Robust Control Optimization with Metaheuristics E-Book

Philippe Feyel

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 331

Veröffentlichungsjahr: 2017

Beschreibung

In the automotive industry, a Control Engineer must design a unique control law that is then tested and validated on a single prototype with a level of reliability high enough to to meet a number of complex specifications on various systems. In order to do this, the Engineer uses an experimental iterative process (Trial and Error phase) which relies heavily on his or her experience. This book looks to optimise the methods for synthesising servo controllers ny making them more direct and thus quicker to design. This is achieved by calculating a final controller to directly tackle the high-end system specs.

Leseprobe

Table of Contents

Cover

Title

Copyright

Preface

Introduction and Motivations

I.1. Developing control engineering in an industrial framework

I.2. The place of optimization

I.3. Notations and definitions

1 Metaheuristics for Controller Optimization

1.1. Introduction

1.2. Evolutionary approaches using differential evolution

1.3. Swarm approaches

1.4. Summary

2 Reformulation of Robust Control Problems for Stochastic Optimization

2.1. Introduction

2.2. H

∞

synthesis

2.3. µ-synthesis

2.4. LPV/LFT synthesis

3 Optimal Tuning of Structured and Robust H

∞

Controllers Against High-level Requirements

3.1. Introduction and motivations

3.2. Loop-shaping H

∞

synthesis

3.3. A generic method for the declination of requirements

3.4. Optimal tuning of weighting functions

3.5. Optimal tuning of the fixed-structure and fixed-order final controller

4 HinfStoch: A Toolbox for Structured and Robust Controller Computation Based on Stochastic Optimization

4.1. Introduction

4.2. Structured multiple plant H

∞

synthesis

4.3. Structured µ-synthesis

4.4. Structured LPV/LFT synthesis

4.5. Structured and robust synthesis against high-level requirements with HinfStoch_ControllerTuning

Appendices

Appendix A: Notions of Coprime Factorizations

A.1. Definitions

A.2. Practical computation of normalized coprime factorizations

A.3. Reconstruction of a transfer function by its coprime factors

Appendix B: Examples of LFT Form Used for Uncertain Systems

B.1. Example of LFT form use for a neglected dynamic

B.2. Example of LFT form use for a parametric uncertainty on a transfer function

B.3. Example of LFT form use for a parametric uncertainty in a state-space representation

B.4. Example of LFT form use for a complex uncertainty

B.5. Example of LFT form use for a closed-loop system

Appendix C: LFT Form Use of an Electromechanical System with Uncertain Flexible Modes

C.1. General model and objective

C.2. State-space representation of the nominal model

C.3. LFT form use for uncertainties

Appendix D: FTM (1D) Computation from a Time Signal

D.1. Computation principle

Appendix E: Choice of Iteration Number for CompLeib Tests

Appendix F: PDE versus DE

Bibliography

Index

End User License Agreement

List of Illustrations

Introduction and Motivations

Figure I.1. Method for developing controllers in viewfinders

Figure I.2. The multimodal function to be minimized

Figure I.3. Extracting local minimums

1 Metaheuristics for Controller Optimization

Figure 1.1. Principle of evolutionary algorithms

Figure 1.2. Darwin’s theory. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 1.3. A flock of birds

Figure 1.4. A school of fish

Figure 1.5. PSO Principle. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 1.6. The particles’ movement in PSO (a) and QPSO (b)

Figure 1.7. A cuckoo egg (gray) in a dunnock nest. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 1.8. Levy flight a), simple random walk b) and Brownian movement c)

2 Reformulation of Robust Control Problems for Stochastic Optimization

Figure 2.1. Standard form for H

∞

synthesis

Figure 2.2. Synthesis by loop-shaping subject to order constraint

Figure 2.3. Loop-shaping synthesis in equivalent four-block form

Figure 2.4. Standard form for H

∞

synthesis

Figure 2.5. Structured and decentralized H

∞

synthesis

Figure 2.6. H

∞

synthesis for multiple plants

Figure 2.7. H

∞

synthesis for multiple plants using non-smooth optimization

Figure 2.8. Transforming the search space

Figure 2.9. Transforming the search space using the ash

10

(Mx) function. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.10. Implementation in cascade form

Figure 2.11. Mean ranks of different algorithms on the test overall (234 tests)

Figure 2.12. Multiple-plant H

∞

synthesis

Figure 2.13. Flexible-mode motorized system

Figure 2.14. H

∞

loop-shaping synthesis

Figure 2.15. Direct multiplicative uncertainty

Figure 2.16. Frequency responses from the different models. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.17. Frequency responses of the corresponding uncertainties

Figure 2.18. Loop-shape and verification of the condition [2.166]. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.19. Nominal open loop. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.20. Open loops on the set of the ball of models. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.21. Open loops on the whole ball of plants after fixed-order multiple-plant evolutionary synthesis

Figure 2.22. Controllers obtained before and after the multiple-plant optimization. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.23. Mixed H2/H

∞

synthesis

Figure 2.24. Control scheme with uncertainty block in LFT form

Figure 2.25. Interpretation of the singular structured value

Figure 2.26. Control scheme for performance robustness analysis

Figure 2.27. D-K iteration: computing K(s) with fixed D(s)

Figure 2.28. Flexible-mode system

Figure 2.29. Loop-shaping H

∞

synthesis

Figure 2.30. LFT form

Figure 2.31. Flexible-mode system with fictive input time constant

Figure 2.32. Performance robustness analysis scheme. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.33. Control scheme for µ-synthesis. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.34. µ-synthesis with non-smooth optimization

Figure 2.35. Synthesis model of LPV/LFT form

Figure 2.36. Closed-loop LPV

Figure 2.37. Equivalent closed-loop LPV

Figure 2.38. Mobile pendulum in the cart

Figure 2.39. Reference of the cart’s motion

Figure 2.40. Control synthesis scheme of the pendulum in the cart

Figure 2.41. H

∞

correction on a nominal system. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.42. Variation of parameters J(t) and l(t)

Figure 2.43. H

∞

correction on a varying system. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.44. LPV correction on a varying system. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 2.45. Structured LPV correction on a varying system. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

3 Optimal Tuning of Structured and Robust H

∞

Controllers Against High-level Requirements

Figure 3.1. High-level system specifications

Figure 3.2. Standard form for control

Figure 3.3. Standard approach

Figure 3.4. Standard plant for robustness analysis

Figure 3.5. Robust stabilization of a system modeled by its coprime factors

Figure 3.6. Loop-shaping approach

Figure 3.7. Iterative tuning process for H

∞

controllers

Figure 3.8. “High-level” optimization

Figure 3.9. Loop-shaping synthesis

Figure 3.10. Four-block interpretation of the method

Figure 3.11. Controller frequency implementation

Figure 3.12. Particular frequency implementation

Figure 3.13. State-feedback/observer implementation

Figure 3.14. Management of requirements priority. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.15. Simplified block diagram of a gyrostabilized platform

Figure 3.16. Frequency representation of the SIMO system. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.17. Observation viewfinder control scheme

Figure 3.18. Temporal acquisition of Γ

vx

, Γ

vy

and Γ

vz

Figure 3.19. Loop-shaping synthesis

Figure 3.20. Optimal tuning of weighting functions. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.21. Dispersed mechanic block. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.22. Dispersed electric block. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.23. Frequency response of the open loop for the ball of plants. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.24. The problem of stability robustness. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.25. Problem of stability margin robustness. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.26. Frequency response of the viewfinder for shooting

Figure 3.27. Viewfinder for shooting control scheme

Figure 3.28. Temporal acquisition of Γ

vx

and Γ

vy

Figure 3.29. Acquisition of shock type disturbances

Figure 3.30. Weighting function optimization. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.31. Ball of plants. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.32. Examination of weighting functions robustness. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.33. Problem of stability margin robustness. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.34. MTF and contrast. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.35. Viewfinder control scheme in cascade form

Figure 3.36. Optimal tuning of the structured controller. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.37. Robustness controller examination. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.38. Problem of stability margin robustness. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.39. Viewfinder for shooting control scheme

Figure 3.40. Controller optimization. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.41. Examination of controller robustness. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.42. Problem of stability margin robustness. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.43. Two-degree-of-freedom controller in the loop-shaping framework

Figure 3.44. Final reconstruction of the two-degree-of-freedom controller

Figure 3.45. Two-degree-of-freedom controller – externalized weights

Figure 3.46. Two-degree-of-freedom correction – disturbance rejection

Figure 3.47. Two-degree-of-freedom correction, general formalism

Figure 3.48. Control scheme for µ-synthesis with scalings (D

o

,D

i

). For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.49. Control schema for µ-synthesis with scalings (D

o

,D

i

). For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 3.50. Standard form for control

4 HinfStoch: A Toolbox for Structured and Robust Controller Computation Based on Stochastic Optimization

Figure 4.1. HinfStoch toolbox

Figure 4.2. Multiple plant H

∞

synthesis

Figure 4.3. Standard form for control

Figure 4.4. Multiple plant formalism

Figure 4.5. µ-Synthesis

Figure 4.6. Closed-loop LPV

Figure 4.7. Servo-loop and high-level specifications

Figure 4.8. Standard form for control

Figure 4.9. Multiple plant framework

Figure 4.10. Decentralized feedback controller

Figure 4.11. Decentralized cascade controller

Figure 4.12. Second-order response. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.13. “iter” display mode

Figure 4.14. “Nominal_graphic” window mode. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.15. Controller in cascade form

Figure 4.16. Multi-objective synthesis. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.17. Synthesis with stable controller. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.18. Distillation column

Figure 4.19. Control scheme of the distillation column

Figure 4.20. Closed-loop step responses. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.21. Singular values of the open loop

Figure 4.22. Control scheme of the distillation column with disturbance rejection. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.23. Step responses of references and disturbance rejection (here unitary). For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.24. Singular values of the open loop

Figure 4.25. HIMAT

Figure 4.26. Singular values of the linearized process

Figure 4.27. HIMAT controller synthesis scheme. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.28. Result of HIMAT controller synthesis. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.29. HIMAT step response. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.30. HIMAT controller. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.31. HDA transfer function

Figure 4.32. Control scheme of the hard disk assembly. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.33. Optimization of HAD controller. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.34.. K

1

controllers (left) and K

2

(right)

Figure 4.35. Step HDA response. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.36. Line of sight stabilization

Figure 4.37. Definition of the standard form. For a color version of this figure, see www.iste.co.uk/feyel/optimization.zip

Figure 4.38. Modified Simulink model ‘modified_rct_helico.slx’

Figure 4.39. Optimization results

Figure 4.40. Final HinfStoch_ControllerTuning’s window for the rotorcraft controller optimization

Appendix B: Examples of LFT Form Used for Uncertain Systems

Figure B.1. Use of LFT form for a neglected dynamic

Figure B.2. LFT form use for a parametric uncertainty on a transfer function

Figure B.3. LFT form use for a neglected dynamic

Figure B.4. LFT form use of an uncertain loop

Appendix C: LFT Form Use of an Electromechanical System with Uncertain Flexible Modes

Figure C.1. Electromechanical system with flexible modes

Figure C.2. LFT form

Guide

Cover

Table of Contents

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Robust Control Optimization with Metaheuristics

Philippe Feyel

First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2017

The rights of Philippe Feyel 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 Control Number: 2016958347

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-042-3

Preface

In industry, control engineers have to design a unique control law valid on a single prototype with a sufficient degree of robustness to satisfy a complex specification on many systems. To this end, the development methodology employed consists of an experimental iterative process (trial and error phase) that is heavily reliant on engineers’ own level of expertise.

In this book, we try to make the methodology for computing controllers that are more efficient and more direct with a less costly development time by calculating a final structured controller using a direct optimization on a high-level specification system.

The complexity of high-level specifications drives us to the use of metaheuristics: these optimization techniques do not require gradient formulation, the only constraint being the possibility of evaluating the specification. Thus, in this work, we propose to reformulate robust control problems for stochastic optimization: we show how to synthesize structured controllers for control problems such as H∞ synthesis, μ-synthesis or LPV/LFT synthesis, showing that the interest of the formulated approach lies in its flexibility and the consideration of exotic complex constraints.

Since evolutionary algorithms have proved to be so effective and competitive, we have used them as the foundation for a new method for synthesizing robust and structured controllers with respect to any form of optimization criteria. The validation of this work was performed on the industrial example of the line of sight stabilization problem in addition to several academic problems.

Philippe FEYEL

November 2016