] 0
φdq = Lm (idqs + idqr ) + r
0 = η0 Rr idqo + 0 φdq = 0
2 η0 Lm idq0 3
[8.31]
dφdq
0
dt
2 η0 Lm Q(θ0 ) 3
[8.32]
idqs + idqr +
0
2 η0 idq0 3
[8.33]
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Control Methods for Electrical Machines
By using the same transformation used to obtain the primary translation of an equivalent power transformer system, we can write global flux equations as: φdq = φdq + φdq = Lf idqs + Lm (idqs + idqr − ˜idq0 ) s
m
f
φdq = φdq = Lm (idqs + idqr − ˜idq0 ) r
[8.34]
m
φ˜dq = η0 Q(θ0 ) φdq 0
m
with
0 ˜idq = − 0
0
2 η0 idq0 , 3
˜ φ = dq 0
3 φ 2 dq0
[8.35]
Also, the current equation of faulty winding is given by: dφ dφ ˜idq = 2 η0 Q(θ0 ) dqm = R0−1 dqm 0 3 Rr dt dt
[8.36]
where Q(θ0 ) is the localization matrix. 8.2.2.2. Equivalent electrical schemes According to equation [8.36], faulty winding is a simple resistance element in parallel with magnetizing inductance and rotor resistance. Because the reference frame is chosen according to rotor speed, it is impossible to translate this element in stator border U dqs . The solution consists of establishing an equivalent induction machine system, with Park’s rotor resistance Rr to the fault resistance R0 . Thus, the equivalent resistance Req referred to the rotor is the stake in parallel with the rotor resistance and fault resistance as: −1 Req = Rr−1 + R0−1
= Rr−1 +
2 η0 Rr−1 Q(θ0 ) 3
[8.37]
By inversion, we obtain the expression of an equivalent resistance matrix: Req = Rr + Rdefect α = Rr − Q(θ0 ) Rr 1+α
[8.38]
where α = 23 η0 . Thus, equivalent rotor resistance in the broken rotor bar case is a series connection of a healthy rotor resistance Rr and fault resistance Rdefect . Figure 8.6 is the resulting rotor fault circuit diagram for induction machines.
Diagnosis of Induction Machines by Parameter Estimation
i dq
U dq
s
Lf
Rs
Z .P ( S / 2 ).I dq
259
s
i dq
r
Rr s
Lm
i dq
R defect m
Figure 8.6. Broken rotor bars model
The angle θ0 allows an absolute localization of the faulty winding, according to the first rotor phase. Indeed, induced bar currents create a nb -phases system and fault angle θ0 is fixed by the initial rotor position according to the stator position. However, when two broken rotor bars occur in a machine, estimation of fault angles θ01 and θ02 allows us to obtain an angular gap Δθ between the broken bars [BAC 02]: Δθ = θ02 − θ01 .
[8.39]
8.2.3. Global stator and rotor fault model In previous sections, two models of stator and rotor faults were presented. For a global simulation and detection of simultaneous stator and rotor faults, we propose the global fault model including: – Park’s model with the electrical parameters (Rs Rr Lm Lf ); – stator fault model with the three additional parameters (ηcck , k = 1 − 3); and – rotor fault model with broken rotor bars parameters (η0 , θ0 ). Figure 8.7 shows a global electrical model of squirrel cage induction motors for stator and rotor fault detection. i dq
i'dq
s
i dq
cc1
U dq
s
Qcc1
i dq
Qcc2
cc2
s
Rs
Lf
Z .P ( S / 2 ).I dq
s
i dq
i dq
Qcc3
cc3
r
Rr Lm
i dq
Rdefect m
Figure 8.7. Stator and rotor fault model for induction motors
260
Control Methods for Electrical Machines
8.2.3.1. State space representation For simulation and identification with the developed approach (seen in Chapter 7), it is necessary to write this fault model in state space representation. If mechanical speed ω is assumed to be quasi-stationary with respect to the dynamics of the electric variables, the model becomes linear but not stationary with fourth order differential equations [BAC 01a]. For simplicity, the state vector is chosen composed of two-phase components of the dq stator currents idqs and the rotor flux φdq . Then, the continuous r time model of the fault induction motor, expressed in the mechanical reference frame, is given by: x(t) ˙ = A(ω) x(t) + B u(t)
[8.40]
y(t) = C x(t) + D u(t)
[8.41]
with x = ids ! u=
A(ω) =
Uds Uqs
iqs
φdr
φqr !
" y=
,
ids iqs
T
: state space vector
" : input and output vector
! − Rs + Req L−1 f − ω P (π/2)
⎡ 1 ⎢ Lf ⎢ ⎢ ⎢ B=⎢ 0 ⎢ ⎢ ⎣ 0 0
Req ⎤
⎡ ⎥ 1 ⎥ ⎢0 1 ⎥ ⎥ ⎢ , C=⎢ Lf ⎥ ⎥ ⎣0 ⎥ 0 ⎦ 0 0 0
Req = Rr · I −
−1 " Req L−1 m − ω P (π/2) Lf −Req L−1 m
⎤T 0 3 1⎥ 2 ηcck ⎥ P (−θ) Q θcck P (θ) ⎥ , D= 3 Rs 0⎦ k=1 0
α Q θ0 1+α
8.2.3.2. Discrete-time model The discrete-time model is deduced from the continuous-time model by second order series expansion of the transition matrix [MOR 99]. By using a second order series expansion and the mechanical reference frame, a sampling period Te around 1 ms can be used. The usual first order series expansion (Euler approximation) requires a very short sampling period to give a stable and accurate model. These approximations by series expansion are more precise with low frequency signals.
Diagnosis of Induction Machines by Parameter Estimation
261
Thus, the discrete-time model is given by: xk+1 = Φk xk + Bdk uk
[8.42]
y k = C xk + D uk
[8.43]
where T2 Te + A2 e 1! 2! T2 B = I · Te + A e 2 · 1!
Φk = eATe = I + A Bdk
[8.44] [8.45]
and xk = x(tk ) and y k = y(tk ). The components of the known input vector uk are the average of the stator voltage between tk and tk+1 . 8.3. Diagnosis procedure Parameter estimation, presented in the previous chapter, is the procedure that allows the determination of the mathematical representation of a real system from experimental data. Two classes of identification techniques can be used to estimate the parameters of continuous time systems: equation error and output error [LJU 87, MOR 99]: – Equation error techniques are based on the minimization of quadratic criterion by ordinary least squares [LJU 87, TRI 88]. The advantage of these techniques is that they are simple and require few calculations. However, there are severe drawbacks, especially for the identification of physical parameters, not acceptable in diagnosis, such as the bias caused by the output noise and the modeling errors. – Output error (OE) techniques are based on iterative minimization of an output error quadratic criterion by a non-linear programming (NLP) algorithm. These techniques require many more calculations and do not converge to a unique optimum. However, OE methods present very attractive features, because the simulation of the output model is based only on the knowledge of the input, so the parameter estimates are unbiased [TRI 88, MOR 99]. Moreover, OE methods can be used to identify non-linear systems. For these advantages, the OE methods are more appropriate for the diagnosis of induction motors [MOR 99]. Parameter identification is based on the definition of a model. For the case of fault diagnosis in induction machines, we consider the previous mathematical model ([8.40] to [8.41]) and we define the parameter vector: θ = Rs
Rr
Lm
Lf
ηcc1
ηcc2
ηcc3
η0
θ0
T
[8.46]
262
Control Methods for Electrical Machines
As soon as a fault occurs, the machine is no longer electrically balanced. Using previous faulty modes, electrical parameters (Rs , Rr , Lm and Lf ) do not change and only the fault parameters (ηcck and η0 ) vary to indicate a fault level according to relations: ˆ cck = ηˆcck · ns . – Number of inter-turn short windings at k th phase: n – Number of broken bars: n ˆ bb = ηˆ03nb . Thus, during industrial operation, the diagnosis procedure by parameter estimation of induction machines requires sequential electrical data acquisitions. Using each set of data, identification algorithm computes a new set of electrical parameters to discover the magnetic state of the machine and new fault parameters to approximate of the number of inter-turn short circuit windings and broken rotor bars. 8.3.1. Parameter estimation Assuming that we have measured K values of input-output (u(t), y ∗ (t) with t = k · Te ), the identification problem is then to estimate the values of the parameters θ. Then, we define the output prediction error: ˆ u) εk = y ∗k − yˆk (θ,
[8.47]
where predicted output yˆk is obtained by numerical simulation of the state space fault model [8.43] and ˆθ is an estimation of true parameter vector θ. As a general rule, parameter estimation using the OE technique is based on the minimization of a quadratic criterion defined in the case of the induction motor as: J=
K k=1
εTk εk =
K
i∗dsk − ˆidsk
2
2 + i∗qsk − ˆiqsk
[8.48]
k=1
Usually, for induction motors, the parameters are well known, so it is very interesting to introduce this information in the estimation process to provide more certainty on the uniqueness of the optimum. For this, we have applied the modification of the classical quadratic criterion [MOR 99, TRI 88], in order to incorporate physical knowledge. 8.3.1.1. Introduction of prior information In order to incorporate physical knowledge or prior information, the classical quadratic criterion has been modified. The solution is to consider a compound criterion Jc mixing prior estimation θ0 (weighted by its covariance matrix M0 ) and the classical criterion J (weighted by the variance of output noise δˆ2 ). Then, the compound criterion is usually defined as: K 2 εds + ε2qs T −1 k k Jc = θˆ − θ0 M0 θˆ − θ 0 + [8.49] δˆ2
Diagnosis of Induction Machines by Parameter Estimation
263
Thus, the optimal parameter vector minimizing Jc is the mean of prior knowledge and experimental estimation weighted by their respective covariance matrix. In reality, we have no knowledge of the fault; indeed, no prior information is introduced on fault parameter. Only electrical parameters (Rs , Rr , Lm and Lf ) are weighted in the compound criterion. Thus, the covariance matrix is defined as: 1 1 1 1 −1 M0 = diag [8.50] 2 , σ 2 , σ 2 , σ 2 , 0, 0, 0, 0, 0 σR Rr Lm Lf s 2 2 2 2 , σR , σL and σL are the variance of parameters respectively, with prior where σR s r m f information Rs , Rr , Lm and Lf .
8.3.1.2. Non-linear programming algorithm We obtain the optimal values of θ by non-linear programming techniques. Practically, we use the Marquardt algorithm [MAR 63] for off-line estimation:
−1 +λ·I · Jθ [8.51] θˆi+1 = θˆi − Jθθ ˆ θ=θ i
with Jc θ
=2·
K
M0−1 (θˆ −
Jcθθ
≈2·
M0−1
θ0 ) −
K +
k=1
T k=1 εk
· σ k,θ
δˆ2
σ k,θ · σ Tk,θ δˆ2
the gradient
the Hessian
λ the monitoring parameter σ k,θ =
∂ yˆ the output sensitivity function. ∂θ
8.3.1.3. Criterion weight designation Prior information is mainly used to avoid aberrant estimates given by minimization of classic criterion. As a consequence, our interest is focused on the optimal choice of θ0 , M0 and δˆ2 . Prior information can result from two origins: – Experiments or motor information given by manufacturers. In this case, θ0 and M0 are obtained by usual electrical tests performed on induction machines (locked rotor, load shedding, etc.) and all material characteristics. – In practice, prior information is given by physical knowledge and partial estimation. Firstly, a set of experiments and identification of only electrical parameters
264
Control Methods for Electrical Machines
with classical criterion J is used in order to constitute an electrical reference value database. The pseudo-covariance matrix M of the parameters and the noise pseudo-variance are σ ˆ 2 defined as: M = δˆ2 (φTd φd + φTq φq )−1 (φd + φq )T (φd + φq )(φTd φd + φTq φq )−1
[8.52]
Jopt [8.53] K −N where K, N and Jopt are respectively the number of data, the number of parameters and the optimal value of experimental criterion. Matrices φd and φq are matrices of output sensitivity functions according to the dq current axis. σ ˆ2 =
Thus, the covariance matrix M0 is obtained by diagonal values of M . To evaluate the noise variance, it is necessary to use δˆ2 > σ ˆ 2 to take into account the effect of modeling errors. The motor used in the experimental investigation is a three phase, 1.1 kW, 4-pole squirrel cage induction machine (Figure 8.8). The data acquisition was done at a sampling period equal to 0.7 ms. Before identification, measured variables are passed through a 4th order Butterworth anti-aliasing filter whose cut-off frequency is 500 Hz.
Speed controller
Data acquisition
Figure 8.8. Motor experimental setup
With a mean of 10 realizations in the healthy case, we obtained the reference of electrical parameters noted θref and the weights of quadratic criterion. Then, for all experiment estimations, we used: T θ ref = 9.81 3.83 0.436 7.62.10−2 0 0 0 0 θ0init M0−1 = diag 5.102 , 65.102 , 17.105 , 107 , 0, 0, 0, 0, 0 noise variance: δˆ2 = 0.22.
Diagnosis of Induction Machines by Parameter Estimation
265
8.3.2. Implementation Experiments are performed in closed loop. The induction machine is driven by field oriented vector algorithm included in a speed control closed-loop and run under different loads with the help of a DC generator mechanically coupled to the motor. The speed excitation is realized with a pseudo-random binary sequence (PRBS) equal to 90 rpm added to the reference of the speed loop equal to 750 rpm. The mechanical position and the three phase voltages and currents are measured and translated in low frequencies by Park’s transformation. Stator windings were modified by adding a number of tappings connected to the stator coils in the 1st and 2nd phases (464 turns by phase). These tappings correspond to 18 inter-turns (3.88%), 29 inter-turns (6.25%), 58 inter-turns (12.5%) and 116 inter-turns (25%). The other end of these external wires is connected to a terminal box, allowing the introduction of shorted turns at several locations and levels in the stator winding. Different rotors, with broken bars, are used to simulate a bar breakage occurring during operation. 8.3.2.1. Estimation results Different tests (10 realizations by experiment) with inter turn short circuit windings and broken rotor bars have been performed. Table 8.1 shows the mean of fault parameter estimates for 10 acquisitions. As observed in Table 8.1, there is good agreement between a real fault and its estimation. All fault parameters vary to indicate the values of inter turn short circuit in the three-stator windings and the number of broken rotor bars. Estimation results Experiments ncc1 , ncc2 , ncc3 (inter-turns), nbb (Δθ)
(mean of 10 realizations) n ˆ cc1
n ˆ cc2
n ˆ cc3
n ˆ bc
5.57
3.52
−0.03
0.08
17.86
−1.11
2.51
0.94
3) 0, 58, 0 (inter-turns), 2 bars (π/2.8)
3.11
54.52
0.28
1.86
4) 18, 58, 0 (inter-turns), 2 bars (2π/28)
16.05
53.31
−2.54
1.88
5) 58, 29, 0 (inter-turns), 2 bars (π/2.8)
53.69
26.87
−2.46
1.82
1) Healthy machine 2) 18, 0, 0 (inter-turns), 1 bar
Table 8.1. Estimation results of stator and rotor faults
266
Control Methods for Electrical Machines
Indeed, a parametric approach gives good estimations of the short circuit turns number n ˆ cck . The estimation error is negligible and does not exceed five turns in each defect situation. At simultaneous faults in several phases (cases 4 and 5), we observe that the estimates of the fault parameters of each phase is a realistic indication of the faults. This proves that each short circuit element explains the fault occurring at its phase and that no significant correlation exists between these elements. Moreover, broken rotor bar estimation n ˆ bb gives a satisfactory indication from the fault. 8.3.2.2. Parameter evolution Figure 8.9 gives, for one realization in faulty situation (case 5), the evolution of electrical and the fault parameters during the estimation procedure. For the electrical state, it shows that optimum values are achieved in only four iterations. However, their variation according to the initial values corresponding to prior information is negligible. For faulty state, it is shown that their variations, contrary to the electrical parameters, are very important. Each fault parameter varies to indicate stator and rotor fault levels occurring in the machine (for example, ncc1 varies to approach the 58 inter-turns in defect present in the 1st phase, while nbb varies to approach 2 broken rotor bars). 11
(windings)
(:)
(:)
(windings)
60
3.86
30
10.5
40
3.85 10
R
s
Rr
3.84
20
ncc1
n
cc2
20
10
9.5 9 0
0.445
1
2
3
4
3.83 0
1
2
3
4
0 0
L
L
0.077
m
2
3
4
0 0
1
0.415 0
f
2
3
4
0.071 0
4
3
4
2
cc3
4
n
bb
2
0.074
1
3
n
1 0.425
2
(bars)
6
0.08
0.435
1
(windings)
(H)
(H)
1
2
3
4
0 0
1
2
3
4
0 0
1
2
Figure 8.9. Estimation of electrical and fault parameters in the faulty case
This comparison is important because it is evident that only fault parameters change when the faults occur according to the prior information principle. Moreover, electrical parameter variations are a function of the temperature and of the magnetic state of the machine, and are independent of the faults. Figure 8.10 presents the evolution of inter turn short circuit estimation in one phase for several experiments and the dispersion of the 10 estimations in different situations of rotor faults. We observe that all the estimation results exhibit the good approximation of the stator and rotor faults.
Diagnosis of Induction Machines by Parameter Estimation Bars
(windings)
58 inter-turn short circuit windings 60
Estimation Real faults
2
45
Estimation Real faults
15
Two broken rotor bars
1.5
One broken rotor bar
29 inter-turn short circuit windings 30
267
18 inter-turn short circuit windings Healthy machine
0
1
0.5
Healthy machine
0
Figure 8.10. Estimation of electrical and fault parameters in the stator and rotor fault case
8.4. Conclusion The output error method associated with the compound criterion is a good tool for identifying continuous model parameters and is therefore very relevant for the diagnosis. In this chapter, we thus proposed a procedure for detection and localization of defects in the induction machine based on parameter estimation and on the use of a general fault model. Two fault models, which are simple to implement, have been presented. The first one allows us to explain stator faults with three short circuit elements, each element dedicated to a stator phase. A new equivalent Park’s rotor resistance has been expressed to allow for the decreasing of the number of rotor bars in faulty situation. Finally, the association of the stator and rotor faulty element with the nominal model on induction machine explains the simultaneous stator and rotor faults. This resulting model allows an extensive monitoring of the induction machine. The proposed model has been validated on an experimental test bench. The identification procedure has allowed, on the one hand, the localization of stator faults at several phases and the determination of their number with a maximum error of six turns and, on the other hand, the quantification of the number of broken rotor bars. Thus, in situations of real defects, the diagnosis procedure by parameter estimation gives a very realistic indication of the imbalance occurring in the machine. The monitoring methods based on parameter estimation have been poorly applied so far in diagnosis of physical systems, especially in electrical engineering. Our experience shows that this method is well suited to fault detection and localization. The association of parameter estimation technique with prior information and a faults model based on common and differential modes seems perfectly adapted to the case of the induction machine.
268
Control Methods for Electrical Machines
8.5. Bibliography [ABE 99] A BED A., BAGHLI L., R AZIK H. and R EZZOUG A., “Modelling induction motors for diagnosis purposes”, EPE’99, p. 1–8, Lausanne, Switzerland, September 1999. [BAC 01a] BACHIR S., T NANI S., P OINOT T. and T RIGEASSOU J.C., “Stator fault diagnosis in induction machines by parameter estimation”, IEEE International SDEMPED’01, p. 235–239, Grado, Italy, September 2001. [BAC 01b] BACHIR S., T NANI S., T RIGEASSOU J.C. and C HAMPENOIS G., “Diagnosis by parameter estimation of stator and rotor faults occuring in induction machines”, EPE’01, Graz, Austria, August 2001. [BAC 02] BACHIR S., Contribution au diagnostic de la machine asynchrone par estimation paramétrique, PhD Thesis, Poitiers University, 2002. [BAC 06] BACHIR S., T NANI S., T RIGEASSOU J.C. and C HAMPENOIS G., “Stator fault diagnosis in induction machines by parameter estimation”, IEEE Transactions on Industrial Electronics, vol. 53, no. 3, p. 963–973, 2006. [FIL 94] F ILLIPPITTI F., F RANCESHINI G., TASSONI C. and VAS P., “Broken bar detection in induction machine: comparaison between current spectrum approach and parameter estimation approach”, IAS’94, p. 94–102, New York, USA, 1994. [INN 94] I NNES A.G. and L ANGMAN R.A., “The detection of broken bars in variable speed induction motors drives”, ICEM’94, December 1994. [LJU 87] L JUNG L., System Identification: Theory for the User, Prentice Hall, USA, 1987. [MAK 97] M AKKI A., A H - JACO A., YAHOUI H. and G RELLET G., “Modelling of capacitor single-phase asynchronous motor under stator and rotor winding faults”, IEEE International SDEMPED’97, p. 191–197, Carry-le-Rouet, France, September 1997. [MAL 99] M ALÉRO M.G., ET AL ., “Electromagnetic torque harmonics for on-line interturn shortcircuits detection in squirrel cage induction motors”, EPE’99, Lausanne, Switzerland, September 1999. [MAN 96] M ANOLAS S.T., T EGOPOULOS J. and PAPADOPOULOS M., “Analysis of squirrel cage induction motors with broken rotor bars”, ICEM’96, p. 19–23, Vigo, Spain, 1996. [MAR 63] M ARQUARDT D., “An Algorithm for least-squares estimation of non-linear parameters”, Soc. Indust. Appl. Math, vol. 11, no. 2, p. 431–441, 1963. [MEN 99] M ENSLER M., Analyse et étude comparative de méthodes d’identification des systèmes à représentation continue. Développement d’une boîte à outil logicielle, PhD Thesis, Nancy I University, 1999. [MOR 99] M OREAU S., Contribution à la modélisation et à l’estimation paramétrique des machines électriques à courant alternatif: Application au diagnostic, PhD Thesis, Poitiers University, 1999. [SCH 99] S CHAEFFER E., Diagnostic des machines asynchrones: modèles et outils paramètriques dédiés à la simulation et à la détection de défauts, PhD Thesis, Nantes University, 1999.
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269
[TRI 88] T RIGEASSOU J.C., Recherche de modèles expérimentaux assistée par ordinateur, Technique et Documentation Lavoisier, Paris, 1988. [TRI 99] T RIGEASSOU J.C., G AUBERT J.P., M OREAU S. and P OINOT T., “Modélisation et identification en génie électrique à partir de résultats expérimentaux”, Journées 3EI’99, Supelec Gif-sur-Yvette, March 1999.
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Chapter 9
Time-based Coordination
9.1. Introduction The coordination of electric motor actions which make up a system animating a complex installation is achieved using the coupling of control laws. Each motor is driven by a relationship (control law), which defines the set points applied to the controllers governing the behavior of the system. Coordination, then, consists of linking various control laws through a common variable to each of them. The variable concerned here is time. Time may act in two ways: either discretely or continuously. In the first case, it appears as a “date” whose occurrence triggers an action. For example, this moment ti can be defined by the task of a particular motor. The occurrence of ti involves for another motor (or others) either a change of a parameter of the control law or the replacement of the actual control law with another planned in advance. The tasks of each motor generate different ti, which coordinate the operating of the system. It should be noted that the value of ti cuts no figure. This is essentially an order relationship on ti set; this is generally the case for sequential systems and hybrid systems. However, the study of these issues is beyond the scope of this book.
Chapter written by Michel DUFAUT and René HUSSON.
272
Control Methods for Electrical Machines
The other way to look at time as a variable of coordination is to see it as a continuous variable. In this case, the variable is quantitative, especially when the duration concept interferes. For example, durations can be imposed on motor tasks in the time constraint form. To respect these constraints, one possibility is to act on the work speed (the rotation speed) of motors. Then, the motors will be coordinated by means of speed references. To better understand these problems, an example is essential: we will use the command-coordinated arm of a robot manipulator. The study of the coordination of articulated arm movements with time requires a minimum knowledge of the functional structure, representation and models of the arm movement [KHE 96].
9.2. Brief description system 9.2.1. Functional structure The standard functional structure of a robot manipulator consists of five major parts: – the mechanical structure; – the motion transceivers; – the actuators and power controlled sources (choppers, hydraulic generators, etc.); – the sensors (internal or proprioceptive sensors and external or exteroceptive sensors); and – the computer (with a large hierarchy of software). The links between these various parts are represented in Figure 9.1. Here we are specifically interested in the mechanical structure, the motion transmission, the actuators and an overview of modeling. The other functions will. however, be presented briefly. The internal sensors group is essentially dedicated to control (position, speed, current, voltage sensors, etc.). They inform the computer on the robot state variables and in particular they provide information on the “robot joint variables” in real time. The external sensors provide information outside the computer on the position and orientation of the “end effector” with respect of the environment. The camera coupled with its vision system is not the only external sensor. There are proximity
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sensors (proximeters) mounted on the hand of the robot that can measure short distances, or pressure sensors that avoid the crushing of objects, held by the gripper.
Figure 9.1. Robot block diagram
The software implemented in the computer generally includes mathematical descriptions of the various models of the robot (geometric, kinematic and dynamic models) to describe the geometric relationship between devices and machines around the robot, and control laws to perform trajectories and different work tasks. Other software called services run the robot. These include software for data acquisition that allow the management of relations between the robot and its working environment: the robot works as an open loop system where the vision system provides partial information (usually 2D-vision processing) to adapt the position and orientation of the end effector with respect to unexpected changes in the working environment.
9.2.1.1. Articulated mechanical device (AMD) 9.2.1.1.1. Joints The articulated mechanical system consists of three parts: the base (called the vehicle for mobile robots), the robot arm, which includes the “arm” and the “forearm”, and the “wrist” that moves the end effector (gripper or tool). This structure is not always anthropomorphic and concerns “manipulator arm” robots. There are other structures like the gantry robot or pendulum systems that have a different structure but retain an entire wrist and terminal body similar to those of the manipulator arm.
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The articulated mechanical structure consists of a set of rigid links connected by joints, to form an open kinematic chain. The word “articulation” is understood in a broad sense, i.e. it does not necessarily mean a rotation. The relative motion between two consecutive links are only rotations and translations with respect to an axis, and are defined by only one variable, rotation Ti or translation di. Only a simple symbol, qi, designates these joint variables, called articular variables, set for the ith joint: qi = VLTL + (1 – VL)di
[9.1]
whereVi is 0 or 1 depending on the type of joint: translation or rotation. The vector qi, combining the values of n joint variables at any given moment, is called the configuration (or attitude) of the robot. The joints are represented by the following symbols. di TL
a) rotation
b) translation
Figure 9.2. Symbolic representation joints
These symbols can achieve the equivalent of the robot scheme, highlighting the motion axes, the joint variables and the general structure of the system, regardless of the shape of the pieces that make it up. To create such a representation, a number must be assigned to each moving part of the system. This number will be the subscript of concerned variables. The numbering is done from the fixed base of the robot, to which the value 0 is assigned, and then step by step, numbering moving parts in ascending order to the end effector (or robot’s hand), which is numbered n. The joint (i) connects the link [i-1] to the link [i]. The number of the connection is the same as its joint variable (see Figure 9.3).
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T3 T2
T4
(5)
(2) (3)
T1 T5
(1) T6
(4)
(0)
(6) a) Six-axis robot
b) Equivalent diagram
Figure 9.3. A robot and its equivalent diagram
9.2.1.1.2. A robot’s degrees of freedom The principal aim of a robot is to be able to carry and move workpieces or tools compatible with its load capacity in a three-dimensional space. In general, the location of an object in free space is determined by six independent variables, or six degrees of freedom: three indicate the position (center of gravity), while the other three indicate the orientation (for example Euler angles). To succeed in placing a body in imposed position and orientation, the robot must also have at least six degrees of freedom. If not, it could not fulfill its task completely. While it is equipped with a number of degrees of freedom greater than six, there will theoretically be infinite solutions to the problem (the robot is said to be redundant). The motorized joints materialize the degrees of freedom of the robot. Of course, the orientation of the axes must be chosen correctly, to obtain three degrees of freedom relative to the position and three degrees of freedom able to perform the orientation. The carrier generally implements the first three, while the wrist implements the last three. However, the range of motion of joints, such as translations or rotations, is limited by mechanical stops. If a joint bumps into its stop, the corresponding degree of freedom disappears, because a constraint is created. Indeed, the number of degrees of freedom of the actual robot is equal to the difference of the original degrees of freedom, and the number of constraints. This motion range limitation should be taken into account during the control: the motor operation must be determined to ensure that the joint works, on average, around the mid-point of its joint variable qi range. As each joint is different from others, the amplitude and time constraints on the control will have to be adjusted for each one.
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9.2.1.1.3. Reference coordinate system The mechanical structure of the robot arm is adapted to the working mode of the robot and in particular to the coordinate system in which it prefers to operate. Thus, the simplest structure is a gantry crane with three translation possibilities along axes of a Cartesian frame (Figure 9.4a). The cylindrical coordinates are also easily implemented using a column that pivots around a vertical axis, on which a rigid arm will translate vertically and horizontally realizing the vertical position and elongation (Figure 9.4b). Generally, an articulated arm without translation is used to operate in spherical coordinates. The three main rotations of the robot arm around a vertical axis and two horizontal axes are enough to define the site, the azimuth and the elongation (as shown in Figure 9.3). X
Y
U T
Z
Z
a) Cartesian robot
b) Cylindrical coordinate robot
Figure 9.4. Coordinate systems robot arm
The wrist movements are identical for most of the robots regardless of the form of their structure. They revolve around three orthogonal intersecting axes and make three rotations, roll (z-axis), pitching (y-axis) and yaw (x-axis), as shown in Figure 9.5. This structure includes various alternatives because of the difficulty in accurately manufacturing three strictly intersecting orthogonal axes. Finally, the mechanical articulated structure, such as machine tools, has a particular configuration called the zero robot. In this configuration each joint variable is at its zero point. Thus, the “zero” of the robot identifies the zero point of all the position sensors. The robot zero point is usually chosen in the middle of joint variable qi range.
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277
z Yaw y
Roll x
Figure 9.5. Wrist rotations
9.3. Some ideas on the manipulator system models 9.3.1. Various model types A robot is usually represented by three kinds of models: – The geometric models describe the position and orientation of the end effector in base coordinates, with respect to joint variables. This type of model involves trigonometric relationships between the rotation joint variables. They are essentially non-linear and lead to many qi solutions for the single Cartesian position and orientation required for the end effector. – The kinematic models describe the linear and angular velocities of the end effector according to joint velocities. This results from the derivation of the geometric model, but is generally more easily reached by geometric considerations. It is linear with respect to velocities, but non-linear with respect to the angles. It also allows us to describe or to control the incremental motions and static forces. – Dynamic models are more complete and express velocities and accelerations according to torques and effort applied by the actuators. They take into account almost all the dynamic parameters of the robot (inertia, viscous friction, etc.) and the linearized equations in state space of this tool can be written from them. The implementation of these models in real time requires fast and powerful computers. – Moreover, geometric and kinematic models have inverse models (the geometric inverse model and the inverse kinematic model), which can solve the following problems: - what position set points should be assigned to the actuators so that the gripper is in a given position and orientation?; - what velocity set points should be assigned to motors so that the end effector moves on the required trajectory with angular and linear imposed velocities?
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9.3.2. Geometric models The establishment of the geometric model of a robot is based on the use of frames attached to each moving link [PAU 82]. The various authors and models employ several types of convention defining these frames. The best known and most used is called the Denavit-Hartenberg convention. The fixed base is the link 0 and the moving links are numbered from 1 to n; there will be n frames R0, R1…. Rn. the frame Ri is attached to ith link. The method is to express the position and orientation of the Ri frame in the Ri-1 frame, (from R1 to Rn) by means of a 4x4 matrix of the form:
A
ªorientation position º « » «3x3 matrix 3x3 vector » « 0 0 0 » w ¬ ¼
G ªn «0 ¬«
G o
G a
0
0
G pº 1 »¼»
[9.2]
The 3x3 sub-matrix defining the orientation consists of 3 vectors whose components are the projection of Ri frame unit vectors onto the axes of the Ri-1 frame, that is to say, the dot products of Ri and Ri-1 frame unit vectors respectively. The coordinates of the Ri origin in Ri-1 define the elements of the position vector. The coefficient w is a scaling factor used when a camera controls the robot. Its role is to ensure the change of scale between the real world and the obtained image. Most of the time w is equal to 1. If Ai is the passage matrix from the Ri-1 frame to the Ri frame, the position and orientation of the Rn frame described in the R0 base frame will be given by the matrix product: Tn = A1.A2….. Ai…. An
[9.3]
Considering there is only one joint between two moving parts (i.e. between two frames), each matrix Ai depends upon only one joint variable qi. The mathematical expressions of Ai(qi) are relatively simple, but the product of n matrices, expressed by Tn, is complex. In general, Tn is also a 4x4 matrix, of which 12 elements involve trigonometric relationships between rotation joint qi variables. If these variables are numerically specified, taking account of the origin of angular sensors, an algebraic matrix X is then defined that corresponds to a specific position and orientation of the end effector in the working space. For example:
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X
G The vector q
ª 0.87 0.5 « 0.5 0.87 « « 0 0 « 0 ¬« 0
0 0 1 0
210 º 364 »» 80 » » 1 ¼»
279
Tn q1 , q2 ,...qi ,...qn
>q1 " qn @ digitally sets the robot configuration.
Figure 9.6. Modeling example of an arm with two moving links. The joint variables here are q1 = T1 and q2 = T2, a1 and a2 are constants (links are assumed rigid, and in this case represent the links lengths)
This model Tn represents the robot. It is a mapping that matches the joint space with the space of operational variables (x, y, zMT, ȥ coordinates of the origin of the end effector frame in the base frame and Euler angles of its axis). Unfortunately this mapping is not bijective due to the presence of sinus and cosines of rotation G angles in the expression of Tn. Indeed, there may be several q vectors for a position and orientation assigned by a given matrix X, (angles are defined modulo 2kS). As a G result, the inverse problem, i.e., the calculation of the q vector corresponding to a given matrix X, must be obtained by a special model, the inverse geometric model, which provides various solutions, among which a choice should be made to define the position set points of actuators. The development of these inverse models is usually complex, but most of them are found in the literature because the number of robot types is limited. Figure 9.7 represents, via block diagrams, the operation of the positioning and orientation of a robot end effector.
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Compensator x, y, z
X
MT\
Calc. of X
Inverse geometric model
G q
Trajectory generator
Power supply and motors
qi
Sensors
Figure 9.7. Diagram for positioning and orientation of the gripper
This figure highlights the need to convert operational data into a matrix including the director cosines of the three axes of the Rn frame. The trajectory G generator operates when switching from one configuration to another (i.e. from qI G to qII ). This requires the path discretization for this transition with respect to an assigned trajectory. During this transition between two configurations, the problem of coordination by time will occur, which is the focus for this chapter.
9.3.3. Kinematic models The kinematic models are less complex to establish than the geometric models: first, because the mapping that matches elements of all joint speeds with overall operational speeds is bijective, and second, because they can be determined by applying the classical geometric relationships of the theoretical mechanics. The direct geometric model and the inverse conversion of the director cosines of the terminal frame into Euler angles give six relations: x
M
G f1 q y G f 4 q T
G G f 2 q z f 3 q G G f 5 q \ f 6 q
We may define: ªDº «¬4 »¼
G C
> x, y, z, M, T,\ @T
G F q
> f1, f2, f3, f4, f5, f6 @T
and by derivation have: G C
G G J q q
>V: @
[9.4]
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G G G where J q is the Jacobian of the vector function F q . Note that J q is not constant with respect to time due to the changes in joint variables, but it is exactly defined at a given moment. It is also worth noting that relationship [9.4] is linear with respect to operational and joint speed. G If we consider small variations of C , we can use relationship [9.4] to describe G small movements around a point of the trajectory, noting J q as locally constant around that point. We then have an incremental relationship: G 'C
G G J q 'q
[9.5]
Relations [9.4] and [9.5] can be calculated either in the R0 frame of the task, which is the general case, or in another any frame [GOR 84]. In particular, it is interesting to express [9.5] in the end effector frame: each small movement is then defined in relation to the last point reached by the end effector. This is a useful method because it reduces the trajectory calculations, although it has the disadvantage of accumulating numerical errors. G When the Jacobian is invertible (i.e. when the J q matrix is regular), we can establish the inverse model:
G q
G 1 J q C
G 'q
G 1 ªV º J q .« » ¬: ¼
G J q 1'C
[9.6a]
[9.6b]
This relationship is used to define the set point speed of the actuators satisfying constraints. For example, the processes (arc welding, object track, etc.) impose linear and angular gripper velocities (or tool) for the travel on a trajectory. To achieve the Jacobian inversion, two approaches are possible: either numerical inversion (using a program like Matlab) or algebraic inversion. The latter (laborious for a 6x6 matrix) is facilitated by the use of computer algebra software. It is still easier as the wrist almost always has three orthogonal and intersecting axes of rotation. In this case, it is possible to write the Jacobian in the special form as follows: G J q
G G ª A q 0 º « G G » «¬C q B q »¼
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In this case, its inversion only requires simple inversions of three 3x3 matrices. We obtain: G ª A 1 q « 1 1 ¬« B CA
G 1 J q
G 0
º G » B q ¼» 1
This form can also highlight the singular configurations, i.e. the values that make this matrix singular. They are the solutions of the system: G Det A( q ) = 0 G Det B( q ) = 0
For the robot control to be valid, we must avoid those configurations that would require infinite joint speeds to solve the problem. Normally, the rank of the matrix is equal to the number of degrees of freedom of the robot. Thus, when we reached a singular configuration, the matrix rank lowers by one unit, which means that it loses a degree of freedom and that the task is not feasible. In the case of redundant robots, with more degrees of freedom than the task, the Jacobian is no longer a square matrix. Its inversion implies the use of pseudo-inverse or other more sophisticated techniques [COI 86]. Relations [9.3] and [9.6b] are also used offline for numerically solving the inverse geometric model, by iterative calculation of the joint variables corresponding to a position and orientation given in the space of the task. The principle of this algorithm is as follows. Delay C
'C
+
qk-1
k
J(q)-1
+
qk
'q k
_ Ck
Data conversion
Xk
T(qk)
Figure 9.8. The principle of the algorithm resolution of the geometric model
The kinematic model or Jacobian model is still used to calculate the quasi-static efforts developed by the robot. For example, during machining operations, the movements are linked to the advance of the tool that cuts metal chippings. The
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movements are therefore very slow, but require significant effort. Applying the D’Alembert principle of virtual work, we can link these efforts to joint efforts. G G If E is the vector of applied forces and W the joint efforts, we can write:
G E
G G T G ª¬ F , * º¼ ; W G G T ªV º E .« G » ¬: ¼
G ªV º «G» ¬: ¼ G
W
T
>W 1 ,W 2 ,"W n @ G G
UW T q
G G G G G J q q ; E T .J q q 1
U
G G
UW T q
G G J T q E
[9.7]
Vertical cutting force: F Resisting torque: *
Figure 9.9. Example of applied forces
The torque and effort set points of actuators for the machining operation can be determined by relationship [9.7].
9.3.4. Dynamic models Dynamic models, which are more complex than the previous models, can be established in two forms: a global form, based on the use of Lagrange equations [PAU 82], and a recursive form resulting from the implementation of the formalism of Newton [McIN 86]. The global form allows, thanks to a local linearization and the principle of small movements, access to the space state equation of the robot. The dynamic parameters of each moving link (forces, accelerations, velocities, etc.) can be quickly determined, step by step using the recursive form.
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Control Methods for Electrical Machines
9.3.4.1. Dynamic global model (Euler-Lagrange formulation) We will not give the details of its development ([PAU 82] or [LEE 82]). However, the model takes the form of a non-linear vector differential equation of the second order, obtained by derivation of the “Lagrangian” of the robot. It is written: L = Ke – Pe, where Ke is the total kinetic energy of the robot and Pe is the potential energy, which reflects the action of gravity on its elements. The Bi generalized effort applied to the (i) joint is calculated from: Bi
d dt
§ wLi ¨ ¨ wq © i
· wL ¸ ¸ wq i ¹
with
Bi
V iW i 1 V i fi
where Wi and fi represent the force or torque developed by the actuator at this joint. G By grouping n (generally n = 6) Bi efforts in a B vector, the differential equation is then written in the form: G B
G G G D q q f ª¬ qi , q j q º¼ G q
[9.8]
G
– D q is the nxn inertia matrix of the system, which groups the inertia of the moving parts (including the inertia of actuator rotors returned to the rotation axis).
– f ª¬ qi , q j , q º¼ is the nx1 Coriolis vector and centrifugal forces that arise
during coupled rotations and viscous friction forces.
G – G q is the nx1 vector of gravitational forces acting on the different parts of the robot. G G G – q, q and q are nx1 vectors of acceleration, velocity and joint positions.
– The moving parts are assumed to be infinitely stiff. Equation [9.8] can be implemented in real time or simulated to determine the control vector. In the caseGof use in real time, it is possible to apply specific control laws to actuators through B, which then becomes a time-variant vector function. The state space linearized around a point of the trajectory is obtained by G equation G G discretization of q, q and q in the form: G
Gq
T
> x1 , x2 " xn @
G
G q
T
> xn 1 , xn 2 ," x2 n @
F
G G T ªG q , G q º ¬ ¼
Equation [9.8] becomes, after a complex linearization:
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G G G G G D0G q V0G q P0G q G B and G q G q
or ªIn «0 ¬«
0 º F D0 »¼»
ª 0 In º ª 0 º «P V » F « »G B 0» ¬In ¼ ¬« 0 ¼
[9.9]
The D0, V0, P0 are nxn matrices, which depend on the current point of trajectory, and In is the n order identity matrix. As D0 is a matrix of inertia, it is symmetric, positive-definite and then regular. Thus, D01 exists and:
F
ª 0 « D 1 P «¬ 0 0
In º ª 0 º F « 1 » U » 1 D0 V0 »¼ «¬ D0 »¼
G G A( q0 ) F B ( q0 )U
[9.10]
The Ȥ vector represents the difference between the nominal trajectory and the actual trajectory; the U vector is a correction couple vector, whose role is to bring back the end effector of the robot on its nominal trajectory; and the vector q0 is the configuration of the robot at the considered point of the trajectory.
9.3.4.2. Recursive dynamic model (Newton-Euler formulation) It is necessary to calculate ([McIN 86] or [LEE 82]) for each moving link the linear and angular velocities of the Ri coordinate frame attached to its ith body; according to the linear and angular velocities of the Ri-1 frame attached to the (i-1)th body. All velocities and accelerations of various frames can be determined by mean recursive calculation (called forward recurrence), going from the base (index 0), of which the velocities are zero, to the end effector (index n). This calculation then gives the speed and acceleration of the center of mass of the link [i]. After establishing the forces and torques that the effector must exercise, the efforts each actuator must develop are calculate step by step, from the end effector to the base. The calculation is done by recurrence (called backward recurrence): the forces exerted on the [i] link by the [i-1] link must compensate for the resultant G dynamic forces Fi (including gravity effects, induced by the acceleration g) and the G fi 1 forces exerted by the link [i+1] on the link [i]. Thus, we can say G G G G G G fi Fi fi 1 mvi fi 1 ; vi is the acceleration of the center of gravity of the link [i] and mi is its mass. Although more difficult to use for control, this dynamic model can be easily implemented in real time, thanks to its recursive structure. In addition, it contains
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information on the forces exerted on the bearings and gives valuable information to manufacturers. One of the other interests of this model is to provide, in detail, all the dynamic data from all moving parts of the robot. It also provides a detailed modeling of the effort to exert for gripping particular objects and the introduction of simple, viscous or dry friction additional forces. The global model cannot provide the same result. Many authors, for example [LEE 85], make complementary use of the two models (global and recursive) for the control: the recursive model determines the various points of the trajectory, taking into account the dynamic parameters, and the global linearized model describes the small movements around each point previously determined. Usually the identification of the dynamic matrices realized in real time [AST 84] in the state space equation of the robot, at each sampling period, is preferred over recalculation from the physical parameters, as it avoids the accumulation of small measurement errors.
9.4. Coordination of motion 9.4.1. Why coordinate movements? The coordination of motion occurs mainly in the robot’s “point to point” control. It defines a few points in the operational space by which the effector of the robot must move with an orientation imposed. There are two kinds of point: stop points and through points. These points generally correspond to joint configurations that are very different from each other. The set points of controlled systems vary according to a sequence of steps whose sign (rotating sense) may also be changed. A sign change in the control automatically leads to discontinuities of velocities and acceleration due to the change in the motion direction. A velocity discontinuity creates a very high acceleration, which then generates a force or torque impulse. This will cause a violent jolt to be applied to the robot structure. This action occurring for each joint will result in vibrations whose frequency may even reach the resonant frequency of the mechanical structure and cause damages. One solution consists of avoiding excitation of the system resonant modes, synchronizing all movements, i.e. they start and stop all at once. An acceleration discontinuity will generate an infinite derivative of the acceleration, causing a phenomenon known as “jerk”, in which the efforts take, in principle, an infinite growth speed. As for velocities, these discontinuities cause
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vibrations and a loss of local control of the structure. That is why in some applications joint variables are forced to follow a variation polynomial law with respect to time [LUH 83]. This caution ensures the variable continuities necessary for the proper operation of the system. Finally, a reason much less compelling than its predecessors, advocating the coordination of movements is the harmonization of machine operation. All movements start and finish together, then they are synchronized with the movement lasting the longest. Thus, only one axis operates at its maximum performance; the others, being slowed, benefit from reduced wear.
9.4.2. Step response of a controlled shaft 9.4.2.1. Motors The separately excited DC motors and self-driven synchronous motors (brushless motors) can be submitted to torque control, rotation velocity control or position control. Indeed, the synchronous self-driven motors, modeled in the d-q reference frame, can be controlled as DC motors if they are equipped with an angular position sensor. The electrical motors used in robotics have a power ranging from 100 watts to a few kilowatts, according to the size of the controlled joint. The maximum torques they can provide range from a few Nm to 100 Nm. They are therefore insufficient to animate a joint directly without the help of a gearbox. The gearbox divides or multiplies the inertia by the square of the reduction ratio depending on the calculation point: from input or output. Special motors, known as low-inertia motors, have to be used to avoid the great effect of the rotating parts inertia. The rotor of these devices is designed so that its geometry may minimize the moment of inertia (glass epoxy disk on which conductors are “printed”, a long cylinder with very small diameter), and the excitation is achieved using lanthanide permanent magnets. The self-driven synchronous motors have rotors consisting of samarium-cobalt permanent magnets, with very low moments of inertia. At equal capacity, their characteristics are slightly higher than that of previous motors, in particular with regard to mass. Most DC motors are power supplied by pulse width modulation (PWM). The efficiency of this power supply type is 80% higher, and the response time is negligible (switching transistor time). It is also possible to make them work in the four quadrants of the voltage-current plan [AUB 81], [KIR 92]. The power supply of self-driven synchronous motors consists of a continuous voltage source, coupled with a switch, which allocates voltage square signals on the
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phase’s circuits of the machine, creating a rotating field from a periodic voltage wave. The rotating speed motor depends on the frequency of the power supply. The control of conduction electronic switches of commutator is produced from an encoder giving the position of the rotor. This process ensures the rotating field position is controlled by the rotor position, by making the phase difference between the reference signal and the signal from the encoder zero [BER 85].
9.4.2.2. Current regulation Current regulation of the robot’s actuators is the heart of the control system. Indeed, the trajectory tracking involves a dynamic control, which will be implemented by a torque regulation. However, the current determines the latter. In the case of a DC servomotor, the equations are: U
ri L di E; dt
E
k:;
*
ki
I m d: f: *c dt
where k is a torque constant, r, L and Im are respectively resistance, inductance and the moment of inertia of the servomotor; E is the back electromotive force voltage, : is the angular velocity and f is the viscous friction coefficient. Assuming the load torque *c is purely inertial, and It is the total inertia calculated on the motor shaft through the gearbox, the transfer function can be written: It p f i(p) | 2 U(p) k 1 W e p 1 W m p
W eW m
LI t rf k 2
;
Wm
Gi (p)
rI t Lf rf k 2
;
[9.11]
W m !! W e
Regarding It, it should be noted that the arm link inertia moved by the actuator is divided by n2 and torques generated by the centrifugal and Coriolis forces are divided by n3. Consequently, the It variations can be assumed weak and It can be considered constant for the first approximation. To achieve the current loop, a compensator C(p) will be added, whose output will be the motor voltage and the input difference between the ir(p) reference current, and the i(p) actual current. U(p) = C(p)[ir(p)-i(p)]
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The C(p) compensator is usually a PI corrector, so the transfer function ir(p)/i(p) will reduce to a simple first order transfer function by a judicious choice of parameters. The schematic diagram is given in Figure 9.10: i( p) ir ( p )
C ( p )Gi ( p) 1 C ( p )Gi ( p )
1 k 1 W e p 1 W m p 1 C ( p) It p f 2
U(p)
ir(p)
C(p)
+
i(p) Gi(p)
Figure 9.10. Current regulation
If the f viscous friction coefficient is ignored, with a PI compensator, the following relationship can be written as: C(p)
Ki
1W pp p
k2 mIt
§ 1 Wm p · ¸ ¨¨ p ¸¹ ©
where Ki = k2/Itm (m is a positive number) and Wp = Wm.. If m is high, the time constant is almost zero and: i (p ) ir (p)
1 |1 W p 1 e 1 m m
Therefore, the current loop can be accelerated so that we can consider the current servo as a system with instantaneous response. This property is fundamental to the dynamic control of robots, which implies that the torques, and thus the currents, are used as control variables. The hydraulic actuators do not own similar property because of the oil compressibility, which (however small) introduces time constants in the pressure transmission during dynamic operation.
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9.4.2.3. Speed control From the viewpoint of regulations, the self-driven synchronous motors can be assimilated to DC motors. We will therefore not distinguish between the two cases, although regulators have a matrix structure in the first case and scalar structure in the second case. From the current regulation, a speed control can be realized using the velocity error as a reference current. The torque is assumed to be purely inertial . If we choose as and defined by the current, and we have *m ki kiref I t :
:(p) : ref (p)
kv :ref : , then the transfer function is:
current set point iref
k . kv k.kv It p
1
1 It k. kv
[9.12] p
This is a first order transfer function and its time constant can be adjusted by the gain kv. The control block diagram is given in Figure 9.11. iref (p)
:ref kv
Current Regulation
i(p)
k It p
:(p)
Figure 9.11. Speed control
9.4.2.4. Position control The position control can be obtained directly from the speed control loop with the previous structure. Just add a position feedback loop that will define a speed set point (see Figure 9.12). Tref
kp kv
:ref
iref (p) kv
Current Regulation
i(p)
Figure 9.12. Position control
k It p
:(p)
1 p
T
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We have:
kp
: ref
kv
T p
T ref p
T ref T and i
ref
= kv(:ref – :) =kp(Tref – T) – kv: leading to:
1 kv I 1 p t p2 kp kk p
[9.13]
This system is always stable because kv and kp are positive. All these transfer functions are valid only if the assumptions stated above are validated. It is necessary to have iref = i, which means that, in the motor power supply, the carrier frequency chopper is much greater than the time constant inverse of the electrical circuit armature. For speed regulation, the hypothesis of a purely inertial load is acceptable only if the robot arm is statically balanced (using counterweight or deformable parallelogram), and with a very high reducing ratio (n > 100) which divides the variable inertia torques by n2 and torques due to centrifugal and Coriolis forces by n 3. The torque control and the speed control are frequently used to track trajectories. The position control is only used to generate the point to point movement as it cannot impose a known path between the start and end points. Transfer function [9.13] of the position joint servo, with a velocity feedback loop and an optimized current feedback loop (i.e. for which the electric time constant is negligible), is second order. The unit step response T(t) will be aperiodic or pseudoperiodic, as shown in Figure 9.13. T
T
1
1
b)
a) t 0
tacc
tfr
t 0
tacc tfr
Figure 9.13. Responses: a) pseudo-periodic; and b) aperiodic
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In both cases the speed is zero at the origin and then increases up to a constant value, before finally tending towards zero, with or without oscillations. The times tacc and tfr indicate, respectively, the end of acceleration and the start of braking.
9.4.3. Speed representation in a point-to-point movement 9.4.3.1. Speed and acceleration limits Consider a single axis, moving from the angular position T1 to position T2. The set point amplitude applied to the joint servo is Tref = T2 – T1 = 'T, and the response is that of the Figure 9.13 for which the asymptote ordinate, 1, is replaced with 'T. After an acceleration phase, the motor continues with constant speed, corresponding to the slope of the response curve. That speed depends only on the motor characteristics; those of transmission, charge (which are fixed) and controller characteristics (which are adjustable). In general, we try to find the fastest response, without overshoot. Then the parameters have to be tuned close to the “critical damping”, which corresponds to the parameters giving two equal roots in transfer function denominator [9.13]. In this case, the damping factor is equal to 1 and the time constant is T = It / kk p . The requirement to be met is kv2
4 I t k p /k . In this
particular case, the position unit step response, velocity and acceleration are respectively:
T t 1
t T t /T T t e T
t t / T T t e T2
1 § t · 1 ¸ et / t 2 ¨ T © T¹
[9.14]
where It is the total inertia of the rotating parts of the motor, transmission and load; *m is the motor torque; *c is the load torque (*c = 0 if the load is inertial, *c 0 if the load is gravitational or due to Coulomb friction); fv is the viscous friction coefficient and imax is the maximum transient current admissible. The equation of motion is mechanical: *m
ItT fvT *c
In steady state (constant velocity), the theoretical rotating limit velocity of the motor without regulation is:
Tmax
kimax *c fv
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This speed is usually very high because fv is very small and in fact the velocity is limited by mechanical considerations (rotor burst). If current, speed and position regulations control the motor, the maximum velocity is given by the slope of the tangent at the inflection point of the unit step response curve under critical state. This differs significantly from the previous value and above all depends on the time constant of the system. Its value, considering that the inflection point ( T 0 [9.14]) lies at tinf = T, is given by: Tinf 1 / Te . Rotating velocity is adjustable through kp and must remain below the velocity limit imposed by the manufacturer. However, the acceleration limits depend mainly on the load. Indeed, if the viscous frictions are ignored, the time taken by the motor to reach its maximum speed is a function of its power and load inertia: the more powerful the motor and the lower the load, the shorter the acceleration time will be. The maximum acceleration of a single shaft can be determined as follows: The transient maximum admissible torque is given by: *mmax
kimax
The acceleration limit is given:
Tmax
kimax *c It
[9.15]
The viscous friction term is neglected (initially zero and remaining small with regard to kimax). If the system is regulated, the acceleration is maximum at starting (t = 0) and then we find, according to [9.14]:
Tmax
1 T2
kk p It
In the case of a multiple link manipulator, such as a robot arm, the problem can be similarly processed by assuming that the coupling terms of the inertia matrix, the centrifugal and Coriolis torques are negligible in equation [9.8]. The values corresponding to the worst configuration are then assigned to the remaining diagonal terms in the matrix D(q), and at the gravity terms G(q), in order to maximize both of these terms. Equations deduced from equation [9.8] are decoupled and can be written:
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Bi
Dii qi Gi
Dii and Gi are the maximum values of the diagonal terms and the gravity vector components, limited according to the worst configuration.
The acceleration limit is given by: qi max
Bi Gi Dii
[9.16]
If we want to take the neglected terms into account, we have to solve an optimization problem under constraint equality.
9.4.3.2. Velocity modeling The assumptions made to carry out these models concern the acceleration and braking duration, which are assumed to be identical, although in practice nonlinearities caused by the Coulomb friction can induce an operating asymmetry in these movements. Similarly, when the velocity reaches its peak it is assumed that it will remain constant until the time of braking tfr. If the speed is represented by a curve as a time function, the area under the curve is equal to the angular distance traveled to get from one point to another. The time taken to complete this distance will be called Tpp. Note that the previous approach concerns the particular case of a rotating motion produced by a DC motor with a close loop adjusted to the critical state. It can be generalized to other servo settings and translation motions, as well as responses to higher-order 2. Therefore, the q generalized variable will be used to describe the movement. At first, we set aside very rough modeling, which consists of ignoring acceleration and braking times, because they give infinite accelerations. The classic representation of the velocity is achieved using a trapezoidal “speed profile”, as shown in Figure 9.14 (also called a “bang-bang acceleration profile”).
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. . q
.. q (t)
q (t)
q(t)
295
.. q
max
max
'q
tacc
tfr
a) position
tfr Tpp
t tfr Tpp
tacc
t
tacc
b) speed
t
c) acceleration
Figure 9.14. Position, velocity and acceleration: profiles
The speed, acceleration and braking are assumed constant in these representations. Thus, it can be deduced that: %q qmax (tfr ) qmax (Tpp tacc ) and qmax qmax / tacc
[9.17]
tacc and Tpp can be extracted from relations [9.17]. However, these results are only valid if a flat portion of the speed curve can be achieved, i.e. if %q tacc qmax or %q
2 qmax . qmax
Otherwise, the shape of the speed profile is an isosceles triangle, for which the 2 surface is given by %q tpp qmax /4 , and height, h = 1 qmaxT pp , as shown in Figure 2 9.15.
.
q(t)
..q(t) ..qmax
.
qmax
h
Tpp t t
.. -q
max
Tpp a) velocity
b) acceleration
Figure 9.15. Triangular speed profile
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The two parameters of the profile can be deduced from Figure 9.15: 'q T pp 2 , and extreme velocity, reached just before the braking phase qmax (below the maximum speed), h
qmax 'q .
The previous modeling or “bang-bang” control implies discontinuity speeds, at angular points of the profiles, whereas in reality speeds are still continuous. To better approximate the real speed, acceleration is now assumed to vary linearly during acceleration and braking phases. In this case, the derivative of the acceleration, the “jerk”, is limited by a maximum value, which will be noted Jemax. We then obtain the profiles shown in Figure 9.16, also known as “smoothed or softened profiles”, for which connections are parabolic. The curve symmetry gives: W1 = W3 – W2 = W5 –W4 = W7 – W6, where W7 = Tpp. The area under the speed curve is equal to the area of the trapezoid that envelops this curve and corresponds to 'q. The value of Tpp can be deduced from the following relations: T pp
q q 'q max max qmax qmax Jemax
[9.18]
(the left side of the trapezoid cut the horizontal axis to W1/2).
. . q
.. ..q
q(t)
q(t)
max
max
W4 W5 W6 W7 W1 W1 W2 W3
W4 W5 W6 W7
a) velocity
W2 W3
.. -q
t
max
b) acceleration
Figure 9.16. Smoothed speed and acceleration profiles
Calculation is more complicated for a triangular profile and requires time to reach maximum acceleration. In this case only, Tpp is the solution of a second degree equation.
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.
q(t)
t2
.
b
297
..q(t) ..q
t3
max
qex
c
W4 W5 W6 t1
t4
t
W1 W2 W3 t
a W1 W2 W3 W4 W5 d W6
Figure 9.17. Smoothed triangular speed profile
The symmetry of the curves in Figure 9.17 lead us to:
W1 = W3 – W2 = W4 – W3 = W6 – W5 ; W6 = Tpp; W3 =Tpp/2; the area under the speed curve equals that of trapezoid abcd and 'q; and the maximum speed reached for this shift qex q W 3 is equal to the area of the first trapezoid of the acceleration curve. From these observations we can deduce: W1
qmax ; J erk
The ab line equation is q t
'q
q W 3
qmax W 3 W1
§ T pp qmax ¨ ¨ 2 J erk ©
· ¸ ¸ ¹
qex ;
W · § qmax¨ t 1 ¸; the abcd area is 2¹ ©
§ T pp qmax q 2qex¨¨ ex 2J erk 2qmax © 2
· ¸ ¸ ¹
and 2 2 T pp
qmax 'q T pp 4 qmax Jemax
0
[9.19]
In order to establish a model that ensures velocity and acceleration continuities, a parabolic speed profile can be also used [KHA 99]. An attribute of this profile is its suitable representation of the motor start-up phase; the parabolic shape is identical to the first terms of expansion in Taylor’s series of the speed for a small t value, from [9.14] we have:
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T
t t 2 t3 " T 2 T 3 2T 4
t e t / T T2
Therefore, qi will be chosen as the form shown in Figure 9.18: qi
at 2 bt c
Figure 9.18. Parabolic velocity and linear acceleration profiles
The speed being zero at t = 0 and t = Tpp, c = 0 and b = -aTpp, the joint variable is in this case: q(t) = at3/3 + bt2/2. The area under the parabolic curve equals 'q. Thus: a
6'q 3 T pp
and b
6'q 2 T pp
providing: q t
§ 2 3 'q¨ 3t2 2t3 ¨T © pp T pp
· ¸ ¸ ¹
q t
§ 2 6'q¨ t2 t 3 ¨T © pp T pp
· ¸ ¸ ¹
q t
§ 6'q¨ 12 23t ¨T © pp T pp
· ¸ [9.20] ¸ ¹
Tpp is then determined from the maximum speed: T pp
3'q 2qmax
The advantage of this formulation is its simplicity. It implies that the “jerk” is constant. A parabolic speed profile with a flat portion can also be developed. Applying these methods, the calculation of Tpp is done without any particular difficulty. This model uses a 3-degree polynomial to represent the movement q(t).
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To further improve the model, some applications use a 5-degree polynomial. The coefficients are determined by two additional conditions: an acceleration zero at the beginning and at the end of movement. Calculations of other factors follow the same approach as the 3-degree polynomial method.
9.4.3.3. Coordination of point-to-point movements for a multi-axis system This coordination aims to simultaneously start and stop all motor rotations that contribute to create a movement. If each motor (i) is independent of the others, it will undertake a rotation 'qi, with its maximum speed. In this set of motors, there is one that will work longer that the others. The corresponding axis is called “the slowest axis” or “the binding axis”. The timing will be made according to this slowest case: the velocities of the different motors will be modified so that the 'qi range is the same, regardless of (i). Figure 9.19 shows some speed profiles for different joint movement. The longest time, corresponding to the binding axis movement, will be noted Tmax (Tmax = Tpp3 in the case of Figure 9.19) and will be determined for a robot arm with n degrees of freedom by: Tmax
Max T ppi, i >1,...n @ i
[9.21]
In order to obtain the same duration Tmax for all the axis control, the speed of the other actuators must be reduced so that all the tasks may last the same length of time. This new reduced speed will be called qiR for the actuator (i), and be determined by: qiR
'qi Tmax tacci
T ppi tacci qi max Tmax tacci
where tacci takes the value qi max / qi max for a trapezoidal profile and qi max / qi max qi max / J erki a softened trapezoidal profile. The coefficient kiR to be used is deduced from this result:
qiR
T ppi / tacci 1
qi max
Tmax / tacci 1
kiR
[9.22]
However, such a reduction may result in a change of speed profile. A joint operating in triangular mode may, after slowing, operate in trapezoidal mode. The following test will make, for each axis:
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Control Methods for Electrical Machines 2
! qi max 'qi qi max
and adapt the calculation method according to the test results.
. q(t) . 'q2max . 'q1max . 'q3max
q(t) 'q2 'q3 'q1 'q4 t Tpp4 Tpp2
Tpp1 Tpp3
t Tpp4 Tpp2Tpp1
a) 4 position step responses
Tpp3
b) Uncoordinated speed profiles
Figure 9.19. Uncoordinated movements
In a particular triangular profile, the speed must not be reduced; but the acceleration Tppi should be reduced to Tmax, and qi max must be reduced to a new value, qiR such that: T ppi
2
'qi qi max
Tmax
2
'qi qiR
i.e. qiR
4'qi
2 T ppi
2 Tmax
2 Tmax
qi max
The acceleration reduction ratio is equal to the square of the ratio of the axis task time, considered to be the binding axis.
9.4.4. Partially specified trajectories
The previous study assumed zero speed at the start points and end points; the end effector trajectory to reach these two points was free. If intermediate points or “via points” are specified, the speed will differ from zero for the joint variable values on
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these points. After converting the point coordinates of the path, in joint coordinates, using the inverse geometric model, a joint trajectory can be defined for each actuator. Therefore, the motion coordination requires path segments joining two via points to be covered at the same time for each articulation. If we call A, B, C and D via points of a path in the workspace, we can keep the same letters for their counterparts in the joint space for each actuator. The movement of the joint (i) is shown in Figure 9.20. Some points should be achieved absolutely (they will be called “through points”), while others may simply be approached without the trajectory passing exactly through these points (case of a bypass for example), and they will be called “approach points”. The control laws of actuators must give a smooth trajectory, for which the speed, acceleration and possibly “jerk” have to be continuous during a speed transition. To this end, the joint variable q(t) is represented by a 4-order polynomial, which ensures the continuity of derivatives until 4th order. The time origin was arbitrarily chosen at the moment where point B is reached (articular configuration is deduced from the inverse model). Points A and D are through points, and points B and C are approach points of the trajectory. For a joint (i), tacc depends on the traveling speed difference of the two consecutive segments of the trajectory and the value of the maximum admissible acceleration. It varies from one point to another. Then the 4-order trajectory is composed of straight lines and curve arcs. During the transition speed between points A and B of Figure 9.20, the joint variable is assumed to take the form: q(t)
a4t 4 a3t 3 a2t 2 a1t a0
To identify these polynomial coefficients, we have the following information: qA
q tacc ; q A
qb
q tacc
q tacc
qB qC T1
q A qB tacc
q BC ; qb
q AB ; qA
q(tacc )
q tacc
0
0
The ai coefficients take the following values: a0
qA
tacc 3q BC 13q AB ; a1 16 a4
q AB qBC ; 2
q BC q AB 3 16tacc
a2
3 q BC q AB ; a3 8tacc
0
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The equations of motion, speed and acceleration are: q t
q q 3 q BC q AB 2 q BC q AB t BC 3 AB t 4 t t acc 3q BC 13q AB q A 8tacc 2 16 16tacc
q BC q AB
3 q BC q AB q q AB t BC 4tacc 2
q t
q t
ª 3 q BC q AB « § t ¨ « ¨© tacc 4tacc ¬
3 4tacc
t3
2 º · ¸ 1» ¸ » ¹ ¼
qi(t) A
a) Motion of joint (i)
qA qB B qC
b
c
C d
D
qD
t
- tacc
tacc
T2
T1 T1-tacc
. q(t) . q
T1+tacc t b) Speed of joint (i)
BC
.
qCD
.
qAB
Figure 9.20. Portion of joint trajectory without stopping
These equations are general and can be used to smooth the speed curve in the neighborhood of all the via points, especially around the point C. To do this, simply replace, in the previous equations, the kinematic parameters of point A with those of C:
qC qBC T1tacc qB, qC qBC, qC 0 and the kinematic parameters of point B with those of D:
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qD qCD
303
qC qD ,qD 0 T2 T1
This can easily be repeated for all approach points of the curve. The modeling of the movement to through points poses no particular problem. Between the transition areas, q(t) varies linearly. For example, between B and C, q t q BC t qB . To synchronize the movement of each joint, it is necessary that the time intervals between the passage of the end effector at points A, B, C, D, etc. of the workspace trajectory are identical to the time intervals between the point counterparts A, B, C, D, etc. of trajectories in the space of variable joints, for each joint (i). The method in section 9.4.3.3 can then be used to calculate these time intervals between points of the broken line in Figure 9.20. The time T1 is then imposed on all axes, at points B and C of each joint trajectory. However, the value of T1 is an approximation because it does not take into account slowdown at the via points B and C, to provide the speed continuity. This operation is repeated for all Tk. The calculation of acceleration time tacc at each via point of the broken line, is performed by noting that the acceleration is maximum for t = 0 (as in point B), because q(t) is parabolic. This gives:
tacc
3 q BC q AB 4 qi max
[9.23]
We must repeat this calculation at all points of the trajectory for the controlled joint. The qi max acceleration involved in [9.23] is given by [9.15]. This method is the more classic and is commonly used. However, it does not lead to a trajectory joint model that goes through all the involved points (in this case there are no longer any approach points). To achieve this goal, a method [LIN 83] consists of representing the joint trajectory between two points as a cubic polynomial. The acceleration is then modeled by a linear time function and the speed as a parabolic function, which ensures the continuity of the main variables. The time between two imposed points of the curve q(t) is calculated in a way similar to the previous case. Determining the coefficients of cubic polynomial is quite laborious, but it does, however, allow the optimization of traveling times. This approach can be compared to the method described in section 9.4.3.2, except that the speed varies continuously.
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9.5. Conclusion
The example of robotics has led to the presentation of the movement coordination for a multi-motor system, using the time variable. The robot is well suited to this statement because it implies in the same task a sufficient number of motors to highlight the main problems to solve, without achieving a discouraging degree of complexity. However, the described methods are not specific to robotics and can be applied without difficulty, to other electromechanical systems that implement complex movements driven by a set of motors, as encountered in machinery manufacturing. These results can also be extended to the problem of synchronizing multiple machines working in a common process. Indeed, there is no constraint regarding the physical location of the engines or the nature of their task. 9.6. Bibliography [AST 84] ASTROM K., WITTENMARK B., Computer Controlled Systems, Theory and Design, Prentice Hall Inc., Englewood Cliffs N.J, 1984. [AUB 81] AUBRY J.F., HUSSON R., IUNG C., PFITSCHER G, La commande des machines électriques par microprocesseurs., Le Point en Automatique, vol 1, J.C. Pruvost, Technical Documentation, Lavoisier Paris, 1981. [BER 85] BERGMANN C., COUREAU P.,LOUIS J.P, Digital Direct Control of a Self-controlled Synchronous Motor with Permanent Magnet, EPE, Brussels, 1985. [COI 86] COIFFET P., La robotique, principes et applications, Hermes, 1986. [FAG 98] FAGES G., Statistiques 97, Robaut, no. 21, pp 28-32, 1998. [GOR 84] GORLA B., RENAUD M., Modèles des robots manipulateurs, Cepadues-Editions, 1984. [KHA 99] KHALIL W., DOMBRE E., Modélisation, identification et commande des robots (2nd edn), Hermes, 1999. [KHE 96] KHEIR N.A., Systems Modelling and Computer Simulation (2nd edn), Marcel Dekker, Inc., 1996. [KIR 92] KIRAT R., Modélisation et commande en position d’un vérin pneumatique contrôlé par un ensemble de quatre électrovannes pilotées en modulation de largeur, Thesis, INSA, 1992. [LAL 94] LALLEMAND J.-P., ZEGHOUL S., Robotique. Aspects fondamentaux: Modélisation mécanique – CAO robotique – Commande, Masson, 1994. [LEE 82] LEE G., “Robot arm kinematics, dynamics and control”, IEEE Computer, vol. 15, pp 62-80, 1982.
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[LEE 85] LEE C. S. G., CHUNG M.J., “Adaptative perturbation control with feed-forward compensation for robot manipulators”, Simulation, pp 127-136, 1985. [LIN 83] LIN C.S., CHANG P.R., LUH J.Y.S., “Formulation and optimisation of cubic polynomial joint trajectories for industrial robots”, IEEE. Trans. on Automatic Control, vol. AC-28, no. 12, December, pp 1066-1073, 1983. [LUH 83] LUH J., “An anatomy of industrial robots and their controls”, IEEE Trans AC, vol. 8, no. 2, pp 133-152, 1983. [McIN 86] McINNIS C., LIU F., “Kinematics and dynamics in robotics: a tutorial based upon classical concepts of vectorial mechanics”, IEEE J. of Robotics and Automation, vol. 2, no. 4, pp 181-186, 1986. [PAU 82] PAUL R., Robots Manipulators. Mathematics, Programming and Control, MIT Press, Cambridge MA, USA, 1982. [SAM 91] SAMSON C.I., LE BORGNE M., ESPIAU B., Robot Control. The Task Function Approach, the Oxford Engineering Science Series, Clarendon Press, Oxford, UK, 1991.
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Chapter 10
Multileaf Collimators
Radiotherapy treatment uses ionizing radiation on cancerous tissues in order to kill all the tumor cells and control the development of the disease. A treatment plan is composed of several radiation beams delivered by a dedicated linear accelerator. In conformal radiotherapy, a multileaf collimator, located between the radiation source and the patient, is used to adjust the geometric outline of the treatment beams to the tumorous region shape. This device is made up of two opposed series of parallel leaves, whose positions determine the outline of the treatment beam. Each leaf position and movement is monitored by a motor, and the accuracy of all these parameters is checked by a real time control system. After a brief introduction to radiotherapy and medical linear accelerators, this chapter will focus on geometric and technical characteristics of different multileaf collimator devices. A specific radiation technique called intensity modulated radiotherapy, based on multileaf collimator use will then be presented, with special focus on the algorithms developed to optimize the coupling of the position of the leaves, which are moving dynamically during the radiation.
10.1. Radiotherapy In this chapter, radiotherapy and the linear accelerator are presented in order to introduce the context of use of the multileaf collimators.
Chapter written by Sabine ELLES and Bruno MAURY.
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Control Methods for Electrical Machines
10.1.1. The medical prescription Based on patient examinations, a radiotherapist will outline the tumorous region that will be irradiated in order to kill cancerous cells and control the lesion extension. The next step consists of determining a treatment plan made of several isocentric radiation beams, usually equi-distributed, so that their geometric intersection fits as close as possible to the tumor shape while avoiding all the surrounding healthy tissue. Numerical computer-based on simulations helps the radiotherapist and physicists with this task, allowing them to visualize beams, patient anatomy and even to simulate the treatment sequence.
Figure 10.1. Treatment plan made of 5 equi-distributed beams: computer simulation of patient positioning and visualization of the beam shape projected on patient’s skin
The prescription quality is then estimated using a dosimetric study, which consists of calculating the radiation dose deposited in all the tissues. Once validated, the treatment is spread over 25 to 30 radiation sessions. 10.1.2. Linear accelerators A linear accelerator produces homogenous and stable electron radiations or Xrays whose energy can reach several tens of MeV. Electrons produced by a gun are sent through a section where they accelerate up to the speed of light. This section is a waveguide under vacuum made of consecutive accelerating cavities, into which a 3 GHz high frequency wave is injected. The HF wave carries the electrons on its crest in the same way as a water wave carries a surfer. Due to power limitation, the HF wave and the electron emissions are pulsed; these pulses are generated by a high voltage power supply charging a pulse-forming network of inductors and capacitors, regularly discharged by a thyratron. The HF wave is obtained either by a set driver (oscillator) – klystron (power amplifier) or a magnetron if less power is needed.
Figure 10.2. Linear accelerator used in radiotherapy
Multileaf Collimators 309
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Control Methods for Electrical Machines
The generated high energy electron beam is then deflected to the patient by steering coils. X-rays are finally produced by inserting a target of tungsten into the beam. The intensity of this beam is made uniform by flattening filters and the additional ion chambers check this intensity. The size of the delivered treatment beam is defined by four paired lead diaphragms and can vary from 4x4 cm2 to 40x40 cm2 defined at a reference point called the isocenter. This point is the intersection of the arm rotation axis and the collimator rotation axis, and is usually located at 100 cm from the source (materialized by the target).
10.2. Multileaf collimators The initial treatment beam outline is rectangular depending on the position of the four lead diaphragms. The next step consists of modifying this outline in order to conform to the shape of the radiation target. This can be achieved with customized lead blocks interposed manually between the radiation source and the patient or with an automated multileaf collimator (see Figure 10.3). The latter is composed of two opposed series of tungsten leaves whose positions are calculated in order to approximate the defined shape.
Figure 10.3. Shape fit with lead blocks and multileaf collimator
In comparison to manufacturing and handling the leads blocks, the use of a multileaf collimator has become widespread because of its ability to simulate a large panel of beam geometries and use security provided by the electronically controlled device. The technical characteristics, command and copy system of different multileaf collimators will be presented in the next section.
Multileaf Collimators
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10.2.1. Geometric characteristics of multileaf collimators The most significant characteristics of a multileaf collimator are the number of leaves and their width. Minimum and maximum treatment beam sizes depend on these parameters, as well as on the accuracy of the shape approximation: the smaller the width of leaves, the better shape the approximation. By convention a leaf pair is composed of a leaf and its opposite. Classic multileaf collimator leaf width varies between 0.5 and 1 cm, whereas micro-multileaf collimator leaf width is less than 0.5 cm. 10.2.1.1. Micro multileaf collimators The micro multileaf collimators are dedicated to the treatment of small tumors that need high precision conformation, for example cerebral tumors. Thus, the number of leaves is reduced and their widths are small. Some available configurations with corresponding maximum beam sizes are: 40 leaf pairs of 1.7 mm (7x7 cm2), 26 leaf pairs of 3, 4.5 and 5.5 mm (10x10 cm2) or 24 leaf pairs of 6.25 mm (15x15 cm2). These devices are usually produced by independent manufacturers and fixed on the accelerator head. They are coupled with a dedicated electronic command system and controlled separately. 10.2.1.2. Classic multileaf collimators These collimators are developed by the linear accelerator manufacturers, integrated in the head of the machines and controlled by a common command system. Almost all beams are developed using a collimator instead of lead blocks. The maximum beam size is identical to that of the diaphragm, 40x40 cm2. The three main available systems are composed respectively of 40 leaf pairs of 10 mm width; 27 leaf pairs of 10 mm and 1 leaf pair of 62.5 mm at the boundary; 40 leaf pairs of 5 mm surrounded by 10 leaf pairs of 10 mm (Figure 10.4). The leaf length is usually around 20 cm for a thickness of 8cm.
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Control Methods for Electrical Machines
Figure 10.4. Examples of beam realization
10.2.2. Technical characteristics The leaf command principle is common to all the manufacturers: each leaf is driven by a DC motor associated with its electronic command, and with a computerized readout system intended to obtain the exact leaf position. A dynamic control software displays the field shape in real time. Each leaf movement is controlled by a motor that commands a drive screw passing through a nut. Ball bearings ensure the sliding between the two next leaves.
Leaf Drive screw Motor /Gearbox
Nut
Figure 10.5. Multileaf collimator and detailed leaf devices
The multileaf collimator control software runs independently from the accelerator. When the field shape is in accordance with what is prescribed, a signal is sent to the linear accelerator and treatment can begin. However, due to treatment security constraints, computing resources tend to be shared between both systems.
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10.2.3. Readout systems for leaf position checking 10.2.3.1. Readout by image analysis In order to control patient positioning before treatment, a linear accelerator provides a simulation option based on a light ray that reproduces the radiation beam geometry exactly. Reflective spots are stuck on the top of each leaf. During simulations or treatments, these spots are illuminated by the light. These signals are sent back towards a camera and the resulting image analysis gives the position of every leaf. Four reference reflectors ensure the system calibration: every leaf position is relative to these marks. The light beam is diverted towards the collimator by a set of mirrors, and the reflection signal must be permanently visible by the camera. The whole optical system combines a stiff mirror, a mylar mirror and a semi-reflective mirror.
Figure 10.6. Readout by image analysis
The mylar mirror is a polyester aluminized film, transparent to the radiation but reflective to light. The 25/75% reflective mirror is made of glass resistive to radiation, sending back the light from the bulb but not the image of the reflective spots towards the camera. The signal resulting from the CCD 2/3 inch camera is first sent to a Matrox image treatment board, and then treated by an Intel 320 computer to display a real-
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time graphic representation of the position of the leaves. If a leaf movement is detected during radiation, the treatment is automatically interrupted. This system requires much maintenance: even the slightest light interference may disturb the recognition of leaves. The optical system is especially sensitive and reflector efficiency deteriorates over time so they need to be replaced regularly. The same problem occurs with the CDD camera when its dark current level becomes too high. Nevertheless, the simplicity of these devices makes maintenance quite easy and the breakdown rate is small. 10.2.3.2. Readout by potentiometers The readout by potentiometers is based on the information concerning the leaf position given by an encoder linked to the controlling motors. A corresponding calibration position is determined by alternately moving every leaf until it cuts an optical beam perpendicular to the leaf movement. To confirm this value, a second security readout is made by a cursor located on each leaf. This cursor slides on a resistive track that simulates a linear potentiometer: a voltage is produced proportional to the position. For this type of collimator both leaf series are fixed on a mobile rack, and potentiometric leave positions are therefore defined relative to rack positions which are measured by encoders. Readout information is sent by optical fiber to the computer system and display graphically in real time.
10.2.4. Leaf command system The main command systems used to control the leaves of a collimator are described below. However, their concepts are subject to industrial confidentiality so only a rough description will be given. 10.2.4.1. Simultaneous command system This system consists of a Windows NT dedicated computer, a VME bus-based controller rack and the electronics of the collimator. The controller rack has three interfaces: for the user computer operating system, for the accelerator computing system and for generating the signals sent to each leaf series. These signals represent the discrepancies between prescribed and actual leaf positions. These latter electronic boards have two optical fiber connections: the first is dedicated to receiving position information, the second to sending command signals. Optical fibers are linked to the collimator via “Transceiver” boards that are in charge
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of decoding the PWM (pulsed width modulation) signal: the wider the impulse, the more important power applied to the motor is. These signals are then sent to the “Driver” circuit in charge to supply power to the motor power and to control the mechanical brake of the rack.
Figure 10.7. Motor circuit
Power is applied to the motor by closing simultaneously two “switches” (A and D for one direction, B and C for the opposite direction). To slow down the motor, A and B are closed simultaneously. In order to protect the motor, the current consumption is monitored by the resistance R included in the circuit. A threshold value between 200 and 450 mA is defined for each leaf. If the current exceeds this limit, the circuit is opened during 12 Psec to reduce the power sent to the motor. The system also integrates a thermal protection. If the current exceeds 1.8 A, the power supply to problematic circuits is cut. For maintenance purposes these kinds of collimators are also equipped with a direct and independent electronic system to drive the leaves. Leaf and movement direction can be selected and security protection overstepped. This feature is mostly used to unlock the collimator in the case of leaf collision. 10.2.4.2. Multiplexed command A multiplexed command system is used for systems with position readout by camera. In this case leaves are not fixed on a rack and their movement amplitude is greater, they can cross the middle point of radiation beam and reach half-beam size on the opposite side. The command system is centralized on an Intel 32 bit computer and serial signals are sent to an interface board linked with the collimator. This signal is composed of
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each leaf command line address followed by the tension value needed to reach a prescribed position. A MTU (multiplex terminal unit) board decodes this information: two digital-to-analog converters generate the appropriate –5V to +5V voltage sent to the leaf motor. A demultiplexing board converts the data of each leaf series to tension and leaf address. Tensions are “memorized” by dedicated sample-and-hold components, and a new value will be taken into account only if it differs from the memorized value. Tension is then amplified and converted to the current directly applied to the motor. Although movement amplitude increases risks of leaf collision, these collimators do not provide independent electronic control system. A software-based tool is available to control and calibrate leaf positions and speed.
Axis length: 7 cm Nominal voltage: 12 V Nominal current: 45 mA Starting current: 300 mA
Diameter: 1.2 cm Gearbox maximum output torque: 4.5 mNM Gearbox reduction ratio: 16:1 Nominal speed: 12,500 t/m for a leaf movement of 12.5 mm/s
Figure 10.8. Motor description and characteristics
10.2.5. Accuracy of command and leaf positioning Some treatment techniques need high precision beam geometry shaping, like intensity modulated radiotherapy which is described in the next section. So, leaf position, movement and synchronization accuracy must be checked periodically. This is done by radiating radiosensitive film following predefined protocols. Some of these are described here. The first test consists of radiating a film with leaves moving at the same speed with a constant gap between opposed leaves. An energy profile is the measured film radiation along the axis movement of a leaf pair. If speeds are identical and stable for all leaves, their energy profiles should be uniform.
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The leaf speed inertia is estimated by momentarily stopping leaves and the beam during radiation. An energy profile showing a peak for a given stopping position is characteristic of a leaf affected by inertia at start up, while a trough is characteristic of a leaf affected by inertia at stopping [LIU 00]. Synchronization is tested by studying the entire collimator behavior in predefined positions located at right angles to movement. Radiation is achieved with all leaves moving at the same speed and pausing for a moment at these points. The alignment of these stopping regions gives an estimation of synchronization [RAM 00]. Due to motor characteristics and the command system accuracy, the speed of movement is stable and inertia effects are negligible. However, a weekly collimator control is recommended to ensure accuracy including calibration checking and adjustment of shifts resulting from routine usage. Studies made for speeds between 0.3 and 3.0 cm.s-1 show that the position errors vary from 0.03 to 0.21 cm, the corresponding radiation energy error is about 2%.
10.3. Intensity modulated radiotherapy Coupling leaf movements is not really essential for basic use of a collimator for shape beam conformation: leaves are positioned according to prescription, the shape is validated and radiation starts. A new radiation technique called intensity modulated radiotherapy is currently in development. This technique takes advantage of the new degree of freedom due to computerized control of the collimator: a higher energy gradient can be realized by increasing the dose delivered to a tumor while better saving the surrounding healthy tissues. The treatment plan definition is also modified and a new concept is born: inverse dosimetry. Instead of defining the radiation beams and estimating the quality of prescription by computing energy distribution, the radiotherapist defines the energy distribution and a specific dosimetric planning system calculates the intensity of the radiation beams that will produce this distribution.
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. . . .
Figure 10.9. Uniform intensity beam: an intensity modulated beam and its profile
Each beam is considered to be a map of elementary beams: in the case of a uniform intensity beam, the contribution of each elementary beams is identical (see Figure 10.9a). However, contributions of elementary beams are different in intensity modulated treatment (see Figure 10.9b). The set of intensities for the whole beam is called the intensities matrix, and the energy profile (see Figure 10.9c) corresponds to the intensities along the axis of movement of a collimator leaf pair. The proposed example (see Figure 10.9) shows a beam of a prostate treatment plan: the volume to radiate is defined as the target, surrounded by the rectum (left) and the bladder (right). These latter organs are located close to the prostate so the intensity modulated technique is particularly necessary. The size of an elementary beam is 0.25x1.00 cm2.
10.3.1. How to realize a modulated intensity beam with a multileaf collimator The prescribed dose to deliver is specified in monitor units (MU). Supposing a constant radiation rate, the energy deposited by a beam is proportional to its exposure time. Modeling an intensity modulated beam means converting the prescribed elementary beam intensities into successive sets of leaf positions so that the total exposure time of each beam corresponds to the intensity matrix. Two techniques have been developed for radiating these beams. The first consists of superimposing a series of static beams created in sequence one by one, radiation being stopped while leaves are positioned for the next beam. The second
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technique is dynamic: leaves are moving continuously while radiation and modulation is obtained via speed variation. The mechanical and technical characteristics of the accelerator and the collimator have to be taken into account while calculating beam intensity matrices: minimum and maximum radiation rates, maximum leaf movement speed and more specific constraints like the minimum distance between opposed leaves or the ability of two opposed adjacent leaves to cross. Leaf and diaphragm movements are sometimes coupled to overcome these constraints. In addition, the sliding between two adjacent leaves is guided by a tongue and groove system that ensure the optimum movement of each leaf. This leads to an radiation artifact along these crack line. For example (see Figure 10.10) radiation of beam (b) and (c) at different times is different from the radiation of beam (d): an over/under radiation appears in the middle of the beam. In order to minimize this problem, elementary beam decomposition algorithms try to optimize the synchronization of the motion of the leaves.
Figure 10.10. Transversal view of leaves: artifact in the sum of two adjacent beams
Finally, the last important constraint is the leakage between opposed closed leaves that occurs when this junction is radiated. It is therefore recommended not to close opposed leaves in the radiation beam. Several intensity matrix decomposition algorithms are presented in the next section corresponding to static and dynamic radiation techniques.
10.3.2. Discretization into static elementary beams Discretization into static elementary beams is usually chosen for collimators with minimal distance between leaf constraints. The first step consists of constructing a simplified intensity matrix made of integer values according to a step of intensity 'I. This new matrix is then processed by a segmentation algorithm.
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10.3.2.1. Collimator without position constraint In the case of a collimator without position constraint, leaf pairs are considered independently. The intensity matrix is interpreted line by line, with each line corresponding to the movement of a leaf pair. The concept of this segmentation method is quite simple: the intensity profile of each leaf pair is extracted from the matrix and represented according to a ('x,'I) binning. 'x is the step for leaf movement and 'I the step for intensity. Two series of points are then positioned on these plots whether the gradient is positive or negative (see points z and { in Figure 10.11). There, numbering defines the way they are rearranged, in turn defining the trajectory of the left and right leaf. The intensity profile is read according to growing, the 'x axis. For each x value each time a z point is found, the left leaf trajectory is incremented by 'x, and respectively each time a { point is found, the right leaf trajectory is incremented [BOR 94, WEB 98].
Figure 10.11. Segmentation from energy profile to leaf positions ('x: step of leaf movement, 'I: step in intensity)
This algorithm coordinates implicitly opposed leaf movement, each moving from left to right during beam radiation. Their trajectories are represented by the two curves surrounding leaf position schema in Figure 10.10. 10.3.2.2. A collimator with position constraints The segmentation algorithm presented here has been developed for a collimator subject to a constraint: opposed adjacent leaves cannot cross. In the example shown in Figure 10.12, the collimator is able to reproduce leaf position geometry (a), but not geometry (b).
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Figure 10.12. Leaf position examples: binary matrix discretization
The intensity matrix is first discretized according to a 'I intensity step and then split into the sum of binary matrices. These binary matrices are then converted into a sum of basic radiation beams. A radiation beam is first discretized as a mxn matrix M (see Figure 10.12), where: – m is the number of leaf pairs; – n is the number of elementary beams according to the axis of leaf movement ('x); – Mi,j is zero if the corresponding elementary beam is masked, and 1 if the beam is radiated. A consecutive set of values equal to 1 is called a segment (see Figure 10.12) and can be considered as a basic radiation beam. The distribution of the different segments composing a binary matrix reproduces the succession of basic beams that will be processed during radiation. Two methods for discretization into binary matrix and segments are presented below. The first is a factorization method and the second is an elementary beam rearrangement process. 10.3.2.2.1. Factorization according to the power of 2 values First the intensity matrix is decomposed according to coefficients equal to the power of 2 values: p
M
¦
2
i
M
i 0
where m is the greatest value, such that: 2
p
d Max ( Mi , j )
and where (Mi)i=0,..,p are binary matrices made up of the values 0 and 1.
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The factorization coefficients are the exposure times of each elementary beam during radiation. However, (Mi)i=0,..,p binary matrices are not necessarily feasible depending on the segment distributions. The conversion of a binary matrix into basic beams is done by examining the matrix from left to right: elementary beams are added to a basic beam until the basic beam is feasible. Once it is no longer possible to add more elementary beams to the current basic beam, the process starts again with the next basic beam until all the elementary beams are taken into account [XIA 98]. It has been proven that this factorization method minimizes the number of basic beams [HAM 00]. The example presented here shows the different steps of the factorization method, and the next intensity matrix: 5 4 4 3 4
M=
7 1 5 2 7
2 3 2 5 9
4 10 9 7 2
7 5 8 3 4
Factorization in power of 2 leads to the following result: M = 8 M8 + 4 M4 + 2 M2 + M1.
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Then binary matrix conversion into basic beams leads to: M = 8 M8 + 4 (M4,1 + M4,2) + 2 (M2,1 + M2,2) + M1,1 + M1,2
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Figure 10.13. Binary matrix decomposition
10.3.2.2.2. Factorization by vertical rearrangement The basic principle of this algorithm is to convert the intensity matrix into a first basic beam as large as possible and a set of other basic beams to refine energy delivery for high irradiated regions (see section 10.3.2.1) [SIO 99]. The optimization criteria selection is the total time needed to create the beam: from leaf positioning and validation to radiation time. The problem to solve is: finding the n coefficients {D1, ..., Dn } Nn and the corresponding feasible binary matrices S1,… Sn such that the total time execution of the following beam is as low as possible: n
M
¦D S
i i
i 1
The solving process is exhaustive: all possible sets of values for the n parameters {D1, ..., Dn } are tested, and the best set according to radiation feasibility and time execution is then chosen.
Figure 10.14. Example of factorization
The first large size basic beams are calculated by factorization of the intensity matrix. The remaining elementary beams are then rearranged into basic beams, each energy profile corresponding to a leaf pair is transformed into profile leaf position according to the same algorithm described in section 10.3.2.1 (see Figure 10.14).
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Figure 10.15. Example of vertical rearrangement for an energy profile and for the complete matrix
This first conversion is done separately for each energy profile. A second 3D rearrangement is then processed level by level, considering all mechanical leaf movement constraints like collisions between opposed adjacent leaves or leaf synchronization, to avoid artifact radiations. This step is done by vertical rearrangement of the 3D profile matrix: columns are shifted up until every level represents a feasible basic beam (see Figure 10.15). When a leaf collision is detected in a level matrix (see Figure 10.16a), the column with the lowest base is pushed up along with all the columns on its right side (see Figure 10.16b), and the collision problem is then solved for this level and the whole rearrangement algorithm restarts. Synchronization of adjacent leaf movement is achieved by trying to respect the following rule: the smallest column should be located as often as possible in the interval defined by the higher column (see Figure 10.16c). It is not possible to fulfill all synchronization requirements but we should avoid radiation artifacts where possible.
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synchronization
Figure 10.16. Vertical rearrangement to solve leaf collision: adjacent beam synchronization
This rearrangement process is done iteratively until all mechanical constraints are satisfied and beam synchronization is optimal.
10.3.3. Discretization into dynamic beams The intensity matrix is discretized into a set of control points according to the leaf axis direction step 'x. This step can vary over time depending on the gradient of the intensities. At the beginning of radiation all leaf pairs are closed and positioned at the left boundary of the beam. They will move together towards the right side of the beam with different speeds until they join each other there. Energy delivered to a given point is proportional to the exposure time to radiation; that is, to the time between when the right leaf crosses this point and exposes the point to radiation, and the time when the left leaf crosses this same point and hides it from radiation. 10.3.3.1. Collimator without position constraint This algorithm is similar to the one presented for discretization into static beams (see section 10.3.2.1) except that the global radiation time is minimized by setting the speed of one leaf from a pair at its maximum value for every point in time. Let vmax be the maximum leaf speed. Both leaves are located on the left side of the beam at the beginning of radiation (x=0). Leaf positions are computed based on an energy profile: if the intensity gradient is positive, the right leaf is moving to its position with speed vmax, while if the gradient is negative, the left leaf is moving to its position with speed vmax [BOY 00]. The sliding of the leaf that is moving at maximum speed is entirely defined by its position at the previous step, while the opposite leaf movement is given by the speed v:
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V(x) = vmax/(1 r vmax(d)/dx)) () stands for intensity) Figure 10.17 shows an example of leaf positions calculated from an energy profile.
Figure 10.17. Energy profile example, and its associated leaf positions
Usually leaf pair movements are synchronized in order to start and stop at the same time either by adapting the vmax value for each pair depending on the total time needed to develop the energy profile, or by re-evaluating leaf movement between the two last control points. 10.3.3.2. Collimator with position constraints This application case is more complex: radiation is realized dynamically although minimal distances between opposed leaves and opposed adjacent leaves must be satisfied.
Figure 10.18. Example of minimal distances between opposed and opposed adjacent leaves
Due to the concomitance of both distance constraints, it is mandatory to synchronize all leaf movements during intensity matrix segmentation. In practice, the four diaphragms that form the boundary of the initial rectangular beam coming out from the accelerator head are also included in the process in order to reduce radiation artifacts between adjacent leaves [CON 98]. The intensity matrix and movement along the leaf axis are discretized according to steps 'I and 'x.
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As in the previous method radiation is made from left to right starting with all the leaf pairs closed at the left boundary of the beam. The algorithm is iterative and checks elementary beams according to the x-axis. To illustrate its mechanism, the segmentation of an intensity matrix is shown (see Figure 10.19). The four initial steps consist of finding the new leaf/diaphragm configuration corresponding to the elementary beams: 1) The left diaphragm position is computed first: as soon as all elementary beams located along diaphragm are radiated, its position is incremented by one step 'x (c, e, f). 2) Left leaves are positioned in order to expose/hide elementary beams located along the left diaphragm. 3) Right leaves are positioned according to the intensity matrix by staying as close as possible to the left diaphragm. 4) The right diaphragm is positioned to agree with the leaf located furthest to the right (a, b, etc.). These steps are followed by a checking process: 1) Minimal distances between left and right diaphragms are checked and their positions are shifted if needed. 2) Minimal distances between leaves are also checked: if the distance between two opposed leaves is too small, the left leaf moves back under the left diaphragm or the right leaf moves forward under the right diaphragm (d, g). Once leaf position geometry is validated, the algorithm checks the radiation of the current x position and the next x position corresponding to elementary beams.
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(a)
(b)
(c) (h)
(d) (i)
(e) (j)
(f)
(g)
Figure 10.19. Radiation segmentation with the corresponding leaf/diaphragm configurations
This segmentation process can sometimes lead to leaf/diaphragm configuration corresponding to infeasible geometries. In this case complete segmentation process is again calculated by changing the positioning of the right leaf rule: instead of staying as close as possible to the left diaphragm (rule 3), the leaves can be positioned one step further. A more hybrid approach used to overcome an unfeasible beam consists of computing segmentation until the algorithm is blocked: this series will correspond to a first treatment step. The algorithm is then applied to the remaining elementary beams and so on until all intensities are treated. The global intensity matrix is segmented into a series of dynamic sequences but the radiation is stopped between each of them.
10.4. Conclusion The multileaf collimators used in radiotherapy are a concrete illustration of coupling of a set of low power motors. These collimators are currently widely used and new radiation techniques based on their uses have been developed. The main benefit is a better focus and a radiation
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control of tumorous regions. The accuracy and speed of leaf positioning ensures the quality of the treatment delivered to patients. Despite their high technicity, breakdown of these devices is rare but periodic maintenances and calibrations are essential as well as a real time security position control system. Technical characteristics of collimators and motors have been presented. The algorithms used to convert intensity matrices into leaf movement are dependent on these properties in the form of leaf position constraints, minimal distance or synchronization constraints. Different algorithms have been described depending on the radiation technique, static or dynamic, and the set of collimator constraints.
10.5. Bibliography [BEA 00] BEAVIS AW., GANNERY PS., WHITTON VJ, XING L., “Slide and shoot: a new method for MLC delivery of IMRT”, Proceedings of the XIIIth International Conference on the Use of Computers in Radiation Therapy, Heidelberg, 2000. [BOR 94] BORTFELD T., KAHLER D., WALDRON T., BOYER A., “X-ray field compensation with multileaf collimators”, Int. J. Radiat. Oncol. Biol. Phys., vol. 28, pp.723-730, 1994. [BOY 00] BOYER AL., XING L., LUXTON G., CHEN Y., “IMRT by dynamic MLC”, Proceedings of the XIIIth International Conference on the Use of Computers in Radiation Therapy, Heidelberg, 2000. [CON 98] CONVERY DJ., WEBB S., “Generation of discrete beam-intensity modulation by dynamic multileaf collimation under minimum leaf separation constraints”, Phys. Med. Biol., vol. 43, pp.2521-2538, 1998. [HAM 00] HAMACHER HW., LENZEN F., “A mixed integer programming approach to the multileaf collimator problem”, Proceedings of the XIIIth International Conference on the Use of Computers in Radiation Therapy, Heidelberg, 2000. [LIU 00] LIU C., XING L., “Dosimetric effects of mechanical inaccuracy of MLC leaf positions on IMRT”, Proceedings of the World Congress on Medical Physics and Biomedical Engineering, Chicago, July, 2000. [MA 98] MA L., BOYER AL., XING L., MA CM., “An optimized leaf-setting algorithm for beam intensity modulation using dynamic multileaf collimators”, Phys. Med. Biol., vol. 43, pp.1629-1643, 1998. [RAM 00] RAMSEY C., SPENCER K., OLIVER A., “Leaf position and dose rate error during conformal dynamic arc treatment”, Proceedings of the World Congress on Medical Physics and Biomedical Engineering, Chicago, July, 2000.
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[SAM 00] SAMANT S., ZHENG W., WU J., BURNHAM B., COFFEY D., SAWANT A., PATRA P., YUNPING Z., SONTAG M., “Automated verification of multileaf collimation (MLC) using a Siemens EPID”, Proceedings of the World Congress on Medical Physics and Biomedical Engineering, Chicago, July, 2000. [SIO 99] SIOCHI RAC., “Minimizing static intensity modulation delivery time using an intensity solid paradigm”, Med. Phys., vol. 43, pp.671-680, 1999. [SPI 94] SPIROU SV., CHUI CS, “Generation of arbitrary intensity profiles by dynamic jams or multileaf collimators”, Med. Phys., vol. 21, pp.1031-1042, 1994. [WEB 98] WEBB S., “Configuration options for intensity-modulated radiation therapy using multiple static fields shaped by a multileaf collimator”, Phys. Med. Biol., vol. 43, pp.241260, 1998. [XIA 98] XIA P., VERHEY LJ., “Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments”, Med. Phys., vol. 25, pp.1424-1435, 1998.
Chapter 11
Position and Velocity Coordination: Control of Machine-Tool Servomotors
All developments presented below aim first at defining the notion of open architecture systems, which is part of the design of new generations of CNC machine-tool. Structures and implementation strategies of control laws are then examined, mainly focusing on the concept of cascaded control loops, as one of the most important in this field. Finally, two specific applications describing the control of machines-tools AC servomotors are proposed to illustrate the impact of advanced control structures.
11.1. Open architecture systems Open architecture-based systems are nowadays designed following a welldefined concept in the machine-tool field.
11.1.1. Historical overview Since the beginning of the 1990s, there have been several initiatives, issued from universities as well as machines-tools builders, aimed at developing structures which may enable machines-tools designers, software vendors and end-users to take advantage of more flexibility. Indeed, all aspects related to modularity as well as the effects of the controllers and communication networks architecture on the system Chapter written by Patrick BOUCHER and Didier DUMUR.
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performance appear to be crucial [PRI 93], [KOR 96]. Among all these projects, the most important are the following: – OSEC (Open System Environment for Manufacturing Consortium) in Japan; – OMAC (Open Modular Architecture Controllers) in the US; – OSACA (Open System Architecture for Controls within Automation systems) in Europe. The major idea of all these research activities tends to provide a simplified implementation and integration through “open” interfaces within a standardized environment [ALT 94]. The final objective of this kind of structure is of course cost reduction and an increase in flexibility, for example by means of software “re-use” or integration of specific user-built modules.
11.1.2. Principle and advantages The design of “open” systems implies the development of dedicated structures which can adapt themselves very quickly to the developments of micro-informatics and communication technologies. From this point of view, the key word of these open architectures is modularity, which imposes the use of small independent modules, connected together by means of communication interfaces. An open system must be structured in such a coherent way that properly implemented applications can work on a wide range of platforms from different builders and can communicate with other applications. Consequently, research in this field looks at increasing the system functionalities, with a better portability and reconfigurability.
11.1.3. Modular architecture example Within the framework of machine-tool axis drive control, the previous approach considers the machine and the CNC functionalities under an open architecture formalism, with multiple independent modules, and processors dedicated to specific tasks, as illustrated in Figure 11.1. This modularity imposes a permanent exchange of information between the various tasks, via the human-machine interface (HMI), while the concept of open architecture in addition allows the simple and fast integration of new independent modules. The modules developed (and more particularly those dedicated to the realtime part) must present a minimal material dependence. Consequently, each task has its own module:
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– data acquisition module (probe for current measurement or incremental encoders for position restitution); – power interface module, providing the voltage references required by the PWM part from the torque information; Graphic display (torque, current, velocity, position)
Automatic design of the tuning parameters
CNC screen User interface Supervision
Current measurement
Position encoder
Power interface
Trajectory generation
Frequency analysis Simulation module
Real-time module Axis control DSP velocity/position Real-time module DSP for torque and flux control
Figure 11.1. Open machine-tool architecture
– the trajectory generation module, elaborating from a pre-specified space trajectory the references for each axis, requiring an additional interpolator when the sampling periods of this module are different from those appearing in the real-time blocks; – the real-time module including for example a DSP independent of the higher levels, dedicated to torque and flux control, generally with very high sampling rates; – the real-time module, also including an independent DSP, dedicated to axis velocity and position control, with, for example, cascaded strategies developed in the following sections; – human-machine interface (HMI), ensuring the link between all the elementary modules and the user module, and coherence between the various real-time parts, etc.;
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– the module dedicated to the design of control law parameters, in order to help non-specialist users; – the simulation module, making it possible to test the impact of the axis drive controllers from the stability, robustness and time-domain behavior point of view (e.g. spatial errors, etc.); – the graphic interface module, which displays all variables of interest as a result of the simulation part, and signal measurements. Consequently, this structure offers the advantage of great flexibility and is in fact a prerequisite to the implementation of axis control structures more powerful than those generally provided by the designers of numerical control, and still mostly based on PID and feedforward actions.
11.2. Structure and implementation of control laws This section specifies the general theoretical framework allowing thereafter the implementation of advanced control laws within the numerical control of machinetools. The cascaded control structure is first examined, and then the formulation of the controllers in polynomial form is detailed.
11.2.1. Cascaded structure The block-diagram of this type of control architecture is given in Figure 11.2. The output of the main loop controller acts as the setpoint for the controller of the secondary loop. The two controllers are linked within a so-called cascaded scheme. The advantages of this two-loop control structure are the following [BOR 93], [RAC 96]: – The transient is faster. Indeed, the phase shift of the main open loop is reduced by the secondary feedback, which allows an increase in the speed of the controlled system, and attenuates the influence of the process non-linearities.
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Disturbances c
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Figure 11.2. Two-loop control structure or cascaded architecture
– The effect of disturbances occurring in Process 2 (Figure 11.2) is reduced as the secondary loop runs faster than the main loop. A step disturbance is at once taken into account by the secondary loop, and its effect on the main loop is thus significantly reduced, as illustrated in Figure 11.3. It must be noted that when the secondary loop becomes faster with respect to the primary loop, the effect of the disturbances on the main output becomes negligible. 1.4
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4
5
0
tr
of S. L. 1
2
5
Figure 11.3. Comparison of a single loop control scheme (left) and a cascaded structure (right) when disturbances occur in the inner loop
– The secondary loop allows the regulation of an auxiliary variable. Another way of understanding the cascaded control is to consider that the auxiliary variable s 2 (t ) has to follow the control signal u1 (t ) issued from the main controller. In particular, when an important transient occurs in the main loop, inducing saturation of the control signal u1 (t ) , the auxiliary variable then has to follow the saturated value of u1 (t ) . This last property enables not only control of s 2 (t ) , but also the imposition
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of a maximum value onto this signal. This results in the limitation of the auxiliary variable during important transients All these properties can be illustrated on the particular case of the control of motor velocity where there is in general a secondary loop dealing with current control (see Figure 11.4). c
+
H R1 (s )
-
H2
u1 +
u R2 (s )
-
Js f K c Ke Js f R Ls
I
Kc Js f
:
m
Figure 11.4. Current-velocity cascaded control of a DC motor
Consequently, the velocity cascaded control also enables control of the current. This is of most interest because, if the proportionality between the motor torque and the current is assumed, restraining the current through the secondary loop action induces a limitation of the torque, which is equivalent to work during transient with constant torques. All these advantages enlighten the numerous industrial use of the cascaded control strategy [DIN 77], [FLA 94], [GOD 07], [MAR 87]. To conclude, this type of control can be generalized to an unspecified number of loops using various auxiliary variables available within the process. The only condition is that the bandwidth of these variables must decrease when moving from the inner to the external loops.
11.2.2. Polynomial structure of controllers Numerical control of continuous systems can be achieved either by transposition of continuous controllers into discrete time structures (e.g. digital PID), or by means of specific discrete time methodologies (e.g. predictive control algorithms developed in section 11.3.2). However, whatever the procedure could be, the resulting architecture can be modeled under a single formalism [LAN 88], for which the equivalent “RST” controller is of polynomial form, according to the scheme given in Figure 11.5.
Position and Velocity Coordination: Control of Machine-Tool Servomotors
1
T ( q 1 )
w
S (q 1 )
+
DAC
Process
337
ADC
u
-
y R (q 1)
Equivalent polynomial controller
Figure 11.5. Polynomial RST structure of a numerical controller
Programming the resulting control law is consequently straightforward, requiring the implementation of a simple finite difference equation:
S ( q 1 )u (t ) T ( q 1 ) w(t ) R( q 1 ) y(t )
[11.1]
1
with q-1 the backward shift operator, or considering the signal samples:
s0 u(t) s1u(t 1) ... t0 w(t) t1w(t 1) ... r0 y(t) r1 y(t 1) ... In addition, this very general RST structure is a two-degrees of freedom architecture, which allows for the specification of the disturbance rejection dynamic 1 1 1 ( R (q ) and S (q ) ) independently from the tracking dynamic ( T (q ) ). This enables for example the specification of response times which could be different during trajectory tracking or disturbance rejection phases [DEL 07].
Let us consider as an example the classical PID controller. The continuous time transfer function of this controller is given by:
C (s)
U (s) H (s)
§ 1 K ¨¨1 Td © Ti s
· s ¸¸ ¹
[11.2]
with s being the Laplace variable. Discretization of this relation with a sampling rate Te gives the discrete time transfer function:
C ( q 1 )
1
U (q )
H (q 1 )
K
T T T T 1 e d (1 2 d ) q 1 d q 2 Ti Te Te Te (1 q 1 )
[11.3]
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Control Methods for Electrical Machines
This transfer function representation of the control law is in fact equivalent to difference equation [11.1] which further enables real-time implementation, with the following notations: R(q 1 )
ª T º T T T K «1 e d q 1 (1 2 d ) d q 2 » Te Te ¬ Ti Te ¼
S ( q 1 ) 1 q 1 T ( q 1 )
R( q 1 )
[11.4]
1 1 In this case, however, the R( q ) and T (q ) polynomials are equal, which cancels one degree of freedom of the general RST structure. To overcome this the
T (q 1 ) prefilter can be modified, choosing for example: T (q 1 )
P(q 1 ) R(q 1 )
[11.5]
where the P polynomial must satisfy additional constraints to fulfill steady state error cancellation. It is thus possible with this structure to consider the derivative action of the PID controller acting only on the measured output: T (q 1 )
K (1
Te q 1 ) Ti
[11.6]
and even cancel proportional and derived actions acting on the error by choosing: T (q 1 )
K
Te Ti
[11.7]
11.2.3. Conclusion
This section has detailed the basic framework inside which efficient motor drives control laws must be implemented. It is in fact a combination of modular and flexible design structures, and cascaded control strategies including RST polynomial controllers. Starting from this global overview, several scenarios, examined below, can be easily added to improve control performances. Even if the theoretical developments presented during these case studies are oriented towards specific applications, they are in fact very general and can be applied to all kinds of motors.
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339
11.3. Application to machines-tools axis drive control
Cascaded control structures with current, velocity and position loops are surely for the time being the most industrially used structure to numerically control machines-tools, including either DC motors, or synchronous or asynchronous motors (brushless motors, field-oriented control).
11.3.1. Classic control scheme
Control structures currently implemented in the CNC machines-tools are essentially based on the scheme in Figure 11.6. K aa (1 z 1 ) 2 K av (1 z 1 ) *
4
+
Hp Kp
+
+ -
+
Hv
+ R1 ( z )
4
u
Ha R2 ( z )
Motor drive
-
-
: I
Figure 11.6. Structure of machine-tool axis control
This scheme shows that servo amplifiers now on the market include a cascaded current/velocity/position control structure and supplementary velocity and acceleration feedforward actions. Controllers of the current and velocity loops are typically PI(D)-type controllers. The controller of the position loop is very often only a simple gain, which requires the addition of these feedforward actions to compensate for acceleration and velocity errors during acceleration and deceleration phases.
11.3.2. Cascaded velocity-position predictive control of synchronous motors
This first case study focuses on the application of predictive control to cascaded velocity-position control of synchronous axis motors for machines-tools. Based on the previous framework, which includes open architecture features, cascaded control
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Control Methods for Electrical Machines
and polynomial controllers, the objective is to illustrate the advantages of an advanced control law, in particular in terms of trajectory tracking. A first part will briefly present the main principles of generalized predictive control (GPC), while a second part will propose the results completed with predictive control laws compared to those obtained with classic controllers. The reader is encouraged to refer to [BIT 90] and [WER 87] for more details about predictive control. 11.3.2.1. Predictive control methodology Predictive control is based on rather old and intuitive ideas [RIC 93], but the interest in this advanced control technique really started in the middle of the 1980s. This increase in interest can be linked to: – generalized predictive control (GPC), D.W. Clarke, 1985; – predictive functional control (PFC), J. Richalet, 1987. The philosophy of predictive control is based on four main ideas, which are in fact similar to all existing methods: elaboration of an anticipative effect through an explicit use of the knowledge of the trajectory to be followed in the future, the definition of a numerical model for prediction, minimization of a quadratic cost function over a finite horizon, and the receding horizon principle. The following developments will consider these four fundamental concepts in the particular case of generalized predictive control [CLA 87a], [CLA 87b], [CLARKE 88] for monovariable systems, looking at the related theoretical and mathematical formalism. 11.3.2.1.1. Definition of the numerical model Several kinds of structures can be adopted; among them is the input/output polynomial form, which will be considered below. In this case, the model is classically devised under the CARIMA (controlled autoregressive integrated moving average) form: A( q 1 ) y (t )
B( q 1 )u (t 1) [ (t )
'( q 1 )
[11.8]
1 1 where '(q ) 1 q , u (t ) and y(t) are respectively the control signal input and 1 system output, [ (t ) is a white noise with zero mean value, q is the backward shift 1 1 operator and A(q ) and B(q ) are polynomials defined by:
Position and Velocity Coordination: Control of Machine-Tool Servomotors
° A(q 1 ) 1 a q 1 "" a q na 1 na ® nb 1 1 B ( q ) b b q " " b °¯ 0 1 nb q
341
[11.9]
This model, also called the “incremental model”, introduces an integral action enabling the cancellation of steady state errors in response to step inputs or disturbances. 11.3.2.1.2. Optimal predictor The predicted output y (t j / t ) can be split into the free response and the forced response [BOU 96], again using the polynomial form to obtain at the end a RST polynomial controller, leading to the following relation: y (t j / t )
F (q 1 ) y (t ) H j (q 1 )'u (t 1) j
free response
G j (q 1 )'u (t j 1) J j (q 1 )[ (t j )
forced response
[11.10]
All unknown polynomials in [11.10] are solutions of the Diophantine equations. Finally, the optimal predictor is defined considering that the best prediction of the noise signal in the future is its mean value (assumed to be equal to zero), leading to: yˆ (t j / t )
F j ( q 1 ) y(t ) H j ( q 1 ) 'u (t 1) G j ( q 1 ) 'u (t j 1)
[11.11]
11.3.2.1.3. Quadratic cost function definition The optimal control law results from the minimization of a quadratic cost function, which includes terms related to future predicted errors and control signal increments:
J
N2
Nu
j N1
j 1
¦ >yˆ (t j) w(t j)@ 2 O ¦ 'u 2 (t j 1)
with: 'u (t j ) { 0 for
j t Nu
For this cost function, the user must select four tuning parameters:
[11.12]
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Control Methods for Electrical Machines
– N1 : minimum prediction horizon; – N 2 : maximum prediction horizon; – N u : control horizon; – O: control weighting factor. 11.3.2.1.4. Design of the equivalent polynomial RST controller Minimization of the previous cost function leads to a polynomial RST controller, in a form similar to the one given in Figure 11.5, so that the resulting control law is formulated using difference equation [11.1]. In this case, however, the T (q ) polynomial is non-causal (positive power of q), which will generate the anticipative aspect of the predictive law. The three polynomials R, S , T are thus calculated offline and uniquely defined since the four tuning parameters are selected. Consequently, the real-time loop runs very fast; this kind of control law enables the selection of small sampling rates and is well-adapted to the control of fast electromechanical systems (machines-tools, high speed machining, etc.) [DUM 98]. 11.3.2.1.5. Choice of the tuning parameters The definition of quadratic cost function [11.12] has shown that the user must select four tuning parameters. This choice may be somehow complicated for nonspecialist users, since no empirical relation exists which could have linked these parameters to indicators usually defined in the control theory, e.g. stability margins and bandwidth. Based on the study of an important number of classical monovariable systems, some rules which consider usual stability and robustness criteria can be emphasized [BOU 95] and are summarized below: – N1: minimum output prediction horizon. The product N1 Te ( Te sampling period) is chosen to be equal to the system delay. – N2: maximum output prediction horizon. The product N2 Te is bounded by the time response value. Increasing N2 improves the stability of the controlled system but decreases its global response time.
Position and Velocity Coordination: Control of Machine-Tool Servomotors
343
– Nu: control horizon. Nu equal to 1 simplifies the calculation part and does not penalize the stability margins (although increasing this value tends to damage the phase margin). – O: control weighting factor. This parameter is related to the static gain of the system through the following empirical relation:
Oopt
tr(G TG)
[11.13]
where the G matrix [BOU 96] built from the coefficients of the model step response is part of the equations involved in the minimization strategy. The parameters choice is most often restricted to a bidirectional search (N2 and O). 11.3.2.2. Implementation As stated in section 11.1 and following the open architecture principle, the main task of the control module is to generate the torque reference that will be used by the close control part of the machine (the current loops have not been changed and are realized in an analog way). Having in mind the cascaded velocity/position scheme, only the position information is available from the incremental encoders, and the velocity information is then calculated from the position information. The predictive controllers are first designed using the human-machine interface, with all available tools for simulation and stability and performance analysis. The resulting polynomials are then conveyed to the axis control DSP module where the control laws are implemented by means of difference equations (RST structure). The axis motor of the benchmark has the following features: – The electrical part of the current loops is equivalent to a first order transfer function (relation between the torque and the torque setpoint) with a time constant of 0.33 ms. – The mechanical part (i.e. transfer function between the torque and the velocity) takes into account the motor inertia, equal to 0.04 kg/m2 and the viscous friction coefficient equal to 0.04 Nm/s 2 . With a sampling rate of Te = 10 ms, the discrete time transfer function, identified between the torque setpoint * and the velocity : , is given by: :(t )
1.1139q 1 1.259q 2 0.0026q 3
* (t )
1 0.99q 1
[11.14]
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Control Methods for Electrical Machines
Using this identified function, the predictive controllers of the two loops are designed considering the following tuning parameters: – velocity loop: N12
1 ; N 22
– position loop: N11
1; N 21
16 ; N u 2
17 ; O2 29 ; O1
28 ; N u1
6, 000
12 .
The setpoint delivered by the trajectory generation module with a 10 ms sampling rate is trapezoidal, corresponding to a four rotation variation of the motor shaft, with a velocity of ±250 rpm. The results obtained according to this setup are given in Figures 11.7 to 11.9. 4.5 Position [rotation]
4 3.5 3 2.5 2 1.5 1 0.5 0 - 0.5
T ime [ s] 0
1
2
3
4
5
6
Figure 11.7. Angular position of the motor and position setpoint
Position and Velocity Coordination: Control of Machine-Tool Servomotors
345
250 Motor velocity and ve loci ty setpoint [rpm]
200 150 100 50 0 -50 -100 -150 -200 -250
Time [s] 0
1
2
3
4
5
6
Figure 11.8. Angular velocity of the motor and velocity setpoint 6 Torque setpoint [N m] 4
2
0
-2
-4 Time [s] -6
0
1
2
3
4
5
6
Figure 11.9. Motor torque setpoint
For comparison purposes, a classical cascaded control structure has also been implemented, following the scheme given in Figure 11.10. It includes a proportional position controller and an IP (integral proportional) velocity controller.
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Control Methods for Electrical Machines
Position setpoint
+
-
Velocity setpoint K pp
+
-
Torque reference
Kiv 1q
1
+
-
K pv Synchronous motor
Velocity Position
Figure 11.10. P/IP cascaded control structure
With the tuning given below, and for the same setpoint, the resulting performances are shown in Figures 11.11 to 11.13: – velocity loop: Kpv = 0,16 ; Kiv = 0.075 rad/s; – position loop: Kpp = 160. 4.5 Posi tion [rotation] 4
3.5 3 2.5 2
1.5 1 0.5 0
-0.5
Time [s] 0
1
2
3
4
5
6
Figure 11.11. Angular position of the motor and position setpoint
Position and Velocity Coordination: Control of Machine-Tool Servomotors
347
250 200
Motor velocity and velocity setpoint [rpm]
150 100 50 0 -5 0 -1 0 0 -1 5 0 -2 0 0 -2 5 0
Time [s] 0
1
2
3
4
5
6
Figure 11.12. Angular velocity of the motor and velocity setpoint
The comparison of these two control structures leads to the following conclusions: – Compared to the P/IP structure, the predictive control strategy provides better results in terms of cancellation of position and velocity errors, response time and overshoot. In fact, the introduction of an integral action within the position controller of the P/IP structure should have cancelled the velocity error, which can be seen in Figure 11.11. In this case, however, an important loss of stability and important overshoots would have resulted, which is not acceptable in the machinetool field. – The interest of the open architecture structure is clearly emphasized, since the modularity considered during the design stage of the machine enables the user to include many different control strategies based on the RST formalism (from classical approaches to more sophisticated approaches).
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Control Methods for Electrical Machines 5 Torque setpoi nt [N m]
4 3 2 1 0 -1 -2 -3 -4
Time [s] 0
1
2
3
4
5
6
Figure 11.13. Motor torque setpoint
This first study has illustrated an original structure which aims at designing machine-tool axis control, for which each task is specified in a modular and independent way. This open architecture provides a better portability, in such a way that new modules can be easily added by the user. Some modules helping nonspecialist users can also be included, in order to simplify the implementation of advanced control laws, e.g. related to the design and tuning of the controller.
11.3.3. Multivariable flux-position predictive control of asynchronous motors
The main interest of the induction machine is related to its wide industrial field of applications, as well as to its reliability, robustness and cost properties. Nevertheless, three main difficulties arise during the control design phase, due to a non-linear dynamic model of the system, unavailable rotor flux variables, rotor resistance fluctuating with temperature variations and inducing important changes in the system dynamic. Several classic control strategies are well-known; amongst them, the field oriented control [BOS 86] is surely the most famous. However, some improvements can still be considered in order to reduce the influence of electrical and mechanical parameter variations and to increase performance during transient phases. This second case study illustrates the abilities of advanced control structures in the framework of flux-position cascaded control.
Position and Velocity Coordination: Control of Machine-Tool Servomotors
349
11.3.3.1. General philosophy This section examines the general control structure that will be applied to the trajectory tracking problem. This structure is in fact the combination of an open loop action which guarantees a “nominal” tracking of a specified reference trajectory, and a feedback term which ensures stability around this reference trajectory. Let us consider a non-linear system for which some variables must follow a planned trajectory. The “flatness” concept for a non-linear system [FLI 95] is a useful tool to solve this problem by imposing a spatial dynamic to the system through a so-called “flat” control signal obtained from the inverse of the system. The application of the “flatness” concept requires a two-step procedure: – First, plan the trajectory to be followed. This means finding a trajectory which satisfies all the constraints, as well as the adequate control signal that may generate this trajectory (open loop design). – Then, stabilize the system around the desired trajectory (feedback control), since the previous open loop control action is not sufficient to deal with disturbances occurring in the physical system. Figure 11.14 below summarizes the two aspects of the developed strategy: first the elaboration of an open loop reference control signal u r from a prespecified output reference trajectory y r , then the design of an additional feedback term u u r based on the difference y r y between the output and the reference trajectory. This second term only acts on the system when the physical output withdraws from the planned behavior y r . yr
yr
+
System inversion
Open loop reference control signal
ur +
Hy
Stabilizing controller
-
+ ubf
u
Non-linear system
Closed-loop additional control action
Figure 11.14. Combination of an open loop reference control signal and a closed loop stabilizing control action
y
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Control Methods for Electrical Machines
This control strategy, which may appear sophisticated and complex, is in fact natural, since it refers to the classical feedforward control scheme, well-known in the machines-tools field (see section 11.3.1). From this point of view, this original non-linear control structure is quite similar to the system currently implemented within most of the CNC machines-tools, and is thus quite familiar in this application field. The realization of the first step is possible only if the system is flat. A system is considered to be differentially flat if a vector y can be found, a so-called flat output, including m fictitious outputs ( y1 , " , y m ) such that the state x and the control signal u can be formulated as functions of y and a finite number of its derivatives, and the flat output y can be formulated as a function of the state x , the control signal u and a finite number of its derivatives: ...) u # ( y, y , y...) and y h(x, u, u , u
More details about the flat systems theory may be obtained from [FLI 95], [BOU 06]. Based on this, the open loop reference control signal u r is elaborated from the pre-specified planed trajectory yr, which must have good derivative properties, using the relation: ur # ( yr , yr , yr ,...)
The achievement of the second step can be obtained via classic or more advanced control laws, without any restriction. In continuity of the application presented in section 11.3.2, only a predictive control strategy will be designed. However, the predictive structure proposed in section 11.3.2.2 must be modified to explicitly take into account, in the algorithm, the input/output errors leading to a socalled reference model strategy [IRV 86]. Therefore, the quadratic cost function will now be:
J
N2
Nu
j N1
j 1
¦ >yˆ (t j) y r (t j)@ 2 O ¦ >'u(t j 1) 'u r (t j 1)@2
with: 'u (t j ) { 'u r (t j ) for
j t Nu
[11.15]
Position and Velocity Coordination: Control of Machine-Tool Servomotors
351
A minimization based on the output and control signal difference with respect to a behavior imposed by a reference model clearly appears. The following sections consider the design of this structure in the particular case of the flux-position multivariable control of the induction machine. 11.3.3.2. Application to the induction machine – design of the control law Following the previous general principles, and starting from planned flux and position trajectories, an open loop reference control signal is first numerically calculated based on the flatness theory. Then, the closed loop stabilization is guaranteed by means of a polynomial cascaded flux-position predictive structure. To achieve this, the procedure will consider the general equations issued from the twophase machine representation. 11.3.3.2.1. Model of the induction machine The three phase-two phase Park’s transformation provides the model of the induction machine in the stator fixed D E reference frame (see for example [LEO 85], [BOS 86]). Assuming that the magnetic circuit remains linear, the equivalent representation of the two phase machine is given by the following fifth-order model: x
f ( x) gu
x
[i s D , i s E , I r D , I r E , Z ]7
u
[u sD , u sE ]7
with:
[11.16]
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Control Methods for Electrical Machines
f ( x)
K ª º «-J i sD T I rD pZKI rE » r « » « » K «-J i sE pZKI rD I rE » T r « » « M » 1 i sD I rD pZI rE » « Tr « Tr » « M » 1 i sE pZI rD I rE » « Tr « Tr » « M » I rD i sE I rE i sD » «p « J Lr » « » 1 « » TL fZ J ¬ ¼
g
K
ª 1 «V L « « «0 ¬
0
0
0
0
0
s
1
V L
M ; V VL s Lr
s
1
M2 ; J L s Lr
º 0 » » » 0 » ¼
7
Rs R M2 r VL s VL s L2r
where isD , isE are the stator current components, IrD , Ir E the rotor fluxes, usD , us E the stator voltages, Ls , Lr the stator and rotor inductances, Rs , Rr the stator and rotor resistances, Z the speed, J the inertia of the machine, M the mutual inductance, f the friction coefficient, p the pole pair number, TL the load torque and finally Tr Lr / Rr the rotor time constant. 11.3.3.2.2. Generation of the reference control signal As previously mentioned, the elaboration of the planned reference trajectories can be successfully achieved only if the system is flat. It has been proved by [MAR 96] and [CHE 96] that y (T , D ) is a flat output, with T the rotor position and D the rotor flux angle. However, it seems much better to consider the usual outputs of the induction machine, i.e. the rotor position T and the rotor flux norm U , which also enables a better control of the machine at high velocity regimes during which the flux of the machine must be decreased. In this case, similar to the flatness approach, it is possible to elaborate the sinusoidal control voltages usD and usE from
Position and Velocity Coordination: Control of Machine-Tool Servomotors
353
the two previous outputs and a finite number of their derivatives and first integrals [MAA 00a]. In that sense, T and U are not completely flat outputs, but the general methodology remains. Compared with the scheme given in Figure 11.14, the elaboration of the reference control signals is now more complex, since these signals are sinusoidal. It has been shown [MAA 00b] that the system inversion leads to the scheme of Figure 11.15, with a linear part producing the continuous signals u1ref and u2ref , related to the reference torque, and a non-linear part finally producing the reference control signals of the stator voltage, and related to Park’s transformation. T ref
u1ref fL
Uref
us D ref f NL
u2 ref
P.W.M.
Voltage Inverter
I. M.
u s E ref
Figure 11.15. Generation of the reference control signals
11.3.3.2.3. Design of the closed loop control action This control term includes the action elaborated in the previous section and brings an additional contribution due to the feedback action of the polynomial predictive controller, as stated in Figure 11.16. Since both the rotor flux norm and the rotor position must be controlled, this figure includes two polynomial RST controllers dedicated to the stabilization of these two outputs around their respective specified trajectories. Note that the rotor flux norm cannot be measured and is reconstructed by means of a rotor flux observer using measurements of the rotor position and two components of the stator current. Some more sophisticated strategies could have been used, e.g. based on classic Kalman filters, but this tends to increase the real-time calculation load.
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Control Methods for Electrical Machines T ref
Open loop reference control signal
u1refBO fL
Uref
u2 refBO +
T ref R1 (q )
+
u sD
+
1
1
1
-
S1 ( q )
+
f NL
P.W.M. us E
Position controller
+ R1 (q 1 )
U ref
R2 (q 1 )
Voltage inverter
T
I. M.
Incremental encoder
1 +
1
-
Flux controller
S2 (q ) R2 (q 1 )
Uˆ
Rotor flux estimation
T is D is E
Figure 11.16. Global control strategy
11.3.3.3. Experimental results The control law is implemented on an experimental benchmark including an induction machine with features: 1.1 kW, 220/380 volts, 50 Hz, 1500 rpm, related to the LGEP (Electrical Engineering Laboratory of Paris) benchmark, with: Rr 3.6 : f 0.005 Nm s Lr 0.47 H Tnom 7 Nm
Rs 8.0 : p 2 Ls 0.47 H Znom 73.3038 rad/s
J 0.015 kg m 2 Ir DE 1.1 Wb M
0.452 +
[11.17]
The system inertia can take two different values 7 10-3 and 15 10-3 kg.m2. Two motor-load connecting systems are available: a rigid one and a semi-rigid one. A variable mechanical unbalance can also be included. Mechanical position measurement is performed by an incremental encoder with an 8,000 increment/rotation resolution. The close control part of the machine including the inverter and the PWM (pulse width modulation) strategy is managed by a specific DSP (digital signal processing) card DSP32C RISC-microprocessor from AT&T. This card enables us to choose a fast 153.2 Ps sampling period for the current loop. For positioning applications, the predictive controllers are designed with a 1 ms sampling rate, i.e. 7 cycles of the current loop, in order to avoid oversampling problems within the polynomial predictive structure.
Position and Velocity Coordination: Control of Machine-Tool Servomotors
355
The selected position setpoint comes from the “fast transitic” specifications, where positioning aspects are crucial and which are also fundamental for machinetool axis control. Two different cycles are considered: “slow” and “fast” cycles. The “slow” cycle takes into account motor velocity around the nominal value, which corresponds to the voltage used to power the inverter. During this cycle, the magnetic features of the machine remain close to their nominal values. This kind of cycle has been chosen because it enables the study of the actuator behavior at very low or zero speeds, with change of the load torque, at nominal speeds and during non-linear position trajectory tracking (here parabolic trajectory). The “fast” cycle enables us to test the actuator behavior in the area where the flux must be decreased (velocity is twice the nominal velocity). This cycle runs with no load torque. Running at low speeds (t [2.5 s; 4.5 s] for the “slow” cycle, t [1.3 s; 2.3 s] for the “fast” cycle) corresponds to a position reference moving from 20S to 21S at a velocity of S/2 rad/s. For both cycles, the position reference is filtered by a second order filter with a damping coefficient equal to 1, and a natural frequency of 50 rad/s for the “slow” cycle and 25 rad/s for the “fast” cycle. Figures 11.17 to 11.19 illustrate the trajectories to be followed. 70
Position [rad] 60
filtered
50 40 30 20
Not filtered 10
Time [s] 0 70
0
2
4
6
8
Position [rad]
60
filtered
50 40 30 20
Not filtered 10 Time [s] 0
0
1
2
3
4
5
6
Figure 11.17. Position trajectories (top “slow” cycle, bottom “fast” cycle)
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Control Methods for Electrical Machines 50
Velocity [rad/s]
40 30 20 10 0 -10 -20 -30 -40
Time [s]
-50
0
2
4
6
8
100 Velocity [rad/s] 80
60 40 20 0 -20 -40
-60 Time [s]
-80
0
1
2
3
4
5
6
Figure 11.18. Velocity trajectories (top “slow” cycle, bottom “fast” cycle) 5
Torque [Nm]
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
Time [s] 0
2
4
6
8
Flux [Wb] 1.2
1 0.8 0.6 0.4
0.2 0
Time [s]
0
1
2
3
4
5
6
Figure 11.19. Top: load torque (“slow” cycle), bottom: flux trajectory (“fast” cycle)
The predictive controllers were designed with the following tuning parameters, satisfying stability and robustness specifications: – Flux loop: N12
1; N 22
20; N u 2
1; O2
0, 0272.
Position and Velocity Coordination: Control of Machine-Tool Servomotors
– Position loop: N11
1; N 21
45; N u1
1; O1
357
0,3918.
Looking for the moment only at the impact of the open loop strategy, which elaborates the reference control signals from the planned position and flux references, Figure 11.20 gives the control signal applied to the machine and the resulting position error for the position trajectory of Figure 11.17 (“fast” cycle) at constant flux. 150 Control signal [V] 100
50
0
-50
-100 Time [s] -150
0
1
2
3
4
5
6
7
3
4
5
6
7
8
0.03 Position error [rad] 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 Time [s]
-0.05
0
1
2
8
Figure 11.20. Reference control signal and position error, “fast” cycle
It can be noticed that this approach solves the trajectory tracking problem, even if the tracking is not completely satisfactory (small static position error due to model mismatch). This motivates the introduction of the closed loop term, with the objective of guaranteeing a good disturbance rejection behavior. Applying the complete strategy (open loop reference trajectory and predictive feedback control) gives the results displayed in Figures 11.21 to 11.22 for the “slow” cycle, and Figures 11.23 and 11.24 for the “fast” cycle.
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Position error [rad]
0.02
0
-0.02
-0.04
-0.06
-0.08
Time [s] 0
2
4
6
8
5
Velocity error [rad/s]
4 3 2 1 0 -1 -2 -3 -4 -5
Time [s] 0
2
4
6
8
Figure 11.21. Position and velocity error, “slow” cycle Flux error [Wb]
0.01
0.005
0
-0.005
-0.01
Time [s] 0
2
4
6
8
300
Control signal [V] 200 100 0 -100 -200 -300
Time [s] 0
2
4
6
8
Figure 11.22. Flux error and control signal applied to the motor, “slow” cycle
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0.03 Position error [rad] 0.02 0.01 0 -0.01 -0.02 -0.03
Time [s] 1
0
2
3
6
5
4
8 Velocity error [rad/s]
6
4 2 0
-2 -4 -6 Time [s]
-8
0
1
2
3
4
5
6
Figure 11.23. Position and velocity error, “fast” cycle
For comparison purposes, Figures 11.25 and 11.26 below illustrate the results obtained for the same scenario with the “fast” cycle, but now the induction machine is controlled using a classic field oriented control strategy [LEO 85], [BOS 86] (torque and flux loops, the objective being to decouple flux and torque signals), and including a cascaded PID controller for the position loop.
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Control Methods for Electrical Machines 0.04 Flux error [Wb] 0.03 0.02 0.01 0
-0.01 -0.02 -0.03 -0.04 Time [s]
-0.05
0
1
2
3
4
5
2
3
4
5
6
250 Control signal [V]
200 150 100
50 0 -50 -100 -150 -200 Time [s]
-250
0
1
6
Figure 11.24. Flux error and control signal applied to the motor, “fast” cycle 0.15 Position error [rad] 0.1 0.05 0 -0.05 -0.1 Time [s] 0
1
2
3
4
5
6
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0.08 Flux error [Wb]
0.06 0.04 0.02 0 -0.02
-0.04 -0.06 Time [s]
-0.08
0
1
2
3
4
5
2
3
4
5
6
250 Control signal [V] 200
150 100 50 0
-50 -100
-150 -200 -250
Time [s]
0
1
6
Figure 11.26. Flux error and control signal applied to the motor, “fast” cycle
Similar experiments have been conducted with a structure combining the open loop reference control signal and PI/PID controllers for the feedback action instead of predictive laws. The results are not displayed here due to space limitation. However, a comparison of these three experiments – PID with open loop reference control, GPC with open loop reference control and field oriented control – enables us to formulate the following comments and remarks: – Compared with the reference control signal applied to the machine without feedback action, the addition of PID or GPC terms cancels steady state errors that take place on the flux and position when working in open loop. – The field oriented control structure guarantees a perfect rotor position trajectory tracking steady state error cancellation. The error transient is more important, with a maximum value reached during the parabolic decreasing phase; it is the same with respect to the rotor flux magnitude, with a significant error dynamic occurring during load variations (“slow” cycle). – PID/GPC strategies combined with the open loop reference control considerably improve the previous results given by more conventional control structures.
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– The contribution of the predictive strategy compared to the PID action is mostly significant for the “fast” cycle, where for example the position error is divided by a factor of 2 during transient phases of the reference trajectories (0.04 rad maximum during transient). Generally speaking, the dynamic errors remain globally the same, but the magnitude during transient decreases. – Position and flux errors at high speeds (flux decrease during the “fast” cycle) become slightly more important compared with those observed during the “slow” cycle; however, the tracking performances remain very good. – The behavior of the machine during flux and position steady state and transients proves the efficiency of the controllers fulfilling the constraints of the machine (e.g. voltage applied to the machine). – Finally, all PID/GPC experiments also including the open loop reference control signal have been realized with at least a 10% variation in the resistance and inductance values compared with the nominal values; the good results obtained in this mismatched case point out the robustness of these control laws with respect to these variations, but also to load torque variations during the “slow” cycle. This control structure provides a simple and efficient solution to problems related to trajectory tracking, which are crucial in the field of machine-tool axis control. All experimental results clearly prove the significant contribution of advanced control laws, in particular the original combination of an open loop reference control action and an additional closed loop term. This combination enables the decoupling of the trajectory tracking and the disturbance rejection problems. It must also be noticed that this strategy is fully compatible with the open architecture design concept, since the closed loop control law is built following the general polynomial RST structure. Consequently, either a PID controller or a GPC law can be implemented based on a plug and play approach.
11.4. Conclusions
Currently, the cascaded control strategy is surely at the present time the most industrially applied structure, and represents the most important part of numerical control structures for machines-tools. The examples presented in the previous sections highlighted its ability to control electromechanical systems, either with DC motors or with synchronous or asynchronous machines. However, the CNC interfaces were, up until now, extremely closed, thus not allowing simple and fast
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implementation of new control structures that could be more efficient than classic controllers. This is the reason why designing machines-tools and CNCs according to an open architecture principle is crucial, and represents a decisive move to enable future implementation of advanced control laws. Among all potential advanced control structures, and focusing on the industrial applications dedicated to robots or machine-tool motor drive control for which the setpoint is fully planned (given by a trajectory generation module), predictive control seems to be well adapted to this domain. Moreover, the predictive control structures developed in the previous parts proved their design and implementation simplicity since, whatever the version (one loop or cascaded), the real-time calculation of the control law is always performed using difference equations issued from the polynomial RST controller formalism. This fundamental feature leads to very fast real-time loops, because the calculations of the design phase are performed offline as soon as the tuning parameters are selected. This control structure is thus an efficient solution for applications with severe specifications in terms of fast sampling rates (e.g. very high speed machining for machine-tool applications). Moreover, coupling these predictive strategies with open loop reference control signals leads to strategies which are in fact very similar to those currently implemented in CNCs, combining feedback and feedforward actions. The advantage of the reference control trajectory elaborated following the flatness theory is that the physical features of the system are taken into account, so that the resulting control signal is feasible and tolerable by the system. This guarantees that the planned trajectory will be successfully reached while fulfilling the system constraints. Finally, one important point, which gives a significant idea of the method complexity, is related to the number of tuning parameters. Advanced control techniques, in particular predictive control, in general require a more important number of tuning parameters, and their choice could be somehow difficult, in particular for non-specialist users. This is the reason why it is absolutely necessary to develop, in parallel with the design and implementation of the control law, an autotuning strategy which must be as transparent as possible to potential users; this is the purpose of the autotuning module of Figure 11.1.
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11.5. Bibliography [ALT 94] ALTINTAS, Y. and MUNASINGHE, W.K., “A hierarchical open-architecture CNC system for machine tools”, Annals of the CIRP, vol. 43, no. 1, pp.349-354, 1994. [BIT 90] BITMEAD, R.R., GEVERS M. and WERTZ, V., Adaptive Optimal Control. The Thinking Man’s GPC, Prentice Hall, NJ, USA, 1990. [BOR 93] BORNE, P., DAUPHIN-TANGUY, G., RICHARD, J.-P., ROTELLA, F. and ZAMBET-TAKIS, I., Analyse et régulation des processus industriels. Tome 1. Régulation continue, Tome 2. Régulation numérique, Éditions Technip, Paris, 1993. [BOS 86] BOSE, B.K., Power Electronics and AC Drives, Prentice Hall, Englewood Cliffs, New Jersey, 1986. [BOU 95] BOUCHER, P. and DUMUR, D., “Predictive motion control”, Journal of Systems Engineering. Special Issue on Motion Control Systems, vol. 5, pp. 148-162, SpringerVerlag, London, 1995. [BOU 96] BOUCHER, P. and DUMUR, D., La Commande Prédictive, Collection Méthodes et Pratiques de l’Ingénieur, Editions Technip, Paris, 1996. [BOU 06] BOUCHER, P. and DUMUR, D., La Commande Prédictive: Avancées et perspectives, H. Abou-Kandil (Ed.), Hermes, 2006. CHE 96] CHELOUAH, A., DELALEAU, E., MARTIN, P. and ROUCHON, P., “Differential flatness and control of induction motors”, Proceedings CESA’96, vol. 1, pp. 80-83, 1996. [CLA 87a] CLARKE, D.W., MOHTADI, C. and TUFFS, P.S., Generalized predictive control, part I “the basic algorithm”, part II “extensions and interpretation”, Automatica, vol. 23-2, pp. 137-160, March, 1987. [CLA 87b] CLARKE, D.W., MOHTADI, C. and TUFFS, P.S., “Properties of generalized predictive control”, Proceedings 10th World Congress IFAC’87, vol. 9, pp. 63-74, Munich, July, 1987. [CLA 88] CLARKE, D.W., “Application of generalized predictive control to industrial processes”, IEEE Control Systems Magazine, pp. 49-55, April, 1988. [DEL 07] DE LARMINAT, Ph., Analysis and Control of Linear Systems, ISTE Ltd., 2007. [DIN 77] DINDELEUX, D., Technique de la régulation industrielle, Eyrolles, 1977. [DUM 98] DUMUR, D. and BOUCHER, P., “A review introduction to linear GPC and applications”, Journal A, vol. 39, no. 4, pp.21-35, December 1998. [FLA 94] FLAUS, J.M., La régulation industrielle, Hermes, 1994. [FLI 95] FLIESS, M., LÉVINE, J., MARTIN, P. and ROUCHON, P. “Flatness and defect of nonlinear systems: introduction theory and examples”, International Journal of Control, vol. 61, no. 6, pp. 1327-1361, 1995.
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[GOD 07] GODOY, E., Régulation industrielle, Dunod, 2007. [IRV 86] IRVING, E., FALINOWER, C.M. and FONTE, C., “Adaptive generalized predictive control with multiple reference model”, Proceedings 2nd ACASP/86, Lund, June, 1986. [KOR 96] KOREN, Y., PASEK, Z.J., GALIP ULSOY, A. and BENCHETRIT, U., “Real-time open control architectures and system performance”, Annals of the CIRP, vol. 45, no. 1, pp.377-380, 1996. [LAN 88] LANDAU, I.D., Identification et commande des systèmes, Hermes, 1988. [LEO 85] LEONHARD, W., Control of Electrical Drives, Springer-Verlag, 1985. [MAA 00a] MAAZIZ, M.K., MENDES, E., and BOUCHER, P., Nonlinear multivariable real-time control strategy of induction machines based on reference control and PI controllers, Proceedings of the 13th International Conference on Electrical Machines ICEM’2000, Espoo, August, 2000. [MAA 00b] MAAZIZ, M.K., MENDES, E. and BOUCHER, P., “A new real-time control strategy for induction motors based on a reference control and RST predictive structure”, 9th International Power Electronics and Motion Control Conference EPE-PEMC’2000, Bratislava, September, 2000. [MAR 87] MARET, L., Régulation automatique, Presses Polytechniques Romandes, 1987. [MAR 96] MARTIN, PH and ROUCHON, P., “Two remarks on induction motors”, Proceedings CESA’96, vol. 1, pp 76-79, Lille, July, 1996. [PRI 93] PRITSCHOW, G., DANIEL, C., JUNGHANS, G. and SPERLING, W., “Open system controllers – a challenge for the future of the machine tool industry”, Annals of the CIRP, vol. 42, no. 1, pp.449-452, 1993. [RAC 96] RACHID, A., Systèmes de régulation, Masson, 1996. [RIC 93] RICHALET, J., Pratique de la Commande Prédictive, Hermes, 1993. [WER 87] WERTZ, V., GOREZ, R. and ZHU, K.Y., “A new generalized predictive controller application to the control of process with uncertain dead-time”, Proceedings 26th Conference on Decision and Control, pp. 2168-2173, Los Angeles, December, 1987.
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List of Authors
Smaïl BACHIR LAII Ecole supérieure d’ingénieurs de Poitiers France
Didier DUMUR Supélec Gif-sur-Yvette France
Patrick BOUCHER Supélec Gif-sur-Yvette France
Sabine ELLES LAPP-CNRS-IN2P3 University of Savoie Annecy-le-Vieux France
Gérard CHAMPENOIS LAII Ecole supérieure d’ingénieurs de Poitiers France
Pascal FONTAINE Ecole nationale supérieure d’électricité et de mécanique Nancy France
Christian CUNAT LEMTA Ecole nationale supérieure d’électricité et de mécanique Nancy France
Jean-François GANGHOFFER LEMTA Ecole nationale supérieure d’électricité et de mécanique Nancy France
Michel DUFAUT CRAN Ecole nationale supérieure d’électricité et de mécanique Nancy France
Mohamed HABOUSSI LEMTA Ecole européenne d’ingénieurs en génie des matériaux Nancy France
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René HUSSON CRAN Ecole nationale supérieure d’électricité et de mécanique Nancy France
Jean-François SCHMITT LEMTA Ecole nationale supérieure d’électricité et de mécanique Nancy France
Frédéric KRATZ LVR University of Orleans France
Slim TNANI LAII Ecole supérieure d’ingénieurs de Poitiers France
Bruno MAURY Fédération nationale des centres de lutte contre le cancer Nancy France Rachid OUTBIB University of Marseille France Thierry POINOT LAII Ecole supérieure d’ingénieurs de Poitiers France Rachid RAHOUADJ LEMTA Ecole nationale supérieure d’électricité et de mécanique Nancy France
Jean-Claude TRIGEASSOU LAII Ecole supérieure d’ingénieurs de Poitiers France Michel ZASADZINSKI CRAN Henri Poincaré University Nancy France
Index
A a priori information, 218, 219, 221, 224, 225, 231, 232, 234, 235, 238, 241, absolute error, 97 acceleration, 284, 285, 286, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303 discontinuity, 286 limit, 293, 294 maximum acceleration, 293, 296 profile, 294 action, 131, 146, 157, 159, 162 of fuzzification, 157 adjustments, 95 affine, 175, 189 algorithmic approach, 132 analog models, 82 analytical dynamics, 46 anticipative effect, 340 aperiodic response, 96 approach points, 301, 303 approximation, 105, 106 articular variables, 274 articulated structure, 276 artifact, 319, 324 assigned trajectory, 280
asymptotic stability, 122, 187, 189, 199 asynchronous motors, 339, 348 attitude, 274 attractivity, 174, 177, 179, 187 condition, 177 domain, 205 axis, 274, 276, 279, 284, 287, 292, 296, 299, 300 control, 299 drive control, 332, 339
B bang-bang control, 153, 181, 294, 296 base, 273, 274, 277, 278, 279, 285 basic radiation beams, 321 Bayesian estimation, 209, beam conformation, 317 bearings, 64 Bellman’s principle, 124 belt, 66, 67, 68, 89, 90, 91 binding axis, 299, 300 Bode method, 105, 107 braking, 292, 294, 295, 296 times, 294 Broïda form, 97, 104 method, 104
370
Control Methods for Electrical Machines
C CARIMA, 340 carrier, 275, 291 cascaded control, 331, 334, 335, 336, 338, 339, 345, 346, 348, 362 casing, 64, 71 characteristic equation, 113 chattering, 151, 179, 180, 206 phenomena, 179, 180, 206 Clausius-Duhem inequality, 78, 79 Cohen-Coon method, 100, 103, 104 collimator, 307, 310, 311, 312, 313, 314, 315, 317, 318, 319, 320, 329, 330 control, 312, 317 multileaf, 307, 310, 311, 312, 318, 328, 329, 330 collision, 324 commutation, 172, 173, 174, 178, 179, 182, 183, 184, 189, 192, 193 surface, 183 compensator, 288, 289 condensation, 66, 68, 69, 89 configuration, 274, 276, 280, 285, 293, 294, 301 conformation, 311 conservation law, 72, 76, 75, 77 control, 169, 170, 171, 172, 173, 179, 180, 181, 182, 185, 186, 187, 188, 189, 190, 191, 193, 194, 198, 199, 201, 204, 205, 206, 207, 208 classic, 339, 348 device, 109, 117 horizon, 142 law, 108, 109, 111, 113, 114, 115, 117, 131, 171, 173, 179, 185, 186, 187, 188, 189, 190 linear, 108, 118, 122, 123, 125, 127, 130 LQG/H2, 127 multivariable, 351 signal, 93, 117
convergence, 208, 209, 210, 213, 215, 217, 219, 223, 225, 232 coordination, 271, 272, 280, 286, 287, 299, 301, 304 of motion, 286 Coriolis forces, 288, 291 torques, 293 vector, 284 covariance matrices, 128, 129 criterion, 209, 210, 212, 213, 215, 216, 217, 219, 220, 221, 222, 223, 224, 225, 229, 230, 231, 232, 236, 238 quadratic, 209, 210, 212, 213, 215, 216, 220, 222, 230 composite, 231, 238 critical gain, 101, 102 current loop, 288, 289 cut-off frequency, 105, 106
D damping factor, 105, 106, 107 DC motors, 287, 290, 312 degrees of freedom, 275, 282, 299 Denavit-Hartenberg, 278 derivative gain, 92 derived action, 92, 94 detectable, 111, 122, 128, 129 diagnosis, 207, 225,, 245, 246, 247, 249, 251, 253, 255, 257, 259, 261, 262, 263, 265, 267, 268, 269 diagram representation, 91 difference equation, 337, 338, 342, 343, 363 discontinuity, 286, 296 discontinuous control, 199, 206 discretization, 319, 321, 325, 326 dissipation, 65 dosimetry, 317
Index dynamic gain, 92 loads, 13 matrix control (DMC), 144, 148, 168 models, 277, 283 radiation, 319 sequences, 328
E effector, 277, 279, 281, 285, 286 eigenmodes of vibration, 51 eigenvalues, 113, 114, 115, 116, 118, 131 elasticity, 83 electronic control, 316 elementary beams, 318, 319, 321, 322, 323, 327, 328 end effector, 272, 273, 274, 277, 278, 279, 281, 285, 300, 303 energy, 69, 72, 76, 77, 78, 79, 83, 85, 89 energy profile, 316, 317, 318, 320, 323, 324, 325, 326 equation -error, 208, 209, 216, of motion, 15, 75 equivalent control, 170, 180, 181, 190, 191, 194, 196, 198 estimation error, 109, 113 estimator, 208, 209, 212, 214, 215, 216, 219, 225, 241 Euler angles, 275, 279, 280 Euler formulation, 285 exponentially stable, 111, 121, 125 exposure time, 318, 325 exteroceptive, 272
371
control, 130 gain, 116 feedforward, 334, 339, 350, 363 field of vectors, 173, 174, 182 Filippov method, 182 finished horizon, 118, 119, 124 finite element method, 64, 68, 72, 83 flat output, 350, 352 flatness, 349, 351, 352, 363, 364 friction, 1, 2, 4, 5, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 64, 82, 90 functional structure, 272 fundamental law, 75, 76 fuzzy controller, 155, 156, 157, 162 logic, 154, 155, 158, 160, 161, 168 sets, 157
G, H gantry crane, 276 gears, 64, 70, 89 generalized predictive control (GPC), 144, 148, 168, 340, 361, 362, 364 gentle response, 93 geometric model, 277, 278, 279, 280, 282, 301 geometric non-linearity, 64 global model, 66, 68 reduced complexity, 66 heuristic (or cognitive) approaches, 132 process, 155
I F fault detection, 225, 232, 238, 240, 241 localization, 232 of fault, 209, 234, 235, 237, 238, 240, 241 feedback, 171, 186, 188, 334, 349, 353, 357, 361, 363
identification, 207, 208, 209, 210, 216, 217, 219, 225, 226, 229, 230, 231, 233, 235, 239, 241 with a priori, 235 induced machine, 209, 225, 226, 228, 229, 230, 241 induction machine, 348, 351, 352, 354, 359
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Control Methods for Electrical Machines
inertia matrix, 284, 293 inference, 157, 159, 162, 163 infinite horizon, 120, 121, 125, 126 time, 120, 123 information, 212, 216, 218, 219, 220, 221, 222, 224, 225, 231, 232, 234, 235, 238, 239, 241 prior, 262, 263, 266, 267 initial error, 116 input vector, 108 integral action, 92 gain, 92, 106, 107 intensity beam, 318 matrix, 318, 319, 320, 321, 322, 323, 325, 326, 327, 328 profile, 320 internal model, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141 invariance condition, 190 inverse model, 277, 281, 301 irradiated, 308, 323 regions, 323 irreversibility, 78
J, K Jacobian, 281, 282 jerk, 286, 296, 298, 301 joint configurations, 286 efforts, 283 space, 279, 301 speeds, 280, 282 trajectory, 301, 302, 303 variables, 272, 274, 277, 281, 282, 287 Kalman filter, 127, 128, 130, 132 kinematic chain, 65 models, 277, 280
L Lagrange equations, 283 leaf axis, 325, 326 collision, 315, 316, 324, 325 movement, 312, 314, 316, 319, 320, 321, 324, 325, 326, 329 pairs, 311, 320, 321, 325, 327 position, 307, 312, 313, 314, 316, 318, 320, 323, 326, 327, 329 speed, 317, 325 leakage, 319 least squares, 207, 208, 209, 210, 216, 241 level matrix, 324 linear in parameters, 213, 216 Lipschitz condition, 173 Luenberger observer, 110, 111 Lyapunov function, 175, 176, 184, 185, 187, 188 method, 170, 184
M margin, 106 Marquardt algorithm, 220, 263 matrix, 69, 80, 278, 279, 280, 281, 282, 285, 290, 293 definite matrix, 119, 121, 124, 126, 127 gain, 111 of transfer, 115 mechanical drive, 63 structure, 272, 274, 276, 286 membership functions, 158, 159, 161, 163 milling machine table, 52, 53 model algorithmic control (MAC), 144 modeling, 207, 209, 212, 218, 222, 224, 225, 226, 227, 234, 235 modulated beam, 318 radiotherapy, 307, 316, 317
Index momentum kinetic, 72 motors, 272, 277, 287, 299, 304, 307, 312, 315, 316, 317
N natural frequency, 105, 107 neural network, 165, 166, 167, 168 control, 164 Newton-Euler formulation, 285 Newton formalism, 283 Newtonian dynamics, 38 noise, 93, 123, 132 white noise, 122, 123, 126, 127, 340 nominal trajectory, 285 non-damped discrete system, 49 non-linear in parameters, 16 non-linearities, 64, 83
O open architecture, 331, 332, 339, 343, 347, 348, 362, 363 operational variables, 279 optimal control, 91, 119, 120, 121, 122, 124, 125, 126, 127, 130, 131 optimization H2, 130 optimization, 323 oscillatory response, 97 output-error, 208, 209, 210, 214, 216, 218, 225, 229, 231, 239, 241 technique, 61, 267 output vector, 108 overshoot, 93, 96, 102, 106, 116
P passage matrix, 278 performance zone, 116, 117 perturbation, 185, 187, 188, 195, 196 phase margin, 105 PID controller, 337, 338, 359, 362 point-to-point, 292, 299 pole placement, 105, 107, 108, 110, 131 position control, 329
373
potentiometers, 314 prediction horizon, 142, 147 predictive control, 142, 143, 144, 149, 336, 339, 340, 347, 348, 350, 363, 364, 365 predictive functional control (PFC), 340 proportional band, 93 proportional control, 155 proportional gain, 92, 101 proprioceptive, 272 pulse width modulation (PWM), 333, 354 PVP, 75, 76
Q, R quadratic cost function, 340, 341, 342, 350 quadratic criterion, 209, 210, 212, 213, 215, 216, 220, 222, 230, radiation, 307, 313, 330 beam, 313, 317 constant radiation, 318 maximum radiation, 319 time, 323, 325 radiotherapy, 307, 309, 328 conformal radiotherapy, 307 treatment, 307 receding horizon, 340 reconstruction error, 127 recursive structure, 285 reference control, 349, 350, 351, 352, 353, 357, 361, 362, 363, 365 regularity function, 195 regularization, 182 Riccati equation, 119, 120, 121, 122, 125 rigid rotor, 42 robot configuration, 279 rotor fault model, 246, 259 RST controller, 342, 363
S self-driven synchronous motors, 287, 290 semi-definite matrix, 127
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sensitivity functions, 210, 213, 214, 215, 216, 217, 218, 221, 230, 238 shaft, 71, 87, 88, 89 singular configurations, 282 sliding condition, 177, 181 control, 149 domain, 175, 205 mode control, 170, 179, 189, 199 surface, 170, 174, 189, 197, 199, 201, 202, 205 space representation, 91 state equation, 283 spectral analysis, 253, 255 speed, 308, 316, 317, 319, 325, 329 profile, 294, 295, 297, 298, 299 stability, 111, 178, 184, 187, 189, 199, 207 state equation, 110, 122 error, 106, 107 feedback, 171, 184, 185 feedback control, 107, 109, 128, 130 observer, 110 -space model, 212, 228 variables, 272 vector, 108, 109, 110, 111, 113, 122 stator fault model, 253, 254, 255, 259 and rotor fault model, 246, 259 step response, 96, 100, 102, 103 strain, 76, 79, 80, 81, 82, 83, 84, 86 structures, 173 sub-structuration, 66 surface dynamics, 170 switching surface, 175, 180, 181, 189 synchronous motors, 339
T target, 310, 318 thermomechanics of continuous mediums, 76
through points, 286, 301, 303 time constant, 92 continuous, 118, 120, 122, 125, 129 discrete, 124, 125, 126, 127, 129 -invariant, 108, 111, 121, 126 rise, 96, 116 settling, 96 torsion, 67, 87, 88, 89 of a cylindrical shaft, 87 tracking error, 93, 109 trajectories, 273, 291, 300, 303, 305 transfer function, 65, 71 transmission power systems, 1 trapezoidal profile, 299 treatment beams, 307 sequence, 308 triangular profile, 296, 300 tumorous, 307, 308, 329
V variable structure, 170, 172, 173, 206, 207 variance matrices, 131 variance, 127, 129, 131, 212, 216, 220, 221, 222, 223, 224, 225, 231, 232, 234, 235, 236, 241, variational principles, 83, 86 vibratory behavior, 51 viscous behavior, 82 damper, 67
W, Z weighting matrices, 128, 130, 131 wrist, 273, 275, 276, 281 zero point, 276 robot, 276 Ziegler-Nichols form, 99 method, 100, 102, 10
