TP-Model Transformation-Based-Control Design Frameworks

Péter Baranyi

TP-Model Transformation-Based-Control Design Frameworks

TP-Model Transformation-Based-Control DesignFrameworks

Péter Baranyi

TP-Model Transformation-Based-Control DesignFrameworks

123

Péter BaranyiTechnology and EconomicsSzecheny Istvan University

and Budapest Univerity of Technologyand Economics

Hungary

ISBN 978-3-319-19604-6 ISBN 978-3-319-19605-3 (eBook)DOI 10.1007/978-3-319-19605-3

Library of Congress Control Number: 2016936784

Springer Cham Heidelberg New York Dordrecht London© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made.

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Preface

“Condita descrescit, vulgata scientia crescit.”My goal in this book is to share the benefits of TP model transformation-based

solutions uncovered through work in my laboratory and to share some of ourexperiences in control design. I hope the frameworks introduced in the book willhelp to radically decrease the amount of analytical work that is performed, oftenunnecessarily, by researchers and engineers working in the field of control designoptimization. If our experience can serve as any basis for generalization, manyexisting analytical approaches can be substituted by more flexible and effectivenumerical methods.

The TP model transformation-based frameworks provide a simple, generic, andflexible way to interface between identification stages and, primarily, linear matrixinequality-based control design theories. Further, they support stability verificationpurposes in general, even in cases where identification and design are based on verydifferent representations. Finally, the presented frameworks lay the foundations forconvex hull manipulation-based control design optimization.

I would like to express my appreciation to my friends Prof. Yeung Yam andProf. Péter Várlaki for their strong support and for their help in shaping, throughmany discussions, a broader scientific and conceptual view behind the TP modeltransformation. I am indebted to the work of young researchers Dr. Béla Takarics,Dr. Péter Galambos, Dr. Ádám Csapó, Patricia Gróf, József Kuti, and SzöllösiAlexandra, who have helped in preparing a large number of experimental casestudies and in extending the TP-tool MATLAB toolbox. I am grateful to AnnaSzemereki for her help in managing all the related research work and projectsthat made it possible for the research group to focus on the research behind thisbook. Finally, I would like to thank our collaborators and graduate students, pastand present, for their inputs and contributions to research on this subject.

Budapest, Hungary Péter BaranyiJanuary 2016

v

Contents

Acronyms and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

The Key Messages of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

Part I Generalized TP Model Transformation

1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 TP Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 TP Model of qLPV Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 TP Model: TS Fuzzy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 HOSVD and Quasi-HOSVD Based Canonical Form

of TP Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Algorithms of the TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Original TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Bi-Linear TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Enriched TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Convex TP Model Transformation: Convex Hull Manipulation . . . 25

2.4.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Pseudo TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 Partial TPC Model Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.7 Multi TP Model Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.7.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

vii

viii Contents

2.8 Generalized TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.9 Interpolation of the Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.9.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.10 Unifying the Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.11 Operations Between TP Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.12 Towards Approximation in Case of Non-TP Functions . . . . . . . . . . . . 61References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Part II TP Model Transformation Based Control Design andOptimalization Frameworks

3 TP Model Transformation is a Gateway BetweenIdentification and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 TP Model Transformation Based Control Design Structure . . . . . . . . . . 69References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 General Stability Verification and Control Design . . . . . . . . . . . . . . . . . . . . . . 735.1 Key Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Decoupling the Design, Optimization, and Stability

Verification: Generalized Design Frameworks. . . . . . . . . . . . . . . . . . . . . . 775.3.1 Multi-Way Convex Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3.2 Main and Independent TP Model Component

Analysis via the HOSVD Based Canonical Form. . . . . . . . . 825.3.3 Convex Hull Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.4 LMI Based System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3.5 Exact System Reconstruction: Unified TP

Model Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.6 LMI Based Stability Verification . . . . . . . . . . . . . . . . . . . . . . . . . . 86

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 TPI Model Transformation for the Class of Non-qLPV Models . . . . . . 876.1 Key Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2 TPI Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.3 Example of Re-identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 TP� Model Transformation for Systems Including Time Delay . . . . . . . 917.1 TP� Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2 Example of the TP� Model Transformation. . . . . . . . . . . . . . . . . . . . . . . . . 92References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Part III Analysis of the TP Model Based Design Frameworksvia a Complex Example

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Contents ix

8 qLPV Model of the 3DoF Prototypical Aeroelastic Wing Section . . . . . 978.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.2 Including Stribeck Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

9 TP Model Based Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039.1 Exact and Convex TP Model of the 3DoF Aeroelastic

Wing Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039.2 Control Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049.3 Selecting LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069.4 Results of the Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.4.1 Controller 1: Asymptotic Stabilization andDecay Rate Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.4.2 Controller 2: Constraint on the Control Value . . . . . . . . . . . . 1079.4.3 Controller 3: State Feedback Control

Including Stribeck Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089.4.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089.4.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

10 Convex Hull Manipulation Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . 11710.1 Convex Hull Manipulation Based Design Framework . . . . . . . . . . . . . 117

10.1.1 Key Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11810.1.2 Step 1: Convex TP Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11810.1.3 Step 2: Convex TP Model Interpolation . . . . . . . . . . . . . . . . . . . 11810.1.4 Step 3: LMI Based Design and Stability Verification . . . . . 120

10.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12010.2.1 Determination of the Feasibility Region. . . . . . . . . . . . . . . . . . . 12010.2.2 Results of the Numerical Simulations . . . . . . . . . . . . . . . . . . . . . 121

11 Complexity Manipulation Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 13111.1 The Control Design Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11.1.1 Main TP Model Component Analysis:HOSVD Based Canonical Form of the Model . . . . . . . . . . . . 132

11.1.2 LMI Based System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.1.3 Exact System Reconstruction: Unified

Weightings in the Polytopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13711.1.4 LMI Based Stability Verification . . . . . . . . . . . . . . . . . . . . . . . . . . 13711.1.5 Maximizing Omega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

11.2 Evaluation of the Benefits of the Proposed Control Design . . . . . . . . 138References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

12 TP Model Manipulation Influences the ControlPerformance and the Feasibility of LMI Based Design. . . . . . . . . . . . . . . . . 14512.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

12.1.1 Initialization of the Numerical Analysis . . . . . . . . . . . . . . . . . . . 145

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12.1.2 Results of the 2D Analysis: Feasibility andConvex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

12.1.3 Results of the 3D Analysis: Feasibility,Convex Hull, and Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

12.1.4 Results of the 4D Analysis: Feasibility,Convex Hull, Complexity, and Parameter Space . . . . . . . . . . 148

12.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15412.2 Control Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

12.2.1 Control Performance Results of the NumericalSimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

12.2.2 Evaluation and Comparison of the DerivedCases and the Best Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Part IV TP Model Based Control Design of the Dual-ExcenterVibration Actuator

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

13 qLPV Model of the Dual Excenter Vibration System . . . . . . . . . . . . . . . . . . 165References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

14 Convex TP Model of the Dual Excenter Vibration System . . . . . . . . . . . . 17114.1 The Quasi-HOSVD Based Canonical Form:

Approximation and Complexity Trade-Off . . . . . . . . . . . . . . . . . . . . . . . . . 17114.2 The Convex TP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

15 Derivation of the Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17915.1 LMI Based Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17915.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Part V Control of the Impedance Model Including VaryingTime Delay via TP� Model Transformation

16 Impedance Control for Force Reflecting Telemanipulation . . . . . . . . . . . . 18716.1 Impedance Control with Feedback Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 18816.2 Control Structure for Stability Preservation. . . . . . . . . . . . . . . . . . . . . . . . . 190References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

17 Impedance Model with Varying Feedback Delay in TPModel Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19517.1 The Quasi-HOSVD Based Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . 195

17.1.1 Exact Quasi-HOSVD Based Canonical Form . . . . . . . . . . . . . 19517.1.2 Executing Trade-off by TP� Model Transformation . . . . . . 198

17.2 Manipulation of the Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19917.2.1 The Vertices of the Exact TP Model . . . . . . . . . . . . . . . . . . . . . . . 20417.2.2 The 5 Vertices of the Reduced TP Model . . . . . . . . . . . . . . . . . 208

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17.2.3 The 4 Vertices of the Reduced TP Model . . . . . . . . . . . . . . . . . 21017.2.4 The 3 Vertices of the Reduced TP Model . . . . . . . . . . . . . . . . . 211

17.3 Validation of the Convex TP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21117.3.1 Constant Time-Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21217.3.2 Varying Time-Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

18 TP� Transformation Based Control Design for ImpedanceControlled Robot Gripper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21718.1 The Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21718.2 Execution of the TP� Model Transformation . . . . . . . . . . . . . . . . . . . . . . . 21818.3 LMI-Based Multi-Objective Controller and Observer Design . . . . . 21818.4 Resulting Controller and Observer Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

18.4.1 Controller-Observer 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22018.4.2 Controller-Observer 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22018.4.3 Controller-Observer 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

18.5 Evaluation and Validation of the Control Design . . . . . . . . . . . . . . . . . . . 221References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Acronyms and Abbreviations

CHOSVD Compact HOSVDCNO Close-to-normalityDoF Degree of freedomHOOI Higher-order orthogonal iterationHOSVD High-order singular value decompositionINO Inverse normalityIRNO INO and RNOLMI Linear matrix inequalityLMIs Linear matrix inequalitiesLPV Linear parameter-varyingLTI Linear time-invariantNN NonnegativenessNO NormalityPDC Parallel distributed compensationqLPV quasi-LPVqNN quasi-NNqSN quasi-SNRHOSVD Reduced HOSVDRNO Relaxed normalitySN Sum normalizationSVD Singular value decompositionTP Tensor productTP model Finite element TP-type polytopic modelTPC Pseudo TP model transformation

xiii

The Key Messages of the Book

The TP (tensor product) model transformation was originally proposed in [3, 12]and summarized in [15] for polytopic representation-based qLPV (quasi-linearparameter varying) control theories. The core of the TP model transformationwas first introduced as an approach to the complexity reduction of fuzzy systems[13, 14, 74]. It transforms a function (which can be given via closed formulasor neural networks, fuzzy logic, etc.) into TP function form whenever such atransformation is possible. If an exact transformation is not possible, then themethod determines a TP function that is an approximation of the given function. TheTP model transformation also provides a trade-off between approximation accuracyand the complexity of the resulting TP function. These properties were investigatedin [7, 15, 51, 69]. The HOSVD (higher-order singular value decomposition)-basedcanonical form of TP models was initiated in [10], and it was also proved that theTP model transformation is capable of numerically reconstructing this form [64].A computationally relaxed variant of the TP model transformation was proposedin [8, 49]. A centralized variant of the transformation was given in [48]. Convexhull manipulation techniques were incorporated into the TP model transformation[4, 15, 71, 74].

Besides serving as a transformation of functions, however, the TP modeltransformation also represents a new concept in qLPV-based control. It is uniquelyeffective in manipulating the convex hull of polytopic forms and, as a result, hasrevealed and proved the fact that convex hull manipulation is a necessary and crucialstep in achieving optimal solutions and decreasing conservativeness in modernLMI (linear matrix inequality)-based control theory [15, 66]. Hence, although theTP model transformation is just transformation (HOSVD of functions) from amathematical point of view, it has nevertheless been successful in establishing aconceptually new direction in control theory and has laid the ground for further newapproaches toward optimality.

Soon, it was extended to TP model transformation-based system control designframework for polytopic model and LMI-based system control through a series of

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xvi The Key Messages of the Book

applications [4–6, 11, 52, 68]. A more complex design framework was proposedin [64]. A MATLAB toolbox was also created for this transformation and relatedfunctionalities; see [9].

Relying on the above properties, a variety of further control solutions have beenproposed in the literature (prime examples can be found in [1, 2, 16, 18, 19, 21,23, 24, 26–34, 38, 40, 42–45, 47, 53–55, 57, 59, 62, 63, 65, 70, 73, 75, 76]. Furtherapplications of the TP model transformation in sliding-mode control were presentedin [35–37, 39, 61, 67, 77–80].

Very recently published papers in the special issue “TP model transformation-based control system design” of the Asian Journal of Control are presenting furthernew directions [11, 17, 20, 22, 25, 41, 42, 46, 50, 56, 58, 60, 72].

In many respects, this book can be seen as a continuation of [15] that focusesprimarily on the TP model transformation, with special attention to its capabilitiesrelated to approximation, complexity reduction, and convex hull manipulation. Afew general thoughts were also given in that book on how these capabilities can beeffectively applied in control design tasks. In the current book, our primary goal is tocover new aspects and frameworks of control design and optimization based on theTP model transformation and its various extensions. The chapters and the complexdynamical control tasks which they cover are organized so as to present and analyzethe beneficial aspects of this family of approaches. Additionally, the book aims toconvey the following messages to the reader:

• Simple TP modelingThe book demonstrates that the TP model transformation provides designers

with a means of automatically transforming qLPV models (given by variousdifferent representations such as closed formulas or softcomputing techniques)to TP model form in a numerically tractable way. The TP form is a polytopicstructure based on which LMI-based design approaches can be directly applied.Based on this capability, researchers and engineers are enabled to apply modernconvex optimization-based approaches formulated in the form of LMIs tocomplex systems without the need to rely on complicated analytical derivations.

• TP model manipulation-based possibilities for optimizationGiven that the TP model transformation provides an efficient method toward

the manipulation of the convex hull defined by the vertices of the TP model,it brings to light the relatively unknown and rarely analyzed fact that LMIs(and the solvers too) in general are highly sensitive to the geometric propertiesof the convex hull, i.e., the number and the relative locations of the vertices.Based on this property, it is clear that besides LMI manipulation, convex hullmanipulation aided by the TP model transformation can be an effective approachin the optimization of control performance. The book also points to the previouslyunknown fact that in searching for the optimal design of various components ofa system (e.g., the controller and the observer), it is often desirable to applydifferent kinds of convex hull manipulations, and what is even more important,the simultaneous use of several TP model representations of a given model(i.e., based on different convex hulls) can lead to significantly improved control

The Key Messages of the Book xvii

performance. Thus, it may, for instance, be effective to design a controllerbased on a TP model representation that is built over a tight convex hull,while designing an observer based on a TP model representation that is builtover a loose convex hull. The book presents design framework for this noveloptimization possibility.

• General framework for stability analysisOne of the main messages of the book is that using the TP model transfor-

mation, it is possible to transform all components of a system to TP modelswith common weighting function structure. Based on this capability it becomespossible to perform LMI-based stability analysis on a wide range of systems.The primary criticism against soft-computing approaches in general is thatno comprehensive framework exists that could be used to prove the stabilityof systems which were designed using a combination of heuristic techniques.Thus, for example, no comprehensive method exists which could be used toderive LMI-based stability verification to prove the stability of a system thatwas designed based on a combination of fuzzy and artificial neural networktechniques (for instance, fuzzy controller and neural network observer arecombined). Using the TP model transformation, however, it becomes possible tocreate a unified polytopic structure for all components of a systems; thus, LMI-based methods are directly applicable. As a result, a general stability verificationmethod can be arrived at, which is capable of answering the criticisms citedearlier.

• Standardized gateway between identification and designBased on the above, the TP model transformation provides a standardized

interface between various heuristic identification tools used for different appli-cations and the well-defined LMI-based design procedures. Thus, TP modeltransformation supports the conversion of any model to TP model form, irrespec-tive of the identification technique which was used to create the model, and hencethe validation of any model can proceed as if it was a TP model from the outset.The TP model transformation is capable of further manipulating the convex hullof the resulting models, as described above. Thus, TP model transformation canbe used as an interface and preprocessing tool for further steps in LMI-basedcontrol design.

• TP model-based design frameworkGiven that the TP model transformation is well-suited to complexity reduc-

tion, convex hull manipulation, as well as the creation of unified TP model forms,it can be used to support the creation of complete control design frameworks.Within the frameworks, the entire design process can be effected on a reducedmodel (which results in reduced design complexity), and various TP modelrepresentations can be used for different system components (resulting in theadvantages described earlier). As all steps of the framework are based on the TPmodel form, important simplifications can be made between neighboring steps,resulting in a compact and relaxed TP model-based control design method andframework. Complexity reduction in this case also allows for an increase in theeffectiveness of LMI techniques.

xviii The Key Messages of the Book

• Gateway to non-qLPV models and time-delayed systemsThe book demonstrates that the TP model transformation can be used to

modify the interpretation of operational parameters such that, for instance, time-delay systems can be converted into qLPV systems (excluding delay) whichconsiders time delay as an external system parameter. Based on this aspect,the applicability of well-established qLPV and LMI-based methods can beeffectively extended to a wide range of problems involving time delay.

Outline of the Book

The book is organized into five main parts. The first part (Part I: Chap. 2)proposes the generalized TP model transformation that includes various TP modelmanipulation techniques. Based on these manipulations, the second part (Part II:Chaps. 3–7) proposes control design frameworks with various beneficial features.The third part (Part III: Chaps. 8–12) of the book gives complex control designexample focusing on one control design problem that have emerged very recently,in order to show how these frameworks can be applied to real-world problems andto study new features of design and effectiveness relevant to their use. The last twoparts (Parts IV and V: Chaps. 13–18) focus on two real-world engineering controlproblems. The chapters are organized as follows:

Chapter 1 recalls some basic definitions and concepts about the TP modeltransformation presented in previous works. It recalls the definition of the TPfunction, as well as the TP model, which introduces the use of TP functions intothe concept of polytopic qLPV models. The chapter also briefly discusses theconceptual similarities and differences between TP models and TS fuzzy models.This discussion makes clear that all model manipulation and LMI design conceptsdiscussed in the book can be applied in fuzzy modeling and control design as well.The chapter also highlights the fact that the HOSVD-based canonical form of qLPVmodels can be extended. Thus, it redefines the previously published HOSVD-basedcanonical form.

Chapter 2 integrates various ideas about the TP model transformation into oneconceptual framework and formulates the framework in terms of the generalizedTP model manipulation. Several new extensions of the TP model transformationare proposed, such as the numerical reconstruction of quasi and “full,” compact andrank-reduced HOSVD-based canonical form of TP models and the bilinear-, multi-, pseudo-, and convex-TP model transformations. All of these extensions togetherform the generalized TP model transformation, which provides an effective tool tofreely and readily manipulate the weighting functions and, hence, the convex hulldefined by the vertexes of the TP models. Further, they provide a means to performTP model-based main component analysis, as well as a host of complexity andaccuracy trade-offs within TP models. All of these techniques form a generalized

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tool for the convex hull manipulation specialized to TP models, which has a crucialrole in polytopic model-based LMI design methods as will be discussed in the nextpart of the book. The chapter also gives some hints on how to use the TP modeltransformation to define and compute operations on and between TP models thatprovide further freedom toward TP model manipulation. Specifically, this chaptershows how to unify the weighting functions of preexisting TP models, i.e., howto find a common weighting function system to which all TP functions can beexactly transformed. The chapter also shows how to interpolate between pairs ofTP models in such a way that their weighting functions are interpolated in a waythat corresponds to a new TP model. This technique is very useful in the case ofconvex hull manipulation-based control performance optimization as will be shownin the context of design examples.

Part II of the book demonstrates that the proposed manipulation forms a new,effective, and necessary optimization step of polytopic models and LMI-basedcontrol design and that it can also be used to decrease conservativeness. Thisalleviates the problem that identification techniques are typically constructed basedon the data and measurement set that is available and based on the type of systemthat is to be identified, irrespective of the kind of representation that is most suitableto the control design framework.

Chapters 3 and 4 demonstrate how the proposed generalized TP model transfor-mation is unique in the sense that it bridges between various soft-computing-basedidentification techniques and the TP model. By association, the model makes itpossible to merge soft-computing-based identification with polytopic model-basedcontrol design approaches. The chapter gives a discussion on the dual role ofthe TP model transformation as a final step of identification on the one handand as a generalized “interface” on the other, which is capable of serving as apreprocessing step prior to the fulfillment of further design requirements (e.g.,convex hull manipulation). This very unique feature of the TP model transformationmakes it a very powerful tool, as it allows for the free combination of availabledesign and identification techniques without the usual drawbacks of having to dealwith incongruent mathematical derivations and representations. These advantagesbecome even more clear when the extended TP model-based control designframework is applied to real-world problems.

Chapter 5 proposes the multi-TP model transformation-based stability verifi-cation framework, which is a tractable and non-heuristic framework that enablesthe stability verification of results obtained through (hybrid) soft-computing-basedcontrol design approaches, i.e., in which different system components can even beformulated in different representations (e.g., the functionality of a controller mightbe expressed through fuzzy rules, the observer might be formulated using a neuralnetwork, etc.). The multi-TP model transformation-based framework can providean answer to the frequently emerging criticisms regarding the lack of mathematicalstability verification techniques in soft-computing-based control design. The entirestability verification technique developed in the chapter can be fully automated andnumerically executed in a reasonable amount of computing time.

Outline of the Book xxi

Chapters 6 and 7 extends the TP model transformation-based design frameworksto non-qLPV systems and time-delayed systems.

In Part III Chap. 8 introduces the most recent version of the 3DoF nonlinearaeroelastic test apparatus (NATA) wing section model generated from real mea-surements. The model was presented and deeply elaborated in a series of paperspublished in the Journal of Guidance, Control, and Dynamics. This chapter takesthe most recent model and extends it with Stribeck friction, a component whichconsiderably increases the dynamic and modeling complexity of the system. Thechapter then focuses on the problem of flutter suppression for the prototypicalaeroelastic wing section. The flat plate airfoil is constrained to have 3DoF, i.e.,plunge, pitch, and trailing-edge surface deflection. The main goal of the chapter is todescribe and prepare a complex example for the next chapters, in order to study theeffectiveness of various design techniques based on the TP model transformation.

Chapter 9 applies the TP model transformation in a very straightforward and“direct” way, i.e., without any convex hull manipulations, to the extended modelof the aeroelastic wing section derived earlier. Different controllers are derived andthe design also includes constraints on the control value and decay rate control. Thechapter also provides an evaluation of the resulting controller based on numericalsimulations. It is shown that the entire design process can be executed in anautomated way with minimal human interaction

Chapter 10 continues further with the control design for the 3DoF aeroelasticmodel. The solution provided in the chapter incorporates convex hull manipulation.Through this aspect, the chapter demonstrates the little known fact that the use ofdifferent TP model representations of the system when designing the controllerand observer can have beneficial effects on the resulting control performance.The chapter shows that by tightening the convex hull of the TP model, theconservativeness of the control design is decreased; however, by loosening theconvex hull, the observer performance is improved. Hence, the chapter suggests thatthe trade-off between these contradictory requirements can be optimized throughconvex hull manipulation so that the performance of the entire system as a wholeis optimal. A further takeaway is that if no compromise is accepted betweenthe requirements, defining separate convex hulls for the controller and observer(tight and loose, respectively), further improvements can be achieved. The chaptershows that the TP model transformation can be used to achieve these objectives.Discussions in the chapter are supported by numerical simulation-based evaluations.

Chapters 11 and 12 evaluates the effectiveness of the convex hull manipulationin control design. This manipulation includes the shape and the complexity of theconvex hull. As in previous chapters, the analysis is applied to the 3DoF aeroelasticwing section.

Part IV presents a complex example of the control design of the dual-excentervibration actuator. Vibration actuators are widely used, for instance, in handhelddevices to provide vibrotactile feedback or silent notification to users. In mostcases, miniature DC motors with an eccentric rotor or the so-called coin-typeshaftless vibration motors are utilized. The common disadvantage of the single rotordesign is that the frequency and the intensity of the generated vibration cannot be

xxii Outline of the Book

adjusted separately. On the other hand, the construction that is composed of twoindependently driven coaxial eccentric rotors—which makes for a strongly couplednonlinear system—allows for the separate control of the frequency and amplitude bythe adjustment of the angular speed and the total eccentricity. The chapter presentsa complete control design approach based on qLPV modeling and LMI-basedsynthesis utilizing the TP model transformation to determine the convex polytopicrepresentation of the parameter-dependent nonlinear system. The design approach isdemonstrated via a concrete numerical example using the parameters of a real dual-excenter prototype device. The control performance is validated through numericalsimulations. This case study goes through a complete nonlinear control problemfrom the modeling phase to the design of the implementation-ready controller whiledrawing generalizable lessons on TP model transformation-based control design.

Part V presents an example that focuses on impedance model control applied toa force feedback telemanipulation system in which time delay leads to significantproblems. Through the example, the chapter studies the use and effectiveness ofthe TP� model transformation. The chapter introduces a widely applied impedancecontrol scheme for force reflecting telemanipulation and then focuses attention onthe theoretical limitations of the stability of such systems under time delay. Stabilityis then analyzed in terms of the maximum time delay under which the impedancemodel still remains stable. Finally a stabilizing controller is designed.

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61. H. Sharina, Z. Guoliang, Y. Yanhua, R. Qiuping, Intelligent tensor product mode transforma-tion -based three-sliding-surface sliding mode controller design, in Proceedings of 34th ControlConference (CCC), Hangzhou, China, 2015 (IEEE, New York, 2015), pp. 3258–3263

62. X. Su, Y. Jia, Self-scheduled robust decoupling control with H1 performance of hypersonicvehicles. Syst. Control Lett. 70, 38–48 (2014)

63. Z. Szabó, P. Gáspár, Sz. Nagy, P. Baranyi, TP model transformation for control-oriented qlpvmodeling. Aust. J. Intell. Inform. Process. Systems 10(2), 36–53 (2008)

64. L. Szeidl, P. Várlaki, HOSVD based canonical form for polytopic models of dynamic systems.J. Adv. Comput. Intell. Intell. Inform. 13(1), 52–60 (2009)

65. K. Széll, A. Czmerk, P. Korondi, Friction with hysteresis loop modeled by tensor product.Automat. J. Control Meas. Electron. Comput. Commun. 55(4), pp. 463–473 (2014) [OnlineISSN 1848-3380]

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xxvi Outline of the Book

67. B. Takarics, TP Model Transformation Based Sliding Mode Control and Friction Compen-sation. Ph.D. thesis, Budapest University of Technology and Economics, Budapest, Hungary,2011

68. B. Takarics, P. Baranyi, Tensor-product-model-based control of a three degrees-of-freedomaeroelastic model. J. Guid. Control. Dyn. 36(5), 1527–1533 (2013)

69. D. Tikk, P. Baranyi, R.J. Patton, Approximation properties of TP model forms and itsconsequences to TPDC design framework. Asian J. Control 9(3), 221–231 (2007)

70. I.-B. Ursache, P.A. Clep, P. Baranyi, R.-E. Precup, S. Preitl, J.K. Tar, On the combinationof tensor product and fuzzy models, in Proceedings of IEEE International Conference onAutomation, Quality and Testing, Robotics (AQTR 2008), Cluj-Napoca, Romania, (IEEE, NewYork, 2008), pp. 48–53

71. P. Várkonyi, D. Tikk, P. Korondi, P. Baranyi, A new algorithm for rno-ino type tensor productmodel representation, in Proceedings of the IEEE 9th International Conference on IntelligentEngineering Systems (2005), pp. 263–266

72. T.T. Wang, W.F. Xie, G.D. Liu, Y. M. Zhao, Quasi-min-max model predictive control forimage-based visual servoing with tensor product model transformation. Asian J. Control 17(2),402–416 (2015)

73. K.K. Wu, Y. Yam, Control stability of TP model transformation design via probabilisticerror bound of plant model, in 2013 IEEE International Conference on Systems, Man,and Cybernetics (SMC), pp. 1259–1264. doi:10.1109/SMC.2013.218 [INSPEC AccessionNumber: 14067688]

74. Y. Yam, P. Baranyi, C.T. Yang, Reduction of fuzzy rule base via singular value decomposition.IEEE Trans. Fuzzy Systems 7(2), 120–132 (1999)

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80. J. Zhu, X. He, W. Gueaieb, Tensor product model transformation based sliding mode designfor LPV systems, in Recent Advantages in Sliding Modes: From Control to IntelligentMechatronics, X. Yu, M. Onder. Studies in Systems, Decision and Control 24, vol. 24(Springer, Berlin, 2015), pp. 277–298

Part IGeneralized TP Model Transformation

Chapter 1Basic Concepts

Abstract This chapter introduces some fundamental concepts and definitions usedthroughout of the book. It shows that all concepts and methodologies developed forTP models in this book can readily be applied in qLPV and LMI based controltheories and TS fuzzy model based concepts. This chapter also discusses theHOSVD and the quasi-HOSVD based canonical form of TP functions that will beused as a basic steps in various frameworks proposed in later chapters.

Keywords TP function • TP model • TS fuzzy model • HOSVD based canonicalform

1.1 Notations

The following notation is used throughout this book.

• a denotes a scalar value;• a denotes a vector;• A denotes a matrix;• A denotes a tensor;• OA denotes an approximation ofA;• A denotes a set;• Nf .x/ the bar over the function denotes that the function is a piece-wise linear

function;• A.n/ is the n-mode (dimension) layout matrix of tensorA [6];• FD.�;G/ denotes the discretized variant of function f .x/ over grid G and space �;• A �n U is interpreted as the n-mode multiplication of tensorA with matrix U in

dimension n.

• AN�

nD1Un denotes a multiple product that can also be written as A �1 U1 �2 U2

�3 � � � �N UN ;• A �

d2DUd denotes a multiple product in which tensor A is multiplied by Ud in

all dimensions d 2 D, such that the set D contains the ordinal numbers of alldimensions in which multiplication occurs.

• AC denotes the pseudo-inverse of matrix A;

© Springer International Publishing Switzerland 2016P. Baranyi, TP-Model Transformation-Based-Control Design Frameworks,DOI 10.1007/978-3-319-19605-3_1

3

4 1 Basic Concepts

• Rn D rankn.A/ denotes the n-mode rank of tensor A, that is, rankn.A/ D

rank.A.n// [6];• Rn D rankn.f .x/; �/ denotes the n-mode rank of functions f .x/ in the space

x 2 � � RN .

The following indexing system and special letters are frequently used in thebook:

• Indices n; i; j; k; l : : : run from 1 to upper bound N; I; J; K; L : : :.• Letter N denotes the dimensionality of vector x 2 RN in f .x/ and the

dimensionality of parameter vector p.t/ 2 RN in system matrix S.p.t//;• Sets N; K; D; : : : contain the number n; k; d; : : : of dimensions (e.g., n 2 N). This

is used to refer to the dimension of the tensor product operation;• Sets N; K and L contain elements N D f1; 2; : : : ; Ng � N, K D f1; 2; : : : ; Kg �

N and L D f1; 2; : : : ; Lg � N respectively;• l (lowercase ‘L’) is used to denote sets of functions as fl.x/, where l 2 L;• K and O denote the dimensionality and the size of the output tensors Y 2

RO1�O2��OK of functions;• � D !1 � !2 � � � � � !N denotes the space of x 2 RN of function f .x/, or

parameter vector p.t/ 2 RN of system matrix S.p.t//;• G denotes a G1 � G2 � � � � � GN sized hyper-rectangular grid.

1.2 TP Function

This section introduces some basic concepts to be used throughout the book. Theseconcepts are deeply discussed in the book [4].

Definition 1.1 (TP Function). The function Y D f .x/ 2 RO1��OK , x 2 RN is aTP function if it has the structure

Y D S �n2N

wn.xn/; (1.1)

where N D f1; : : : ; Ng � N, wn.xn/ D�

wn;1.xn/ � � � wn;In.xn/�

such that weightingfunctions wn;i.xn/ 2 Œ�1; 1� and S 2 RI1��IN is the core tensor whose all elementsare tensors with the size of O1 � � � � � OK . Thus tensor S can also be considered as atensor of scalar elements with the size of I1 � � � � IN � O1 � � � � � OK . The operationdenoted by � is the tensor product (the original notation—˝—as introduced byLathauwer et al. [6] is slightly modified here to emphasize the higher-level structureof the core tensor: namely, that its elements are tensors or LTI system matrices aswill be discussed later).

Remark 1.1. Note that not all functions f .x/ have TP function structure in whichthe size of the core tensor is bounded, cf. [10].

1.3 TP Model of qLPV Systems 5

Remark 1.2. The above Eq. (1.1) without tensor operation takes the form of:

Y D †I1i1D1†

I2i2D1 � � � †

INiND1

�…N

nD1wn;in.xn/�

Si1;i2;:::;iN ; (1.2)

where Si1;i2;:::;iN 2 RO1��OK are the elements of tensor S.

Definition 1.2 (Convex TP Function). TP function (1.1) is convex if its weightingfunctions have no negative value and for 8n; xn W wn.xn/1 D 1.

In such case we mark the weighting functions as wCon .xn/ as the output Y is

always within the convex hull defined by the O1 �� OK sized elements of the coretensor. Therefore these elements are also referred as vertexes. If we impose furthercharacteristics on the weighting functions we can define various special types of theconvex hull defined by the vertexes. Such requirements are discussed and detailed in[1, 11, 13] and in the book [4], see for instance the SN (Sum-normalized), NN (Non-Negativeness), NO (Normalized), CNO (Close to Normalized), RNO (Relaxed NO),INO (Inverse NO), and IRNO (Inverse RNO) weighting function types leading todifferent tightness and shapes of the convex hull.

1.3 TP Model of qLPV Systems

In later sections, we will focus on the role of the TP model transformation in controldesign; therefore, a brief introduction to the above discussed TP function in thecontext of dynamic systems modeling is also in order here. Let us consider a linearparameter-varying (LPV) state-space model:

�Px.t/y.t/

�D S.p.t//

�x.t/u.t/

�; (1.3)

with input u.t/, output y.t/, and state vector x.t/. The system matrix S.p.t// 2

RO1�O2 is a parameter-varying object, where p.t/ 2 � � RN is a time varyingN-dimensional parameter vector which is an element of closed hypercube

� D !1 � !2 � � � � � !N D Œ!min1 ; !max

1 � � Œ!min2 ; !max

2 � � � � � � Œ!minN ; !max

N � � RN :

(1.4)

p.t/ can also include some elements of x.t/, and hence this model belongs tothe class of nonlinear systems. This kind of form is often referred to as a quasiLPV (qLPV) model. Further parameter dependent channels, which represent variouscontrol performance requirements, can be inserted into S.p.t//.

Definition 1.3 (TP Type Polytopic Model: TP Model). The qLPV model (1.3)can then be defined using a TP function structure as follows:

�Px.t/y.t/

�D S �

n2Nwn.pn.t//

�x.t/u.t/

�: (1.5)

6 1 Basic Concepts

The N C 2-dimensional core tensor S 2 RI1�I2��IN�O1�O2 is constructed from theLTI system matrices Si1;i2;:::;iN 2 RO1�O2 .

If we have weighting functions wCon .pn.t// for all n as

�Px.t/y.t/

�D S �

n2NwCo

n .pn.t//

�x.t/u.t/

�; (1.6)

then the TP model becomes a polytopic representation, and in consequence S.p.t//is always within cof8n; in W Si1;i2;:::;iN g, where Si1;i2;:::;iN are referred as the vertex LTIsystems. Thus the TP model is a higher structured polytopic representation as it canalways be given as:

S.p.t// D

RX

rD1

wr.p.t//Sr; (1.7)

where vertexes Sr are equivalent to the vertexes stored in tensor S as Sr D Si1;i2;:::;inand wr.p.t// D …N

nD1wn;in.xn/ where r is the linear equivalent of multidimensionalindexes i1; i2; : : : ; iN .

The advantage of this convex TP model form is that a large set of LMI basedsystem control design theories can immediately be applied to this kind of TPmodel type polytopic form, see later. This TP model form can readily be furthermanipulated to enhance the multi-objective optimization, see later.

1.4 TP Model: TS Fuzzy Model

Let us recall the transfer function of the Takagi–Sugeno fuzzy operator and product-sum-gravity defuzzyfication based fuzzy model [12]. Assume that we have a set offuzzy rules with N inputs such as:

IF A1;i1 AND A2;i2 AND � � �

� � � AND AN;iN THEN Bi1;i2;:::;iN: (1.8)

Let wn;i.xn/ 2 Œ0; 1�, xn 2 R be the membership function of antecedent fuzzy setAn;i, i D 1; : : : ; In on input universe Xn, n 2 N. Let the observation fuzzy sets besingleton sets with elements xn, and the consequent fuzzy sets Bi1;i2;:::;iN be singletonsas well, represented by their single elements bi1;i2;:::;iN on the output universe Y . Ifthe output of the TS fuzzy model is not a scalar value, but a vector, matrix, oreven a tensor Y of dimensions O1 � O2 � � � � � OK , then the consequent fuzzysets represent vectors, matrices, or tensors, respectively. The consequent sets canalso be represented by parametrized functions as f .bi1;i2;:::;iN ; x0/. For the sake ofsimplicity, and without the loss of generality, we do not distinguish between what theconsequent sets symbolize, and we simply assume that the consequents are assignedto parameters arranged into tensors Bi1;i2;:::;iN 2 RO1�O2��OK . In order to have a

1.4 TP Model: TS Fuzzy Model 7

more general form, we turn to multi-output fuzzy rule bases. Let the number ofoutputs be denoted by l 2 L, such that each of the outputs are assigned to Yl 2

RO1�O2��OK . We merge the output tensors into a single tensorY along the K C1thdimension, and whenever we need to extract a single output, we can work withseparate partitions of Y 2 RO1�O2��OK�L. In the same way, we construct B fromBi1;i2;:::;iN ;l assigned to the outputs for further discussion.

A very typical requirement in fuzzy modeling is the Ruspini-partition:

Definition 1.4 (Ruspini-Partition). The antecedent membership functions ofdimension n are given in Ruspini-partitions if they satisfy

8n; xn W

InX

iD1

wn;i.xn/ D 1: (1.9)

Membership functions which satisfy this property are denoted by wRPn;i .xn/.

Having Ruspini partition, we arrive at the following general transfer function forTS fuzzy models:

Y D †I1i1D1†

I2i2D1 � � � †

INiND1

�…N

nD1wRPn;in.xn/

�Bi1;i2;:::;iN ; (1.10)

where x 2 RN . This transfer function is specifically a Tensor Product (TP) function;therefore, it can be given in the form of:

Y D B �n2N

wn.xn/; (1.11)

where wn.xn/ D�

wRPn;1.xn/ � � � wRP

n;In.xn/

. Since the membership functions have

no negative value this leads to convex form; thus these membership functions arewCo

n;in.xn/.

We can conclude that the concepts of TP models and the TS models overlap,since in some aspects the TS fuzzy model is a special case of the TP models, i.e.,the weighting functions of the TP model can take negative value and the TP modelmay represent any parametrized structure or database, etc. But in other aspects theTP model is a special case of TS fuzzy models, since in case of the TS fuzzy modeleach consequent set may have different types of functions, and these functions maynot have the same parameter structure that would be required to construct a coretensor as in the case of TP model.

In conclusion, all concepts and methodologies developed for TP models inthis book can readily be applied in TS fuzzy model based concepts of fuzzytheories (e.g., Parallel distributed compensation (PDC) theory proposed in thebook by Tanaka and Wang [9]).

8 1 Basic Concepts

1.5 HOSVD and Quasi-HOSVD Based Canonical Formof TP Functions

Paper [2] redefines the previously published HOSVD based canonical form [3, 4, 8],and highlights the fact that the previously published HOSVD based canonical formdoes not decompose all dimensions of the core tensor; thus, it is quasi-HOSVDonly. In order to resolve this shortcoming, a “full” canonical form is presented herebased on the paper [2]. The key idea is that the quasi-HOSVD based canonicalform is resulted when the HOSVD is executed only for dimensions assigned tothe variables xn, and “full” HOSVD is resulted when HOSVD is also executed ondimensions assigned to the dimensions of the vertex elements. Since the HOSVDbased canonical form determines the contribution (in decreasing order) of thecomponents via the higher order singular values (or via the singular values bydimensions) we can view this canonical form as a tool for main TP function orTP model component analysis. If we discard those components which have smallercontribution we arrive at a trade-off between complexity and accuracy.

Papers [8] prove that the TP model transformation is capable of numericallyreconstruct the HOSVD and the quasi-HOSVD based canonical form and derivesconverge theorems according to numerical setting of the TP model transformation.

Theorem 1.1 (HOSVD Based Canonical Form of TP Functions). For brevityone may say HOSVD of TP functions. A TP function f .x/, x 2 � � RN withoutput Y 2 RO1�O2��OK has the HOSVD canonical form:

Y D

�S �

n2Nwn.xn/

��

NC.k2K/Tk; (1.12)

where K D f1; : : : ; Kg � N in which

1. singular functions wn;in.xn/ 2 Œ�1; 1�, in D 1; : : : ; In contained in singularvectors wn.xn/ form an orthonormal system in the sense of

8n W

Z max.!n/

min.!n/

wn;i.xn/wn;j.xn/dxn D ıi;j; (1.13)

where 1 � i; j � In and ıi;j is the Kronecker-function (ıi;j D 1, if i D j andıi;j D 0, if i ¤ j).

2. Transformation matrices Tk are unitary matrices.3. Core tensor S is a real tensor and its subtensors SinD˛ (which can be obtained

by fixing the nth index to ˛) have the properties of

(a) all-orthogonality: two subtensors SinD˛ and SinDˇ are orthogonal for allpossible values of n; ˛ and ˇ subject to ˛ ¤ ˇ: hSinD˛SinDˇi D 0;

when ˛ ¤ ˇ;

(b) ordering: kSinD1k � kSinD2k � � � � � kSinDInk � 0; for all possible valuesof n.

1.5 HOSVD and Quasi-HOSVD Based Canonical Form of TP Functions 9

Based on the analogy of the HOSVD of tensors [6], we refer to the Frobenius-normskSinDik, symbolized by �

.n/i , as the n-mode singular values of the TP function.

Proof 1. The proof of the existence of the HOSVD based canonical form in caseswhere a TP function has a single scalar output is given in [8]. The existence of theHOSVD based canonical form for non-scalar functions can be proved in the sameway, the only difference being that in dimensions larger than N, we simply have theoriginal tensor HOSVD.

Proof 2. The proof of uniqueness of the HOSVD based canonical form in caseswhere a TP function has a single scalar output is given in [8]. The uniqueness ofthe HOSVD based canonical form for general cases can be proved in the sameway, the only difference being that in dimensions larger than N of the core tensor,the HOSVD itself guarantees the unique decomposition [6]. Thus, this propertyof the HOSVD of tensors is true for the HOSVD based canonical form as well.As a matter of fact, the decomposition is unique to the extent of the signs of thesingular functions and the columns of the transformation matrices, which can besystematically switched, just like in the case of HOSVD of tensors. If there are equalsingular values on any dimension, then the HOSVD based canonical form is notunique. In this case the n-mode singular functions or vectors corresponding to thesame n-mode singular value can be replaced by orthonormal linear combinations.This property is proved in the original paper on the HOSVD concept itself, see [6].

Remark 1.3. Transformation matrix Tk transforms to the minimal RNC1 � RNC2 �

: : :�RNCK (Rn D rankn.S/) orthonormal subspace that is structured via higher ordersingular values. This transformation indicates whether or not the vertex componentshave linear dependencies. Thus we may deal only with the linearly independentcomponents, and once we have the canonical core components, we can restructurethe expected tensor.

Remark 1.4. Note that the HOSVD based canonical form cannot be interpreted as aTS fuzzy model directly as its weighting functions may assume negative values thatis not interpretable as membership values of fuzzy sets.

Definition 1.5 (n-Mode Rank of TP Function or TP Model). The n-mode rankof a TP function, where x 2 �, denoted by Rn D rankn.f .x/; �/ is the numberof nonzero singular values in the nth dimension, thus Rn D rankn.f .x/; �/ D

rankn.S/. We can also indicate the rank of the vertex components in dimensionsn D N C k 2 K.

Definition 1.6 (CHOSVD/RHOSVD-Based Canonical Form). This definition isabout the complexity trade-off property of the HOSVD. We arrive at the compactHOSVD (CHOSVD) of TP functions or TP models, when we keep the first Rn,(n D 1; : : : ; N C K) nonzero singular values only in all dimensions as discussedabove. Accordingly, the size of the core tensor is R1 � R2 � � � � � RNCK , where Rn

is the n-mode rank of the TP function or TP model. We have rank reduced/relaxedHOSVD (RHOSVD), when we keep J1 � J2 �� JN nonzero singular values only,where 8n W Jn � Rn and 9n W Jn < Rn. The RHOSVD canonical form of TP function

10 1 Basic Concepts

or TP model is only an approximation, where the error (in L2 norm) can be derivedbased on the sum of the discarded singular values as in the case of the HOSVDof tensors (see the proof in [6]). Using higher order orthogonal iteration (HOOI)we can further tune the core tensor to decrease the error [5, 7]. A comprehensiveanalysis on the approximation properties are given in [10].

Theorem 1.2 (Quasi-HOSVD-Based Canonical Form). If we multiply by T inthe HOSVD canonical form (1.12), then we arrive at the quasi-HOSVD canonicalform (we do not execute SVD on dimensions n > N):

Y D S �n2N

wn.xn/: (1.14)

Note that even in case we use quasi-HOSVD based canonical forms, thedecomposition still gives, obviously, the higher order ranking of the componentsfor main component analysis.

References


2. P. Baranyi, The generalized TP model transformation for TS fuzzy model manipulation andgeneralized stability verification. IEEE Trans. Fuzzy Syst. 22(4), 934–948 (2014)

3. P. Baranyi, L. Szeidl, P. Várlaki, Y. Yam, Definition of the HOSVD based canonical formof polytopic dynamic models, in Proceedings of the 2006 IEEE International Conference onMechatronics, Budapest, 3–5 July 2006, pp. 660–665

4. P. Baranyi, Y. Yam, P. Varlaki, Tensor Product Model Transformation in Polytopic Model-Based Control (CRC/Taylor & Francis Group, Boca Raton/London, 2013)

5. L. De Lathauwer, B. De Moor, J. Vandewalle, Dimensionality reduction in higher-order-onlyICA, in Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics,1997, Banff, Alberta (1997), pp. 316–320

6. L. De Lathauwer, B. De Moor, J. Vandewalle, A multilinear singular value decomposition.SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)

7. M. Ishteva, L. De Lathauwer, P. Absil, S. Van Huffel, Dimensionality reduction for higher-order tensors: algorithms and applications. Int. J. Pure Appl. Math. 42(3), 337 (2008)

8. L. Szeidl, P. Várlaki, HOSVD based canonical form for polytopic models of dynamic systems.J. Adv. Comput. Intell. Intell. Inf. 13(1), 52–60 (2009)

9. K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear MatrixInequality Approach (Wiley-Interscience, New York, 2001)


11. P. Várkonyi, D. Tikk, P. Korondi, P. Baranyi, A new algorithm for RNO-INO type tensorproduct model representation, in Proceedings of the IEEE 9th International Conference onIntelligent Engineering Systems (2005), pp. 263–266

12. Y. Yam, Fuzzy approximation via grid point sampling and singular value decomposition. IEEETrans. Syst. Man Cybern. B Cybern. 27(6), 933–951 (1997)

13. Y. Yam, P. Baranyi, C.T. Yang, Reduction of fuzzy rule base via singular value decomposition.IEEE Trans. Fuzzy Syst. 7(2), 120–132 (1999)

Chapter 2Algorithms of the TP Model Transformation

Abstract This chapter proposes the generalized TP model transformation thatincludes various TP model manipulation facilities into one conceptuar framework.The generalized TP model transformation includes extensions such as the HOSVDand quasi HOSVD canonical form, the Bilinear-, Multi, Pseudo, and convex TPmodel transformation which all serves the goal to have a Transformation techniquethat is capable of freely manipulating all components of the TP model according tovarious conditions.

Keywords Bi-linear- • Pseudo- • Muti- • Generalised TP model transformtion

2.1 Original TP Model Transformation

This section recalls the TP model transformation from [1, 3, 6] and restructures itin order to have a core algorithm that can readily assume further extensions to beintroduced in the chapter.

Definition 2.1 (Discretization Space �). � is a space in which we intend toperform the discretization of a given function f .x/.

Definition 2.2 (Discretized Function). Tensor F D.�;G/ 2 RG1��GN�O1��OK isthe discretized variant of function Y D f .x/ 2 RO1��OK in the discretizationspace �, and over the discretization grid G D G1 � � � � � GN (Gn denotesthe number of gridpoints on dimensions n 2 N). Vector gn defines the posi-tions (typically, but not necessarily equidistantly located) of the grid as gn D�

gn;1 D !minn � � � gn;Gn D !max

n

�by dimensions. Thus, the O1 � : : : � OK sized

elements Fm1;m2;:::;mN of tensor F D.�;G/ (such that mn D 1; : : : ; Gn) are:

Fm1;m2;:::;mN D f .x/; (2.1)

where x D�

g1;m1 � � � gN;mN

�.

If we have a vector w.x/ containing weighting functions wi.x/, (i D 1; : : : ; I) as

w.x/ D�

w1.x/ � � � wI.x/�

(2.2)


11

12 2 Algorithms of the TP Model Transformation

then WD.!;G/ 2 RG�I is a matrix whose column vectors are the discretized variantsof the functions wi.x/ as:

WD.!;G/ D�

.wD.!;G/1 /T � � � .w

D.!;G/I /T

; (2.3)

where

wD.!;G/i D

�wi.g1/ � � � wi.gG/

�: (2.4)

Thus

WD.!;G/ D

0

B@

w1.g1/ � � � wI.g1/:::

: : ::::

w1.gG/ � � � wI.gG/

1

CA : (2.5)

Lemma 1. The discretization of a given f .x/ D B �n2N

wn.xn/ simplifies to the

discretization of the weighting functions as:

FD.�;G/ D B �n2N

WD.!n;Gn/n (2.6)

Note that the result of HOSVD has the same structure as the discretized TPfunctions. Thus, the key idea is that executing HOSVD on the discretized functionF D.�;G/, we obtain the discretized form of the HOSVD based canonical form of theTP function:

Algorithm 1 (TP Model Transformation). Let us assume a function given asY D B �

n2Nvn.xn/, x 2 � � RN. The goal of the algorithm is to numerically

reconstruct the HOSVD based canonical TP function:

Y D

�S �

n2Nwn.xn/

��

NC.k2K/Tk (2.7)

in �:

• STEP 0: Numerical initialization: Define discretization grid G fit to �.• STEP 1: Discretization: Determine FD.�;G/.• STEP 2: Reconstruct the core tensor of the model: Determine S and Un by

executing compact HOSVD (CHOSVD) on FD.�;G/ (in case of rank reductionor complexity trade-off RHOSVD is executed in this step). This results in

FD.�;G/ D OS �n2f1;:::;NCKg

Un (2.8)

Thus OTk D UNCk (k 2 K).

2.1 Original TP Model Transformation 13

• STEP 3: Determine wn.xn/: Let OWD.Gn;!n/

n D Un. Weighting functions Own.xn/ in

Y D

�OS �

n2NOwn.xn/

��

NCk2KOTk (2.9)

can be reconstructed over any point in !n. For instance, let us calculate theweighting functions Owd.xd/ on dimension d over a given point xd. Let us define anew discretization grid G0 as G1 � � � � � Gd�1 � 1 � GdC1 � � � � � GN and restrictthe discretization space to xd as �0 D !1 � � � � � !d�1 � xd � !dC1 � � � � � !N,then define FD.G0;�0/. Then for xd:

Owd.xd/ D FD.G0;�0/

.d/

�Q.d/

�C: (2.10)

where

Q D

�OS �

n2N;n¤dUn.xn/

��

NCk2KOTk: (2.11)

lower case “./.d/” denotes the n-mode layout of that dimension.• STEP +1: Transformation error: This step is a numerical checking of the

accuracy of the resulting TP function over a huge number of random points in �.

Proof 3. Szeidl et al. [8] proves that the TP model transformation numericallyreconstructs the HOSVD canonical form in case of scalar output functions, i.e.,that if Gn ! 1 then OS ! S and Un D OW

D.!n;Gn/n ! W

D.!n;Gn/n . If we consider

matrices OTk resulting from SVD in the same way as matrices Un are consideredfor n 2 N in the proof presented in paper [8] (but without transforming themto functions), we arrive at a proof of the claim that the TP model transformationnumerically reconstructs the HOSVD based canonical form ( OT ! T, as well as aquasi-HOSVD based canonical form when we multiply by OTk (see Theorems 1.1and 1.2). Paper [8] also gives theorems for the speed of the convergence forthe numerical reconstruction depending on whether we use equidistant or non-equidistant rectangular grids for discretization.

Remark 2.1. The numerical implementation limits the grid density, as 8n 2

N W Gn ! Gmaxn < 1. Furthermore, the computational load of HOSVD can

easily explode as Gn and N grow larger. These factors form the bottlenecksof this algorithm. However, we can still say that the TP model transformationnumerically reconstructs S and the weighting functions. Further, papers [4, 7]propose very effective computational complexity reduction techniques for the TPmodel transformation, especially for cases when it is executed on qLPV models.

Remark 2.2. The paper of Szeidl et al. [8] also derives theorems for the smallestgrid density necessary for finding all the ranks of the TP function or model, thus, thediscretization density should be set according to Szeidl’s theorems and based on thefact that the maximum rank is determined by the number of the weighting functions


by dimensions of the given TP function or model. If we do not see the structure ofthe given TP function or model to be transformed, or the given model is not a TPfunction or model (see later), then we can practically use a grid with the highestdensity made possible by the numerical implementation. If we have limitations forthe maximum number of weighting functions and vertexes to be accepted, thenwe may apply Szeidl’s theorem as in the case where we have information aboutthe maximum rank. As a matter of fact, this may lead to an approximation if theaccepted number of weighting functions is less than the rank of the TP function.

Remark 2.3. If the density of the discretization grid is not sufficiently high to findthe rank of the given TP function, then the TP model transformation results in anapproximation only.

In the case of the above two remarks the transformation works as in the case ofnon-TP functions (see details later and also [9]).

2.1.1 Numerical Example

The example of this section presents a HOSVD based canonical from (e.g.,transformed from the result of an identification) of a very simple qLPV state-spacemodel. This numerical example will be used in this part of the book to study thedifferent features of the generalized TP model transformation and the convex hullmanipulation techniques to be discussed in the next sections. All examples in thebook are computed using the TP-tool MATLAB toolbox [5].

Let us assume that we have the following qLPV model:

�Px.t/y.t/

�D S.p.t//

�x.t/u.t/

�; (2.12)

where p.t/ 2 � � R2, � D Œ�5; 5� � Œ�5; 5� and

S.p.t// D

�p2

1.t/ p22sin.2p2.t//

2 p1.t/ C p2.t/

�: (2.13)

Executing the TP model transformation over G D 137 � 137 results in a verysimple quasi-HOSVD based canonical form:

S.p.t// D S �n2N

wn.pn.t//; (2.14)

where S 2 R3�3�2�2 and N D f1; 2g. The normalized (and the original)singular values of the first dimension are �1;1 D 13:00166.1781:23/, �1;2 D

4:54219.622:28/ and �1;3 D 2:90790.398:382/. The singular values of the seconddimension are �2;1 D 11:88067.1627:65/, �2;2 D 7:05744.966:869/ and �2;3 D

2:67823.366:918/.

2.1 Original TP Model Transformation 15

This two-dimensional case can be easily given without tensor operations as:

S.p.t// D

3X

iD1

3X

jD1

w1;i.p1.t//w2;j.p2.t//Si;j D

9X

rD1

wr.p.t//Sr; (2.15)

where vertexes Sr 2 R2�2 stored in S are equivalent to Si;j and wr.p.t// D

w1;i.p1.t//w2;j.p2.t//, where r D 1; : : : ; 9 is a one-dimensional linear indexing of2 dimensional index i; j.

The vertex systems of the HOSVD based canonical form are:

S1;1 D 1000 �

1:5199 �0:0000

0:2379 �0:0000

�(2.16)

S2;1 D

324:4647 0:0000

�135:9956 �0:0000

�(2.17)

S3;1 D

�0:0000 0:0000

�0:0000 �398:3823

�(2.18)

S1;2 D

0:0000 826:6437

0:0000 145:6048

�(2.19)

S2;2 D

0:0000 �472:6149

�0:0000 �83:2463

�(2.20)

S3;2 D 10�12 �

�0:0000 �0:2981

�0:0001 �0:2593

�(2.21)

S1;3 D

0:0000 55:2557

�0:0000 �313:7040

�(2.22)

S2;3 D

0:0000 �31:5912

�0:0000 179:3532

�(2.23)

S3;3 D 10�13 �

�0:0000 �0:1992

�0:0008 0:6839

�(2.24)

and the assigned weighting functions are shown in Fig. 2.1.If we would like to generate the “full” HOSVD based canonical form, we can

execute the HOSVD on all dimensions of S. This results in:

S.p.t// D S0 �n2N

wn.pn.t// �3 T1 �4 T2 (2.25)

T1 D

�0:987937 �0:154851

�0:154851 0:987937

�


0.1

0.05

0

-0.05

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

-0.1

-0.15

-5

0.1

0.15

0.2

0.05

0

-0.05

-0.1

-0.15

-0.2

-4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

P2

w3

w3

w2

w2

w1

w1

Fig. 2.1 Weighting functions of the exact HOSVD and the quasi-HOSVD based canonical form

2.2 Bi-Linear TP Model Transformation 17

T2 D

�0:999837 �0:018028

�0:018028 0:999837

�

and the new vertexes are:

S01;1 D

131:9268 �0:0000

17:6940 0:0000

�(2.26)

S02;1 D

31:8852 0:0000

�9:3233 �0:0000

�(2.27)

S03;1 D

�0:0000 �0:0000

0:0000 �31:9142

�(2.28)

S01;2 D

�0:0000 �82:8854

�0:0000 5:1783

�(2.29)

S02;2 D

�0:0000 43:6737

0:0000 �2:7285

�(2.30)

S03;2 D 10�12 �

0:0000 �0:1790

�0:0000 0:1688

�(2.31)

S01;3 D

0:0000 �1:7340

0:0000 �27:7556

�(2.32)

S02;3 D

0:0000 0:9137

�0:0000 14:6249

�(2.33)

S03;3 D 10�14 �

�0:0000 �0:0079

0:0000 �0:1421

�(2.34)

Obviously the weighting functions do not change (Fig. 2.1).In order to perform complexity trade-off we can execute RHOSVD during the

TP model transformation and discard the singular values in increasing order, namely�2;3, �1;3, �1;2, �2;2. Figures 2.2, 2.3, 2.4, and 2.5 show the weighting functions ofthe relaxed quasi and “full” HOSVD based canonical forms.

2.2 Bi-Linear TP Model Transformation

In various engineering cases we have different accuracy requirements for differentcomponents of the TP function; hence, it is not always necessary to find all pointsof the weighting functions in Step 3 of the TP model transformation. For instance,in case of robust control design the precise core tensor S is important to the extent


0.1

0.05

0

-0.05

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

-0.1

-0.15

-5

0.1

0.15

0.05

0

-0.05

-0.1

-0.15

-4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

P2

w3

w2

w2

w1

w1

Fig. 2.2 Weighting functions of the RHOSVD based canonical form where 5 singular values arekept


0.1

0.05

0

-0.05

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

-0.1

-0.15

-5

0.1

0.15

0.05

0

-0.05

-0.1

-0.15

-4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

P2

w2

w2

w1

w1



-0.06

-0.04

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

-0.16

-0.14

-0.12

-0.1

-0.08

-5

0.1

0.15

0.05

0

-0.05

-0.1

-0.15

-4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

P2

w2

w1

w1



-0.06

-0.04

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

-0.16

-0.14

-0.12

-0.1

-0.08

-5

-0.0854

-0.0854

-0.0854

-0.0854

-0.0854

-0.0854

-0.0854

-0.0854

-0.0854

-4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

x2

w1

w1



that the control design is based on it; however, in the final implementation of theTP controller, we can accept a good piece-wise approximation of the weightingfunctions. This idea leads to a practically useful engineering implementation, wherewe simply use the piece-wise linear variant of the weighting function in thecontroller.

Definition 2.3 (Piece-Wise Linear Weighting Function System Denoted byNw.x/). Function Nw.x/, including functions Nwi.x/, is defined by matrix U and gridG over x 2 ! in such a way that U D NWD.!;G/. A linear interpolation betweenneighboring values in each column of U fully defines the piece-wise linear functionsNwi.x/.

Algorithm 2 (Bi-Linear TP Model Transformation). The Bi-linear TP model

transformation results in a bi-linear approximation f .x/ � SN�

nD1Nwn.xn/ of the given

function fit to a given grid G. It differs only in Step 3 as:STEP 3: Nwn.xn/ is directly defined by Un (determined in Step 2) and grid G.


Let us take the simple dynamic TP model from the previous section and executethe Bi-linear TP model transformation over a sparse grid G ( G1 � G2 D 10 � 10)to define the quasi-HOSVD based canonical form. Figure 2.6 shows the piece-wiselinear weighting functions of the resulting TP model.

The vertexes are:

S1;1 D

131:9268 �0:0000

17:6940 0:0000

�(2.35)

S2;1 D

31:8852 0:0000

�9:3233 �0:0000

�(2.36)

S3;1 D

�0:0000 0:0000

�0:0000 �31:9142

�(2.37)

S1;2 D

�0:0000 �82:8854

�0:0000 5:1783

�(2.38)

S2;2 D

�0:0000 43:6737

0:0000 �2:7285

�(2.39)

S3;2 D 10�13 �

0:0000 0:0329

0:0000 0:1332

�(2.40)


−5 −4 −3 −2 −1 0 1 2 3 4 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

p1

Wei

ghtin

g fu

nctio

ns

w2

w1

w3

−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

p2

Wei

ghtin

g fu

nctio

ns

w2

w1

w3

Fig. 2.6 Weighting functions of the bi-linear canonical form


S1;3 D

0:0000 �1:7340

0 �27:7556

�(2.41)

S2;3 D

0:0000 0:9137

0 14:6249

�(2.42)

S3;3 D 10�14 �

�0:0000 0:0069

�0:0000 �0:7105

�(2.43)

2.3 Enriched TP Model Transformation

If the grid density is sufficient to find the precise core tensor, but is too sparseto determine the weighting functions Nwn.xn/ with good resolution, then we maycombine the third steps of the TP model transformation and the bi-linear TPmodel transformation. Step 3 of the TP model transformation does not require theexecution of HOSVD. Only the memory available limits the off-line storage of anumber of points of wn.xn/ in Step 3, which can readily be calculated over any x.Therefore, we may simply determine Hn new gridpoints on dimension n in Step 3 ofthe TP model transformation, where Hn can be considerably larger than Gmax

n , andwe can determine W

D.!n;Hn/n in Step 3 of the TP model transformation, which leads

to a better resolution of Nwn.xn/.

Remark 2.4. The enriched TP model transformation can be used as a tool forrelaxing the computational complexity of the TP model transformation. If weknow the ranks of the given function in each dimension (i.e., if we know the TPstructure of the given function) we can set the minimal grid density accordingly[8]. Alternatively, we may increase the grid density gradually until we find all thenonzero ranks (i.e., the point after which only the number of the zero singular valuesare found to increase with Gn); following this point, we do not need to furtherincrease the grid density, but may instead use the enriched TP model transformationto define a high resolution for the weighting functions. If we are not sure whetherwe have found all ranks, we may still proceed further with the enriched TP modeltransformation. In either case, the step where we numerically check the solution(Step +1) will indicate us whether the system has a sufficient number of weightingfunctions per dimension. This helps in cases where N is large, so that we can set avery sparse grid per dimension in order to be able to execute the computationallyexpensive HOSVD then we can define the higher resolution of the piece-wiseweighting functions.

2.4 Convex TP Model Transformation: Convex Hull Manipulation 25


Let us continue the example of the previous section and increase the grid densityto 20-by-20 and to 500-by-500. Using the enriched TP model transformation, wehave the weighting functions shown in Figs. 2.7 and 2.8. It is important to remarkthat the vertexes do not change in obvious ways in the present case (only Step 3of the TP model transformation is executed for the extra, dense grid); it is mostlythe resolution of the weighting functions that is improved without having to executeHOSVD on a huge tensor that would be obtained through discretization over a dense20-by-20 or 500-by-500 grid. Note that the weighting functions will converge tothe weighting functions defined by the points taken over the 10 � 10 grid. This isdifferent from the case where the density of the discretization grid is increasedin the first step of the TP model transformation, and the weighting functionsconverge to the singular weighting functions of the HOSVD based canonicalform.

2.4 Convex TP Model Transformation: Convex HullManipulation

The goal is to transform the given function to a convex TP function S �n2N

wCon .xn/.

This section focuses on the step where the weighting functions can be manipulatedand where the already published convex hull generation methods can be incorpo-rated in the algorithm of the TP model transformation.

Algorithm 3 (Convex TP Model Transformation). Let us assume a given TPfunction f .x/, x 2 � � RN. The goal is to numerically reconstruct TP functionf .x/ D S �

n2NwCo

n .xn/ and to include a complexity trade-off if needed. The steps of

this transformation are the same as in the TP model transformation. Only Step 2is extended by the convex hull generation. We also add evaluation Step +2 to thealgorithm to be executed after Step 3 or Step +1.

• STEP 2: Reconstruct the core of the TP structure: Determine S and Un viaHOSVD, then use the SN and NN transformations introduced by Yam in [11],which transform Un to UCo

n , which will be considered as WCo;D.!n;Gn/ in Step 3;

further, define S for FD.�;G/ D S �n2N

UCon . For instance S D FD.�;G/ �

n2N

�UCo

n

�C,

where “+” means pseudo inverse.• STEP +2: The weighting functions are wCo

n .xn/ in the case of the bi-linear TPmodel transformation; however one has to check this between the grid if allpoints of the weighting functions are recalculated.


−5 −4 −3 −2 −1 0 1 2 3 4 5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

p1

Wei

ghtin

g fu

nctio

ns

w2

w1

w3

−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

p2

Wei

ghtin

g fu

nctio

ns

w2

w1

w3

Fig. 2.7 Results of the enriched TP model transformation when G is increased from 10 to 20


−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

p1

Wei

ghtin

g fu

nctio

ns

w2

w1

w3

−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

p2

Wei

ghtin

g fu

nctio

ns

w2

w1

w3

Fig. 2.8 Results of the enriched TP model transformation when G is increased from 20 to 500


Further types of convex TP functions can be generated by using Normalized(NO), Close to NO (CNO), Inverse NO (INO), Relaxed NO (RNO), Inverse RNO(IRNO), etc. transformations in the same way as SN and NN transformation isexecuted in Step 2 above, see for instance [2, 6, 10, 11].


Let us continue the example of the previous section. Using the convex TP modeltransformation the following models are derived:

2.4.1.1 SNNN Type TP Model

The vertexes of the model are:

S1;1 D

122:9804 �62:4297

2:0000 28:5317

�(2.44)

S2;1 D

�4:5476 �62:4297

2:0000 77:5678

�(2.45)

S3;1 D

�4:5476 �62:4297

2:0000 28:5317

�(2.46)

S1;2 D

122:9804 124:7210

2:0000 22:1938

�(2.47)

S2;2 D

�4:5476 124:7210

2:0000 71:2299

�(2.48)

S3;2 D

�4:5476 124:7210

2:0000 22:1938

�(2.49)

S1;3 D

122:9804 �13:0489

2:0000 �17:7209

�(2.50)

S2;3 D

�4:5476 �13:0489

2:0000 31:3152

�(2.51)

S3;3 D

�4:5476 �13:0489

2:0000 �17:7209

�(2.52)


The weighting functions of the resulting TP model are shown in Fig. 2.9. Theweighting functions of the relaxed SNNN type TP models are shown in Figs. 2.10,2.11, and 2.12.

Remark 2.5. Figure 2.12 shows an interesting case when the number of weightingfunctions is still 4 (the number of singular values kept is 3). This means thatthe SNNN transformation increased the number of weighting functions to achieveconvexity. Thus, in this case we do not have complexity reduction, but the approx-imation accuracy is degraded because of the rank reduction (the added weightingfunctions are not linearly independent). As a result, this solution actually has nomeaning in the sense of TP model relaxation. The previous solution has a betterapproximation accuracy with the same complexity!

2.4.1.2 CNO Type TP Model


S1;1 D

�18:5023 8:2775

2:0000 �12:9462

�(2.53)

S2;1 D

25:0000 8:2775

2:0000 �7:8897

�(2.54)

S3;1 D

25:0000 8:2775

2:0000 �18:0028

�(2.55)

S1;2 D

�18:5023 �21:1903

2:0000 �2:1129

�(2.56)

S2;2 D

25:0000 �21:1903

2:0000 2:9436

�(2.57)

S3;2 D

25:0000 �21:1903

2:0000 �7:1695

�(2.58)

S1;3 D

�18:5023 77:4772

2:0000 90:3550

�(2.59)

S2;3 D

25:0000 77:4772

2:0000 95:4115

�(2.60)

S3;3 D

25:0000 77:4772

2:0000 85:2984

�(2.61)

The weighting functions of the resulting TP model are shown in Fig. 2.13.


Wei

ghtin

g fu

nctio

ns

-5

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Wei

ghtin

g fu

nctio

ns

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w1

w3

w3

w2

w2

Fig. 2.9 Weighting functions of the SNNN type exact TP model


Wei

ghtin

g fu

nctio

ns

-5

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Wei

ghtin

g fu

nctio

ns

1

0.7

0.8

0.9

0.5

0.6

0.4

0.3

0.2

0.1

0-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w1

w3

w2

w2

Fig. 2.10 Weighting functions of the SNNN type relaxed TP model where 5 singular values arekept


Wei

ghtin

g fu

nctio

ns

-5

0.8

0.9

1

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Wei

ghtin

g fu

nctio

ns

1

0.7

0.8

0.9

0.5

0.6

0.4

0.3

0.2

0.1

0-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w1

w2

w2



Wei

ghtin

g fu

nctio

ns

-5

0.8

0.9

1

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Wei

ghtin

g fu

nctio

ns

1

0.7

0.8

0.9

0.5

0.6

0.4

0.3

0.2

0.1

0-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w1

w2

w2



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P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w3

w1

w3

w2

w2

Fig. 2.13 Weighting functions of the CNO type exact TP model

2.5 Pseudo TP Model Transformation 35

The weighting functions of the relaxed CNO type TP models are shown inFigs. 2.14, 2.15, and 2.16. We have the same situation with the number of weightingfunctions as in the previous case when 3 singular values are kept.

2.4.1.3 IRNO Type TP Model


S1;1 D

48:2074 �32:7238

2:0000 6:2024

�(2.62)

S2;1 D

7:8944 �32:7238

2:0000 18:9717

�(2.63)

S3;1 D

�6:8121 �32:7238

2:0000 �0:0777

�(2.64)

S1;2 D

48:2074 40:0422

2:0000 �0:4252

�(2.65)

S2;2 D

7:8944 40:0422

2:0000 12:3441

�(2.66)

S3;2 D

�6:8121 40:0422

2:0000 �6:7053

�(2.67)

S1;3 D

48:2074 �7:3183

2:0000 �12:8303

�(2.68)

S2;3 D

7:8944 �7:3183

2:0000 �0:0610

�(2.69)

S3;3 D

�6:8121 �7:3183

2:0000 �19:1104

�(2.70)

The weighting functions of the resulting TP model are shown in Fig. 2.17.The weighting functions of the relaxed IRNO type TP models are shown in

Figs. 2.18, 2.19, and 2.20. We have the same situation with the number of weightingfunctions as in the previous case when 3 singular values are kept.

2.5 Pseudo TP Model Transformation

We may want to find an equivalent TP function with a predefined weighting functionsystem, namely, to transform a given function to a TP function with given weightingfunctions. For such purposes, we propose the pseudo TP model transformation asfollows:


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0-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w3

w1

w2

w2

Fig. 2.14 Weighting functions of the CNO type relaxed TP model where 5 singular values arekept


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0-4 -3 -2 -1 0 1 2 3 4 5

P2

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P1

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P2

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0-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w3

w1

w2

w2

w3

Fig. 2.17 Weighting functions of the IRNO type exact TP model


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0-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w1

w3

w2

w2

Fig. 2.18 Weighting functions of the IRNO type relaxed TP model where 5 singular values arekept


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P2

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P1

w1

w1

w2

w2


2.6 Partial TPC Model Transformation 43

Algorithm 4 (Pseudo TP Model Transformation, TPC Model Transformationfor Short). Assume a given function Y D f .x/, x 2 � � RN and weightingfunction system wn.xn/. The goal is to determine S such that f .x/ D S �

n2Nwn.xn/,

or if this is not possible, then the goal is to find Of .x/ D S �n2N

wn.xn/, where Of .x/ is

the best or at least a good approximation under the rank constraints implicitly givenby wn.xn/ (e.g., the number of linearly independent weighting functions may be lessin dimension n than rankn.f .x//). Steps 0 and +1 are the same as in the TP modeltransformation, and only the following steps are extended:

• STEP 1: Discretization: Determine FD.�;G/ and WD.!n;Gn/n .

• STEP 2: Determine the core tensor:

S D FD.�;G/ �n2N

�WD.!n;Gn/

n

�C: (2.71)

If WD.!n;Gn/n introduces rank reduction, then we arrive at Of .x/ D S �

n2Nwn.xn/.

This works like in the case of complexity trade-off via TP model transformation.If we have predefined transformations T:

S D Q �NCk2K

TCk ; (2.72)

where

Q D FD.G;�/ �n2N

�WD.!n;Gn/

n

�C: (2.73)

• STEP +2: Checking the weighting functions: Once we have the core tensor S wemay recalculate the weighting functions between the points of WD.!n;Gn;/

n throughStep 3 and numerically compare to the given wn.xn/.

2.6 Partial TPC Model Transformation

Algorithm 5 (Partial TPC Model Transformation). Assume a given TP functionY D f .x/, x 2 � � RN. Further, assume a given weighting function system wd.xd/,d 2 D � N. The goal is to determine S such that f .x/ D S �

n2Nwn.xn/, where

weighting functions wn.xn/, n … D, are the same as in the case of the TP modeltransformation. If this is not possible, or if we need a complexity trade-off, thenthe goal is to find Of .x/ D S �

n2Nwn.xn/, where Of .x/ is the best, or at least a good

approximation under the rank constraint implicitly given by wd.xd/. Steps 0,1,3,+1,and +2 are the same as in the case of the TPC model transformation:


STEP 2: Determine the core tensor as K D FD.�;G/ �d2D

�W

D.!d ;Gd/d

C

. Execute

HOSVD on K in all dimensions n … D to obtain:

K D S �n2Nn…D

Un: (2.74)

Let WD.!n;Gn/n D Un, n … D, in which case:

FD.G;�/ D

S �n2Nn…D

Un

!

�d2D

WD.!d ;Gn/d D (2.75)

D S �n2N

WD.!n;Gn/n : (2.76)


Let us use the model discussed in the previous examples. We assume that thefollowing weighting function systems

w1.p1.t// D�0:5sin.p1.t/=5/ 0:000025p2

1.t/ 0:005p1.t/�

; (2.77)

and

w2.p2.t// D�0:2p2.t/ �0:2p2.t/ 20

�: (2.78)

are given, see Fig. 2.21.The TPC model transformation results in the following vertexes

S1;1 D

�0:0000 0:0295

�0:0000 0:0343

�(2.79)

S2;1 D

�0:0000 �1:1354

0:0000 �1:3181

�(2.80)

S3;1 D

�0:0000 0:9292

�0:0000 1:0787

�(2.81)

S1;2 D

0:0000 �0:0295

0:0000 �0:0343

�(2.82)

S2;2 D

0:0000 1:1354

�0:0000 1:3181

�(2.83)


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-0.4

25

20

15

10

5

0

-5

-10

-15

-20

-25

-0.3

-0.2

-0.1

0

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-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

w1

w1

w2

w3

w3

w2

Fig. 2.21 Predefined weighting function systems


S3;2 D

0:0000 �0:9292

0:0000 �1:0787

�(2.84)

S1;3 D

0:2399 �0:0407

0:0186 �0:0946

�(2.85)

S2;3 D

1:5845 1:5668

�0:7171 3:6380

�(2.86)

S3;3 D

0:0914 �1:2822

0:5868 �2:2419

�(2.87)

Let us keep the weighting function system on the first dimension and assume thatwe need a CNO type system of weighting functions in the second dimension. Thepartial TPC model transformation results in the following vertexes:

S1;1 D

4:7985 �0:2571

0:3728 0:9948

�(2.88)

S2;1 D

31:6892 9:8893

�14:3410 �38:2666

�(2.89)

S3;1 D

1:8274 �8:0930

11:7362 46:0220

�(2.90)

S1;2 D

4:7985 5:2362

0:3728 �1:0247

�(2.91)

S2;2 D

31:6892 �201:4098

�14:3410 39:4138

�(2.92)

S3;2 D

1:8274 164:8268

11:7362 �17:5490

�(2.93)

S1;3 D

4:7985 �29:8745

0:3728 �12:1163

�(2.94)

S2;3 D 1000 �

0:0317 1:1491

�0:0143 0:4661

�(2.95)

S3;3 D

1:8274 �940:4026

11:7362 �366:6966

�: (2.96)

The related weighting functions of the second dimension are shown in Fig. 2.22.If we have the same predefined weighting functions on the first dimension and weneed IRNO type weighting function system on the second dimension we obtain theresults shown on second image of Fig. 2.22.


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-4 -3 -2 -1 0 1 2 3 4 5

P2

-5 -4 -3 -2 -1 0 1 2 3 4 5

P2

w1

w1

w2

w3

w3w2

Fig. 2.22 The CNO and the IRNO type weighting functions obtained by the partial TPC modeltransformation


2.7 Multi TP Model Transformation

We may want to transform a set of functions simultaneously to a set of TP functionswith the same weighting functions system. First we investigate a simple case, whenthe functions have outputs with same size, then we discuss the general case wheneach functions may have different sized output.

Algorithm 6 (Multi TP Model Transformation-Simple Case). Let us assumethat we have parameter dependent scalar, vector, matrix, or tensor functionsY D fl.x/, l 2 L, x 2 � � RN. An important property is that they havethe same size and dimensionality as 8l W fl.x/ 2 RO1�O2��OK . The goal is tofind their TP function representations over the same weighting function system as8l W fl.x/ D Sl �

n2Nwn.xn/.

• STEP 0: Define discretization grid G fit to �.• Step 1: Construct an N C K C 1 dimensional tensorH storing all the discretized

function FD.�;G/l into the .N C K C 1/th dimension.

• Step 2: Execute Step 2 as in previous algorithms, but on tensor H that result in(SVD is not executed in the N C K C 1th dimension)

H DM �n2N

wn.xn/: (2.97)

Then decompose tensorM in dimension N C K C 1 into tensors Sl just opposeto the construction done in Step 1. This results in:

FD.�;G/l D Sl �

n2Nwn.xn/: (2.98)

The more complex case:

Algorithm 7 (Multi TP Model Transformation). Let us assume that we haveparameter dependent scalar, vector, matrix, or tensor functions Y D fl.x/, l 2 L,x 2 � � RN. An important property is that they may have different sizes anddimensionality as fl.x/ 2 ROl;1�Ol;2��Ol;Kl . The goal is to find their TP functionrepresentations over the same weighting function system as fl.x/ D Sl �

n2Nwn.xn/:

• STEP 0: Define discretization grid G fit to �.• STEP 1: Discretization: store all the output elements of all fl.x/ in a vector that

is actually a construction of the function v.x/ D�h1.x/ h2.x/ : : : hZ.x/

�, where

Z DPL

lD1 …KlkD1Ol;kl ; or directly arrange the discretized values according to this

ordering that yields the N C 1 dimensional tensor FD.�;G/ of size G1 � G2 � � � � �

GN � Z.• STEP 2–3: These two steps are the same as in the case of the TP model

transformation (including trade-off and convex manipulation etc). As a resultwe have v.x/ D B �

n2Nwn.xn/, where B is N C 1 dimensional.

2.7 Multi TP Model Transformation 49

• STEP 4: By repartitioning tensor B in the N C 1th dimension, in a fashionopposite to Step 1, we obtain tensors Sl containing elements with size ofOl;1 � Ol;2 � � � � � Ol;Kl . Thus we have fl.x/ D Sl �

n2Nwn.xn/.

• STEP +1 and +2: These steps have the same error checking role as in the caseof the previously discussed variants of the TP model transformation.


We assume that we have two different system matrices. The first one S1.p.t// istaken from the previous example, the second one is:

S2.p.t// D

0

@p2

1 p22 � sin.p2/ p1 C p2 11

p22 � cos.2 � p2/ 7 8 p1 C 0:625 � p2

2 5 12460 p1

1

A : (2.99)

If we execute the Multi-TP model transformation with CNO transformation onthese two models then we obtain weighting functions as shown in Fig. 2.23. Thecommon rank of the two systems is 3 in the first dimension and 5 in the seconddimension. Note that if we execute the TP model transformation on S2.p.t// only,we will obtain three weighting functions on the first and only four weightingfunctions on the second dimension.

The vertexes of S1.p.t// are:

S1;1 D 103 �

0:0025 0:0165

0:0002 0:0019

�(2.100)

S2;1 D 103 �

0:0025 0:0165

0:0002 0:0030

�(2.101)

S3;1 D 103 �

�0:0019 0:0165

0:0002 0:0024

�(2.102)

S1;2 D 103 �

0:0025 �0:0007

0:0002 �0:0014

�(2.103)

S2;2 D 103 �

0:0025 �0:0007

0:0002 �0:0004

�(2.104)

S3;2 D 103 �

�0:0019 �0:0007

0:0002 �0:0009

�(2.105)

S1;3 D 103 �

0:0025 �0:0013

0:0002 0:0000

�(2.106)


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-5 -4 -3 -2 -1 0 1 2 3 4 5

P1

-5 -4 -3 -2 -1 0 1 2 3 4 5

P2

w1

w3

w3

w2

w4 w5

w1

w2

Fig. 2.23 The CNO weighting functions obtained by the multi TP model transformation

2.7 Multi TP Model Transformation 51

S2;3 D 103 �

0:0025 �0:0013

0:0002 0:0010

�(2.107)

S3;3 D 103 �

�0:0019 �0:0013

0:0002 0:0005

�(2.108)

S1;4 D 103 �

0:0025 �0:0028

0:0002 �0:0004

�(2.109)

S2;4 D 103 �

0:0025 �0:0028

0:0002 0:0006

�(2.110)

S3;4 D 103 �

�0:0019 �0:0028

0:0002 0:0001

�(2.111)

S1;5 D 103 �

0:0025 �0:0026

0:0002 �0:0014

�(2.112)

S2;5 D 103 �

0:0025 �0:0026

0:0002 �0:0004

�(2.113)

S3;5 D 103 �

�0:0019 �0:0026

0:0002 �0:0009

�(2.114)

The vertexes of S2.p.t// are:

S1;1 D 103 �

2

40:0025 �0:0107 0:0019 0:0011

�0:0022 0:0007 0:0008 0:0010

0:0002 0:0005 1:2460 �0:0005

3

5 (2.115)

S2;1 D 103 �

2

40:0025 �0:0107 0:0030 0:0011

�0:0022 0:0007 0:0008 0:0020

0:0002 0:0005 1:2460 0:0005

3

5 (2.116)

S3;1 D 103 �

2

4�0:0019 �0:0107 0:0024 0:0011

�0:0022 0:0007 0:0008 0:0015

0:0002 0:0005 1:2460 0:0000

3

5 (2.117)

S1;2 D 103 �

2

40:0025 0:0044 �0:0014 0:0011

�0:0023 0:0007 0:0008 �0:0011

0:0002 0:0005 1:2460 �0:0005

3

5 (2.118)

S2;2 D 103 �

2

40:0025 0:0044 �0:0004 0:0011

�0:0023 0:0007 0:0008 �0:0001

0:0002 0:0005 1:2460 0:0005

3

5 (2.119)


S3;2 D 103 �

2

4�0:0019 0:0044 �0:0009 0:0011

�0:0023 0:0007 0:0008 �0:0006

0:0002 0:0005 1:2460 0:0000

3

5 (2.120)

S1;3 D 103 �

2

40:0025 �0:0026 0:0000 0:0011

�0:0023 0:0007 0:0008 �0:0002

0:0002 0:0005 1:2460 �0:0005

3

5 (2.121)

S2;3 D 103 �

2

40:0025 �0:0026 0:0010 0:0011

�0:0023 0:0007 0:0008 0:0008

0:0002 0:0005 1:2460 0:0005

3

5 (2.122)

S3;3 D 103 �

2

4�0:0019 �0:0026 0:0005 0:0011

�0:0023 0:0007 0:0008 0:0003

0:0002 0:0005 1:2460 0:0000

3

5 (2.123)

S1;4 D 103 �

2

40:0025 0:0020 �0:0004 0:0011

0:0017 0:0007 0:0008 �0:0004

0:0002 0:0005 1:2460 �0:0005

3

5 (2.124)

S2;4 D 103 �

2

40:0025 0:0020 0:0006 0:0011

0:0017 0:0007 0:0008 0:0006

0:0002 0:0005 1:2460 0:0005

3

5 (2.125)

S3;4 D 103 �

2

4�0:0019 0:0020 0:0001 0:0011

0:0017 0:0007 0:0008 0:0001

0:0002 0:0005 1:2460 0:0000

3

5 (2.126)

S1;5 D 103 �

2

40:0025 0:0016 �0:0014 0:0011

0:0015 0:0007 0:0008 �0:0011

0:0002 0:0005 1:2460 �0:0005

3

5 (2.127)

S2;5 D 103 �

2

40:0025 0:0016 �0:0004 0:0011

0:0015 0:0007 0:0008 �0:0001

0:0002 0:0005 1:2460 0:0005

3

5 (2.128)

S3;5 D 103 �

2

4�0:0019 0:0016 �0:0009 0:0011

0:0015 0:0007 0:0008 �0:0006

0:0002 0:0005 1:2460 0:0000

3

5 (2.129)

2.8 Generalized TP Model Transformation

This section provides a summary of the above sections and describes the summa-rized algorithm of the TP model transformation.

2.8 Generalized TP Model Transformation 53

Let us assume a set of given functions fl.x/ 2 ROl;1��Ol;Kl , x 2 � � RN ,l 2 L. The goal is to find and further manipulate the TP model representations of thefunctions in a given �. It is irrelevant whether these functions are given using closedformulae or soft-computing techniques (e.g., fuzzy logic, neural network basedmethods, etc.); the only requirement is that their discretized variant F D.�;G/

l shouldbe available. If there are no exact TP model representations for any of the functions,then the goal is to find the best TP model representations of those functions byachieving a trade-off between approximation accuracy and the complexity of thecore tensor.

It can be further assumed that a set of predefined weighting functions wd.xd/

are given for dimensions d 2 D N, or that predefined characteristics definedby the specific points of the functions wh.xh/ expected for dimensions h 2 H N(obviously D\H D 0) are given in the form W

D.!h;Gh/h . For such cases, the following

transformation is proposed:

Algorithm 8 (Summarized TP Model Transformation). We assume that Y D

fl.x/ 2 ROl;1��Ol;Kl , x 2 � � RN, wd.xd/, d 2 D N, WD.!h;Gh/h , h 2 H N,

D \ H D 0 and � are given and 8l W F D.�;G/l exist. The transformation results in

fl.x/ D Sl �n2N

wn.xn/:

• STEP 1: Discretization over G:

– Determine F D.�;G/l 2 RG1��GN�Ol;1��Ol;Kl andWD.!d ;Gd/

d 2 RGd�Id .

– Rearrange the Ol;1 � � � � � Ol;Kl sized elements of tensor FD.�;G/l into vectors

of tensor H 2 RG1��GN�Z, Z DPL

lD1 …KlkD1Ol;kl .

• STEP 2: Determination of the TP structure

– Incorporate the predefined weighting functions or characteristics as

S0 D

�H �

d2D

�W

D.!d ;Gd/d

C�

�h2H

�W

D.!h;Gh/h

C

(2.130)

– Execute CHOSVD, specifically by discarding all zero singular values, indimensions n 2 N; n … D [ H of S0:

S0 D S00 �n2N;n…D[H

Un: (2.131)

– Let WD.!n;Gn/n D Un (n 2 N; n … D [ H), in which the fully discretized

structure of the TP model can be expressed as:

H D S00 �n2N

WD.!n;Gn/n : (2.132)


– S00 can be partitioned in dimension N C 1 according Step 1. Storing theseelements in tensor Sl in a fashion opposite to Step 1, tensor F D.�;G/

l isobtained:

F D.�;G/l D Sl �

n2NWD.!n;Gn/

n : (2.133)

– When a complexity trade-off is necessary, RHOSVD is performed by dis-carding nonzero singular values and the corresponding weighting functions.This leads to an approximation of the discretized tensor, which impliesthat the resulting TP model will only be an approximation. Obviously, thetransformation is also not exact if the rank of any WD.!n;Gn/

n , n 2 D [ H is lessthan the d-mode rank ofH .

• STEP 3: Determination of the weighting functionsThis step is the same as in the original TP model transformation. It determines

all points of the weighting functions of wn.xn/ or the piece-wise linear variantNwn.xn/ from the discretized variants WD.!n;Gn/

n .• STEP +1 and +2: Error check of the resulting TP function and the weighting

functions The numerical computation of the TP model transformation, thereduction of the number of singular values, the recalculation of the contin-uous weighting functions, and the use of piece-wise weighting functions allintroduce errors into the resulting TP model. These errors can be theoreticallybounded based on the discarded singular values, but can also be numericallyapproximated by measurement over a large number of random points in �. Thepredefined weighting functions in dimensions d 2 D can be recalculated in thethird step and checked for accuracy. Such a step can serve as a kind of evaluationof the transformation.

2.9 Interpolation of the Weighting Functions

Since the type of convexity of the TP model influences the LMI based design (seelater), it naturally follows that the control performance can be optimized throughvarious manipulations of the TP model. Typically, controller design benefits fromthe use of a tight convex hull (cf. [6]) that is able to decrease the conservativenessof the solution. However, it is not very well known in the control literature thatthe effectiveness of the observer and the resulting control performance can alsobe improved in cases by loosening the convex hull. Therefore, an optimal convexhull exists between these two opposing directions. In the following, a simple TPmodel manipulation technique is proposed for interpolation between two convex TPmodels, which captures the transition of the convex hull, for instance, between tightand loose forms. The key idea behind the interpolation is based on the interpolation

2.9 Interpolation of the Weighting Functions 55

of the weighting functions. Assuming that two different TP model representationsof a given model are available:

S.p/ D A �n2N

wAn .pn/ D B �

n2NwB

n .pn/: (2.134)

A linear interpolation characterized by a parameter � 2 Œ0; 1� can be applied asfollows:

w�n .pn/ D �wA

n .pn/ C .1 � �/wBn .pn/: (2.135)

Executing the TPC model transformation on the given model with the predefinedweighting functions w�

n .pn/, the following form is obtained:

S.p/ D V �n2N

w�n .pn/ D A �

n2NwA

n .pn/ D B �n2N

wBn .pn/: (2.136)

Note that this technique interpolates the weighting functions and, hence, thevertexes of the interpolated TP model are not the linear interpolation of the vertexesof the given two TP models.


Let us examine the following simple example. We assume the following TP modelS.p.t// D

P3rD1 wr.p.t//Sr given in HOSVD based canonical form, where p.t/ 2

� D Œ0; 0:04� and

S1 D

�11480:824 �11929:594 �8:6483351

11489:012 11921:288 8:6422240

�

S2 D

�132:28962 127:48303 2:1133932

130:26178 �125:36528 �2:1102187

�

S3 D

0:2018141 �0:1981534 �0:4225654

0:2117763 �0:2086436 0:4221598

�:

The weighting functions are depicted in Fig. 2.24.Let us execute the convex TP model transformation on the given TP model with

SN and NN transformation. This leads to a TP model where the vertexes define aconvex hull around the given qLPV model. The weighting functions are given inFig. 2.25. The vertexes are:

Ssnnn1 D

935:16726 1064:7034 2:0443571

�937:13030 �1062:7079 �2:0421822

�


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Fig. 2.24 Weighting function system of the HOSVD canonical form

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Fig. 2.25 SNNN type weighting function system

2.9 Interpolation of the Weighting Functions 57

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2.26 CNO type weighting function system

Ssnnn2 D

1159:1814 836:83060 �2:2077200

�1157:0535 �839:07308 2:2038245

�

Ssnnn3 D

959:93584 1040:8290 0:9979833

�960:88224 �1039:8563 �0:9970246

�

We also execute the TP model transformation with CNO transformation thatleads to a weighting functions system such that the vertexes form a tight convexhull around the given model,

S.p.t// D

3X

rD1

wSNNr .p.t//SSNN

r D

3X

rD1

wCNOr .p.t//SCNO

r : (2.137)

The weighting functions are given in Fig. 2.26. The vertexes are:

Scno1 D

978:02837 1021:7823 0:9348855

�978:91888 �1020:8823 �0:9341519

�

Scno2 D

996:04859 1003:8310 0:4273445

�996:44795 �1003:4311 �0:4272558

�

Scno3 D

959:58489 1041:1860 1:0046280

�960:53769 �1040:2067 �1:0036599

�


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Fig. 2.27 Interpolated weighting functions at � D 0:53

We define the linear interpolation between the weighting function systems for allp.t/ 2 � such that:

w�n .p.t// D �wCNO

n .p.t// C .1 � �/wSNNn .p.t//; (2.138)

where � 2 Œ0; 1�. This means that we are tightening the convex hull with �. Then,using the TPC model transformation, we determine the interpolated and exact TPmodel such that:

S.p.t// D

3X

rD1

w�r .p.t//S�

r : (2.139)

Figure 2.27 shows the case when � D 0:53.The vertexes are:

S�1 D

980:74020 1018:7800 0:9897259

�981:68530 �1017:8291 �0:9889653

�

S�2 D

1018:4915 980:84818 0:06826419

�1018:5465 �980:80845 �0:06872040

�

S�3 D

959:61303 1041:1578 1:0039480

�960:56517 �1040:1791 �1:0029807

�

2.10 Unifying the Weighting Functions 59

2.10 Unifying the Weighting Functions

Let us assume that a set of TP functions are given in the form:

Yl D Sl �n2N

wl;n.xn/; (2.140)

l 2 L. The goal is to find TP model representations in �:

Yl D Tl �n2N

wUn .xn/; (2.141)

where the TP models have the same weighting functions systems (superscript “U”means unified). Thus the goal is to ensure:

8l W Sl �n2N

wl;n.xn/ D Tl �n2N

wUn .xn/; (2.142)

One obvious way to fulfill these criteria is to execute the Multi-TP model trans-formation. However, such an approach would result in a significant computationalload if HOSVD is executed on a large-sized discretized tensor constructed from allof the individual TP functions. We may have a simplified way here as we know theTP structure of the given TP functions; hence, we can easily determine W

D.!n;Gn/l;n .

In order to find the unified set of weighting functions, the matrices of the discretizedweighting functions can be stored in the form:

Hn D�W

D.!n;Gn/1;n � � � W

D.!n;Gn/L;n

; (2.143)

and compact SVD (or reduced, if complexity reduction is on purpose) can beexecuted on H as

Hn D UnDnVTn D UnLn: (2.144)

If needed, the manipulation of the type of the unified weighting functions(defining SN, NN, CNO, etc. type unified functions) can be integrated at this pointwith the execution of SVD to obtain such a Un that leads to the desired weightingfunctions. The discretized unified weighting functions, then, have the followingform:

WU;D.!n;Gn/n D Un; (2.145)

Given these weighting functions, there are two ways to proceed. The firstapproach consists in executing the TPC model transformation on the given setof TP functions using the predefined weighting functions wU

n .xn/ available inthe discretized variant W

U;D.!n;Gn/n . An alternative approach further relaxes the

computational requirements. Since we have all the transformation matrices in Ln


we can directly derive the core tensors to the unified weighting functions. Thus, wedefine the partitions of the matrix Ln according to the number of the columns ofblocks WD.!n;Gn/

l;n in Hn as follows:

Ln D�Vn;1 � � � Vn;L

�: (2.146)

Thus

Hn D�W

D.!n;Gn/1;n � � � W

D.!n;Gn/L;n

D Un

�Vn;1 � � � Vn;L

�(2.147)

that means

WD.!n;Gn/l;n D UnVn;l: (2.148)

The core tensors can be derived using these transformation matrices as:

Sl �n2N

WD.!n;Gn/l;n D Sl �

n2NUnVn;l D (2.149)

D

�Sl �

n2NVn;l

��

n2NUn D Tl �

n2NUn D (2.150)

D Tl �n2N

WU;D.!n;Gn/n : (2.151)

Thus

Tl D Sl �n2N

Vn;l: (2.152)

Finally, the third step of the generalized TP model transformation can beexecuted to determine the continuous weighting functions to one of the pairs offl.x/ and Sl, since we have unified weighting functions.

2.11 Operations Between TP Functions

Once a set of unified weighting functions are obtained, the addition of TP functionscan be performed easily by adding together the corresponding core tensors:

S �n2N

wn.xn/ D

LX

lD1

�Al �

n2Nwn.xn/

�; (2.153)

2.12 Towards Approximation in Case of Non-TP Functions 61

thus

S D

LX

lD1

Al: (2.154)

Further operations are defined between TP models in the following form:

S.x/ D f .A.x/; B.x// D f .A �n2N

wAn .xn/;B �

n2NwB

n .xn//; (2.155)

can also be numerically reconstructed:

S.x/ D S �n2N

wn.xn/; (2.156)

by executing the TP model transformation on f .A.x/; B.x//. Thus, the TP modeltransformation is executed on the whole function. The result will be exact if itis contained within the TP functions, namely, if the rank of S.x/ is bounded bydimensions. For instance, adding TP functions can easily be derived using the TPmodel transformation, even in cases when these functions are given using differentsoft-computing representations (analytical operations between fuzzy and neuralnetwork based representations would be very hard as if not impossible). While theseclaims are quite trivial, they are not applied nearly as much in the literature as theirsignificance would suggest.

2.12 Towards Approximation in Case of Non-TP Functions

The TP model transformation works even in cases where the entire TP modelstructure of the given function or model are hidden. The only requirement of thepresented algorithms is that the model at hand should be discretizable over G. Inthe case of functions which have a TP model structure (with bounded number ofcomponents), once we find all the ranks through the TP model transformation, thenirrespective of how many extra gridpoints we add to the discretization, the numberof the nonzero singular values will not increase upon the execution of HOSVD.If we have a function that has no TP model representation (with bounded numberof components), then the rank of the discretized tensor will increase (at least inone dimension) with the density of G, such that the rank will always be Gn. Sincethe computational power available limits Gn, it becomes irrelevant in engineeringapplications whether the given function is a TP function with a higher rank thanGn, or if it is a function that does not have an exact TP function representation.We are faced with the same uncertainty when we have a limitation on the numberof resulting weighting functions and we have to execute RHOSVD in any case.If we find that the given function and the resulting TP function or model areequivalent in a numerical sense, then we may suppose that we have found all the


ranks. Therefore, it should be kept in mind that in a mathematical sense, we arealways dealing with approximations unless we perform further analysis; however,from an engineering perspective, the possible cases will be numerically equivalent(limitations are imposed only by the available computational resources).

References

1. P. Baranyi, TP model transformation as a way to LMI-based controller design. IEEE TransInd. Electron. 51(2), 387–400 (2004)



4. P. Baranyi, Z. Petres, P. Korondi, Y. Yam, H. Hashimoto, Complexity relaxation of the tensorproduct model transformation for higher dimensional problems. Asian J. Control 9(2), 195–200(2007)

5. P. Baranyi, Z. Petres, Sz. Nagy, TPtool — tensor product MATLAB toolbox. Website (2007).http://tp-control.hu/

6. P. Baranyi, Y. Yam, P. Varlaki, Tensor Product Model Transformation in Polytopic Model-Based Control (CRC/Taylor & Francis Group, Boca Raton/London, 2013)

7. Sz. Nagy, Z. Petres, P. Baranyi, H. Hashimoto, Computational relaxed TP model transforma-tion: restricting the computation to subspaces of the dynamic model. Asian J. Control 11(5),461–475 (2009)

8. L. Szeidl, P. Várlaki, HOSVD based canonical form for polytopic models of dynamic systems.J. Adv. Comput. Intell. Intell. Infor. 13(1), 52–60 (2009)


10. P. Várkonyi, D. Tikk, P. Korondi, P. Baranyi, A new algorithm for RNO-INO type tensorproduct model representation, in Proceedings of the IEEE 9th International Conference onIntelligent Engineering Systems (2005), pp. 263–266

11. Y. Yam, P. Baranyi, C.T. Yang, Reduction of fuzzy rule base via singular value decomposition.IEEE Trans. Fuzzy Syst. 7(2), 120–132 (1999)


Part IITP Model Transformation Based ControlDesign and Optimalization Frameworks

Chapter 3TP Model Transformation is a Gateway BetweenIdentification and Design

Abstract This chapter shows that the TP model transformation can be regarded as ageneric interface between identification and LMI based controler design. This let usfreely select identification technique without suffering of the resulting complicatedrepresentation form.

Keywords Gateway • Interface

Identification and modeling techniques based on fuzzy theory, neural networks,genetic algorithms, or any combination of these approaches (referred to as soft-computing techniques) are extremely powerful in solving modern model identifica-tion engineering tasks, especially in cases where the derivation of closed formulaethrough the consideration of physical and engineering laws would prove to bemuch too difficult. As a result, a number of different identification techniqueshave emerged. However, due to differences in the structure of these identificationtechniques as well as in the unique and often problem-dependent representationsthey use, finding ways to bridge between them and well-developed system designframeworks is in general not a trivial task (one could argue, though, that problemscan also be encountered when highly complex closed analytical formulae, that arenot readily amenable to further manipulation, are applied in control design).

The goal of developing a gateway to TP models (or TS fuzzy models, seeChap. 1) is motivated by the fact that polytopic or affine model based design iswell developed, has a number of frameworks, and has been widely adopted towardsfinding routine-like solutions to engineering problems. The transfer functions ofthe TP model are in reality a higher-order, structured polytopic representationand, hence, are well adapted to modern control theories that rely on polytopicrepresentations and LMI based approaches. This means that as soon as a TP modelrepresentation of a problem is obtained, the mathematical approaches of convexoptimization can immediately be applied, together with a wide range of moderncontrol design theories. In this regard, the reader is referred to the early papers ofGahinet, Bokor, Chilai, Boyd, and Apkarian, as in [1–9]. These authors pioneeredthe use of polytopic models and LMI based design; while others, including Sheerer,Balas, and Packard, have made significant contributions towards creating a holisticframework of control design based on these results [10–12]. From the perspective


65

66 3 TP Model Transformation is a Gateway Between Identification and Design

of fuzzy theory, the reader is referred to the early papers and books of Tanakaet al. about Parallel Distributed Compensation (PDC) design [13–16], which clearlyshows the conceptual similarities and common points between these theories.

The measurement-based identification of functions and systems, as well as thederivation of corresponding models via physical considerations typically (i.e., withthe exception of some special cases), entails considerably larger errors than theTP model transformation; hence, in the practical domain of engineering, the TPmodel transformation can be executed without checking whether the result of theidentification has a TP structure or not, and the resulting (often, approximative)TP structure can be used to replace the result of the identification for validationpurposes. If validation yields positive results, the TP model can just as well beaccepted as the output of the identification phase, irrespective of the means withwhich it was arrived at. Note that in many cases the relatively small (but nonzero)singular values—found during the execution of the TP model transformation—mayactually represent noise in the identification process, and hence, RHOSVD canbe considered in the TP model transformation as a noise filtering step within theidentification phase. Note also that further manipulations of the TP model includingtransformation to various types of TP models would still remain exact—a pointwhich is clearly important with respect to the requirements of control design.

In conclusion, the TP model transformation can be regarded both as afinal step of identification, and as a generalized “interface” that serves as apreprocessing step prior to the further steps addressing design requirementsthrough approaches such as TP model manipulation based optimization.

References

1. P. Apkarian, P. Gahinet, A convex characterization of gain-scheduled H1 controllers. IEEETrans. Autom. Control 40(5), 853–864 (1995)

2. P. Apkarian, P. Gahinet, G. Becker, Self-scheduled H1 linear parameter-varying systems, inProceedings of the 1994 American Control Conference, Baltimore, MD, vol. 1 (1994), pp. 856–860

3. S. Boyd, V. Balakrishnan, P. Kabamba, A bisection method for computing the H1 norm of atransfer matrix and related problems. Math. Control Signals Syst. 2(3), 207–219 (1989)

4. J.C. Doyle, K. Glover, P.P. Khargonekar, B.A. Francis, State-space solutions to standard H2

and H1 control problems. IEEE Trans. Autom. Control 34(8), 831–847 (1989)5. E. Feron, P. Apkarian, P. Gahinet, S-procedure for the analysis of control systems with

parametric uncertainties via parameter-dependent Lyapunov functions, in Proceedings of the1995 American Control Conference, Seattle, Washington, vol. 1 (1995), pp. 968–972

6. P. Gahinet, Explicit controller formulas for LMI-based H1 synthesis, in Proceedings of the1994 American Control Conference, Baltimore, MD, vol. 3 (1994), pp. 2396–2400

7. P. Gahinet, A.J. Laub, Reliable computation of �opt in singular H1 control, in Proceedingsof the 33rd IEEE Conference on Decision and Control, 1994, Orlando, FL, vol. 2 (1994),pp. 1527–1532

8. I. Kaminer, P.P. Khargonekar, M.A. Rotea, Mixed H2/H1 control for discrete-time systemsvia convex optimization. Automatica 29(1), 57–70 (1993)

References 67

9. A. Nemirovskii, P. Gahinet, The projective method for solving linear matrix inequalities,in Proceedings of the 1994 American Control Conference, Baltimore, MD, vol. 1 (1994),pp. 840–844

10. C. Scherer, H1-optimization without assumptions on finite or infinite zeros. SIAM J. ControlOptim. 30(1), 143–166 (1992)

11. C. Scherer, S. Weiland, Linear matrix inequalities in control. Lecture Notes, Dutch Institutefor Systems and Control, Delft, The Netherlands, 2000. http://www.cs.ele.tue.nl/sweiland/lmi.htm

12. P. Seiler, G.J. Balas, A. Packard, Linear parameter-varying control for the x-53 activeaeroelastic wing, in Control of Linear Parameter Varying Systems with Applications, ed. by J.Mohammadpour, C.W. Scherer (Springer US, Boston, MA, 2012), pp. 483–512

13. K. Tanaka, M. Sugeno, Stability analysis and design of fuzzy control systems. Fuzzy SetsSyst. 45(2), 135–156 (1992)

14. K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear MatrixInequality Approach (Wiley, New York, 2001)

15. K. Tanaka, T. Ikeda, H.O. Wang, Fuzzy regulators and fuzzy observers: relaxed stabilityconditions and LMI-based designs. IEEE Trans. Fuzzy Syst. 6(2), 250–265 (1998)

16. H.O. Wang, K. Tanaka, M. Griffin, Parallel distributed compensation of nonlinear systems byTakagi-Sugeno fuzzy model, in Proceedings of the International Joint Conference of the 4thIEEE International Conference on Fuzzy Systems and the 2nd International Fuzzy EngineeringSymposium, 1995, Yokohama, Japan, vol. 2 (1995), pp. 531–538

http://www.cs.ele.tue.nl/sweiland/lmi.htm

http://www.cs.ele.tue.nl/sweiland/lmi.htm

Chapter 4TP Model Transformation Based Control DesignStructure

Abstract This chapter recalls the first TP model transformation based designmethod that is detailed in [3] and initiated in papers [1, 2].

Keywords TP model based control • qLPV • LMI

Let us assume that we have a qLPV model (see Chap. 1) formulated as:

�Px.t/y.t/

�D S.p.t//

�x.t/u.t/

�; (4.1)

where p.t/ 2 � � RN . In this book, we use the following control schema:

u.t/ D � .F.p.t/// x0.t/; (4.2)

where x0.t/ is the combination of vector x.t/ and Ox.t/ that is estimated by theobserver structure as:

�OPx.t/Oy.t/

�D S.p.t//

�Ox.t/u.t/

�C

�K.p.t//

0

�.y.t/ � Oy.t// : (4.3)

Using the TP model representation we have:

�Px.t/y.t/

�D S �

n2NwCo

n .pn.t//

�x.t/u.t/

�; (4.4)

where the control value is:

u.t/ D �

�F �

n2NwCo

n .pn.t//

�x0.t/ (4.5)


69

70 4 TP Model Transformation Based Control Design Structure

and the observer structure is:�

OPx.t/Oy.t/

�D S.p.t//

�Ox.t/u.t/

�C

K �

n2NwCo

n .pn.t//

0

!

.y.t/ � Oy.t// : (4.6)

such that:

S.p.t// D S �n2N

wCon .pn.t//; (4.7)

F.p.t// D F �n2N

wCon .pn.t//; (4.8)

K.p.t// D K �n2N

wCon .pn.t//: (4.9)

The TP model manipulation based design and optimization approach consists ofthree main steps:

• Identification in qLPV formThis step determines S.p.t// through either closed formulae or any other form,

e.g., as a result of soft computing based identification (for example, the matrixfunction S.p.t// might be represented by neural network).

• TP model manipulation based optimization—convex hull manipulationThis step transforms S.p.t// into a TP model based polytopic representation

S �n2N

wCon .pn.t//.

First, the HOSVD based canonical form is generated whenever complexityand accuracy trade-offs are required, i.e., especially in cases where exact TPmodel representations do not exist. Since S.p.t// has an infinite number ofconvex TP model representations (represented by �) expressed as:

S.p.t// D S.�/ �n2N

wCo.�/n .pn.t//; (4.10)

an important point in this step is to find the proper convex TP model representa-tion that leads to the best solution in the third step. Specifically, an optimizationprocess is executed iteratively to find the tensor S which leads to the best controlperformance. An important point here is that the TP model transformation is avery effective tool—as described in the previous chapters—for the manipulationof the entire TP model through the manipulation of the univariate weightingfunctions through matrix operations (irrespective of the way in which S.p.t//is represented). From the point of view of control theory and from a geometricalperspective, this manipulation influences the convex hull defined by the vertexesstored in S as described in [3].

• Linear Matrix Inequality based designThe third step substitutes the vertex systems stored in S to LMIs constructed

according to the desired control performance. Solving the LMIs, we obtain theelements of F and K . In order to find the best solution, we may return to thesecond step and systematically manipulate the TP model, namely by modifyits vertexes. The last part of the book provides various examples of TP modelmanipulation based optimization.

References 71

It is important to emhpasize that the TP model transformation is applicable toany control theories where such a polytopic form is used. Any theory in general thatuses TP models can benefit from TP model manipulation.

References

1. P. Baranyi, TP model transformation as a way to LMI-based controller design. IEEE Trans. Ind.Electron. 51(2), 387–400 (2004)


3. P. Baranyi, Y. Yam, P. Varlaki, Tensor Product Model Transformation in Polytopic Model-BasedControl (CRC/Taylor & Francis, Boca Raton, FL/London, 2013)

Chapter 5General Stability Verificationand Control Design

Abstract The key message of this chapter is that using the Multi TP modeltransformation it is possible to perform LMI-based stability analysis on a wide rangeof systems including soft-computing or hybrid components based systems as well.This fact lightens the stability criticism against soft-computing based approaches ingeneral.

Keywords Stability

5.1 Key Idea

If a system has various different components given in different representations (e.g.,the model is given by equations, the controller by fuzzy logic rules, and the observerby a neural network), we say that the system has a hybrid representation. In suchcases, it is extremely difficult to derive stability proofs. However, the Multi-TPmodel transformation can be used to find the convex TP model representationof all components, such that all TP models have the same weighting functionsystem, in order to apply LMI based stability or performance analysis in astraightforward way [1]. As a matter of fact, if there is no exact TP modelrepresentation for a component, then relevant complexity trade-offs should becarefully executed while considering the best achievable accuracy, and potentiallyperforming validation along the way as discussed in the previous section. Note thatthis idea can also be applied towards systems with time delay, as discussed later inChap. 7 and Part V.

Let us assume that we have a qLPV model based control solution in thecontroller-observer schema discussed in the previous section. In this case, func-tions F.p.t//, K.p.t// to S.p.t// are already determined (we may also focus ona schema that has further components Ck.p.t//). Further, we assume that thesolution is already verified based on physical tests and/or simulations; howeverthere is no stability proof, because the components were either derived throughheuristic approaches (e.g., based on trial-and-error simulations or other sources ofinspiration), or derived through different approaches (for example, the controllermight have been derived based on a Mamdani, min-max fuzzy operator basedmodel described over an incomplete rule base, while the observer might have been


73

74 5 General Stability Verification and Control Design

designed using neural network based approaches). It is important to realize that oneof the main criticisms against soft-computing based control design is that there areno frameworks available for the verification of their stability (hence, traditionallysolutions obtained using such approaches underwent heuristics-based verification,and often there was no mathematically rigorous proof of their adequacy towardssolving the problem).

One generalized framework for stability verification can be based on the Multi-TP model transformation. All the components S.p.t// F.p.t//, K.p.t// and Ck.p.t//of the control schema at hand can be transformed into a unified weighting functionsystem by the Multi-TP model transformation as:

S.p.t// D S �n2N

wCon .pn.t//; (5.1)

F.p.t// D F �n2N

wCon .pn.t//; (5.2)

K.p.t// D K �n2N

wCon .pn.t//; (5.3)

Ck.p.t// D Ck �n2N

wCon .pn.t//: (5.4)

Once we have all the vertexes of all components, we may use LMI techniques tocheck for stability or even to prove various properties of control performance.

From a different aspect, this framework allows us to use design techniques thatare heuristic in nature—and which server our design purposes well—even thoughthey are not amenable to traditional proofs of stability and performance. We cando this exactly because all of our components can, as a final step, be transformedinto a common structure, which in turn can be supplied to LMI-based stability andperformance verification tools.

5.2 Example

We investigate the example of the 3DoF aeroelastic wing section, on which a further,in-depth discussion can be found in Part III of this book. As the goal of the exampleis to demonstrate the general outlines of the verification step, the physical anddynamic aspects of the model itself are not treated here. Instead, the qLPV modelof the wing section, described in Chap. 8, is adopted as a starting point.

The goal of the design task is to stabilize the pitch and plunge motions of the wingby controlling the dynamics of the trailing edge actuator. Challenges relevant to thistask include strong nonlinearities as well as various other phenomena such as limitcycle oscillations and even chaotic behaviors emerging in the uncontrolled case. Forthe current example, we assume that the observer based output feedback design isalready complete, and a fuzzy logic controller as well as a neural network observerare also available to us in the structure given in Fig. 5.1 for � D Œ�0:3; 0:3�rad �

Œ8; 20�m=s. Specifically, we have:

5.2 Example 75

qLPV model of theControlled plant

S(p(t))

T-S FuzzyController F(p(t))

Neural networkObserver K(p(t))

u y

x

-

if A then A1if B then B1if C then C1

InputHidden

Output

S(p(t))u(t)

x(t)

y(t)

x(t)

Fig. 5.1 Observer based output feedback hybrid control structure

• A fuzzy logic controller that has two inputs representing the elements ofthe parameter vector as ˛ and U. Nine antecedent fuzzy sets are given oneach input dimension, see Fig. 5.2. Thus, 81 linguistic fuzzy rules describe thecontrol law. The output of the fuzzy controller is calculated via the TS fuzzymodel strategy. The antecedent sets are depicted in Fig. 5.2. The outputs of thefuzzy engine are the feedback gains F.p.t//. The output of the controller is�F.p.t//Ox.t/, where the elements of Ox.t/ are estimated by the observer, except ˛,which is directly measurable:

• A neural network observer consisting of three layers, each of which contains25 neurons. Each element of K.p.t// is computed by one neural network, so atotal of six neural networks are needed.

We further assume that the performance of the controlled system is acceptable(see the results for a very critical wind speed in Figs. 5.3 and 5.4, where the pitchand the plunge are shown alongside the controlled trailing edge, which has a directeffect on the dynamic motion of the wing and the control value of the dynamicsof the trailing edge). The one assumption we do not make—and which is the keyadvantage of our verification approach—is the existence of a stability proof withrespect to the system. As a result, this example will show that the Multi-TP modeltransformation is capable of transforming the given system components to convexTP models, such that the weighting function systems become the same; further, itwill show that based on this property, stability can be tested for via a feasibility testwith respect to a set of LMIs. Additionally, by substituting the consequent systemsinto the LMIs, some indications on performance can also be obtained.

First of all, we execute the Multi-TP model transformation (with grid M D 137�

137) on the system model, the observer, and the controller. As a result, we find that


Fig. 5.2 Antecedentmembership functions of thefuzzy controller

0 20 40 60 80 100 120

0

0.2

0.4

0.6

0.8

1

Input 1

Membership function plots

µ1,1 µ1,2 µ1,3 µ1,4 µ1,5 µ1,6 µ1,7 µ1,8 µ1,9

0 20 40 60 80 100 120

0

0.2

0.4

0.6

0.8

1

Input 2

Membership function plots

µ2,1 µ2,2 µ2,3 µ2,4 µ2,5 µ2,6 µ2,7 µ2,8 µ2,9

all of these components can be exactly expressed using 2 � 3 vertices and the sameCNO-type weighting function system as follows:

S.p.t// D S �n2f1;2g

wCNOn .pn.t//; (5.5)

F.p.t// D F �n2f1;2g

wCNOn .pn.t//; (5.6)

K.p.t// D K �n2f1;2g

wCNOn .pn.t//: (5.7)

5.3 Decoupling the Design, Optimization, and Stability Verification: : : 77

Fig. 5.3 Time response ofthe controlled system forU D 14:4 m/s

0 0.5 1 1.5 2−20

−15

−10

−5

0

5x 10

−3

Time [s]

Plu

nge,

h [m

]

0 0.5 1 1.5 2

−0.2

−0.15

−0.1

−0.05

0

Time [s]

Pitc

h, a

lpha

[rad

]

The CNO weighting functions are shown in Fig. 5.5. Once we have SrD1::6,FrD1::6 and KrD1::6, we can use, for instance, quadratic stability analysis. In our case,when we substitute these vertices into the MATLAB quad stab function, we see thatstability is guaranteed. The execution of the Multi-TP model transformation and thequad stab functions in MATLAB take a few minutes, in contrast to the analyticalderivations which would otherwise be necessary.

5.3 Decoupling the Design, Optimization, and StabilityVerification: Generalized Design Frameworks

This section proposes a design framework that allows for the combination of variousdesign strategies. The role of the TP model transformation within the framework isto provide a straightforward, non-heuristic way to combine these strategies whilesupporting a flexible, multi-parametric convex hull manipulation based approachtowards optimization.


Fig. 5.4 Time response ofthe controlled system forU D 14:4 m/s

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

Time [s]

Tra

iling

edg

e, b

eta

[rad

]

0 0.5 1 1.5 2−2

−1

0

1

2

3

4

5

6

Time [s]

Con

trol

val

ue

The general stability verification strategy discussed in the previous sectiondecouples the questions of design and stability, leading to significantly less con-straints imposed on the design phase. Various design techniques, which may havespecial features and beneficial representations for different components of thesystem at hand, can be applied to the derivation of the controller, observer, andother components. In this way, very different approaches can be incorporated intothe design phase; even ones that does not yet have a fully developed mathematicalbackground in terms of, e.g., stability proof, but are nevertheless very powerful inthe way they help achieve the desired performance and optimization objectives. Insome cases, one may even decide to modify the model itself, so that a non-exactrepresentation is used as long as all stability and performance validations justifythis decision.


8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Mem

bers

hip

func

tions

Free stream velocity: U [m/s]

−0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

Mem

bers

hip

func

tions

Pitch angle: alpha [rad]

Fig. 5.5 CNO type weighting functions of the dimensions U and ˛

5.3.1 Multi-Way Convex Manipulation

Based on the above, a TP model based design framework is proposed here. Theframework allows us to design each system component separately. Furthermore,each component is derived based on different TP model representations of the givenmodel. Very little attention is given in the literature to the fact that the derivation ofeach component may require different polytopic model representations if the goal isto achieve the best solution. The following part deals with concrete design problemsas examples and shows that when different convex hulls—hence, different TP model


Fig. 5.6 Using only onepolytopic representation. Redarrow representsmanipulation on the model

Fig. 5.7 Two different polytopic representations are used. Left side shows the controller and theright shows the observer design. Red arrow represents the manipulation of the polytopic model.LMI(S,F,K) at the bottom represents the final stability checking

manipulation techniques—are applied to the design of the controller and observer,better results can be achieved. Expressed simply, this means that a given modelshould be transformed to different TP models suitable for the derivation of differentcomponents. This is in contrast to the widely accepted design methods in which theentire design procedure is executed on one polytopic representation of the model asillustrated in Fig. 5.6.

The design framework presented in the following supports separate TP modelmanipulation for each individual system component as illustrated in Fig. 5.7.TP model manipulation includes complexity relaxation as well as convex hullmanipulation. In the case of complexity relaxation, the resulting TP model is notexact, i.e., it does not equivalently represent the given model. As a consequence,any further design step will derive solutions to the non-exact model and not to theoriginal model; and the stability verification executed in the final step is requiredto be able to compensate for this difference in complexity with respect to the


original model. At the same time, convex hull manipulation allows for the improvedeffectiveness of methodologies integrated into the framework. For instance, it allowsfor LMI based methods to be flexibly combined with design approaches, such asfuzzy logic and neural network based modeling, that use different representationalconcepts.

The key idea is to split the design process into four key steps. The first step findsthe main and independent components of the model via its HOSVD based canonicalform, in order to relax the LMIs used in further design steps and to influence theresulting control performance. Examples discussed later show that the complexitymanipulation of the TP model may also lead to an improvement of the resultingcontrol system. In the second step, all components of the entire controller systemare derived (e.g., controller, observer, etc.) separately. This separated derivation ismotivated by the fact that the optimization of the controller and observer requiresthe convex hull manipulation step to fulfill contradictory objectives. Thus, differentTP model representations (i.e., with different numbers of vertexes and types ofconvex hulls) of the model can be used to derive different components of the system.Interestingly, the ideas presented here are applicable not only to LMI-based design.In theory, any combination of any methods are allowed here, if they are deemed toincrease the power of the design. As a result, it is possible to have a hybrid systemwhere the derived components are available in different representations, as long as aproof can be obtained for the stability or other performance measures of the systemin the final step of generalized stability verification as discussed earlier. Specifically,the Multi-TP model transformation can be used to define all components as TPmodels over the same weighting function system and the stability of the entirecontrol solution (with respect to the original, exact model) can be analyzed viaLMIs.

Thus the control design procedure has the following key steps:

• (1) Main TP component analysis: We determine the HOSVD based canonicalform of the given model by TP model transformation and select the main TPmodel, so that we may continue the design with the relaxed model only.

• (2) LMI based design: We manipulate the TP model representations for thecontroller and observer (or any further components) design separately, andexecute the LMI design.

• (3) Exact system reconstruction: We integrate the controller and the observer(or any further components) with the original system model (i.e., not only forthe main TP model component of the given system model) and determine thepolytopic representation of the whole system over the same weighting functionssystem.

• (4) Stability verification: We apply LMIs to check the stability of the wholesystem.

In the following discussions, further details are provided on these four steps.


5.3.2 Main and Independent TP Model Component Analysisvia the HOSVD Based Canonical Form

We execute the TP model transformation on the given model to generate its HOSVDbased canonical form. This step results in the core tensor SE and singular matricesUE

n , where we use superscript “E” to emphasize that this is for the exact TP modelrepresentation:

S.p.t// D SE �n2N

wEn .pn.t//: (5.8)

In order to define the main TP model we execute RHOSVD in the TP modeltransformation (we discard vertexes assigned to smaller singular values) thatresults in:

OS.p.t// D SMTP �n2N

wMTPn .pn.t//; (5.9)

where superscript “MTP” denotes the fact that this is only the main component ofthe given model.

Remark 5.1. Actually, we do not need to determine the continuous weightingfunctions wE

n .pn.t// and wMTPn .pn.t// in this step since we will continue the

design procedure using the vertex models only. Therefore, practically we do notexecute the third step of the TP model transformation. It is enough to carryfurther only the singular matrices UE

n and UMTPn (resulting from Step 2 of the TP

model transformation) and only the unified continuous weighting functions will bedetermined later.

5.3.3 Convex Hull Manipulation

In order to use LMI-based design theories, we need to transform the above HOSVDbased canonical form of the main TP model to convex TP model. Practically, thismeans that we continue with the second step of the convex TP model transformationthat results in UMTP�Co

n , where superscript “MTP-Co” means that we have a convexform. Since the LMI design is very sensitive to the type of the convex hull definedby the vertexes, and this sensitivity differs by component (e.g., for the observer andcontroller, see examples later) we may specify different convex hulls to be usedto derive different components of the system such as controller and observer etc.as discussed in the previous section. For the sake of simplicity, let us derive twodifferent models, one for the controller design:

OS.p.t// D SController �n2N

wFn .pn.t//; (5.10)


and the other for the observer design:

OS.p.t// D SObserver �n2N

wKn .pn.t//: (5.11)

Again, the continuous weighting functions are still not needed here. We cancontinue with UF

n and UKn resulting from Step 2 of the TP model transformation.

5.3.4 LMI Based System Design

As an example let us simplify the required control design performance to asymptoticstability using the design framework discussed in Chap. 4 (note that various furtherconstraints can be considered through properly selected LMIs). We search for asolution in the observer based control system structure.

We search for a controller and observer to the main TP models (5.10) and (5.11)in TP model form:

F.p.t// D F �n2N

wFn .pn.t//; (5.12)

and

K.p.t// D K �n2N

wKn .pn.t//: (5.13)

The goal of LMI-based design is to derive controller vertex gains Fi1;i2;:::;iN storedin F and observer vertex gains Ki1;i2;:::;iN stored in K from SController and SObserver

respectively. For instance, we may take the following very simple LMIs from [2]which are designed to the controller–observer schema discussed in Chap. 4: TheseLMIs use the linear index equivalent of the multi way indexing of the vertexes ofthe TP model (for more details see Eqs. 1.6 and 1.7). Here

Sr D

�Ar Br

Cr Dr

�:

Theorem 5.1 (Asymptotically Stable Controller). Assume the polytopicmodel (5.8) with controller u.t/ D �F.p.t//x.t/. This control structure is globallyand asymptotically stable if there exist P1 > 0 and Mr (r D 1; : : : ; R where R is thenumber of LTI vertex systems) satisfying equations

P1ATr � MT

r BTr C ArP1 � BrMr < 0;

P1ATr � MT

s BTr C AsP1 � BrMs C P1AT

s � MTr BT

s C AsP1 � BsMr < 0;


for r < s � R, except pairs .r; s/ such that 8p.t/ W wr.p.t//ws.p.t// D 0, and whereMr D FrP1. The feedback gains can then be obtained from the solution of the aboveLMIs as Fr D MrP�1

1 .

Theorem 5.2 (Asymptotically Stable Observer). Let us assume that we have apolytopic model (5.8) with observer structure (5.13). This output-feedback controlstructure is globally and asymptotically stable if there exist P2 > 0 and Nr (r D

1; : : : ; R, where R is the number of LTI vertex systems) satisfying equations:

ATr P2 � CT

r NTr C P2Ar � NrCr < 0;

ATr P2 � CT

s NTr C P2Ar � NrCs C AT

s P2 � CTr NT

s C P2As � NsCr < 0:

for r < s � R, except pairs .r; s/ such that 8p.t/ W wr.p.t//ws.p.t// D 0, andwhere N2;r D P2Kr. The observer gains can then be obtained from the solution ofthe above LMIs as Kr D P�1

2 N2;r.

An important point here is that we derive the controller and observer separately,as the stability does not need to be guaranteed in this step (it will be checked inthe next step). Thus, we solve the LMIs in the above theorems separately. One caneasily find examples where the LMI solvers do not indicate feasibility for all LMIsof both theorems, but finds a solution when the theorems are considered separately.

5.3.5 Exact System Reconstruction: Unified TP Model Forms

In this step, the goal is to determine exact TP model representations of thecontrol system upon which LMI-based stability analysis can be executed. Actually,whatever the design technique used in the previous step, the Multi-TP modeltransformation can be used to verify stability. We have a shortcut here if we havethe controller and observer (and other components) in TP model form, as in suchcases we can simply unify the weighting functions according to Chap. 2. Thus, wetransform vertices SE of the exact TP model, and the vertices of the controller andobserver to the same weighting function system without decreasing their ranks (i.e.,without discarding nonzero singular values). Since we have matrices UE

n of the exactTP model and matrices UF and UK in the present case, we may use them directly inthe unification step.

The goal is to find a common Un for all components such that the weightingfunctions guarantee that the TP models will be convex. Formally, the goal is to find:

S0 �n2N

Un D SE �n2N

UEn ; (5.14)

F 0 �n2N

Un D F �n2N

UFn ; (5.15)

K 0 �n2N

Un D K �n2N

UKn : (5.16)


We construct matrix H based on Sect. 2.10:

Hn D�UE

n UFn UK

n

�: (5.17)

Afterwards, we execute the compact SVD (we keep all nonzero singular values)on matrix Hn and execute, for instance, a CNO transformation:

Hn D UCon DnVT

n : (5.18)

We partition the product DnVnT into matrices

DnVnT D

�TS

n TFn TK

n

�: (5.19)

according to the sizes (number of columns) of USn, UF

n , and UKn , we have

�UE

n UFn UK

n

�D UCo

n

�TS

n TFn TK

n

�: (5.20)

Having obtained the above determined common matrix, we can use the TPC

model transformation to transform F.p.t// and K.p.t// to the common weightingfunction system (obviously only the discretized version of the common weightingfunctions is given). However, we can use a shortcut in this step as well, as we havethe transformations matrices to directly derive the new core tensor as:

SE �n2N

UEn D SE �

n2N

�UCo

n TSn

�D

�S �

n2NTS

n

��

n2NUCo

n D S0 �n2N

UCon ; (5.21)

F �n2N

UFn D F �

n2N

�UCo

n TFn

�D

�F �

n2NTF

n

��

n2NUCo

n D F 0 �n2N

UCon ; (5.22)

K �n2N

UKn D K �

n2N

�UCo

n TKn

�D

�K �

n2NTK

n

��

n2NUCo

n D K 0 �n2N

UCon : (5.23)

Finally we have the vertices of the model, controller, and observer over the samediscretized weighting system. If we need to determine the continuous weightingfunctions to

F.p.t// D F 0 �n2N

wCon .pn.t//; (5.24)

or

K.p.t// D K 0 �n2N

wCon .pn.t//; (5.25)

we can execute the third step of the TP model transformation to derive thecontinuous weighting functions to:

S.p.t// D S0 �n2N

wCon .pn.t//: (5.26)


Remark 5.2. It should be emphasized here again that we could have used Multi-TP model transformation to transform the given model, controller, and observer tocommon weighting function based TP models. In that case, however, we would havealso needed to execute HOSVD on the entire, complex system (the dimensionalityof the whole discretized system may be larger). In the current case, we did not needto do this, as the TP form of all components was already available, and hence onlythe unification of the weighting functions was necessary.

5.3.6 LMI Based Stability Verification

As a result of the previous steps, we have all the vertices of all components. Wecan simply substitute the vertices into proper LMIs and easily check whether or notthey are a solution. As a matter of fact, we have many more vertices than duringthe design phase, since we use the exact model now; and further, the exact model,the controller, and observer have the same complexity (e.g., number of vertices) asdoes the exact TP model due to the unified weighting function system. On the otherhand, solving the LMIs (even simultaneously) in this step requires significantly lessresources, since we only have to check whether the vertices are a solution, namelywhether there is a common P.

References

1. P. Baranyi, The generalized TP model transformation for TS fuzzy model manipulation andgeneralized stability verification. IEEE Trans. Fuzzy Syst. 22(4), 934–948 (2014)

2. K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix InequalityApproach (Wiley, New York, 2001)

Chapter 6TPI Model Transformation for the Classof Non-qLPV Models

Abstract The goal of this chapter is to extend the TP model transformationto a set of control problems where the identified model is not given in qLPVform. The chapter shows that replacing the discretisation step of the TP modeltransformation by an identification executed in each iscretisation grid-points leadsto a transformation that does not requires qLPV form.

Keywords TPI model transformation

6.1 Key Idea

When we generate the TP model:

�Px.t/y.t/

�D S �

n2NwCo

n .pn.t//

�x.t/u.t/

�; (6.1)

using the TP model transformation, we assume that S.p.t// is available for thediscretization of the qLPV structure as:

�Px.t/y.t/

�D S.p.t//

�x.t/u.t/

�: (6.2)

Therefore, the TP model transformation discussed in the previous sections isdirectly applicable to qLPV models. However, there are cases where the qLPVstructure is not given, and the result of the identification phase is available only as:

Px.t/ D f .x.t/; u.t/; p.t//: (6.3)

In such cases there is no S.p.t// to be discretized. As a matter of fact, if we canreplace (6.3) with an LTI state-space system in each point of the discretizationgrid G (i.e., if it is possible to identify LTI systems in all points of discretizationgrid G), then we can reconstruct SD.�;G/. Thus, the further steps of the TP model


87

88 6 TPI Model Transformation for the Class of Non-qLPV Models

transformation can be executed to find the TP model structure. Given that we maynot always accept the TP model approximation fit to the linearized LTI modelsassigned to G, we may need to validate the resulting TP model and if it is acceptablethen the transformed TP model can be accepted as the output of the identification.The reconstruction of the continuous weighting functions includes discretizationagain for a given G0. In order to avoid this we need to replace the direct sampling ofS.p.t// with the same identification step used above.

6.2 TPI Model Transformation

Based on this idea we can extend the TP model transformation to the TPI modeltransformation, where superscript “I” stands for “Identification based” instead of“discretization based.” This transformation differs only in Steps 1 and 3 as:

Algorithm 9 (TPI Model Transformation).

• Step 1: Let the element Hi1;i2;i3;:::;iN of SD.�;G/ be the identified LTI model of thegiven system:

Hi1;i2;i3;:::;iN D Identification

�Pxy

�D f .x.t/; u.t/; p D .g1;i1 ; g2;i2 ; : : : ; gN;iN /

�;

(6.4)where Identification denotes the identification that results in an LTI state spacemodel for the given parameter set to the points of the discretization grid. Such areidentification based discretization would be very time consuming.

• Step 3: In order to calculate the weighting functions Owd.xd/ on dimension d overa given point xd, we define a new discretization grid G0 as G1 � : : : � Gd�1 �

1 � GdC1 � : : : � GN and restrict the discretization space to xd as �0 D !1

� : : : � !d�1 � xd � !dC1 � : : : � !N, then reidentify (in the same way as in Step1) SD.G0;�0/. Then for xd:

Owd.xd/ D SD.G0;�0/

.d/

�Q.d/

�C; (6.5)

where

Q D

OS �

n2Nn¤d

wn.xn/

!

; (6.6)

where OS is obtained in the second step, see Eq. (2.8)

Remark 6.1. In the first step of the TPI model transformation we reidentified thegiven system. During this reidentification step, we may change the parameterdomain and the state-space structure or anything that can theoretically be changed.

Reference 89

Of course, the resulting TP model must be validated. The next chapter will presentsuch a specialized application, in which the parameter domain is extended in orderto transform the entire dynamical representation.

6.3 Example of Re-identification

In this section we recall a simple system identification method that is selected fromthe literature as an example for replacing the discretization step to define the TPI

model transformation. We emphasize once again that a huge variety of differentidentification techniques are available in the related literature. The reason why webriefly introduce only one method below is that this will be utilized in the designexample parts of the book.

Equation Error (EE) methods and Output Error (OE) approaches can be differen-tiated among system identification methods. In this section, the Output error methodis described briefly.

The general structure of the output-error model is:

y.t/ DB.s/

F.s/u.t � nk/ C e.t/ (6.7)

B.s/ D b1 C b2q�1 C � � � C bnbq�nbC1 (6.8)

F.s/ D 1 C f1q�1 C � � � C fnf q�nf ; (6.9)

where nb and nf are the orders of the output-error model.OE methods minimize an objective function of output error that is usually

quadratic. This is the error between the original system and the estimated model.In the case of continuous-time identification, the resulting model is given in theform of a continuous time transfer function:

G.s/ DB.s/

F.s/D

bnbsnb�1 C bnb�1snb�2 C � � � C b1

snf C fnf snf �1 C � � � C f1: (6.10)

The coefficients of the polynomials are estimated using a predictionerror/maximum likelihood method. An all-encompassing work on the identificationof continuous-time systems was published by Garnier and Wang [1].

Once we have identified G.s/ in (6.10), we can readily generate its state-spacerepresentation denoted by H in (6.4) over the new parameter space.

Reference

1. H. Garnier, L. Wang, Identification of Continuous-Time Models from Sampled Data (Springer,Berlin, 2008)

Chapter 7TP� Model Transformation for SystemsIncluding Time Delay

Abstract The TP� model transformation is essentially a special application of TPI

that can be used in control problems relevant to systems with time-delay. Due tothe practical importance of such systems, this chapter focuses specifically on TP�

and its use in qLPV models. We will see that this allows for a large class of LMIbased design theories originally developed for qLPV models to be extended towardsthe domain of time-delay systems, thus providing an important gateway betweenthese different control fields. The contents of this chapter are based on the work ofGalambos et al. [1–3].

Keywords TP� model transfromation • Time delay

7.1 TP� Model Transformation

The key idea in this chapter is to apply the TPI model transformation in a specialway—referred to as the TP� model transformation—to develop a compact, reliable,and numerically appealing method for expressing time-delay systems in qLPVmodel form. In this way, time delay is transformed into a common parameter (i.e.,an external parameter with respect to the system description) of the polytopic model,which allows for the direct application of modern multi-objective LMI-based controldesign theories. This approach leads to a conceptual extension of modern controltheories developed for non-delayed systems and represented in qLPV form towardsa set of control problems with time delay.

By identifying or approximating a delayed system (having a delay parameterdenoted by � ) with a non-delayed system over every discretization points alongdimension � , it is possible to create a so-called redefinition-based discretizationof the original system. In this case, the discretization gridpoints are defined bythe elements of the parameter vector and � , so that the discretization results in aparameter-independent, non-delayed, simple LTI system over each gridpoint.


91

92 7 TP� Model Transformation for Systems Including Time Delay

7.2 Example of the TP� Model Transformation

Let us assume that we have a time-delay qLPV model given in by the followingdelay-differential algebraic equation [4]:

Px.t/ D A.p/.t/x.t/ C B1.p.t//u.t/ C B2.p.t//w.t/ (7.1)

y.t/ D C1.p.t//x.t/ C D11.p.t//u.t/ C D12.p.t//w.t/

z.t/ D C2.p.t//x.t/ C D21.p.t//u.t/ C D22.p.t//w.t/;

where

w.t/ D Œz1.t � �.t//; : : : ; zN.t � �.t//�T : (7.2)

To maintain generality, we assume in our following discussions that all elementsof the system matrices can be parameter-dependent, irrespective of whether theparameters behind them have physical meaning or not. The equation can thus berewritten in the following compact matrix form:

2

4Px.t/y.t/z.t/

3

5 D S.p.t//

2

4x.t/u.t/w.t/

3

5 ; (7.3)

where

S.p/ D

2

4A.p.t// B1.p/ B2.p/

C1.p.t// D21.p.t// D12.p.t//C2.p.t// D21.p.t// D22.p.t//

3

5 : (7.4)

In the following step, we execute the TPI model transformation to transform theabove delayed system (7.1) and (7.2) into:

Px.t/y.t/

�D S.p0.t//

x.t/u.t/

�; (7.5)

where

S.p0.t// D S �n2N

w�p0

n .t//�

(7.6)

and

p0.t/ D Œp.t/; �.t/�T ; p0.t/ 2 � � RN : (7.7)

References 93

As a result, �.t/ is transformed into a generic element of the parameter vectorof a non-delayed polytopic qLPV model. Part V presents an example in which thismethod is used to control a system with a quasi-random �.t/ that otherwise causesunstable motions.

References

1. P. Galambos, P. Baranyi, Representing the model of impedance controlled robot interactionwith feedback delay in polytopic LPV form: TP model transformation based approach. ActaPolytech. Hung. 10(1), 139–157 (2013)

2. P. Galambos, P. Baranyi, TP-tau model transformation: a systematic modelling framework tohandle internal time delays in control systems. Asian J. Control 17(2), 486–496 (2015)

3. P. Galambos, P. Baranyi, G. Arz, Tensor product model transformation-based control designfor force reflecting tele-grasping under time delay. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci.228(4), 765–777 (2014)

4. L.F. Shampine, P. Gahinet, Delay-differential-algebraic equations in control theory. Appl.Numer. Math. 56(3), 574–588 (2006)

Part IIIAnalysis of the TP Model Based Design

Frameworks via a Complex Example

The goal of this part of the book is to study the use of, and to examine theeffectiveness of, the theoretical methods presented earlier. This will be done usingthe recently extended model of the 3 Degree of Freedom (3DoF) aeroelastic wingsection that includes strong nonlinear characteristics as well as friction.

Active control of aeroelasticity has been in the focus of aerospace and controlengineering for several decades. An introduction to this topic can be found in [8].The aeroelastic wing section problem has traditionally been used for theoreticalas well as experimental analysis of aeroelastic behavior. A number of relatedstudies can be found in a series of papers published in the Journal of Guidance,Dynamics and Control. The model used in this chapter originates from the 3DoFNonlinear Aeroelastic Test Apparatus (NATA) model investigated with unsteadyaerodynamics in [4, 5]. The model has 3 degrees of freedom and is extendedwith both structural nonlinearities and friction [15]. The goal is to design astate variable feedback-controller and an observer-based output feedback controllerwhich guarantee asymptotic stability via a single trailing-edge control surface.

The challenge here lies in the strong nonlinearities and various other phenomenawhich characterize the wing section, such as limit cycle oscillation and even chaoticbehavior emerging in the uncontrolled case. Several active controllers have beendeveloped in [6, 7, 9–14]. The TP model transformation based control design ofvarious aeroelastic wing section models (2DoF, 3DoF, and 3DoF model includingfriction) is detailed in [1–3, 15].

References

1. P. Baranyi, Output feedback control of two-dimensional aeroelastic system. J. Guid. ControlDyn. 29(3), 762–767 (2006)

2. P. Baranyi, Tensor-product model-based control of two-dimensional aeroelastic system.J. Guid. Control Dyn. 29(2), 391–400 (2006)

3. P. Baranyi, B. Takarics, Aeroelastic wing section control via relaxed tensor product modeltransformation framework. J. Guid. Control Dyn. 37(5), 1671–1678 (2014)

96 III Analysis of the TP Model Based Design Frameworks via a Complex Example

4. J.J. Block, H. Gilliat, Active control of an aeroelastic structure, in AIAA Meeting Papers onDisc, Reno, NV, Jan 1997. American Institute of Aeronautics and Astronautics, Inc, pp. 1–11

5. J.J. Block, T.W. Strganac, Applied active control for a nonlinear aeroelastic structure. J. Guid.Control Dyn. 21, 838–845 (1998)

6. K.W. Lee, S.N. Singh, Multi-input noncertainty-equivalent adaptive control of an aeroelasticsystem. J. Guid. Control Dyn. 33, 1451–1460 (2010)

7. V. Mukhopadhyay, Transonic flutter suppression control law design and wind-tunnel testresults. J. Guid. Navig. Control 23(5), 930–937 (2000)

8. V. Mukhopadhyay, Historical perspective on analysis and control of aeroelastic responses.J. Guid. Control Dyn. 26, 673–684 (2003)

9. G. Platanitis, T. Strganac, Suppression of control reversal using leading- and trailing-edgecontrol surfaces. J. Guid. Control Dyn. 28, 452–460 (2005)

10. Z. Prime, B. Cazzolato, C. Doolan, A mixed H2/H1 scheduling control scheme for atwo degree-of-freedom aeroelastic system under varying airspeed and gust conditions, inProceedings of the AIAA Guidance, Navigation and Control Conference, Honolulu, Hawaii,2008, pp. 18–21

11. Z. Prime, B. Cazzolato, C. Doolan, T. Strganac, Linear-parameter-varying control of animproved three-degree-of-freedom aeroelastic model. J. Guid. Control Dyn. 33(2), 615–619(2010)

12. K.K. Reddy, J. Chen, A. Behal, P. Marzocca, Multi-input/multi-output adaptive outputfeedback control design for aeroelastic vibration suppression. J. Guid. Control Dyn. 30,1040–1048 (2007)

13. S.N. Singh, L. Wang, Output feedback form and adaptive stabilization of a nonlinearaeroelastic system. J. Guid. Control Dyn. 25, 725–732 (2002)

14. T.W. Strganac, J. Ko, D.E. Thompson, Identification and control of limit cycle oscillations inaeroelastic systems. J. Guid. Control Dyn. 23, 1127–1133 (2000)

15. B. Takarics, P. Baranyi, Tensor-product-model-based control of a three degrees-of-freedomaeroelastic model. J. Guid. Control Dyn. 36(5), 1527–1533 (2013)

Chapter 8qLPV Model of the 3DoF PrototypicalAeroelastic Wing Section

Abstract This chapter focuses on the most recent version of the 3DoF nonlinearaeroelastic wing section model generated from a real measurement and includingStribeck friction model. The goal is to describe and prepare a complex example forthe next chapters, in order to study the usability and the effectiveness of the TPmodel transformation based design frameworks.

Keywords Aeroelastic wing section • qLPV model

8.1 Equations of Motion

We consider the problem of flutter suppression for the prototypical aeroelastic wingsection as shown in Fig. 8.1.

The variables related to the wing section are defined below:

• h = plunging displacement• ˛ = pitching displacement• x˛ = the non-dimensional distance between elastic axis and the center of mass• m = the mass of the wing• I˛ = the mass moment of inertia• b = semi-chord of the wing• c˛ = the pitch structural damping coefficient• ch = the plunge structural damping coefficients• kh = the plunge structural spring constant• k˛.˛/ = nonlinear stiffness contribution• L = aerodynamic force• M = aerodynamic moment• ˇ = control surface deflection• � = air density• U = free stream velocity• cl˛ = lift coefficients per angle of attack• cm˛ = moment coefficients per angle of attack• clˇ = lift coefficients per control surface deflection


97

98 8 qLPV Model of the 3DoF Prototypical Aeroelastic Wing Section

hk

c=2*b

M

Lc.g.

Uxkh

b

a*b midchordelastic axis

h

Deflected position

Equilibrium position

Fig. 8.1 The 3DoF aeroelastic wing section

• cmˇ= moment coefficients per control surface deflection

• a = non-dimensional distance from the midchord to the elastic axis

One of the most recent models of the 3DoF aeroelastic wing section based onreal measurements, which we adopt in this investigation, was presented and deeplyelaborated in [1]. The flat plate airfoil is constrained to have 3DoF: plunge h, pitch˛, and trailing-edge surface deflection ˇ. The equations of motion can be written as:

0

B@

mh C m˛ C mˇ maxab C mˇrˇ C mˇxˇ mˇrˇ

maxab C mˇrˇ C mˇxˇOI˛ C OIˇ C mˇr2

ˇ C 2xˇmˇrˇOIˇ C xˇmˇrˇ

mˇrˇOIˇ C xˇmˇrˇ

OIˇmx˛b I˛

1

CA

0

@RhRR

1

AC

(8.1)0

@ch 0 0

0 c˛ 0

0 0 cˇservo

1

A

0

@PhPP

1

AC

0

@kh 0 0

0 k˛.˛/ 0

0 0 kˇservo

1

A

0

@h˛

ˇ

1

A D

0

@�LM

kˇservoˇdes

1

A : (8.2)

k˛.˛/ is obtained in [1] by curve-fitting on the measured displacement-moment datafor a nonlinear spring k˛.˛/ D 25:55 � 103:19˛ C 543:24˛2. It is important toemphasize that the order of the polynomial defining k˛.˛/ does not influence thenumerical execution of the control design methodology, see later. Hence, one canapply a higher-order polynomial to model the nonlinearity of the spring.

8.1 Equations of Motion 99

Quasi-steady aerodynamic force L and moment M are assumed to be:

L D �U2bCl˛

˛ CPh

UC

�1

2� a

�b

P

U

!

C �U2bclˇ ˇ (8.3)

M D �U2b2Cm˛;eff:

˛ CPh

UC

�1

2� a

�b

P

U

!

C �U2bCmˇ;eff:ˇ: (8.4)

The values of L and M above are accurate for the low-velocity regime.Based on [1], it is assumed that the trailing-edge servo-motor dynamics can be

represented using a second-order system of the form:

OIˇR C cˇservo

P C kˇservoˇ D kˇservo uˇ: (8.5)

By combining Eqs. (8.1), (8.3), and (8.5), we obtain:

0

B@

mh C m˛ C mˇ maxab C mˇrˇ C mˇxˇ mˇrˇ

maxab C mˇrˇ C mˇxˇOI˛ C OIˇ C mˇr2

ˇ C 2xˇmˇrˇOIˇ C xˇmˇrˇ

mˇrˇOIˇ C xˇmˇrˇ

OIˇmx˛b I˛

1

CA

„ ƒ‚ …Meom

0

@RhRR

1

AC

(8.6)

C

0

@ch C �bSCl˛ U

�12

� a�

b�bSCl˛ U 0

��b2SCm˛;eff U c˛ ��

12

� a�

b�b2SCm˛;eff U 0

0 0 cˇservo

1

A

„ ƒ‚ …Ceom

0

@PhPP

1

AC

C

0

@kh �bSCl˛ U2 �bSClˇ U2

0 k˛.˛/ � �b2SCm˛;eff U2 ��b2SCmˇ;eff U

2

0 0 kˇservo

1

A

„ ƒ‚ …Keom

0

@h˛

ˇ

1

A D

0

@0

0

kˇservo

1

A

„ ƒ‚ …Feom

u;

where Meom, Ceom, Keom, and Feom are the mass, damping, stiffness, and forcingmatrices of the equation of motion.

The above equation can be transformed to state-space qLPV form:

�Px.t/y.t/

�D S.p.t//

�x.t/u.t/

�; (8.7)

with input u.t/ D uˇ 2 R, measurable output y.t/ D ˛ 2 R, and state vector

x.t/ D�x1.t/ x2.t/ x3.t/ x4.t/ x5.t/ x6.t/

�TD�Ph P P h ˛ ˇ

�T2 R6. The system

matrix

100 8 qLPV Model of the 3DoF Prototypical Aeroelastic Wing Section

S.p.t// D

�A.p.t// B.p.t//C.p.t// D.p.t//

�2 R7�7 (8.8)

is a parameter-varying object, where p.t/ D�U.t/ ˛.t/

�T2 �. p.t/ includes ˛, an

element of x.t/; therefore, (8.8) belongs to the class of qLPV systems.The elements of S.p.t// are:

A.p.t// D

��M�1

eomCeom.p.t// �M�1eomKeom.p.t//

�I 0

�; B D

�M�1

eomFeom

0

�;

C D�0 0 0 0 1 0

�and D D 0: (8.9)

The details and definition of each system parameter can be found in [1] and theyhave the following values:

mh D 6:516 kg; m˛ D 6:7 kg; mˇ D 0:537 kg; x˛ D 0:21; xˇ D 0:233; rˇ D 0 m;a D �0:673 m; b D 0:1905 m; OI˛ D 0:126 kgm2; OIˇ D 10�5; ch D 27:43 Nms/rad;c˛ D 0:215 Nms/rad; cˇservo D 4:18210�4 Nms/rad; kh D 2844; kˇservo D 7:6608

10�3; � D 1:225 kg=m3; Cl˛ D 6:757; Cm˛;eff D �1:17; Clˇ D 3:774; Cmˇ;eff D

�2:1; S D 0:5945 m.

8.2 Including Stribeck Friction

The damping of the aeroelastic wing model in (8.5) has a linear viscous term.However, in many cases nonlinear friction models give more realistic descriptionsof the physical phenomena; thus the linear viscous term is replaced by a Stribeckfriction model given in the following form:

Ff .t/ D �

0

BBBB@

Fc C.Fs � Fc/

1 C

�v

vs

�2!

1

CCCCA

sign.v.t// � Fvv; (8.10)

where cˇservoC D 4:18210�4 Nm is the Coulomb friction term, cˇservoS D 1:2 �cˇservoC

is the Stribeck friction term, and PStribeck D 0:0075 rad/s is the Stribeck velocity.

The values of these parameters were chosen based on engineering considerationsin order to obtain a realistic friction model expected to be valid in the interval ofP 2 Œ�1:5; 1:5� rad/s. Note that when the Stribeck friction is substituted into the

system matrix, the parameter space � must be extended with dimension x3.t/ D P.It is worth mentioning here that other nonlinear friction models can also be

implemented, which can be given in analytical or soft computing forms, as wellas in the form of raw data sets. The only requirement is that the system matrix mustbe discretizable in order to apply the TP model transformation.

Reference 101

Reference


Chapter 9TP Model Based Control Design

Abstract This chapter discusses a very simple use of the TP model transformationwithout any convex hull manipulation to derive a controller to the 3DoF aeroelasticwing section. More complex design approaches are discussed later.

The design procedure applied in this chapter has two steps. First, a TP modelis derived that defines a tight convex hull. In the second step, the LMI designframework is directly executed on the vertex LTI components of the TP model.Following a description of these two steps, the chapter discusses and evaluates theresulting control performance via numerical simulations. The chapter is based onthe works [6].

Keywords TP model transformation • Control design • LMI

9.1 Exact and Convex TP Model of the 3DoF AeroelasticWing Section

The TP model transformation (generating CNO type weighting functions) isexecuted on the qLPV state-space model (8.9). Two TP models are derived here. Thedifference is that the first one does not include Stribeck friction, while the secondone does.

Let the transformation space � be defined with intervals U 2 Œ8; 20� m/s and ˛ 2

Œ�0:3; 0:3� rad. Let the grid density be defined as G1 � G2, G1 D G2 D 137. The TPmodel transformation shows the rank of the discretized tensor SD 2 RG1�G2�7�7,which is 3 in the first dimension and 2 in the second dimension. The resultingweighting functions w1;i.U/, i D 1 : : : 3, and w2;j.˛/, j D 1 : : : 2, are shown inFig. 9.1. We conclude that the 3DoF aeroelastic model (8.9) has a finite element TPtype polytopic model form with 6 vertex LTI models:

S.p.t// D S �n2f1;2g

wn.pn.t//; (9.1)

where w1.p1.t// 2 R3 and w2.p2.t// 2 R2.When friction is included in the model, the parameter space � is extended by

one dimension x3.t/ D P 2 Œ�1:5; 1:5� rad/s. Let the grid density on that dimensionbe G3 D 138 (an even number for the grid in the third dimension is chosen to


103

104 9 TP Model Based Control Design

avoid division by zero during discretization). The ranks of the discretized tensorSD 2 RG1�G2�G3�7�7 are 3, 2, and 2 in the first, second, and third dimensions,respectively. The number of vertexes becomes 3 � 2 � 2 D 12. The weightingfunctions can be seen in Fig. 9.1. Thus, the 3DoF model with friction also has a TPmodel form as:

S.p.t// D S �n2f1;2;3g

wn.pn.t//; (9.2)

where w1.p1.t// 2 R3, w2.p2.t// 2 R2 and w3.p3.t// 2 R2.

9.2 Control Structure

It is assumed that not all of the state variables of the 3DoF aeroelastic wingsection model are measurable (for example, sometimes only the pitch angle ˛

is measurable); therefore, an observer based output feedback design structure isapplied. The observer is required to satisfy x.t/ � Ox.t/ ! 0 as t ! 1 (whereOx.t/ denotes the state-vector estimated by the observer). Since p.t/ does not containvalues from the estimated state-vector Ox.t/, the following simple control structure isapplied here:

OPx.t/ D A.p.t//Ox.t/ C B.p.t//u.t/ C K.p.t//.y.t/ � Oy.t//

Oy.t/ D C.p.t//Ox.t/;

where u.t/ D �F.p.t//x.t/. Applying TP model type polytopic forms, the followingstructure is obtained:

OPx.t/ D A �n2N

wn.pn.t//Ox.t/ C B �n2N

wn.pn.t//u.t/ CK �n2N

wn.pn.t//.y.t/ � Oy.t//

Oy.t/ D C �n2N

wn.pn.t//Ox.t/

u.t/ D �

�F �

n2Nwn.pn.t//

�x.t/: (9.3)

The goal of the design is to determine gains F and K in such a way that thestability of the output-feedback control structure is guaranteed. The LTI feedbackgains Fi1;i2;:::;iN and observer gains Ki1;i2;:::;iN stored in tensor F and K are calledvertex feedback gains and vertex observer gains, respectively.

9.2 Control Structure 105

Fig. 9.1 CNO typeweighting functions ofdimensions ˛ and U. P is forthe case where nonlinearfriction is included

0.9

0.8

0.7

0.6

0.5

0.4

0.3Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

0.2

0.1

8 10

−0.2 −0.1 0 0.1 0.2 0.3

−1−1.5 −0.5 0 0.5 1 1.5

12 14 16 18 200

0.8

1

0.6

0.4

0.2

0

Wei

ghtin

g fu

nctio

ns

0.8

1

0.6

0.4

0.2

0


9.3 Selecting LMIs

There are a number of LMI theorems available for observer and controller design toderive the vertex gainsK of the observer and the feedback gains F of the controllersatisfying the following control performance requirements:

• Asymptotic stability;• Decay rate for the controller;• Constraints on the control value.

This example recalls the LMI theorems as applied in the control design of the2DoF aeroelastic wing section in [1, 2]. These LMIs are derived in [7].

Theorem 9.1 (Globally and Asymptotically Stable Observer and Controller).Let us assume that the polytopic model structure with controller and observerare given as in (9.3). This output-feedback control structure is globally andasymptotically stable if there exist P1 > 0; P2 > 0 and M1;r; N2;r .r D 1; : : : ; Rand R is the number of LTI vertex systems) satisfying equations:

P1ATr � MT

1;rBTr C ArP1 � BrM1;r < 0;

ATr P2 � CT

r NT2;r C P2Ar � N2;rCr < 0;

P1ATr � MT

1;sBTr C AsP1 � BrM1;s C P1AT

s � MT1;rB

Ts C AsP1 � BsM1;r < 0;

ATr P2 � CT

s NT2;r C P2Ar � N2;rCs C AT

s P2 � CTr NT

2;s C P2As � N2;sCr < 0

for r < s � R, except the pairs .r; s/ such that 8p.t/ W wr.p.t//ws.p.t// D 0, andwhere M1;r D FrP1 and N2;r D P2Kr. The feedback gains and the observer gainscan then be obtained from the solution of the above LMIs as Fr D M1;rP�1

1 andKr D P�1

2 N2;r.

Theorem 9.2 (Globally and Asymptotically Stable Observer and Controllerwith Decay Rate). Assume the polytopic model structure with controller andobserver as given in (9.3). This output-feedback control structure is globally andasymptotically stable if there exist P1 > 0; P2 > 0 and M1;r; N2;r .r D 1; : : : ; Rand R is the number of LTI vertex systems) satisfying equations:

P1ATr � MT

1;rBTr C ArP1 � BrM1;r C 2˛P1 < 0;

ATr P2 � CT

r NT2;r C P2Ar � N2;rCr C 2˛P2 < 0;

P1ATr � BsM1;r � MT

1;sBTr C AsP1 � BrM1;s C P1AT

s � MT1;rB

Ts C AsP1 C 4˛

P1 < 0;

ATr P2 � CT

s NT2;r C P2Ar � N2;rCs C AT

s P2 � CTr NT

2;s C P2As � N2;sCr C 4˛P2 < 0;

9.4 Results of the Control Design 107

for r < s � R, except the pairs .r; s/ such that 8p.t/ W wr.p.t//ws.p.t// D 0, andwhere M1;r D FrP1 and N2;r D P2Kr. The feedback gains and the observer gainscan then be obtained from the solution of the above LMIs as Fr D M1;rP�1

1 andKr D P�1

2 N2;r.

Theorem 9.3 (Constraint on the Control Value). Assume that kx.0/k � , wherex.0/ is unknown, but the upper bound is known. The constraint ku.t/k2 � isenforced at all times t � 0 if the LMIs

2I � X�

X MTr

Mr 2I

�� 0

hold.

Remark 9.1. For a large set of initial states, we can set to be a large quantityeven if x.0/ is unknown. However, one should note that a large could lead toconservative designs.

Further constraints can be incorporated by adding LMI terms as discussed in [7].

9.4 Results of the Control Design

The above LMI-based control design can be immediately applied to the TP modelof the 3DoF aeroelastic wing section model (9.1) and (9.2) to derive the followingsolutions:

9.4.1 Controller 1: Asymptotic Stabilization and Decay RateControl

By applying Theorem 9.2, one finds that ˛ D 0 gives the best controllerperformance for the present model. This simply means that the LMIs in Theorem 9.2become equivalent to the LMIs of Theorem 9.1 in the present case.

9.4.2 Controller 2: Constraint on the Control Value

In order to limit the bounds of the control values, the equations of Theorem 9.3were solved simultaneously with the LMIs of the controller and observer design.The minimal L2 bound of the control value that still guarantees feasible LMIs was


searched for in the case of Controller 2.1-“min.” For comparison, Controller 2.2-“max” was also derived, where a ten times larger bound limit of the control signalwas applied.

9.4.3 Controller 3: State Feedback Control Including StribeckFriction

State feedback Controller 3 was designed by Theorem 5.1 for the TP modelrepresenting the 3DoF wing section model including friction.

9.4.4 Simulation

This section presents numerical experiments to demonstrate the performance of thedesigned stabilizing control solution. Critical free stream velocity U D 14:1 m/s ischosen in order to be comparable to other published results. Open loop simulationwas performed at the beginning of each test to let the oscillations fully develop.However, in the figures, only that range of the simulation is in focus where thecontroller is on. Two simulation cases are compared for each controller.

• Case 1—perturbed system is to test the robustness of the solution. Case 1includes:

– random noise normally distributed with a variance of 10 % added to themeasured output signal;

– 3 ms constant time delay representing the computational delay;– modified nominal values of masses and inertia by ˙15 %;– saturation of the control value.

• Case 2—ideal reference case represents the ideal simulation cases without theperturbations listed in Case 1.

When using Controller 3, the Case 1 simulation has a saturation of the controlsignal as the only perturbation.

Figures 9.2, 9.3, 9.4, and 9.5 show the time response of the controlled system forController 2.1 and 3, respectively.

A simulation for Controller 2.1 with sinusoidally varying free stream velocitywas also performed; results can be seen in Figs. 9.6 and 9.7.


−20

−0.250 0.2 0.4 0.6

−0.2

−0.15

−0.1

−0.05

0.05

0 0.5 1 1.5 2

Case 1Case 2

Case 1Case 2

0

−15

−10

−5

0

x 10-3

5

Fig. 9.2 Time response—A of Controller 2.1 for U D 14:1 m/s

9.4.5 Evaluation

All of the designed controllers are able to asymptotically stabilize the state variablesof the 3DoF aeroelastic wing section model with linear and nonlinear friction.Controller 2.1 out of Controller 1, 2.1, and 2.2 has the smallest control signalamplitude in Case 2 and desaturates in 0:5 s, while the others desaturate in 0:9 s


Fig. 9.3 Time response—Aof Controller 2.1 forU D 14:1 m/s

in Case 1. The settling times are similar for all of the controllers. Thus, it canbe concluded that Controller 2.1 has the most favorable properties; therefore thesimulation results of Controller 2.1 are given in Fig. 9.2.

Note that very simple LMI theorems have been applied so far. If one would like togo for a higher control performance, various choices of performance specificationscould be attempted through more powerful LMI design theorems and further convexhull manipulation. Former solutions of the 3DoF aeroelastic control problem do notfocus on considerations other than stability.

• StabilityAn important issue should be addressed here. The applied LMIs guarantee

that the resulting controller is stable. However, the TP model transformationis a numerical method that can be performed over an arbitrarily, but bounded


Fig. 9.4 Time response—Bof Controller 3 forU D 14:1 m/s

domain �. Therefore, the stability ensured by the applied LMIs is restrictedto �. Note that the accuracy of the given model is also bounded in reality forlow speeds. One may extend � and execute the design method again (such anapplication will be discussed later). Controller 3 has an additional dimension as� W Œ�0:3; 0:3� � Œ8; 20� � Œ�1:5; 1:5�.

• Performance discussionThe control performance discussion focuses on two objectives. These are the

maximal control values and the settling time for each controller. The evaluationis summarized in Table 9.1.

It can be concluded that Controller 2.1 out of the first three designedcontrollers has the best performance according to these objectives. Controller3 has a performance that is similar to Controller 1. However, Controller 3 hasto stabilize a considerably more complex system with an additional nonlinearitycaused by the friction component.


Fig. 9.5 Time response—Bof Controller 3 forU D 14:1 m/s

• Comparison with other results found in recent technical literatureThe control performance can be compared with results presented in [4], where

the LQR controller was designed for the same 3DoF aeroelastic wing section. Weobserve that the controllers derived with the TP type polytopic model and LMIdesign produce considerably faster responses in Case 2, but the cost of this is ahigher control value. Case 1, which adopts a more realistic physical environment,saturates the control signal, resulting in a settling time that is somewhat longerthan the results found in [4]. It also has to be mentioned that the LPV model in[4] is considerably simpler. It has a nonlinearity only in one dimension, namelyin U, and the controller designed in that paper is not an output controller, butrather a full state feedback controller (without observer).


Fig. 9.6 Time response—C of Controller 2.1 with sinusoidally varying free stream velocity

A similar model was examined in [3] in which an LQR based output feedbackcontroller was designed. The control performance is similar to the performanceof Controller 2.1. However, simulation Case 1 of Controller 2.1 also includestime delay, parameter uncertainties, and noise on the measured output signal.

Based on the above-mentioned criteria, the control performance obtained hereis similar to the controller presented in [2], which should be of no surprise, giventhat the same LMIs and the same control design methodology was used. On theother hand, it has to be emphasized that the present controller is designed for the


Fig. 9.7 Time response—C of Controller 2.1 with sinusoidally varying free stream velocity

3DoF model, rather than the 2DoF model and the results of Case 1 simulationsinclude time delay, noise on the measured signal, control signal saturation, andparameter uncertainties.

Multi-input/multi-output control designs are used in papers [5, 8]. However,the actuator dynamics are not included in the models in those cases.

References 115

Table 9.1 Maximal control values and the settling times for the designedcontrol solutions

Maximal control value (rad) Settling time (s)

Controller 1.1 Case 1: 5; Case 2: �350 Case 1: 1.5 ; Case 2: 1.5

Controller 2.1 Case 1: 5; Case 2: �15 Case 1: 1.5 ; Case 2: 1

Controller 2.2 Case 1: 5; Case 2: �60 Case 1: 1.5 ; Case 2: 1

Controller 3 Case 1: 5; Case 2: �14,500 Case 1: 1.5 ; Case 2: 1.5

References

1. P. Baranyi, Output feedback control of two-dimensional aeroelastic system. J. Guid. ControlDyn. 29(3), 762–767 (2006)

2. P. Baranyi, Tensor-product model-based control of two-dimensional aeroelastic system.J. Guid. Control Dyn. 29(2), 391–400 (2006)

3. N. Bhoir, S.N. Singh, Output feedback nonlinear control of an aeroelastic system with unsteadyaerodynamics. Aerosp. Sci. Technol. 8(3), 195–205 (2004)


5. K.K. Reddy, J. Chen, A. Behal, P. Marzocca, Multi-input/multi-output adaptive outputfeedback control design for aeroelastic vibration suppression. J. Guid. Control Dyn. 30,1040–1048 (2007)

6. B. Takarics, P. Baranyi, Tensor-product-model-based control of a three degrees-of-freedomaeroelastic model. J. Guid. Control Dyn. 36(5), 1527–1533 (2013)

7. K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix InequalityApproach (Wiley-Interscience, New York, 2001)

8. Z. Wang, A. Behal, P. Marzocca, Model-free control design for multi-input multi-outputaeroelastic system subject to external disturbance. J. Guid. Control Dyn. 34, 446–458 (2011)

Chapter 10Convex Hull Manipulation Based Optimization

Abstract This chapter presents various studies about the benefits of using the TPmodel manipulation. First, it focuses on a simple study on a 3DoF wing sectioncontrol problem without friction, and point out the fact that the derivation andmanipulation of different TP models (obtained by manipulating the convex hull ofthe representation) for the controller and observer design leads to further powerfulopportunities in terms of optimization. Next chapters investigate the fact that thecomplexity relaxation of the TP model derived from the 3DoF model excludingfriction leads to control performance improvement even in cases when the relaxedTP model is not exact. Finally, both convex hull manipulation and complexityrelaxation of the TP model are combined, and a comprehensive investigation isprovided on the benefits of using TP model manipulation based control design. Inthe final version of this example, the 3DoF model will include friction, which resultsin a considerably more complex model with a 3-dimensional parameter vector. Acomprehensive analysis is provided on the effects of TP model manipulation on thestability region and on control performance.

Keywords Convex hull • Optimisation

10.1 Convex Hull Manipulation Based Design Framework

This section provides a simple example on how convex hull manipulation influ-ences the feasibility of the LMI design and the resulting control performance.Furthermore, the section shows that the separate manipulation of the convex hullsfor the controller and observer design can further improve the resulting controlperformance. This fact is almost unknown (or at least very rarely addressed) in thecontrol literature, so that controller and observer design is typically based on thesame TP model.

To demonstrate our point, we generate various TP model type polytopic rep-resentations of the qLPV model of the 3DoF aeroelastic wing section introducedin the previous chapter. Then, the LMI-based controller design process is executedon each TP model and finally we compare the resulting control performances basedon two simple objectives: the L2 norm and the maximum of the control value andthe settling time of the closed loop.


117

118 10 Convex Hull Manipulation Based Optimization

10.1.1 Key Steps

Convex hull manipulation can be implemented in three steps as follows:

• Step 1: Convex TP models: A tight (CNO) and a loose (SNNN, IRNO) convexTP models are determined.

• Step 2: Convex TP model interpolation: TP models are interpolated betweenthe TP models derived in the first step.

• Step 3: LMI based output feedback design: LMI based control design is exe-cuted on the interpolated TP models. Finally the generalized stability verificationframework is applied.

10.1.2 Step 1: Convex TP Models

This step recalls the CNO type TP model of the 3DoF aeroelastic model (excludingfriction) from the previous section, see Fig. 9.1. We also derive the IRNO and SNNNtype TP models here. In both cases the TP model transformation is derived intransformation space �, defined as U 2 Œ8; 20� m/s and ˛ 2 Œ�0:3; 0:3� radwith a grid density is defined as G1 � G2, G1 D 137 and G2 D 137 (the same � andG as earlier). As a result, we have three TP models:

S.p.t// D SCNO �n2f1;2g

wCNOn .pn.t// ;

S.p.t// D SIRNO �n2f1;2g

wIRNOn .pn.t// ;

S.p.t// D SSNNN �n2f1;2g

wSNNNn .pn.t// ;

where SCNO, SIRNO and SSNNN 2 R3�2�7�7. In each case the n D 1 dimension hasthree weighting functions and the n D 2 dimension has 2. The weighting functionsare displayed in Figs. 10.1 and 10.2.

10.1.3 Step 2: Convex TP Model Interpolation

Interpolation between two TP models is done through the interpolation of theweighting functions which actually leads to an interpolation of the tight and looseconvex hulls defined by the vertexes of the TP models. According to Sect. 2.9:

10.1 Convex Hull Manipulation Based Design Framework 119

Fig. 10.1 SNNN typeweighting functions of thedimensions U and ˛

1. Determination of the weighting functions: The weighting functions are gen-erated between a loose hull (SNNN or IRNO) and a tight hull (CNO) type byapplying linear interpolation.

w�n .pn.t// D � � wCNO

n .pn.t// C .1 � �/ � wSNNN;IRNOn .pn.t//; (10.1)

where � is a coefficient and its value goes from 0 to 1.2. Executing the TPC model transformation to find the vertexes: Since we have

the discretized variants of the CNO, SNNN, and IRNO weighting functions(resulted by the Second step of TP model transformation in the previous step)the discretized interpolated weighting function can directly be determined as:

W�;D.!n;Gn/n D � � UCNO

n C .1 � �/ � USNNN;IRNOn : (10.2)

As a result of TPC an interpolated TP model is obtained as:

S.p/ D S� �n2f1;2g

w�n .pn.t// :


Fig. 10.2 IRNO typeweighting functions of thedimensions ˛ and U

10.1.4 Step 3: LMI Based Design and Stability Verification

In order to be comparable to the previous results, we execute the same designstrategy as here as before. As a result, we have the controller and observer gains inTP model form as F�.p.t// and K�.p.t//. Finally we transform the whole system toTP model representation over common weighting functions and use LMIs to verifystability.

10.2 Numerical Simulations

10.2.1 Determination of the Feasibility Region

Based on the above described convex hull manipulation strategy we derive differentTP models as follows. First we define z D 137 equidistantly located points onthe interval of � 2 Œ0; 1�. Then, we interpolate the TP models and perform LMIdesign each point. When z D 1, namely � D 0, then the derived TP model is

10.2 Numerical Simulations 121

SNNN type. If z D 137, namely � D 1 then the resulting TP model is CNO type(that defines a tight convex hull). This means that as z goes from 1 to 137, weare gradually modifying the convex hull defined by the vertexes of the TP model.We derive different TP models for the controller and observer design. This meansthat we can use 137 different TP models in the observer and in the controllerdesign phases, which leads to 137 � 137 different solutions. Comparing thesesolutions to each other, we can select a controller-observer pair that leads to the bestcontrol performance in terms of our performance objectives, such as settling time,maximum control value and L2 norm of the control signal during the evaluation.

We also execute interpolation between IRNO and CNO type TP models. TheIRNO type hull can be considered in this example as tight hull in contrast to SNNN.For this reason, the maximum value of z that represents the convex hulls generatedbetween IRNO and CNO type hulls can be smaller to simplify the numericalcomputation of the example. In the end, zcontroller D 31 � zobserver D 31 .D 961/

pairs are used for control design and for further control performance investigations.

10.2.2 Results of the Numerical Simulations

1. Case 1: SNNN (z=1) ! CNO (z=137)The results can be seen in Fig. 10.3. The feasibility region in which the LMIs

are feasible starts with min zcontroller D 121 and min zobserver D 1. This means thatonly very tight convex hulls defined by the TP model lead to feasible LMIs. Atthe same time, however, a stable observer can be achieved on polytopes defining

Fig. 10.3 L2 norm of the control signal


very large (SNNN, z=1) convex hulls. The minimum L2 norm is obtained usingzcontroller D 137 (CNO) and zobserver D 1

min kuc1k D 793:

Figures 10.4 and 10.5 show the results of the simulation and the trajectory ofthe control signal. We can observe that in the cases of zobserver D 1 (SNNN) andzcontroller D 137 (CNO), the maximum value is max uc1 D 276 rad, and the L2

norm is: kuc1k D 793. If both zobserver D zcontroller D 137 are set, the simulatedvalues are higher: max uc1 D 330 rad and kuc1k D 988. See also Figs. 10.6and 10.7.

2. Case 2: IRNO (z=1) ! CNO (z=31)The process is exactly the same as in Case 1, with the only difference that

an IRNO type polytope is chosen instead of an SNNN type one for the largeboundary hull. The numerical experiment leads to a very similar result, i.e., thecontroller hull is best if it is as tight as possible. The feasibility region is very thin(min zcontroller D 25 out of zcontroller D 31), but in this case the large hull might alsobe applied to satisfy the stability of the observer. In a way similar to Case 1: thebest performance is obtained by the utilization of the largest hull (zobserver D 1)possible for observer and the tightest one (zcontroller D 31) for controller design(Fig. 10.8):

min kuc2k D 893:

It can be observed in Figs. 10.9 and 10.10 that in the case of zobserver D 2

(CIRNO=close to IRNO) and zcontroller D 31 (CNO) the maximum control valueis max uc2 D 295 rad, and the L2 norm is: kuc2k D 893. If both zobserver D

zcontroller D 31 are set the simulated values are higher: uc2 D 330 rad and kuc2k D

988. See Fig. 10.8.3. Case 3: For comparison let Case 3 represent the results achieved in the previous

chapter, where the whole design process was executed on CNO type TP modelonly (excluding friction).

Table 10.1 shows the resulting performances. The settling time remains the samein each case; however, the L2 norms and the maximum control values are differentin order. The best performance values were obtained in Case 1.


Fig. 10.4 Time response of controller for U D 14:1 m/s. Here zcontroller D 137-CNO andzobserver D 1-SNNN


Fig. 10.5 Time response of controller for U D 14:1 m/s. Here zcontroller D 137-CNO andzobserver D 1-SNNN


Fig. 10.6 Time response of controller for U D 14:1 m/s. Here zcontroller D 137-CNO andzobserver D 137-CNO


Fig. 10.7 Time response of controller for U D 14:1 m/s. Here zcontroller D 137-CNO andzobserver D 137-CNO


Fig. 10.8 L2 norm of the control signal


Fig. 10.9 Time response of controller for U D 14:1 m/s. Here zcontroller D 31-CNO andzobserver D 3- close to IRNO


Fig. 10.10 Time response of controller for U D 14:1 m/s. Here zcontroller D 31-CNO andzobserver D 3- close to IRNO

Table 10.1 Control performance in the different cases

Case L2 norm Maximal control value (rad) Settling time (s)

1 793 �276 1.5

2 893 �295 1.5

3 1043 �350 1.5

Chapter 11Complexity Manipulation Based Optimization

Abstract The goal of this chapter is study the benefits of the manipulation of TPmodel complexity on the control design example of the 3DoF aeroelastic wingsection including friction. From a control perspective, the goal is to increase theparameter domain of the controller in contrast to the results obtained earlier. Themotivation behind this goal is that one of the parameters is wind speed, and it is anatural practical requirement that the range of this parameter should be increasedwhile maintaining the validity of the controller. It will be shown that the complexityrelaxation of the TP model allows us, for instance, to considerably increase theparameter domain in which the LMIs are still feasible and still obtain a viablecontroller. The results of this chapter are based on the work [1].

Keywords Optimisation • Complexity

11.1 The Control Design Framework

Based on Sect. 5.3, we proceed through the following steps:

• Main TP component analysis: We determine the HOSVD based canonical formof the 3DoF wing section model through the TP model transformation and selectthe main TP model.

• LMI based design: First, the main TP model is transformed to close-to-normalized (CNO) type. Then, we design the controller and observer with respectto the main TP model only. In order to make our results comparable to previoussections, we use the same LMI theorems as earlier.

• Exact system reconstruction: We integrate the controller and the observer withthe original system model and determine the polytopic representation of thewhole system (i.e., not only for the main TP model) over the same weightingfunction system.

• Stability verification: We apply LMIs to check the stability of the whole system.• Maximizing �: We increase � until the above 4-step procedure leads to feasible

LMIs in the stability verification step.


131

132 11 Complexity Manipulation Based Optimization

11.1.1 Main TP Model Component Analysis: HOSVD BasedCanonical Form of the Model

In order to execute the TP model transformation, we set G1 � G2 � G3 D 137 �

137 � 138 and � D Œ8; 20� m=s � Œ�0:3; 0:3� rad � Œ�1:5; 1:5� rad=s. The secondstep of the TP model transformation results in the quasi-HOSVD based structure ofthe given model. In our case the singular values are:

• Dimension of U.t/: 1521.40437, 38.57478, 0.03038• Dimension of ˛.t/: 1505.96433, 219.61492• Dimension of ˇ.t/: 1437.04339, 501.06441

This means that 3 � 2 � 2 D 12 vertexes are needed for the exact TP modelrepresentation. This step yields the core tensor SE and singular matrices UE

n . We usesuperscript “E” to emphasize that this is for the exact TP model representation.

In the third step of the TP model transformation, weighting functions wEn .pn.t//

are determined based on singular matrices UEn :

UEn ! wE

n .pn.t//: (11.1)

Thus we arrive at the quasi-HOSVD based canonical form:

S.p.t// D SE �n2f1;2g

wEn .pn.t//: (11.2)

Based on the singular values we can define the main TP model component of themodel. Let us keep only the two largest singular values per dimension. The resultingmain TP model will have 8 vertexes and will only be an approximation:

OS.p.t// D SMTP �n2f1;2g

wMTPn .pn.t//; (11.3)

where superscript “MTP” denotes the fact that these are the components of the mainTP model of the system.

In order to use LMI-based design theories, we need to transform the abovequasi-HOSVD based canonical form of the main TP model to a convex TP model.Practically, this means that we continue with the second step of the TP modeltransformation by executing the CNO transformation. This step results in theconvex TP structure of the main TP type polytopic model, namely, core tensorSMTP�CNO and matrices UMTP�CNO

n . The CNO transformation leads to a tight convexrepresentation:

OS.p.t// D SMTP�CNO �n2f1;2g

wMTP�CNOn .pn.t//: (11.4)

In order to show the weighting functions of the quasi-HOSVD based canonicalform of the exact TP model in Fig. 11.1, the main TP model in Fig. 11.2, the CNO

11.1 The Control Design Framework 133

0.15

0.1

0.05

0.0

−0.05

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

−0.1

−0.15

−0.2

−0.05

0.05

0

−0.1

−0.15

−0.2

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

8 10 12 14 16 18 20 22 24 26

Fig. 11.1 Weighting functions of the exact quasi-HOSVD based canonical form

type weighting functions of the exact TP model in Fig. 11.3, and the main TP modelin Fig. 11.4, we execute the third step of the TP model transformation for all thesevariants.

11.1.2 LMI Based System Design

Let us simplify the required control design performance to asymptotic stability. Notethat various further constraints can be considered through properly selected LMIs


0.15

0.1

0.05

0.0

−0.05

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

−0.1

−0.05

0.05

0

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−0.15

−0.2

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

8 10 12 14 16 18 20 22 24 26

Fig. 11.2 Weighting functions of the main component of the quasi-HOSVD based canonical form

as was demonstrated earlier. We search for a solution in the same observer-basedcontrol system structure as discussed in the previous sections:

u.t/ D �F.p.t//x0.t/; (11.5)

where x0.t/ is the combination of state vector x.t/ and Ox.t/, where

OPx.t/ D A.p.t//Ox.t/ C B.p.t//u.t/ C K.p.t//.y.t/ � Oy.t//

Oy.t/ D C.p.t//Ox.t/:


0.9

1

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Wei

ghtin

g fu

nctio

ns0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Wei

ghtin

g fu

nctio

ns

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

8 10 12 14 16 18 20 22 24 26

Fig. 11.3 CNO type weighting functions of the exact TP model

We search for a controller and observer to the main TP model

OS.p.t// D SMTP�CNO �n2f1;2g

wMTP�CNOn .pn.t// (11.6)

in TP model form as:

F.p.t// D F �n2f1;2g

wMTP�CNOn .pn.t// (11.7)

K.p.t// D K �n2f1;2g

wMTP�CNOn .pn.t//: (11.8)


8 10 12 14 16 18 20 22 24 260

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1W

eigh

ting

func

tions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wei

ghtin

g fu

nctio

ns

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

Fig. 11.4 CNO type weighting functions of the main TP model component

The results of the LMI-based design are the controller vertex gains Fi1;i2;:::;iNstored in F and the observer vertex gains Ki1;i2;:::;iN stored in K . We derive theobserver and controller using the same LMI based design theorems as in theprevious section.

An important additional benefit here is that we can derive the controller andobserver separately, as the stability does not need to be guaranteed in this step(it will be checked in the next step). Thus, we solve the LMIs in the above theoremsseparately that considerably relaxes the computation load.


11.1.3 Exact System Reconstruction: Unified Weightingsin the Polytopes

In this step the goal is to determine exact TP model representations of the controlsystem upon which LMI-based stability analysis can be executed. To this end, wetransform the exact TP model of the wing section, the controller, and the observer toTP model form over the same weighting functions without decreasing their ranks(i.e., without discarding nonzero singular values as in the case of the main TPmodel). We unify the weighting functions according to Sect. 2.10. As a result, wehave the vertexes of the model, controller, and observer over the same weightingfunction system. Figure 11.5 shows the CNO type unified weighting functionsystem. Note that the number of the vertexes in the TP form of the controller andobserver will increase, since the rank of the exact TP model of the wing section ondimension assigned to U.t/ is three. Thus the unification of the whole system withthe exact model leads to a system where the parameter dimension U.t/ has rankthree.

11.1.4 LMI Based Stability Verification

As a result of the previous steps we have all the vertexes of all components. We cansimply substitute the vertexes into proper LMIs and easily check whether or not theyare a solution. An additional benefit of the relaxation is that the LMI computationof this step is much relaxed, as we only have to check whether a common P exists.

11.1.5 Maximizing Omega

We set � in the first step of the design procedure, i.e., when we determine the quasi-HOSVD based canonical form to select the main TP model. Then, we check whetheror not the above design process leads to a solution, in other words, to a stable controlsystem. We subsequently begin to gradually increase �, especially in terms of theinterval of the wind speed.

The above steps require a few minutes on a regular computer, for instance usingthe TPtool MATLAB toolbox [2]. Thus, we can readily find the maximum domainof �, and we can define the maximum range of the wind speed within which thecontroller is still valid.


8 10 12 14 16 18 20 22 24 26

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

0.1

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1

Wei

ghtin

g fu

nctio

nsW

eigh

ting

func

tions

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

Fig. 11.5 CNO type weighting functions of all system components

11.2 Evaluation of the Benefits of the ProposedControl Design

The goal of this section is to analyze the benefits that can be obtained using the aboveintroduced main TP model component based design. A very important characteristicof the control problem investigated here is the interval of the external parameterU.t/. Typically, the larger the interval we have, the more powerful controller weneed. If we use the control design procedure based on the TP type polytopicmodel, as in the previous sections (i.e., in which the solution uses the exact TP

11.2 Evaluation of the Benefits of the Proposed Control Design 139

5 6 7 80

5

10

15

20

25

30

Umin

Um

ax

Feasible region in U

Case 1

Case 2, 3

Fig. 11.6 Maximum range for U.t/ where the design is feasible

type polytopic model, not only the main component during the design phase), thenthe maximum interval of U.t/ is strongly limited. Simply stated, the LMIs are notfeasible if the interval is larger than a certain threshold.

Figure 11.6 shows the maximum area for three cases. Case 1 is the maximuminterval achievable by the design solution without complexity relaxation, i.e., whenthe design is based on the exact model. Case 2 is the solution obtained using thecurrent relaxed method, but only with controller design. Case 3 shows the solutionwhen controller and observer are derived using the current relaxed method. Thehorizontal axis on the figure is the lower bound of U.t/, and the vertical axis showsthe maximum U.t/ where the design is still feasible. Cases 2 and 3 show that the areais considerably increased if we base the design on the main TP model component.We can see that the proposed design almost doubled the admissible interval.

Figures 11.7, 11.8, 11.9, and 11.10 show the controlled system for the criticalwind speed Ucrit.t/ D 11:4 m=s. To investigate the control performance in the wholerange, Figs. 11.11, 11.12, 11.13, 11.14, 11.15, 11.16, 11.17, and 11.18 plot thedynamics for the extremities of the interval of U.t/. For U.t/max, only Cases 2 and3 are given, since design Case 1 is not feasible. We can observe that Case 1 showsbetter performance in terms of the maximum control value and smooth stabilizationwith minimal overshoot, etc. This comes from the fact that Case 1 uses the exactmodel for the design. Thus, the price of having a larger interval of feasibility for thecontroller and observer is that the control performance is sightly degraded (in anycase, this should be quite obvious in general).


Fig. 11.7 Pitch for criticalwind speed U.t/ D 11:4 m=s

0

−0.2

−0.15Pitc

h, a

lpha

[rad

]

−0.1

−0.05

0

0.5 1

Time [s]

1.5 2

Case 1Case 2Case 3

Fig. 11.8 Plunge for criticalwind speed U.t/ D 11:4 m=s

0−20

−15

Plu

nge,

h [m

]

−10

−5

0

5

x 10−3

0.5 1

Time [s]

1.5 2

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Fig. 11.9 Trailing edge forcritical wind speedU.t/ D 11:4 m=s

0

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iling

edg

e, b

eta

[rad

]

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Fig. 11.10 Magnified controlvalue for critical wind speedU.t/ D 11:4 m=s

1

−8000

−6000

−4000

−2000

Con

trol

val

ue −

mag

nifie

d

0

2 3

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4

x 10−3

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Fig. 11.11 Pitch for windspeed U.t/ D 7 m=s

0

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Pitc

h, a

lpha

[rad

]

−0.1

−0.05

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1.5 2

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Fig. 11.12 Plunge for windspeed U.t/ D 7 m=s

0−20

−15

Plu

nge,

h [m

]

−10

−5

0

5

x 10−3

0.5 1

Time [s]

1.5 2

Case 1Case 2Case 3


Fig. 11.13 Trailing edge forwind speed U.t/ D 7 m=s

0

−2T

raili

ng e

dge,

bet

a [r

ad]

−1

0

1

2

0.5 1

Time [s]

1.5 2

Case 1Case 2Case 3

Fig. 11.14 Controlvalue—magnified for windspeed U.t/ D 7 m=s

−8000

−6000

Con

trol

val

ue −

mag

nifie

d

−4000

−2000

0

x 10−3

1 2

Time [s]

3 4

Case 1Case 2Case 3

Fig. 11.15 Pitch for windspeed U.t/ D 24 m=s

0

−0.15

−0.2

Pitc

h, a

lpha

[rad

] −0.05

−0.1

0

0.5 1

Time [s]

1.5 2

Case 2Case 3


Fig. 11.16 Plunge for windspeed U.t/ D 24 m=s

0−0.02

Plu

nge,

h [r

ad]

−0.015

−0.01

−0.005

0.005

0.01

0

0.5 1

Time [s]

1.5 2

Case 2Case 3

Fig. 11.17 Trailing edge forwind speed U.t/ D 24 m=s

0−2

Tra

iling

edg

e, b

eta

[rad

]

−1.5

−1

−0.5

0.5

1

0

0.5 1

Time [s]

1.5 2

Case 2Case 3

Fig. 11.18 Controlvalue—magnified for windspeed U.t/ D 24 m=s

−4000

Con

trol

val

ue −

mag

nifie

d

−3000

−2000

−1000

0

1 2

Time [s]

3

x 10−3

4

Case 2Case 3


References

1. P. Baranyi, B. Takarics, Aeroelastic wing section control via relaxed tensor product modeltransformation framework. J. Guid. Control. Dyn. 37(5), 1671–1678 (2014)

2. P. Baranyi, Z. Petres, Sz. Nagy, TPtool — Tensor Product MATLAB Toolbox. Website, 2007.http://tp-control.hu/


Chapter 12TP Model Manipulation Influences the ControlPerformance and the Feasibility of LMIBased Design

Abstract The goal of this section is to summarize the results of the previoussections and deliver a comprehensive analysis of the TP model manipulation via thecontrol design of the 3DoF model including friction. All the variations of the convexhull manipulation and the complexity relaxation of the TP model of the aeroelasticwing section are taken into consideration in the following analysis. This chapter isbased on the work [1].

Keywords Control performance • LMI feasiblity • Convex hull manipulation

12.1 Feasibility

The goal of this section is to show that the manipulation of the vertexes of the TPmodel influences the feasibility of the linear matrix inequality (LMI) based design

(a) the position of the vertexes of the TP model type polytopic representationstrongly influences the feasibility of LMI based control design.

(b) the complexity of the TP model, namely the number of the vertexes containedin the TP model, also strongly influences the feasibility of LMI based controldesign.

(c) statements (a) and (b) are valid both for the controller and observer systemelements but in a separate, different way.

(d) the position and number of the vertexes of the polytopic TP model representa-tion also influence the size of the achievable parameter space where the LMIbased design is feasible.

12.1.1 Initialization of the Numerical Analysis

According to the previous sections we define the exact TP model and the relaxed TPmodel of the wing section. The relaxed TP model has two weighting functions ineach dimension. The CNO and an SNNNN variant of the exact and the relaxed TPmodel are also derived. Then we interpolate TP models between the SNNN and the


145

146 12 TP Model Manipulation Influences the Control Performance and the Feasibility. . .

CNO type TP models. One TP model is interpolated for the controller design andone for the observer design. Thus we have four TP models:

• Interpolated exact TP model for controller design• Interpolated exact TP model for observer design• Interpolated relaxed TP model for controller design• Interpolated relaxed TP model for observer design

The TP models are interpolated over 50 equidistantly located points of theinterpolation parameter � 2 Œ0; 1�. Thus we have different 50 TP models as atransition between the SNNN and CNO type TP models. Then we execute controllerand observer design through the same LMIs used in the previous sections. Weperform the same analysis with the relaxed TP model representation.

12.1.2 Results of the 2D Analysis: Feasibility and Convex Hull

Figure 12.1 illustrates the relation between feasibility and the convex hull in caseof the controller and observer design based on the exact TP model. The x-axisillustrates the convex hull, namely the transition from the loose convex hull (SNNN,� D 0) to the tight convex hull (CNO, � D 1) corresponding to the interpolationparameter �. The y-axis illustrates the feasibility with a line, if the LMI based designresulted in a feasible solution. The value y D 0 illustrates the case if the design didnot yield in a feasible solution.

The controller was designed on the parameter space U.t/ D Œ6 16� .m=s/. Theresults in Fig. 12.1 in case of the controller show a strong correlation between thefeasibility and the convex hull: the feasible LMI designs appear near the tight, CNOtype convex hull than the loose, SNNN type convex hull.

The observer was designed on the parameter space U.t/ D Œ6 400� .m=s/. Thereason to select this unrealistic and large interval of the external parameter windspeed U.t/ is to be able to indicate the influence of the convex hull manipulation onthe feasibility of the observer, which could be detected through this region, since theobserver design in the interval U.t/ D Œ6 16� .m=s/ is always feasible. The resultsin Fig. 12.2 in case of the observer show also a relation between the feasibility andthe convex hull: a non-feasible segment can be detected at a convex hull transitionalsection.

In this context as a conclusion of this section it can be stated that the convexhull of the polytopic TP model representation strongly influences the feasibility ofLMI based designs, which is valid both for the controller and observer cases, in aseparate, different way.

12.1 Feasibility 147

SNNN CNO

Not feasible

Feasible

Controller feasibility regions − exact model

λcontroller

Fig. 12.1 Controller feasibility for external parameter wind speed U.t/ D Œ6 16� .m=s/

SNNN CNO

Not feasible

Feasible

Observer feasibility regions − exact model

λobserver

Fig. 12.2 Observer feasibility for external parameter wind speed U.t/ D Œ6 400� .m=s/


12.1.3 Results of the 3D Analysis: Feasibility, Convex Hull,and Complexity

The investigation of the complexity is further continued for different intervals ofthe external parameter wind speed U.t/. As a result Figs. 12.3 and 12.4 present therelation between feasibility, the convex hull, and complexity for different intervalsof parameter U.t/. The x-axis illustrates the convex hull similar to the previoussection, the y-axis denotes the complexity of the model with the exact and relaxedTP model cases, and the z-axis represents the feasibility, also similar to the previoussection. Figure 12.3 illustrates the results of the controller, with the parameterinterval U.t/ D Œ6 16� .m=s/ and U.t/ D Œ8 33� .m=s/ and Fig. 12.4 illustrates theresults of the observer design with the parameter interval U.t/ D Œ6 400� .m=s/ andU.t/ D Œ8 425� .m=s/. The results show that the complexity of the TP model alsointerferes with the feasibility.

In case of the controller the feasible designs fall in number if the TP model isa relaxed model containing fewer vertexes. However, considering the convex hull,the feasible designs remain similarly near the tight, CNO type convex hull than theloose, SNNN type convex hull both for the relaxed and exact TP model cases.

In case of the observer the feasible designs appear in a larger number if the TPmodel is a relaxed model containing fewer vertexes in the present parameter intervalU.t/ D Œ6 400� .m=s/. Considering the convex hull the results show similar to theprevious section a relation between the convex hull and feasibility: further feasibleand non-feasible segments can be detected at different convex hull transitionalsections.

Based on these results as a conclusion the same phenomenon can be observedas in the previous section with an additional information: besides the convex hull ofthe polytopic TP model representation the complexity also influences the feasibilityregions of LMI based designs, moreover as a further conclusion this is valid bothfor the controller and observer cases, also in a separate, different way.

12.1.4 Results of the 4D Analysis: Feasibility, Convex Hull,Complexity, and Parameter Space

Continuing the investigation further and incorporating the parameter space intothe graphical illustrations as a result the Figs. 12.5, 12.6, 12.7, 12.8, 12.9, and12.10 illustrate the relation between feasibility, the convex hull, complexity, andthe parameter space with the external parameter wind speed U.t/. The axes of thefigures are the same as on the figures of the previous sections; the difference isthat the z-axis provides additional information about the external parameter windspeed U.t/, namely it also illustrates the maximal achievable value Umax of theinterval: a line denotes if the LMI based design is feasible and the height indicatesthe value of Umax. Figures 12.5, 12.6, and 12.7 illustrate the case of the controllerwith the parameter intervals U.t/ 2 Œ4 Umax� .m=s/, U.t/ 2 Œ6 Umax� .m=s/ andU.t/ 2 Œ8 Umax� .m=s/ and Figs. 12.8, 12.9, and 12.10 illustrate the case of the


SNNN

Exact model

Relaxed model

Not feasibleCNO

Feasible

Controller feasibility with U(t) = [6 16] [m/s]

λ

U(t

) [m

/s] a

nd fe

asib

ility

SNNN

Exact model

Relaxed model

Not feasibleCNO

Feasible

Controller feasibility with U(t) = [8 33] [m/s]

λ

U(t

) [m

/s] a

nd fe

asib

ility

Fig. 12.3 Controller feasibility for external parameter wind speed U.t/ D Œ6 16� .m=s/ andU.t/ D Œ8 33� .m=s/


SNNN

CNO

Exact model

Relaxed model

Not feasible

Feasible

Observer feasibility with U(t) = [6 400] [m/s]

λ

U(t

) [m

/s] a

nd fe

asib

ility

Fig. 12.4 Observer feasibility for external parameter wind speed U.t/ D Œ6 400� .m=s/

SNNN

Exact model

Relaxed model

0CNO

20

40

60

Controller feasibility U(t) = [4 64] [m/s]

λ

U(t

) [m

/s] a

nd fe

asib

ility

Fig. 12.5 Controller feasibility regions and achievable external parameter wind speed investigatedover the interval U.t/ 2 Œ4 Umax� .m=s/


SNNN

Exact model

Relaxed model

0CNO

20

40

60


λ

U(t

) [m

/s] a

nd fe

asib

ility

Fig. 12.6 Controller feasibility regions and achievable external parameter wind speed investigatedover the interval U.t/ 2 Œ6 Umax� .m=s/ results

SNNN

CNO Exact model

Relaxed model

0

20

40

60


λ

U(t

) [m

/s] a

nd fe

asib

ility

Fig. 12.7 Controller feasibility regions and achievable external parameter wind speed investigatedover the interval U.t/ 2 Œ8 Umax� .m=s/ results


SNNN

CNO

Exact model

Relaxed model

0

500

Observer feasibility U(t) = [4 600] [m/s]

λ

U(t

) [m

/s] a

nd fe

asib

ility

Fig. 12.8 Observer feasibility regions and achievable external parameter wind speed investigatedover the interval U.t/ 2 Œ4 Umax� .m=s/

SNNN

CNO

Exact model

Relaxed model

0

500


λ

U(t

) [m

/s] a

nd fe

asib

ility



SNNN

CNO

Exact model

Relaxed model

0

500


λ

U(t)

[m/s

] and

feas

ibili

ty


observer with the parameter intervals U.t/ 2 Œ4 Umax� .m=s/, U.t/ 2 Œ6 Umax� .m=s/and U.t/ 2 Œ8 Umax� .m=s/. The investigation was executed in an iterative manner:an LMI based design is executed for the current value Umax of the parameter intervaland the feasibility of the design is checked. If the design is feasible, the maximalvalue Umax of the interval is increased, the LMI based design is repeated and thefeasibility is checked. This is executed until the design is still feasible. The resultsshow that the size of the parameter space of the TP model also interferes with thefeasibility.

It can be seen in case of the controller that the previously stated phenomenaconsidering the convex hull and complexity are also valid, with an additionalstatement: the feasible designs fall in number if the TP model is a relaxed modelcontaining fewer vertexes, the feasible LMI designs appear in larger number nearthe tight, CNO type convex hull, but the feasible results also appear with a higherachievable parameter interval value Umax near the tight, CNO type convex hull. Incase of the controller the achievable parameter interval value Umax reaches the valueof 64 .m=s/ at the tight, CNO type convex hull, whereas it only reaches a smallervalue at the transitional cases further from the tight, CNO type convex hull.

In case of the observer it can be determined that the relation between theconvex hull and feasibility show a similarity to the previous section: feasibleand non-feasible segments can be detected at different convex hull transitionalsections. Considering the complexity the feasible designs appear not necessarilyin a larger number if the TP model is a relaxed model containing fewer vertexes butnevertheless a difference in the feasibility regions can be detected between the exactand relaxed TP model cases.


In this context as a conclusion the following can be stated: the convex hulland complexity of the polytopic TP model representation strongly influences thefeasibility regions of LMI based designs in a different way for controller andobserver, and it also has a strong effect on the achievable external parameter windspeed Umax where the LMI based design is feasible.

12.1.5 Summary

This section provided a proof based on a control design example of a qLPV state-space model that the manipulation of the vertexes of the polytopic TP modelrepresentation strongly influences the feasibility of the LMI based design. Theattributes of the vertexes influencing the feasibility regions of the LMI basedcontrol design include the position and the number of the vertexes contained in themodel. Furthermore the section shows that the vertexes of the polytopic TP modelrepresentation also influence the size of the achievable parameter space where theLMI based design is feasible. These statements are valid both for the feasibility ofthe controller’s and the observer’s LMI based design, but the influence differs in itscharacteristics for the controller and the observer system components.

12.2 Control Performance

In this section two TP models are interpolated between SNNNN and CNO typeexact TP model representations of the aeroelastic model (3DoF, including friction).One is for controller design and the other is for observer design. Using the sameLMIs as above we derive the controller and observer separately and finally we checkthe stability via LMIs and compare the resulting control performances to select thebest solution. Thus a two parametric manipulation space is defined by �controller and�observer. Because of the computational load we examine only 10 � 10 points (10 TPmodels are interpolated) in this space.

12.2.1 Control Performance Results of the NumericalSimulation

In this section the results of the numerical simulation of the designed TP modelsare given and analyzed. Figures 12.11, 12.12, 12.13, and 12.14 show the resultsof the numerical simulation of some feasible TP models for free stream velocityU D 14:1 m=s and simulation duration of 2 s. Additionally, for comparison ofthe control performance, Tables 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 12.7, and 12.8

12.2 Control Performance 155

0 0.5 1 1.5 2−20

−15

−10

−5

0

5 x 10−3 Plunge, h [m]

Time [s]

Plu

nge,

h [m

]

0 0.5 1 1.5 2−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1Pitch, α [rad]

Time [s]

Pitc

h, α

[rad

]

0 0.5 1 1.5 2−5

−4

−3

−2

−1

0

1

2

3

4Trailing edge, β [rad]

Time [s]

Tra

iling

edg

e, β

[rad

]

0 0.5 1 1.5 2−10000

−8000

−6000

−4000

−2000

0

2000Control value

Time [s]

Con

trol

val

ue

Fig. 12.11 Case 1.1. Controller: 11:11 % SNNN and 88:88 % CNO TP model representation,Observer: 77:77 % SNNN and 22:22 % CNO TP model representation

provide measured values of each signal. In case of the controller, the TP modelwith SNNN type convex hull and some interpolated cases did not lead to a feasibledesign; therefore the control performance investigation in these cases could not beexecuted.

Regarding the feasible cases, two sets of results are provided. The first set ofresults—represented by Figs. 12.11 and 12.12 and Tables 12.1, 12.2, 12.3, and 12.4and denoted by case 1.1 and 1.2—consists of a TP model based controller with a11:11 % SNNN, 88:88 % CNO TP model accompanied by two different observers:a 77:77 % SNNN, 22:22 % CNO TP model, and a 11:11 % SNNN, 88:88 % CNOTP model. The second set of results—represented by Figs. 12.13 and 12.14 andTables 12.5, 12.6, 12.7, and 12.8 and denoted by case 2.1 and 2.2—consists of a TPmodel based controller with a 0 % SNNN, 100 % CNO TP model accompanied bytwo different observers: a 77:77 % SNNN, 22:22 % CNO TP model and a 22:22 %SNNN, 77:77 % CNO TP model.


0 0.5 1 1.5 2−0.02

−0.015

−0.01

−0.005

0

0.005

0.01Plunge, h [m]

Time [s]

Plu

nge,

h [m

]

0 0.5 1 1.5 2−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1Pitch, α [rad]

Time [s]

Pitc

h,α

[rad

]

0 0.5 1 1.5 2−0.4

−0.2

0

0.2

0.4

0.6

0.8


Time [s]

Tra

iling

edg

e, β

[rad

]

0 0.5 1 1.5 2−10000

−8000

−6000

−4000

−2000

0

2000Control value

Time [s]

Con

trol

val

ue

Fig. 12.12 Case 1.2. Controller: 11:11 % SNNN and 88:88 % CNO TP model representation,Observer: 11:11 % SNNN and 88:88 % CNO TP model representation

12.2.2 Evaluation and Comparison of the Derived Casesand the Best Solution

Analyzing the results the following comparisons can be made.Considering the signal initial values: the best initial values for the trailing edge

signal is illustrated by case 1.3 with the result of �0:01817. In case of pitch, plunge,and control signals, case 2.2 achieved the best values with �0:2147, �0:01848, and�53:04, respectively.

Regarding the signal end values: the best signal end values for trailing edgewas achieved by case 2.2 with 0:001369. For pitch and plunge by case 1.1 with0:0009113 and 0:00008623 each, and for the control signal by case 1.3 with thevalue of �0:05067.

Considering settling time: the fastest settling time for trailing edge was providedby case 2.1 with the duration of 0:1423 s. Regarding the pitch signal the fastestsettling time was achieved by case 2.2 with 0:3174 s. Last, for the plunge and controlsignals the fastest settling time was reached by case 1.3 with 1:6652 and 0:0097 s,respectively.

Regarding overshoot: the smallest overshoot for the trailing edge signal wasprovided by case 1.3 with a value of 0:8729, for the pitch signal by case 1.1 with


0 0.5 1 1.5 2−20

−15

−10

−5

0

5 x 10−3 Plunge, h [m]

Time [s]

Plu

nge,

h [m

]

0 0.5 1 1.5 2−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1Pitch, α [rad]

Time [s]

Pitc

h, α

[rad

]

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2


Time [s]

Tra

iling

edg

e, β

[rad

]

0 0.5 1 1.5 2−200

−100

0

100

200

300

400

500

600Control value

Time [s]

Con

trol

val

ue

Fig. 12.13 Case 2.1. Controller: 0 % SNNN and 100 % CNO TP model representation, Observer:77:77 % SNNN and 22:22 % CNO TP model representation

0:001597, for the plunge signal by case 2.1 with 0:003191 and for the control signalby case 2.2 with 12:77.

Regarding undershoot: the smallest undershoot for the trailing edge signal wasachieved by case 2.2 with �0:2229. Considering the pitch signal, case 1.1 providedthe smallest undershoot value with �0:003284. In case of the plunge signal, case 2.1provided the best undershoot value with �0:006221. Finally, regarding the controlsignal case 2.2 achieved the smallest undershoot with �3:076.

Comparing these results it can be determined that the interpolated case 2.2provided the majority of the best results indicating thereby the overall best solutionfrom the derived cases. The succeeding results are represented through case 2.1,case 1.2, and case 1.1.


0 0.5 1 1.5 2−0.02

−0.015

−0.01

−0.005

0

0.005

0.01Plunge, h [m]

Time [s]

Plu

nge,

h [m

]

0 0.5 1 1.5 2 2.5 3−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1Pitch, α [rad]

Time [s]

Pitc

h, α

[rad

]

0 0.5 1 1.5 2−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Trailing edge, β [rad]

Time [s]

Tra

iling

edg

e, β

[rad

]

0 0.5 1 1.5 2−60

−50

−40

−30

−20

−10

0

10

20Control value

Time [s]

Con

trol

val

ue

Fig. 12.14 Case 2.2. Controller: 0 % SNNN and 100 % CNO TP model representation, Observer:22:22 % SNNN and 77:77 % CNO TP model representation

Table 12.1 Comparison of trailing edge signals

77:77 % SNNN 11:11 % SNNN

22:22 % CNO TP model 88:88 % CNO TP model

(case 1.1) (case 1.2)

Signal initial value �4.8 �0.01817

Signal end value 0.002226 0.007864

Settling time 0.1504 1.1099

Overshoot 3.332 0.8729

Undershoot �4.8 �0.3863


Table 12.2 Comparison of pitch signals

77:77 % SNNN 11:11 % SNNN


(case 1.1) (case 1.2)


Signal end value �0.0009113 �0.003387


Overshoot 0.001597 0.02318

Undershoot �0.003284 �0.04933

Table 12.3 Comparison of the plunge signals

77:77 % SNNN 11:11 % SNNN


(case 1.1) (case 1.2)


Signal end value �0.00008623 �0.0003498


Overshoot 0.3266 0.009024

Undershoot �0.006981 �0.008674

Table 12.4 Comparison of control values

77:77 % SNNN 11:11 % SNNN


(case 1.1) (case 1.2)

Signal initial value �10000 �10000

Signal end value 0.06755 �0.05067



Undershoot �10000 �4.897

Table 12.5 Comparison of trailing edge signals

77:77 % SNNN 22:22 % SNNN


(case 2.1) (case 2.2)


Signal end value 0.002983 0.001369





Table 12.6 Comparison of pitch signals

77:77 % SNNN 22:22 % SNNN


(case 2.1) (case 2.2)


Signal end value �0.001514 �0.002425


Overshoot 0.001598 0.005278

Undershoot �0.004087 �0.02245

Table 12.7 Comparison of the plunge signals

77:77 % SNNN 22:22 % SNNN


(case 2.1) (case 2.2)


Signal end value 0.0002076 �0.0004004


Overshoot 0.003191 0.00563

Undershoot �0.006221 �0.007401

Table 12.8 Comparison of control values

77:77 % SNNN 22:22 % SNNN


(case 2.1) (case 2.2)


Signal end value 0.06121 �0.1718




Reference

1. A. Szollosi, P. Baranyi, Influence of the tensor product model representation of qLPVmodels on the feasibility of linear matrix inequality. Asian J. Control 18(5), 1–15 (2016).doi:10.1002/asjc.1238

Part IVTP Model Based Control Design of the

Dual-Excenter Vibration Actuator

This part of the book is based on the results published in papers by Kuti et al. [11].Vibration capabilities are essential in a variety of assistive and rehabilitation

applications [1, 14, 19] and used in most hand-held personal informatics devices[2, 16–18], e.g., vibration-based notifications or sophisticated tactile feedback.Vibration capability in such devices is usually implemented using eccentric rotatingmass (ERM) actuators (1DoF approach) composed of a DC micromotor withan eccentric rotor, or the so-called shaftless vibration motor [12]. A commondisadvantage of these solutions is that the frequency and intensity of vibrations arecoupled (resulting in 1 degree of freedom), which causes the generated vibrationsto be less rich than in the case of 2-DoF approaches. Linear Resonant Actuators(LRAs), though widely used, are effective only when they operate at device-specificresonant frequencies.

In this section, we direct our focus on dual-excenter vibration actuators whichcontain two coaxial eccentric rotors driven by miniature DC motors [15]. This set-up enables the separate control of frequency and intensity through the modificationof the angular velocity of and offset angle between the rotors. The generaldynamics and control of such mechanical systems are detailed for example in[10, 13]. In the following we consider the configure with two coaxially locatedeccentric rotors driven by DC motors mounted on a common suspension [15].The investigated mechanical system is nonlinear and parameter dependent. Inparticular, several environmental parameters, some of which are not even completelymeasurable (without significant noise), bring uncertainty into the system. Undersuch limitations, possibilities for mathematically rigorous treatment are also limited.Nevertheless, as we will see, the TP model transformation based design frameworkprovides important advantages where such attempts are concerned.

We begin by noting that in the case of partially unknown system states, potentialcontrol solutions are restricted to static and dynamic output feedback schemes. Thefollowing works provide an important background relevant to such schemes:

162 IV TP Model Based Control Design of the Dual-Excenter Vibration Actuator

• Chadli in [4] proposed a computationally inexpensive LMI system for constantoutput matrices, later extending the solution to cases where state observers areavailable together with a set of robust criteria.

• Guelton proposed a relaxation to such problems through the application of fuzzydescriptor redundancies and fuzzy Lyapunov functions with static, dynamic, andlater robust H1 requirements [3, 8, 9].

• Chang adapted the above approaches to discrete-time systems [6]. Some furtherdynamic output feedback design methods prescribing robustness against param-eter uncertainties and noise cancellation were proposed in [5, 7].

References

1. A.U. Alahakone, S.M.N.A. Senanayake, A real-time system with assistive feedback forpostural control in rehabilitation. IEEE/ASME Trans. Mechatron. 15(2), 226–233 (2010)

2. S. Azenkot, R.E. Ladner, J.O. Wobbrock, Smartphone haptic feedback for nonvisual wayfind-ing, in The Proceedings of the 13th International ACM SIGACCESS Conference on Computersand Accessibility (ACM, New York, 2011), pp. 281–282

3. T. Bouarar, K. Guelton, N. Manamanni, Static output feedback controller design for Takagi-Sugeno systems - a fuzzy Lyapunov LMI approach, in Proceedings of the 48th IEEEConference on Decision and Control, 2009 held jointly with the 2009 28th Chinese ControlConference, CDC/CCC 2009 (2009), pp. 4150–4155

4. M. Chadli, D. Maquin, J. Ragot, et al., Static output feedback for Takagi-Sugeno systems:an LMI approach, in 10th Mediterranean Conference on Control and Automation, MED’2002(2002)

5. J.-L. Chang, Dynamic output feedback disturbance rejection controller design. Asian J. Control15(2), 606–613 (2013)

6. X.-H. Chang, G.-H. Yang, X.-P. Liu, H1 fuzzy static output feedback control of T-S fuzzysystems based on fuzzy Lyapunov approach. Asian J. Control 11(1), 89–93 (2009)

7. B. Ding, New formulation of dynamic output feedback robust model predictive control withguaranteed quadratic boundedness. Asian J. Control 15(1), 302–309 (2013)

8. K. Guelton, T. Bouarar, N. Manamanni, Fuzzy Lyapunov LMI based output feedbackstabilization of Takagi-Sugeno systems using descriptor redundancy, in IEEE InternationalConference on Fuzzy Systems, 2008, FUZZ-IEEE 2008 (IEEE World Congress on Computa-tional Intelligence) (IEEE, New York, 2008), pp. 1212–1218

9. K. Guelton, T. Bouarar, N. Manamanni, Robust dynamic output feedback fuzzy Lyapunovstabilization of Takagi–Sugeno systems - a descriptor redundancy approach. Fuzzy Sets Syst.160(19), 2796–2811 (2009)

10. Q.K. Han, B.C. Wen, Stability and bifurcation of self-synchronization of a vibratory screenerexcited by two eccentric motors. Adv. Theor. Appl. Mech. 1(3), 107–119 (2008)

11. J. Kuti, P. Galambos, A. Miklos, Output feedback control of a dual-excenter vibration actuatorvia qLPV model and TP model transformation. Asian J. Control 17(2), 432–442 (2015)

12. M. Levin, A. Woo, Tactile-feedback solutions for an enhanced user experience. Inf. Disp.25(10), 18–21 (2009)

13. X. Liu, C. Wang, C. Zhao, B. Wen, Observation and control of phase difference for a vibratorymachine of plane motion, in 2010 International Conference on Computer, Mechatronics,Control and Electronic Engineering (CMCE), vol. 4 (2010), pp. 330–334

14. S. Mann, J. Huang, R. Janzen, R. Lo, V. Rampersad, A. Chen, T. Doha, Blind navigation with awearable range camera and vibrotactile helmet, in Proceedings of the 19th ACM InternationalConference on Multimedia (ACM, New York, 2011), pp. 1325–1328

References 163

15. A. Miklós, Z. Szabó, Vibrator with DC motor driven eccentric rotors. Period. Polytech. - Mech.Eng. 56(1), 49–53 (2012)

16. H. Nishino, R. Goto, T. Kagawa, K. Yoshida, K. Utsumiya, J. Hirooka, T. Osada, N. Nagatomo,E. Aoki, A touch screen interface design with tactile feedback, in 2011 InternationalConference on Complex, Intelligent and Software Intensive Systems (CISIS) (IEEE, New York,2011), pp. 53–60

17. H. Qian, R. Kuber, A. Sears, Towards developing perceivable tactile feedback for mobiledevices. Int. J. Hum. Comput. Stud. 69(11), 705–719 (2011)

18. J. Rantala, K. Salminen, R. Raisamo, V. Surakka, Touch gestures in communicating emotionalintention via vibrotactile stimulation. Int. J. Hum. Comput. Stud. 71(6), 679–690 (2013)

19. J. van der Linden, E. Schoonderwaldt, J. Bird, R. Johnson, Musicjacket—combining motioncapture and vibrotactile feedback to teach violin bowing. IEEE Trans. Instrum. Meas. 60(1),104–113 (2011)

Chapter 13qLPV Model of the Dual ExcenterVibration System

Abstract This chapter introduce the mechanical model and corresponding equationof motion based upon which recent work has been carried out, e.g., in [2]. The goalof the chapter is to derive the qLPV model form of the dual excenter vibrationsystem that is ready for the execution of the TP model transformation and thecontroller design in the next chapters.

Keywords qLPV • Equation of motions • Dual excenter vibration

Figure 13.1 illustrates the 2-D mechanical system that contains two independentlydriven, coaxial, eccentric rotors. In this model we assume that the environment isisotropic, such that k represents the stiffness and c the damping coefficient in bothdirections. The two rotors are driven around the axis at point C by torques T1 andT2. The eccentricity is characterized by the mass of the rotors (m0) and the distancee of the center of mass from the rotation axis. J0 denotes the rotor’s moment ofinertia with respect to the axis C. In our discussions, ' will be used to represent themean position of the rotors, while ı is interpreted as half of the phase difference(2ı D '1 �'2) between the rotors. The position of the vibrating system is describedusing two-dimensional Descartes coordinates x; y with respect to the balanced state.To summarize, the following notations are used:

k; c stiffness and damping of the environmente eccentricity of the rotorsm0 mass of a rotorm mass of the moving systemM mass of the moving system and rotorsJ0 mass moment of inertia of the rotor about C axis' angular position of the center of eccentric massı half of the phase angle between the rotorsx; y Descartes coordinates of the vibrating system


165

166 13 qLPV Model of the Dual Excenter Vibration System

R

m

T1

T2

OC

e

m ,J0 0

ck

c

k

x

y

Fig. 13.1 Mechanical model of the vibrating system

The equation of motion can be written as follows:

M.q/ Rq D v.q; Pq; T1; T2; k; c/; (13.1)

where

q D Œ x y ' ı �T ; (13.2)

M.q/ D

2

664

M 0 �2em0 cos ı sin ' �2em0 sin ı cos '

0 M 2em0 cos ı cos ' �2em0 sin ı sin '

�2em0 cos ı sin ' 2em0 cos ı cos ' 2J0 0

�2em0 sin ı cos ' �2em0 sin ı sin ' 0 2J0

3

775 ;

(13.3)

v.q; Pq; T1; T2; k; c/ D

2

664

�cPx � kx C 2. Pı2 C P'2/em0 cos ı cos ' � 4 Pı P'em0 sin ı sin '

�cPy � ky C 2. Pı2 C P'2/em0 cos ı sin ' C 4 Pı P'em0 sin ı cos '

T1 C T2

T1 � T2

3

775 :

(13.4)

13 qLPV Model of the Dual Excenter Vibration System 167

The key feature of this configuration is that the vibrations are generated bytwo actuators, meaning that the system has two degrees of freedom (amplitudeand frequency). The total eccentricity of the system is due to the offset. Throughthe effects of T1 and T2 the system can oscillate around equilibrium point O. Asevidenced by the nonlinear and parameter-dependent equation of motion (13.1),the rotor movements are not independent due to the dynamic coupling effect ofthe suspension. The amplitude of the generated vibration depends on the phasedifference, angular velocity, and suspension parameters. Later, the mean phase anglewill be left out from the qLPV formulation, due to the decision to select generalcoordinates in the equation of motion.

Miklos et al. has shown that the motion of the eccentric rotors tends to be selfsynchronized by the effect of the coupled dynamics [2]. This means that in certainsituations the rotors, driven by different torques, settle at the same angular velocitywith a stable offset in steady state. Liu et al. indicated similar results in twin-rotormachinery [1]. This behavior can be explained through an analysis of the steadystate of the system in two different cases:

• Balanced steady state with angular velocity P' D ! W const and offset ı D �=2:

x D y D 0;

T1 D T2 D 0: (13.5)

• Unbalanced limit cycle with angular velocity P' D ! W const and offset ı ¤ �=2:

x2 C y2 D R2 W const;

R D Rmax.!/ cos ı;

T1 C T2 D c!R2max.!/ cos2 ı;

T1 � T2 D1

2.k � ms!

2/ sin.2ı/R2max.!/; (13.6)

where

Rmax.!/ D 2em0

!r

c2 C�

k�ms!2

!

2: (13.7)

The achievable maximum amplitude (Rmax.!/) is in linear proportion to theangular velocity and has a local maximum at the resonance frequency. The actualamplitude is, in addition, commensurate with cos.ı/ as demonstrated in Eq. (13.6).

In the case of R > 0, kinetic energy is dissipated through the damping, andtherefore an offset other than ı D �=2 can be maintained only if control torquesT1 C T2 > 0 at all times.


For the compensation of the self-synchronizing effect (excluding balanced andzero offset cases), a difference between the torques T1�T2 ¤ 0 is necessary. Further,the difference T1 � T2 will change its sign at the resonance frequency, owing to thefact that the self-synchronizing effect forces the system into a state of zero offset andmaximum vibration, while at frequencies higher than the frequency of resonance,the balanced steady state is stable.

Extending the nonlinear system detailed in Eq. (13.1) specifically with thesimplified dynamics of the DC motors, we may write the qLPV form in a waythat aligns well with the specific requirements for further control synthesis. Theeccentric rotors are in this case driven by permanent magnet DC motors. Within theqLPV model, the armature inductance is neglected as the electric time constant issmaller than the mechanical time constant by more than one order of magnitude.Hence, the simplified motor equation can be written as:

T1;2 Dkt

Ra.U1;2 � ke P'1;2/; (13.8)

that substituted into the equation of motions results in

T1 C T2 Dkt

Ra.U1 C U2 � 2ke P'/; (13.9)

T1 � T2 Dkt

Ra.U1 � U2 � 2ke

Pı/: (13.10)

In our case, the goal of control design is to ensure that the target states arestable and can be reached quickly enough under a wide range of angular velocity( P') and offset (ı) conditions. As a matter of fact, this targets the limit cycles ifthe motion is described using Descartes coordinates x; y whenever ı ¤ �=2. As aresult, the stability of such cases would be more amenable to investigation usingpolar coordinates, which otherwise would introduce additional complexities as wellas a singularity at the origin. For this reason, only the stability of the ı D �=2 andP' D ! 2 Œ!min; !max� states are considered. These states are equilibrium points ifT1 D T2 D 0, that is, U1 D U2 D Ueq.!/, where Ueq.!/ D ke!.

In the first step of the qLPV formulation the equation of motion (13.1) is rewrittento first order differential form:

Px.t/ D f .x.t/; u.t//; (13.11)

where omitting the ' angular coordinate, the state variables are:

x.t/ D�

Px Py P' Pı x y ı�T

;

13 qLPV Model of the Dual Excenter Vibration System 169

the input u.t/ D�

U1 U2

�T, while the right side of (13.1) is

f .x.t/; u.t// D

2

664

M�1.x.t//v.x.t/; u.t//PxPyPı

3

775 :

In the equilibrium states, Px D f .xeq; ueq.xeq// D 0. The state and the input vectorin the investigated equilibrium points can be expressed as:

xeq D�0 0 ! 0 0 0 �=2

�T;

ueq.xeq/ D Œ ke! ke! �T :

We neglect the nonlinearity of the state variable Pı, and instead use the approxi-mation Pı2 C P'2 � P'2. Considering implementation constraints, this approximationcan be justified from several aspects, for instance, from the point of view that thelimitations of the rotation sensors (encoders) impose upper bounds on the applicableangular velocities, while the computation of Pı introduces significant noise with largetime delay at low angular velocities, leading to the realization that Pı could not in anycase be ideally considered in the control algorithm.

Since ' is a cyclic coordinate and not involved in the feedback, only the' D 45ı location is considered. With these simplifications and further algebraicmanipulations, the following qLPV model is obtained:

Px.t/ D A.p.t//.x.t/ � xeq/ C B.p.t//.u.t/ � ueq.xeq//; (13.12)

y.t/ � yeq D C.x.t/ � xeq/; (13.13)

where

y.t/ D�

P' Pı ı�T

; (13.14)

A.p.t// D

2

6664

M�1.p.t//A0.p.t//1 0 0 0 0 0

0 1 0 0 0 0

0 0 0 1 0 0

3

7775

; (13.15)

B.p.t// D

2

664

M�1.p.t//B0

0 0

0 0

0 0

3

775 ; (13.16)


C D

2

40 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 0 0 1

3

5 ; (13.17)

A0.p.t// D

2

6664

�c 0 a1.p.t// cos.'/ �a2.p.t// sin.'/ �k 0 a3.p.t// cos.'/

0 �c a1.p.t// sin.'/ a2.p.t// cos.'/ 0 �k a3.p.t// sin.'/

0 0 � 2kektRa

0 0 0 0

0 0 0 � 2kektRa

0 0 0

3

7775

;

(13.18)

B0 D

2

6664

0 0

0 0ktRa

ktRa

ktRa

� ktRa

3

7775

; (13.19)

and

a1.p/ D 2em0. P' C !/ cos.ı/a2.p/ D 4em0 P' sin.ı/

a3.p/ D em0!2 limı0!ı

cos.ı0/

ı0 � �=2: (13.20)

The state variables of the resulting qLPV model are the state differences relativeto the target equilibrium point (�x.t/ D x.t/ � xeq), that is, they represent the errorthat should be governed into zero.

Clearly, the system matrix S.p.t// D Œ A.p.t// B.p.t// � of the qLPV model

depends on parameters p.t/ D Œ ı P' ! �, that is, the system matrices depend notonly on the actual state ( P'; ı) but equally on the target angular velocity (!) that isnecessary to reach the equilibrium state. For practical reasons, the system output iscomposed of the error of the measurable state variables ı, Pı and P', which resultsin a proportional compensator for the angular velocity and proportional-derivativeregulation law for the offset.

References

1. X. Liu, C. Wang, C. Zhao, B. Wen, Observation and control of phase difference for a vibratorymachine of plane motion, in 2010 International Conference on Computer, Mechatronics, Controland Electronic Engineering (CMCE), vol. 4 (2010), pp. 330–334

2. Á. Miklós, Zs. Szabó, Vibrator with DC motor driven eccentric rotors. Period. Polytech. Mech.Eng. 56(1), 49–53 (2012)

Chapter 14Convex TP Model of the Dual ExcenterVibration System

Abstract In this chapter the goal is to derive and prepare the TP model of thecontrol system to the Dual Excenter for LMI based control design. We applythe TP model transformation on the qLPV model derived earlier in (13.12). Forthe numerical execution of the TP model transformation we use the values of theparameters as given in Table 14.1.

Keywords Convex TP model • HOSVD based canonical form • Complexitytrade-off

14.1 The Quasi-HOSVD Based Canonical Form:Approximation and Complexity Trade-Off

We execute the TP model transformation on the parameter-dependent S.p.t//matrix, where the parameters are p1.t/ D ı; p2.t/ D P' and p3.t/ D ! overthe discretization domain of ı D Œ0::�� rad, P' D Œ10::1000� rad/s and ! D

Œ100::1000� rad/s. The discretization grid is defined by 21, 23, and 23 equidistantlylocated grid, respectively. Since the HOSVD of the discretized system tensorindicates full-rankness in each dimension, see Fig. 14.1, the convex polytopic modelis formed by keeping most significant TP model components (3-2-2 singular valuesin the dimension of ı; P' and !). This results in a good approximation of the originalmodel while preserving a reasonable amount of complexity. The approximationerror is less than 0:01 % that is considerably less than the error caused by the thedispensable physical simplification of the model discussed in the previous chapter.


171

172 14 Convex TP Model of the Dual Excenter Vibration System

Table 14.1 Parameters of the prototypical system

Name Value Unit

Eccentricity of the rotors: e 2:09 mm

Mass of a rotor: m0 1:6 g

Mass of the moving system and rotors: M 45:3 g

Mass moment of inertia of the rotor about C axis: J0 598 kg mm2

Damping of the environment: c 30::200 Ns/m

Stiffness of the environment: k 1000::15000 N/m

Motor voltages: U1; U2 �10:: C 10 V

Voltage resolution: �U 0:05 V

Armature resistance: Ra 11:3 �

Armature inductance: L 0:19 � 10�3 H

Torque and speed constant: ke; kt 5:08 � 10�3 Vs/rad

Fig. 14.1 The singularvalues of each dimension

0

1010

100

10−10

1010

100

10−10

1010

100

10−10

5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 25

14.2 The Convex TP Model

Without further convex hull manipulation, we simply select the CNO-type convexTP model representation here. As a result we obtain the weighting functions shownin Fig. 14.2.

14.2 The Convex TP Model 173

0 0.5 1 1.5 2 2.5 30

0.5

1

Angle of offset ( ) [rad]

200 400 600 800 10000

0.5

1

200 400 600 800 10000

0.5

1

Desired angular velocity (ω) [rad/s]

wei

ghts

(w

3(ω))

wei

ghts

(w

1())

wei

ghts

(w

2())

Current angular velocity ( ) [rad/s]

Fig. 14.2 CNO-type weighting functions

The transformation results in the form:

S.p.t// D S�n2f1;2;3gwCNOn .pn.t// D

I1D3X

i1D1

I2D2X

i2D1

I3D2X

i3D1

w1;i1 .ı/ �w2;i2 . P'/ �w3;i3 .!/Si1;i2;i3 ;

(14.1)

where

S1;1;1 D

2

6666666664

�2660 14:564 �0:12021 0

14:564 �2660 �0:12889 0

1:1402e C 05 �1:1402e C 05 �38:559 0:06872

�0:097037 �0:097037 �3:7768 �38:139

1 0 0 0

0 1 0 0

0 0 0 1


�1:995e C 05 1092:3 �66:997 �0:42716 �0:42716

1092:3 �1:995e C 05 �66:997 0:42716 0:42716

8:5516e C 06 �8:5516e C 06 0 3795:1 3795:1

�7:2777 �7:2777 471:1 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S2;1;1 D

2

6666666664

�2660 14:564 0:12021 0

14:564 �2660 0:12889 0

�1:1402e C 05 1:1402e C 05 �38:559 �0:068719

�0:19909 �0:19909 3:7767 �38:139

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 1092:3 �66:997 0:42715 0:42715

1092:3 �1:995e C 05 �66:997 �0:42715 �0:42715

�8:5515e C 06 8:5515e C 06 0 3795:1 3795:1

�14:931 �14:931 471:1 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S3;1;1 D

2

6666666664

�2660 �45:033 0 �0:015651

�45:033 �2660 0 �0:0049438

�4:7229 4:7229 �37:699 0

�2:7054e C 05 �2:7054e C 05 0:00015644 �38:997

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 �3377:5 �165:78 1:0135 �1:0135

�3377:5 �1:995e C 05 �165:78 1:0135 �1:0135

�354:22 354:22 0 3710:5 3710:5

�2:029e C 07 �2:029e C 07 �21290 3838:3 �3838:3

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775


S1;2;1 D

2

6666666664

�2660 14:564 �0:2322 �0:00074553

14:564 �2660 �0:24088 0:00074553

1:1402e C 05 �1:1402e C 05 �38:559 6:872

�0:097037 �0:097037 �7:1729 �38:139

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 1092:3 �66:997 �0:42716 �0:42716

1092:3 �1:995e C 05 �66:997 0:42716 0:42716

8:5516e C 06 �8:5516e C 06 0 3795:1 3795:1

�7:2777 �7:2777 471:1 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S2;2;1 D

2

6666666664

�2660 14:564 0:2322 �0:00074574

14:564 �2660 0:24088 0:00074572

�1:1402e C 05 1:1402e C 05 �38:559 �6:8719

�0:19909 �0:19909 7:1728 �38:139

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 1092:3 �66:997 0:42715 0:42715

1092:3 �1:995e C 05 �66:997 �0:42715 �0:42715

�8:5515e C 06 8:5515e C 06 0 3795:1 3795:1

�14:931 �14:931 471:1 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S3;2;1 D

2

6666666664

�2660 �45:033 0 �0:54565

�45:033 �2660 0 0:52505

�4:7229 4:7229 �37:699 �0:00028465

�2:7054e C 05 �2:7054e C 05 0:00029711 �38:997

1 0 0 0

0 1 0 0

0 0 0 1


�1:995e C 05 �3377:5 �165:78 1:0135 �1:0135

�3377:5 �1:995e C 05 �165:78 1:0135 �1:0135

�354:22 354:22 0 3710:5 3710:5

�2:029e C 07 �2:029e C 07 �21290 3838:3 �3838:3

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S1;1;2 D

2

6666666664

�2660 14:564 �0:023095 0

14:564 �2660 �0:031775 0

1:1402e C 05 �1:1402e C 05 �38:559 0:06872

�0:097037 �0:097037 �0:83194 �38:139

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 1092:3 �0:66996 �0:42716 �0:42716

1092:3 �1:995e C 05 �0:66996 0:42716 0:42716

8:5516e C 06 �8:5516e C 06 0 3795:1 3795:1

�7:2777 �7:2777 4:711 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S2;1;2 D

2

6666666664

�2660 14:564 0:023095 0

14:564 �2660 0:031775 0

�1:1402e C 05 1:1402e C 05 �38:559 �0:068719

�0:19909 �0:19909 0:83193 �38:139

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 1092:3 �0:66997 0:42715 0:42715

1092:3 �1:995e C 05 �0:66997 �0:42715 �0:42715

�8:5515e C 06 8:5515e C 06 0 3795:1 3795:1

�14:931 �14:931 4:7109 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775


S3;1;2 D

2

6666666664

�2660 �45:033 0 �0:015651

�45:033 �2660 0 �0:0049438

�4:7229 4:7229 �37:699 0

�2:7054e C 05 �2:7054e C 05 3:446e � 05 �38:997

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 �3377:5 �1:6578 1:0135 �1:0135

�3377:5 �1:995e C 05 �1:6578 1:0135 �1:0135

�354:22 354:22 0 3710:5 3710:5

�2:029e C 07 �2:029e C 07 �212:9 3838:3 �3838:3

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S1;2;2 D

2

6666666664

�2660 14:564 �0:13509 �0:00074553

14:564 �2660 �0:14377 0:00074553

1:1402e C 05 �1:1402e C 05 �38:559 6:872

�0:097037 �0:097037 �4:2281 �38:139

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 1092:3 �0:66996 �0:42716 �0:42716

1092:3 �1:995e C 05 �0:66996 0:42716 0:42716

8:5516e C 06 �8:5516e C 06 0 3795:1 3795:1

�7:2777 �7:2777 4:711 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S2;2;2 D

2

6666666664

�2660 14:564 0:13509 �0:00074574

14:564 �2660 0:14377 0:00074572

�1:1402e C 05 1:1402e C 05 �38:559 �6:8719

�0:19909 �0:19909 4:228 �38:139

1 0 0 0

0 1 0 0

0 0 0 1


�1:995e C 05 1092:3 �0:66997 0:42715 0:42715

1092:3 �1:995e C 05 �0:66997 �0:42715 �0:42715

�8:5515e C 06 8:5515e C 06 0 3795:1 3795:1

�14:931 �14:931 4:7109 3753:8 �3753:8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

S3;2;2 D

2

6666666664

�2660 �45:033 0 �0:54565

�45:033 �2660 0 0:52505

�4:7229 4:7229 �37:699 �0:00028465

�2:7054e C 05 �2:7054e C 05 0:00017513 �38:997

1 0 0 0

0 1 0 0

0 0 0 1

�1:995e C 05 �3377:5 �1:6578 1:0135 �1:0135

�3377:5 �1:995e C 05 �1:6578 1:0135 �1:0135

�354:22 354:22 0 3710:5 3710:5

�2:029e C 07 �2:029e C 07 �212:9 3838:3 �3838:3

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3

7777777775

:

Chapter 15Derivation of the Controller

Abstract This chapter derives the control system based on the CNO type convexTP model obtained in the previous chapter.

Keywords Control design • LMI

15.1 LMI Based Controller Design

Rewriting the TP model (14.1) with linear indexing, we have

�Px.t/ D

IX

iD1

hi.p.t// ŒAi�x.t/ C Bi�u.t/�

�y.t/ D C�x.t/; (15.1)

where I D I1I2I3 and hi.p.t// D w1;i1 .ı/ � w2;i2 . P'/ � w3;i3 .!/, while Ai and Bi

are the corresponding partitions of vertices Si1;i2;i3 . The notation � denotes that thedescription is relative to desired equilibrium point.

A static output feedback controller design can be applied to this model based onthe work of Chadli [1]. In this case, the output feedback control law is formulated as:

�u.t/ D

IX

iD1

hi.p.t//Ki�y.t/: (15.2)

Let us recall the corresponding LMI design from the work of Chadli [1]:


179

180 15 Derivation of the Controller

Theorem 15.1. The �x D 0 equilibrium of the TP model (15.1) is globallyasymptotically stable via the output feedback control law (15.2), if there exist thematrices Ni; X; M such that for all i D 1::r, j > i s.t. hi \ hj ¤ ;

X > 0;

Gii C GTii < 0;

Gij C Gji C .Gij C Gji/T < 0;

CX D MC; (15.3)

where Gij D AiX C BiNjC.The feedback gains are obtained by

Ki D NiM�1: (15.4)

Remark 15.1. If matrix C is full row rank, there exist a non-singular matrix M D

CXCT.CCT/�1; and we can compute the gains as Ki D NiCCT.CXCT/�1 [1].

Substituting the vertices of the model into the above LMIs, the LMI solverprovides the following solution:

�u D K.ı; P'; !/

2

4P' � !

Pı

ı � ıd

3

5 ;

K.ı; P'; !/ D K �1 w1.ı/ �2 w2. P'/ �3 w3.!/; (15.5)

where:

Ki1;i2;i3 D

2

4Kf1;1g

i1;i2;i3Kf1;2g

i1;i2;i3Kf1;3g

i1;i2;i3

Kf2;1gi1;i2;i3

Kf2;2gi1;i2;i3

Kf2;3gi1;i2;i3

3

5 : (15.6)

Taking advantage of the symmetry of the system, and the fact that P' is influencedby U1 C U2, while ı and Pı are affected by U1 � U2, the gain values are remapped asfollows:

Ki1;i2;i3 D

2

4Kf1g

i1;i2;i3Kf2g

i1;i2;i3Kf3g

i1;i2;i3

Kf1gi1;i2;i3

�Kf2gi1;i2;i3

�Kf3gi1;i2;i3

3

5 ; (15.7)

15.1 LMI Based Controller Design 181

where

Kf1gi1;i2;i3

DKf1;1g

i1;i2;i3C Kf2;1g

i1;i2;i3

2;

Kf2gi1;i2;i3

DKf1;2g

i1;i2;i3� Kf2;2g

i1;i2;i3

2;

Kf3gi1;i2;i3

DKf1;3g

i1;i2;i3� Kf2;3g

i1;i2;i3

2:

The so obtained controller vertices are:

K1;1;1 D

0:003506 �0:250003 �26:207428

0:003506 0:250003 26:207428

�

K1;1;2 D

0:003539 �0:284493 �31:785550

0:003539 0:284493 31:785550

�

K1;2;1 D

0:003473 �0:247071 �25:836215

0:003473 0:247071 25:836215

�

K1;2;2 D

0:003539 �0:279088 �31:106840

0:003539 0:279088 31:106840

�

K2;1;1 D

�0:003618 �0:131127 �15:560288

�0:003618 0:131127 15:560288

�

K2;1;2 D

�0:003614 �0:126636 �15:245802

�0:003614 0:126636 15:245802

�

K2;2;1 D

�0:003688 �0:131169 �15:565050

�0:003688 0:131169 15:565050

�

K2;2;2 D

�0:003652 �0:126642 �15:246567

�0:003652 0:126642 15:246567

�

K3;1;1 D

�0:003619 �0:131126 �15:560209

�0:003619 0:131126 15:560209

�

K3;1;2 D

�0:003614 �0:126634 �15:245690

�0:003614 0:126634 15:245690

�

K3;2;1 D

�0:003688 �0:131168 �15:564996

�0:003688 0:131168 15:564996

�

K3;2;2 D

�0:003653 �0:126642 �15:246478

�0:003653 0:126642 15:246478

�:


Figure 15.1 shows the nonlinear controller gains within the investigated parame-ter domain.

15.2 Simulation

The simulation described here takes the real characteristics (delay, inaccuracy,stability issues) of the measurement (using n D 3 optical sensors) into consider-ation, as well as the quantized and bounded control signals (U1.t/, U2.t/) and theinductance and commutation in the motor model. The controller sampling time isset to Ts D 1 ms.

The control signal is computed as:

u.t/ D K.ı; P'; !/�y.t/ C ueq.xeq/: (15.8)

In the above case neither the time delay and noise associated with the sensorsystem, nor the discrete time fashion of the control loop was modeled in the designphase. Simulations can show whether or not these attributes influence the controlquality significantly at low angular velocities.

For small values of ! the estimated signals have relatively large time-delays andthe relative change-rates of the angular velocity can also be high. This causes thesystem to be sensitive to instability, resulting in a system that is unstable overall.Such negative effects can be reduced, for instance, by utilizing Kalman-filtering.Since the system parameters are only partly known, we consider only the motorcharacteristics in the filter formulae:

OR'.t C Ts/ D ktU1 C U2 � 2ke OP'.t/

2J0Ra

OP'.t C Ts/ D�

OP'.t/ C OR'.t C Ts/Ts

.1 � G/ C NP'.t/G

ORı.t C Ts/ D ktU1 � U2 � 2ke

OPı.t/

2J0Ra

OPı.t C Ts/ D�

OPı.t/ CORı.t C Ts/Ts

.1 � G/ C

NPı.t/G

Oı.t C Ts/ D�

Oı.t/ COPı.t C Ts/Ts

.1 � G/ C Nı.t/G; (15.9)

where Ts is the sampling time.The overall system extended with the Kalman-filter shows stable and favorable

behavior in terms of settling time and overshoot within the parameter domain! D 200::1000 and ı D 0::� . Some simulation results are shown in Fig. 15.2,where the system’s behavior is investigated at different circular frequencies! D 200; 600; 1000 rad/s. The simulation shows that the controller is capable

15.2 Simulation 183

Fig. 15.1 Values of thecontroller gains as thefunction of ı and P' atconstant ! D 500 rad/s


Offse

t (

) [ra

d]

Fig. 15.2 Simulation results

of stabilizing the ı D �=2 (balanced) equilibrium state and can govern the offsetinto different ı ¤ �=2 values (vibration with R ¤ 0 amplitude) with reasonablesettling time, overshoot, and oscillation.

In the case of low angular speeds, unreliable sensor information can also lead toproblems of destabilization with respect to the step command ı, but stable behaviorcan nevertheless be ensured using a suitable limitation of the command signals’derivatives, as visible in the first offset change in Fig. 15.2.

The change of reference angular frequency ! does not affect the offset, while thefast variation of the desired ı causes spikes in the angular speed that is explained bythe bounded control signal U1 and U2.

Clearly, in case of ı ¤ �=2 the system settles with steady state error in ı and!, because these are not considered in the design as equilibrium. However, therelatively small steady state error does not affect the usability of the system inthe potential application field. The system is tolerant to a wide range of differentenvironmental parameters (k and c); therefore the utilization of robust designapproaches which consider parametric uncertainties is not needed for this problem.

According to the simulations, the maximum settling time is 70 ms in P' and 30 msin ı, which allows for the utilization of such a vibration actuator in vibrotactilefeedback to provide separately adjustable frequency and intensity.

Reference

1. M. Chadli, D. Maquin, J. Ragot et al., Static output feedback for Takagi-Sugeno systems: an LMIapproach, in 10th Mediterranean Conference on Control and Automation, MED’2002 (2002)

Part VControl of the Impedance Model Including

Varying Time Delay via TP�

Model Transformation

Chapter 16Impedance Control for Force ReflectingTelemanipulation

Abstract This Chapter introduces the impedance model that is used in the controlof the force reflecting telemanipulation. The next chapters will use this model toshow the effectiveness of the TP� model transformation based design.

Keywords Impedance control • Force feedback • Time delay

This chapter focuses on a class of impedance model based robot control schemesthat are conceptually illustrated in Fig. 16.1. Results presented in the chapter arebased on the work of Galambos et al. [5–7].

In recent years, the importance of impedance control has become clear in severalfields of robotics such as human–robot interaction, telemanipulation, and roboticsurgery. At the same time, trends in robotics are turning to logically or spatiallydistributed robot control systems, where time-delays are inherent and unavoidable.This tendency is leading to undesirable effects on control performance, which isin turn creating strong motivation for the investigation of time-delay sensitivity inimpedance-controlled robots.

Since the work of Hogan [9–11], which led to the emergence of the conceptof impedance control in application-oriented fields, this control strategy has growninto one of the key technologies of modern robot control. Some areas of roboticsin which impedance control is widely used are telerobotics, dexterous manipulation[22, 24], and flexible joint robots [15].

Time delay—caused by communication jitters in computer networks linkingvarious distributed system elements—usually have an unfavorable influence onclosed loop control systems. Control of time-delay systems is a permanent challenge[17, 20]. Internet based teleoperation is a typical area where communication delayscause relevant problems [4, 18, 21, 23]. Among several other approaches, impedancecontrol is an important strategy in bilateral telemanipulation [1, 2, 14, 19]. Thestability and performance of haptic rendering are also affected by the delays whichoccur in the control loop [8].

Figure 16.2 illustrates the operation of impedance control algorithms in teleop-eration scenario. A common property of these interaction control systems is that the


187

188 16 Impedance Control for Force Reflecting Telemanipulation

Fig. 16.1 Schematicstructure of impedancecontrol of robots

positioncontroller

Robot

Impedance model

Fig. 16.2 Scheme of coupledimpedance force reflectingalgorithm for bilateraltelemanipulation

Master pos.controller

Masterrobot

Impedance model

IP network

Slave pos.controller

Slaverobot

time-delay experienced while the acting force is measured and transmitted to theimpedance model hinders control performance, and over a critical delay, the systembecomes unstable.

16.1 Impedance Control with Feedback Delay

This section investigates a single degree-of-freedom model relevant to the two-fingered parallel jaw tele-grasping problem.

Impedance models can be understood as an encapsulation of dynamic relation-ships between a force and the resulting displacement. Such models are typicallyexpressed through a virtual mass-spring-damper system. In this section, nonlinearfriction components (coulomb friction, stiction) are also considered as part ofthe impedance model. In general cases the task-space impedance model can bedescribed as:

M Rq C B Pq C Kq C C. Pq; q/ D F; (16.1)

16.1 Impedance Control with Feedback Delay 189

Fig. 16.3 Mass-Spring-Damper system

where M, B, K are symmetric, positive-definite matrices describing the mass,damping, and stiffness parameters; and C contains nonlinear friction terms of theimpedance model, while F denotes external forces.

Remark 16.1. In some applications, the end effector path is prescribed and thedisplacement results from the impedance model which is added to the predefinedpath. In this way the robot motion becomes compliant.

Let us consider the single degree of freedom mechanical system depicted inFig. 16.3 as our impedance model. Mass m and viscous damping b are virtualproperties defining the desired dynamics of the manipulator, while k denotes thestiffness of the robot’s environment. In real cases, the environment is usually morecomplicated, but this simplified model is suitable for the investigation of time-delayeffects.

Virtual parameters have to be chosen according to the accuracy $ robustnesstrade-off [13]: the lower the mass and the damping, the faster and more accurate isthe tracking performance, and the better are the robustness properties of the systemwith respect to feedback delays.

The equation of motion of this system is as follows:

Rx.t/ DFh.t/

m�

b

mPx.t/ �

Fe.t/

m: (16.2)

The non-delayed system can be represented by a standard LTI state space model:

Px.t/ D Ax.t/ C Bu.t/ (16.3)

y.t/ D Cx.t/ C Du.t/;

where the elements are:

x.t/ D

Pxx

�u.t/ D Fh.t/

A D

� b

m � km

1 0

�B D

1m0

�

C D�0 1�

D D 0:


Fig. 16.4 Impedance modelwith feedback delay Network delay

&remote env.

Introducing the time-delay � in the interaction (overall delay of the forcemonitoring due to the lag of the signal processing and/or network delays) leads to:

Rx.t/ DFh.t/

m�

b

mPx.t/ �

Fe.t � �.t//

m: (16.4)

Substituting the interaction force (Fe) with the elastic force (kx) in the formula,we obtain:

Rx.t/ DFh.t/

m�

b

mPx.t/ �

k

mx.t � �.t//: (16.5)

If we consider this more deeply, we may realize that this equation represents amass-spring-damper system in which the effects of the spring are delays through�.t/ (see Fig. 16.4).

Figure 16.5 shows the effects of the � feedback delay. When the delay increases,the step response of the system becomes more susceptible to oscillation. Expresseddifferently, the so-called pseudo-damping ratio decreases until the system becomesunstable.

16.2 Control Structure for Stability Preservation

This section introduces recent approaches towards the preservation of stabilityin impedance control-based bilateral telemanipulation, and describes a controlstructure which is appropriate for the impedance model in the telemanipulationscenario described in Sect. 16. In this control structure, the impedance model underfeedback delay can be embedded in such a way that its stabilization design problemreadily leads to a more class of control theories developed for control signal design.This approach slightly reinterprets the previously applied stabilization techniques

16.2 Control Structure for Stability Preservation 191

Fig. 16.5 Step response of the model with various delay values (model parameters: m D 1 kg,b D 100 Ns/m, k D 1000 N/m). Excitation function defines as Fh.t 5 0/ D 0, Fh.t > 0/ D 1

which are based on the adaptive tuning of the impedance model’s parameters. Hencethis structure effectively extends the class of control design theories applicable tostable impedance control design.

In the past two decades a large variety of approaches has been introduced whichaddress stability issues of telemanipulators in the presence of time-delay. Hokayemand Spong published a comprehensive survey [12] that introduces most of theapproaches which can be categorized as passivity based, prediction based, slidingmode based, and others [26].

The widest group of approaches includes those methods that are based onenergy related considerations which culminate in the so-called passivity theory.To be more specific, such approaches directly or indirectly manipulate the so-called energy tanks of the dynamic system in order to guarantee its stability.Among these directions, adaptive tuning of the applied impedance parameters hasa special significance (Fig. 16.6). Dubey et al. [3] published a variable dampingimpedance control method to enhance the quality of master–slave force reflectingtelemanipulation. Wen-Hong Zhu and Salcudean introduced an adaptive controllerin [25] in which the master–slave system behaves essentially as a linearly dampedfree-floating mass, while the mass and damping parameters change according to theestimated dynamics of the environment. In 2004 Love and Book published anotheradaptive impedance control algorithm [16] which sets the master impedance based



Masterrobot

Impedance model

Stabilization byparameter tuning


Slaverobot

IP network

Fig. 16.6 Stabilization of impedance control based force reflecting telemanipulation by parametertuning

on the estimated time-varying, position-dependent representation of the remoteenvironment. In their solution, environment estimation and impedance adaptationare executed simultaneously and in real time.

A general problem is that there are no simple methods to find the appropriatedissipation model that makes the system stable but transparent enough for com-fortable use. Thus, these methods are usually very conservative, making the overallteleoperator system more dissipative (less transparent) than it would in reality berequired.

Another approach is to apply a structure that manipulates the dissipative char-acteristics of the system indirectly by an external damper force that is additionalto the damping which is included in the impedance model itself. This structure isillustrated in Fig. 16.7.

Based on this structure, we can formulate the following equation of motion ofthe impedance model to be stabilized by the appropriate design of Fc.t/.

Rx.t/ DFh.t/

mC

Fc.t/

m�

b

mPx.t/ �

k

mx.t � �.t// (16.6)

The theoretical contribution of this work is focused on a design methodology thatis appropriate to find the delay dependent control law that maintain the stability ofthe impedance model without unnecessary degradation of the transparency.

References 193

Observer Controller


Masterrobot

Impedance model

IP network


Slaverobot

Fig. 16.7 Control scheme for the stabilization of force reflecting telemanipulation under time-delay

References

1. S.H. Ahn, K.H. Lee, Y.K. Kim, H.R. Kim, A bilateral compliance control for timedelayed systems, in SICE-ICASE International Joint Conference, Los Alamitos, CA (2006),pp. 3048–3052

2. H.C. Cho, J.H. Park, Stable bilateral teleoperation under a time delay using a robust impedancecontrol. Mechatronics 15(5), 611–625 (2005)

3. R.V. Dubey, T.F. Chan, S.E. Everett, Variable damping impedance control of a bilateraltelerobotic system, IEEE Control Systems 17(1), 37–45 (1997)

4. P. Fraisse, A. Lelevé, Teleoperation over IP network: Network delay regulation and adaptivecontrol. Auton. Robot. 15(3), 225–235 (2003)

5. P. Galambos, P. Baranyi, Representing the model of impedance controlled robot interactionwith feedback delay in polytopic LPV form: TP model transformation based approach. ActaPolytech. Hung. 10(1), 139–157 (2013)

6. P. Galambos, P. Baranyi, TP-tau model transformation: a systematic modelling framework tohandle internal time delays in control systems. Asian J. Control 17(2), 486–496 (2015)

7. P. Galambos, P. Baranyi, G. Arz, Tensor product model transformation-based control designfor force reflecting tele-grasping under time delay. Proc. IME C J. Mech. Eng. Sci. 228(4),765–777 (2014)

8. S. Hirche, A. Bauer, M. Buss, Transparency of haptic telepresence systems with constant timedelay, in Proceedings of 2005 IEEE Conference on Control Applications, 2005. CCA 2005(2005), pp. 328–333

9. N. Hogan, Impedance control: An approach to manipulation: part I—Theory. J. Dyn. Syst.Meas. Control. 107(1), 1–7 (1985)


10. N. Hogan, Impedance control: an approach to manipulation: part II—implementation. J. Dyn.Syst. Meas. Control. 107(1), 8–16 (1985)

11. N. Hogan, Impedance control: an approach to manipulation: part III—applications. J. Dyn.Syst. Meas. Control. 107(1), 17–24 (1985)

12. P.F. Hokayem, M.W. Spong, Bilateral teleoperation: an historical survey. Automatica 42(12),2035–2057 (2006)

13. S.H. Kang, M. Jin, P.H. Chang, A solution to the accuracy/robustness dilemmain impedance control. IEEE/ASME Trans. Mechatron. 14(3), 282–294 (2009).doi:10.1109/TMECH.2008.2005524

14. W.S. Kim, B. Hannaford, A.K. Bejczy, Force-reflection and shared compliant control inoperating telemanipulators with time delay. IEEE Trans. Robot. Autom. 8(2), 176–185 (1992)

15. A. Kugi, C. Ott, A. Albu-Schaffer, G. Hirzinger, On the Passivity-Based impedance control offlexible joint robots. IEEE Trans. Robot. 24(2), 416–429 (2008)

16. L.J. Love, W.J. Book, Force reflecting teleoperation with adaptive impedance control. IEEETrans. Syst. Man Cybern. B Cybern. 34(1), 159–165 (2004)

17. R. Matusu, R. Prokop, Control of systems with time-varying delay: a comparison study, inProceedings of the 12th WSEAS International Conference on Automatic Control, Modelling& Simulation, ACMOS’10, Catania, Italy (World Scientific and Engineering Academy andSociety, Bulgaria, 2010), pp. 125–130

18. S. Munir, W.J. Book, Internet-based teleoperation using wave variables with prediction.IEEE/ASME Trans. Mechatron. 7(2), 124–133 (2002)

19. M. Otsuka, N. Matsumoto, T. Idogaki, K. Kosuge, T. Itoh, Bilateral telemanipulator systemwith communication time delay based on force-sum-driven virtual internal models, in Pro-ceedings of 1995 IEEE International Conference on Robotics and Automation, Nagoya, Japan(1995), pp. 344–350

20. L. Pekar, Root locus analysis of a retarded quasipolynomial. WSEAS Trans. System Control6(7), 79–91 (2011)

21. I.G. Polushin, P.X. Liu, C.-H. Lung, A Force-Reflection algorithm for improved transparencyin bilateral teleoperation with communication delay. IEEE/ASME Trans. Mechatron. 12(3),361–374 (2007)

22. J. Pomares, G.J. Garcia, F. Torres, Impedance control for fusing multisensorial systems inrobotic manipulation tasks, in Proceedings of the 2005 WSEAS International Conference onDynamical Systems and Control, CONTROL’05, Stevens Point, WI (World Scientific andEngineering Academy and Society, Bulagaria, 2005), pp. 357–362

23. A.C. Smith, K. Hashtrudi-Zaad, Smith predictor type control architectures for time delayedteleoperation. Int. J. Rob. Res. 25(8), 797–818 (2006)

24. M. Tarbouchi, M.R. Strawson, H. Benabdallah, Impedance control of a manipulator using afuzzy model reference learning controller, in Proceedings of the 10th WSEAS InternationalConference on Automatic Control, Modelling Simulation, Stevens Point, WI (World Scientificand Engineering Academy and Society, Bulgaria, 2008), pp. 119–126

25. W.-H. Zhu, S.E. Salcudean, Stability guaranteed teleoperation: an adaptive motion/forcecontrol approach. IEEE Trans. Autom. Control 45(11), 1951–1969 (2000)

26. J. Zhu, X. He, W. Gueaieb, Trends in the control schemes for bilateral teleoperation withtime delay, in Autonomous and Intelligent Systems, ed. by M. Kamel, F. Karray, W. Gueaieb,A. Khamis. Lecture Notes in Computer Science, vol. 6752 (Springer, Berlin/Heidelberg, 2011),pp. 146–155

Chapter 17Impedance Model with Varying Feedback Delayin TP Model Form

Abstract Our goal in this section is to develop and manipulate a variety of convexpolytopic structures for the impedance model with feedback delay. Further, weaim to find tradeoffs between complexity and accuracy of the resulting TP model.Further considerations include effectiveness of LMI based design techniques, andconservativeness of the resulting controller, both of which are investigated inSect. 18. Through this process, we will essentially be validating the applicabilityof the TP� model transformation.

Keywords Time delay • TP� model transformation • Complexity trade-off

17.1 The Quasi-HOSVD Based Canonical Form

In this section we apply the TP� model transformation to the task of expressing theinvestigated impedance model from Eq. (16.5) in a quasi-HOSVD based canonicalform.

The model parameters are given in Table 17.1:It is important to note that the main properties of the polytopic structure are

not significantly affected by the model parameters within a wide range of inputparameter space with practical relevance, as long as the TP� model transformationis performed using the reidentification technique described in Sect. 7.2.

17.1.1 Exact Quasi-HOSVD Based Canonical Form

Following the TP� model transformation on the impedance model, the followingminimal size LPV representation is obtained with six LTI vertex models:

S.�.t// D

6X

rD1

wr.�.t//Sr: (17.1)


195

196 17 Impedance Model with Varying Feedback Delay in TP Model Form

Table 17.1 Parameters ofthe impedance model

Description Parameter Value Units

Mass m 1 kg

Viscous damping b 100 Ns/m

Stiffness of the environment k 2000 N=m

Delay interval � 0-0:07 s

Singular values are as follows:

�1 D 2:3414 � 104; �2 D 3:5305 � 102; �3 D 1:0331; �4 D 2:2164 � 10�2;

�5 D 1:2964 � 10�3; �6 D 6:9808 � 10�5:

Once again, it should be noted that the different model parameters have nosubstantial effect on either the resulting singular values or the rank of the modeland the underlying polytopic structure. As a result, this example properly shows theTP structure of the representation.

The consecutive singular values decrease exponentially by a factor of two ordersof magnitude, see Fig. 17.1. The vertices, then, are:

SHOSVD1 D

"�1:1515 � 104 �1:1896 � 104 �6:3944

1:1521 � 104 1:1890 � 104 6:3906

#

(17.2)

SHOSVD2 D

"�1:8082 � 102 1:7522 � 102 3:0496

1:7781 � 102 �1:7209 � 102 �3:0457

#

(17.3)

SHOSVD3 D

"2:8404 � 10�1 �2:7537 � 10�1 �6:0693 � 10�1

2:9928 � 10�1 �2:9109 � 10�1 6:0668 � 10�1

#

(17.4)

SHOSVD4 D

"8:7316 � 10�3 �9:6607 � 10�3 8:6616 � 10�3

8:7412 � 10�3 �9:6704 � 10�3 �8:7600 � 10�3

#

(17.5)

SHOSVD5 D

"6:6570 � 10�4 6:2594 � 10�4 9:5164 � 10�5

6:6535 � 10�4 6:2629 � 10�4 3:9904 � 10�5

#

(17.6)

SHOSVD6 D

"�2:5581 � 10�6 �2:5893 � 10�6 4:9197 � 10�5

�2:5570 � 10�6 �2:5904 � 10�6 4:9258 � 10�5

#

: (17.7)

17.1 The Quasi-HOSVD Based Canonical Form 197

Fig. 17.1 Singular values of the HOSVD based canonical form

Figure 17.2 shows the weighting functions wr.�.t// over a range of � values.The smoothness of the weighting functions shows that the applied reidentificationmethod is stable along the investigated range of �.t/. This means that the appliedidentification algorithm does not show drastically changing behavior betweendifferent local solutions (in terms of local minima). It is worth bearing in mind that ifthe identification method was to switch between different solutions, additional ranksmight appear in the HOSVD canonical form. By neglecting these extra singularvalues, HOSVD is capable of (smoothly) approximating the ruggedness of thesolution space in a least-square sense, in a way similar to how SVD can be usedfor noise filtering in digital signal processing [1]. However, if fluctuation amongneighboring solutions is large, such approximations do not always lead to viablesolutions.


Fig. 17.2 Weighting functions of the HOSVD based canonical form

17.1.2 Executing Trade-off by TP� Model Transformation

The goal of this section is to highlight the correlation between accuracy and numberof vertices used.

Considering that the significance of the vertex models decreases uniformly,(Fig. 17.1), there is no theoretically appealing point from which to remove lesssignificant vertices whenever the goal is to reduce the complexity of the model.However the accuracy of the reduced TP model can be evaluated as:

r D��SD.�;G/ � SD.�;G/

Approxr

��L2

: (17.8)

Modeling errors are as follows (according to the singular values):

1 D 3:53 � 102; 2 D 1:0333; 3 D 2:22 � 10�2; 4 D 1:3 � 10�3;

5 D 6:9808 � 10�5; 6 D 1:1272 � 10�11 � 0 (numerically zero)

As matrix ŒAB�r contains elements in the order of magnitude 103, and due tothe definition of r, 6 is much larger than 10�15. Such small values are typicallyconsidered as zero (at least in the context of numerical implementations), providedthat all matrix elements are in the range of 101.

In order to support further investigations and ensure that the obtained TP modelis not undersampled, we define the following measure:

17.2 Manipulation of the Convex Hull 199

Definition 17.1 ( RND1000r ).

RND1000r D��SD.�;G0/ � SD.�;G0/

Approxr

��L2

; (17.9)

where G0 denotes a grid with 1000 randomly generated grid points over �. GridG0 is not equidistant and G0

TG D ; (G is the grid used in the TP� model

transformation).

The measure RND1000r compares the reidentified and the approximated systems in1000 randomly generated points while taking into account only the first r vertices ofthe HOSVD based model. RND1000r shows the model accuracy better than alternativemeasures in real situations where arbitrary variations in delay occur. The obtained RND1000r s values are listed below:

RND10001 D 9:6325 � 102 (17.10)

RND10002 D 2:7698 (17.11)

RND10003 D 1:2099 � 10�1 (17.12)

RND10004 D 1:0519 � 10�1 (17.13)

RND10005 D 1:0514 � 10�1 (17.14)

RND10006 D 1:0514 � 10�1: (17.15)

The values for r D 4, r D 5, r D 6 are almost identical and begin to increaseonly at r D 3. Figure 17.3 shows the RND1000r data on a logarithmic scale; hence theprominence of the step at r D 3 is evident.

These results support the hypothesis that the number of nonzero singular valuesdo not increase, even when the density of G increases without bounds. Thus,results show that the representation is minimal and exact, and we may concludethat the applied discretization is not under-sampled, and that our decision to applycomplexity reduction by neglecting these particular less significant vertices is wellestablished. Beyond this purely numerical comparison, the dynamic accuracy of theTP model will also be investigated in Sect. 17.3, following the calculation of TPmodel representations that are suitable for LMI based design approaches.

17.2 Manipulation of the Convex Hull

As emphasized earlier, LMIs are highly sensitive to the shape of the convex hull.In this section, various types of convex hulls of the delayed impedance modelare generated. For the sake of completeness, the exact TP type polytopic modeland the non-exact TP model with 5–3 vertices are examined. The computation of


Fig. 17.3 Accuracy-complexity trade-off: modeling error ( RND1000r ) as a function of dimension-

ality of HOSVD-based canonical form

various TP models takes reasonable time on a regular computer. In the followingsubsections, IRNO, SNNN, and CNO type convex hulls are given by the verticesand the weighting functions. The convex hulls are described through the followingstructures:

• The exact TP model

– SNNN type convex hull (Fig. 17.4)– IRNO type convex hull (Fig. 17.5)– CNO type convex hull (Fig. 17.6)

• Reduced TP model with 5 vertices



Fig. 17.4 Weighting functions of SNNN type convex hull of the exact TP model

Fig. 17.5 Weighting functions of IRNO type convex hull of the exact TP model


Fig. 17.6 Weighting functions of CNO type convex hull of the exact TP model

Fig. 17.7 Weighting functions of SNNN type convex hull of the reduced TP model with 5 vertices


Fig. 17.8 Weighting functions of IRNO type convex hull of the reduced TP model with 5 vertices

Fig. 17.9 Weighting functions of CNO type convex hull of the reduced TP model with 5 vertices


Fig. 17.10 Weighting functions of SNNN type convex hull of the reduced TP model with 4vertices





17.2.1 The Vertices of the Exact TP Model

17.2.1.1 SNNN Type Convex Hull

Ssnnn1 D

9:8260 � 102 1:0206 � 103 �6:1076 � 10�1

�9:8207 � 102 �1:0211 � 103 6:1092 � 10�1

�

Ssnnn2 D

9:6385 � 102 1:0388 � 103 2:7141 � 10�1

�9:6410 � 102 �1:0385 � 103 �2:7127 � 10�1

�





Fig. 17.13 Weighting functions of SNNN type convex hull of the reduced TP model with 3vertices




Ssnnn3 D

9:0316 � 102 1:0976 � 103 2:4409

�9:0555 � 102 �1:0952 � 103 �2:4383

�

Ssnnn4 D

8:7268 � 102 1:1271 � 103 3:3613

�8:7601 � 102 �1:1238 � 103 �3:3583

�

Ssnnn5 D

1:4307 � 103 5:5785 � 102 �6:8959

�1:4239 � 103 �5:6496 � 102 6:8865

�

Ssnnn6 D

9:2741 � 102 1:0741 � 103 1:5495

�9:2893 � 102 �1:0725 � 103 �1:5480

�:

17.2.1.2 IRNO Type Convex Hull

Sirno1 D

9:8112 � 102 1:0197 � 103 3:3566 � 10�1

�9:8148 � 102 �1:0193 � 103 �3:3532 � 10�1

�

Sirno2 D

9:5875 � 102 1:0435 � 103 4:9710 � 10�1

�9:5923 � 102 �1:0430 � 103 �4:9670 � 10�1

�

Sirno3 D

9:4525 � 102 1:0554 � 103 1:3775

�9:466 � 102 �1:0540 � 103 �1:3761

�


Sirno4 D

9:3838 � 102 1:0623 � 103 1:4845

�9:3985 � 102 �1:0608 � 103 �1:4832

�

Sirno5 D

1:1100 � 103 8:8698 � 102 �1:5891

�1:1084 � 103 �8:8863 � 102 1:5866

�

Sirno6 D

8:9945 � 102 1:1029 � 103 1:9874

�9:0140 � 102 �1:1009 � 103 �1:9853

�:

17.2.1.3 CNO Type Convex Hull

Scno1 D

9:5246 � 102 1:0486 � 103 1:0530

�9:5350 � 102 �1:0475 � 103 �1:0520

�

Scno2 D

9:8994 � 102 1:0101 � 103 4:7248 � 10�1

�9:9040 � 102 �1:0096 � 103 �4:7217 � 10�1

�

Scno3 D

1:0169 � 103 9:8246 � 102 1:2070 � 10�2

�1:0169 � 103 �9:8248 � 102 �1:2565 � 10�2

�

Scno4 D

1:0049 � 103 9:9497 � 102 8:5882 � 10�2

�1:0050 � 103 �9:9488 � 102 �8:6082 � 10�2

�

Scno5 D

9:5135 � 102 1:0492 � 103 1:1945

�9:5253 � 102 �1:0480 � 103 �1:1934

�

Scno6 D

9:5001 � 102 1:0513 � 103 1:0027

�9:5099 � 102 �1:0503 � 103 �1:0017

�:

17.2.2 The 5 Vertices of the Reduced TP Model


Ssnnn51 D

9:4975 � 102 1:0516 � 103 1:0061

�9:5074 � 102 �1:0506 � 103 �1:0051

�

Ssnnn52 D

9:5019 � 102 1:0509 � 103 1:0864

�9:5126 � 102 �1:0498 � 103 �1:0853

�

Ssnnn53 D

9:7368 � 102 1:0262 � 103 8:6877 � 10�1

�9:7454 � 102 �1:0254 � 103 �8:6819 � 10�1

�


Ssnnn54 D

1:0024 � 103 9:9751 � 102 1:5039 � 10�1

�1:0025 � 103 �9:9736 � 102 �1:5050 � 10�1

�

Ssnnn55 D

1:0140 � 103 9:8548 � 102 4:6375 � 10�2

�1:0140 � 103 �9:8546 � 102 �4:6693 � 10�2

�:


Sirno51 D

9:7734 � 102 1:0230 � 103 6:0370 � 10�1

�9:7795 � 102 �1:0223 � 103 �6:0321 � 10�1

�

Sirno52 D

9:6627 � 102 1:0353 � 103 5:0519 � 10�1

�9:6678 � 102 �1:0347 � 103 �5:0454 � 10�1

�

Sirno53 D

9:5116 � 102 1:0504 � 103 9:2294 � 10�1

�9:5206 � 102 �1:0495 � 103 �9:2220 � 10�1

�

Sirno54 D

1:0828 � 103 9:1455 � 102 �1:0172

�1:0818 � 103 �9:1562 � 102 1:0154

�

Sirno55 D

9:0530 � 102 1:0962 � 103 2:1220

�9:0738 � 102 �1:0940 � 103 �2:1198

�:


Scno51 D

9:4984 � 102 1:0515 � 103 1:0050

�9:5083 � 102 �1:0505 � 103 �1:0040

�

Scno52 D

9:7541 � 102 1:0245 � 103 8:3214 � 10�1

�9:7623 � 102 �1:0236 � 103 �8:3163 � 10�1

�

Scno53 D

9:5280 � 102 1:0482 � 103 1:0526

�9:5384 � 102 �1:0471 � 103 �1:0515

�

Scno54 D

1:0021 � 103 9:9773 � 102 1:5382 � 10�1

�1:0023 � 103 �9:9758 � 102 �1:5393 � 10�1

�

Scno55 D

1:0028 � 103 9:9689 � 102 2:3357 � 10�1

�1:0030 � 103 �9:9668 � 102 �2:3363 � 10�1

�:




Ssnnn41 D

1:0009 � 103 9:9880 � 102 2:9652 � 10�1

�1:0011 � 103 �9:9852 � 102 �2:9658 � 10�1

�

Ssnnn42 D

1:0005 � 103 9:9947 � 102 1:8125 � 10�1

�1:0006 � 103 �9:9928 � 102 �1:8130 � 10�1

�

Ssnnn43 D

9:6070 � 102 1:0397 � 103 1:0297

�9:6172 � 102 �1:0387 � 103 �1:0288

�

Ssnnn44 D

9:5016 � 102 1:0512 � 103 1:0004

�9:5114 � 102 �1:0502 � 103 �9:9936 � 10�1

�:


Sirno41 D

9:4288 � 102 1:0592 � 103 9:7691 � 10�1

�9:4384 � 102 �1:0582 � 103 �9:7584 � 10�1

�

Sirno42 D

9:7528 � 102 1:0245 � 103 9:1178 � 10�1

�9:7616 � 102 �1:0236 � 103 �9:1119 � 10�1

�

Sirno43 D

1:0524 � 103 9:4607 � 102 �6:4437 � 10�1

�1:0517 � 103 �9:4674 � 102 6:4320 � 10�1

�

Sirno44 D

9:4577 � 102 1:0555 � 103 1:1012

�9:4686 � 102 �1:0544 � 103 �1:1001

�:


Scno41 D

1:0018 � 103 9:9785 � 102 2:8284 � 10�1

�1:0021 � 103 �9:9758 � 102 �2:8292 � 10�1

�

Scno42 D

1:0004 � 103 9:9948 � 102 1:8152 � 10�1

�1:0006 � 103 �9:9930 � 102 �1:8157 � 10�1

�

Scno43 D

9:6003 � 102 1:0404 � 103 1:0393

�9:6105 � 102 �1:0393 � 103 �1:0384

�

Scno44 D

9:5018 � 102 1:0512 � 103 1:0000

�9:5116 � 102 �1:0502 � 103 �9:9904 � 10�1

�:

17.3 Validation of the Convex TP Model 211



Ssnnn31 D

9:7873 � 102 1:0209 � 103 8:6643 � 10�1

�9:7958 � 102 �1:0200 � 103 �8:6591 � 10�1

�

Ssnnn32 D

9:4943 � 102 1:0519 � 103 1:0101

�9:5043 � 102 �1:0509 � 103 �1:0091

�

Ssnnn33 D

1:0005 � 103 9:9947 � 102 1:8248 � 10�1

�1:0007 � 103 �9:9929 � 102 �1:8255 � 10�1

�:


Sirno31 D

9:6225 � 102 1:0392 � 103 6:5354 � 10�1

�9:6290 � 102 �1:0385 � 103 �6:5284 � 10�1

�

Sirno32 D

1:0288 � 103 9:7002 � 102 �1:6177 � 10�1

�1:0287 � 103 �9:7021 � 102 1:6113 � 10�1

�

Sirno33 D

9:3329 � 102 1:0679 � 103 1:4790

�9:3475 � 102 �1:0664 � 103 �1:4776

�:


Scno31 D

9:7873 � 102 1:0209 � 103 8:6644 � 10�1

�9:7957 � 102 �1:0200 � 103 �8:6591 � 10�1

�

Scno32 D

9:4943 � 102 1:0519 � 103 1:0101

�9:5043 � 102 �1:0509 � 103 �1:0091

�

Scno33 D

1:0005 � 103 9:9947 � 102 1:8247 � 10�1

�1:0007 � 103 �9:9929 � 102 �1:8254 � 10�1

�:

17.3 Validation of the Convex TP Model

Our goal in this section is to illustrate the dynamical accuracy of these various TPmodels. Here the model accuracy is investigated by means of the difference betweenstep responses of the TP models and that of the original delayed model. Due to thefact that a detailed treatment of this subject would extend beyond the scope limitsof this book, only a small set of practically interesting validations are described.


The comparison is broken into two parts, which focus on cases with constant andvarying time-delays, respectively.

17.3.1 Constant Time-Delay

First, the dynamic accuracy of the quasi-HOSVD based canonical forms ofthe delayed impedance model is considered at different degrees of complexity.Figure 17.16 shows the step responses of the compared models at an arbitrarilychosen constant delay value (� D 0:05567). A 1 N force step was used as aninput signal 0:1 s into the simulation. Figure 17.16a–f shows the step responsesof TP models with different numbers of less significant vertices left out. The timeplots confirm the results of the modeling error analysis performed in Sect. 17.1.2.As the values of RND1000r suggested, the TP models show similarly acceptablelevels of accuracy with 6, 5, and 4 vertices, while the model accuracy begins todeteriorate once the number of remaining vertices is 3 or less. As a matter of fact,TP models with only 1 or 2 vertices cannot adequately describe the dynamics ofthe original delayed system, as a result of which there is no motivation, theoreticalor otherwise, to even consider such models. Further transformations applied toCNO type TP model representations do not introduce additional errors; hence theresulting performance is equivalent to those of quasi-HOSVD based canonical TPmodels.

For the sake of quantitative comparison, theL2 norm of the position error and themaximum error is computed at four arbitrarily chosen � values (neither of which areon discretization grid G) while running the 1 s long simulation scenario describedearlier. Results are shown in Table 17.2.

It is a noteworthy observation that the investigated TP models have a minorsteady-state error (see Fig. 17.16), which often causes problems in control design.The impedance model is understood as the dynamical relationship between theforce and the resulting velocity. The applied reidentification method often results inmodels where the residual velocity is not 0 m/s but some small value. In the figures,the position is displayed as the integral of the model output, and thus, the nonzeroresidual velocity results in a drift of position value. As demonstrated in Chap. 18,this type of model inaccuracy does not imply the failure of control design. In ourcase, due to the observer-based control approach applied, the TP model does notdirectly appear in the controller, and the state observer receives the velocity fromthe original system. In other applications, where the model is directly used in thecontrol algorithm, a dead-zone filter around zero can be a handy solution.

17.3 Validation of the Convex TP Model 213

Fig. 17.16 Comparison ofthe original delayed modeland the HOSVD-basedcanonical form of the TP�

model at different levels ofcomplexity. (a) Canonicalform with 6 vertices (exactmodel). (b) Canonical formwith 5 vertices. (c) Canonicalform with 4 vertices.(d) Canonical form with 3vertices. (e) Canonical formwith 2 vertices. (f) Canonicalform with 1 vertices

a

b

c


Fig. 17.16 (continued) e

f

Table 17.2 Quantitativecomparison of the originaldelayed model and the CNOtype TP� model with 3vertices.

L2 error Max error

� D 0:01375 s 2:6279 � 10�5 9:8521 � 10�7

� D 0:02941 s 4:0380 � 10�5 5:9765 � 10�6

� D 0:04752 s 4:3281 � 10�5 1:0500 � 10�5

� D 0:06393 s 1:0851 � 10�4 1:3048 � 10�5

17.3.2 Varying Time-Delay

The models developed in the chapter can be compared under varying delays as well.In a set of recent experiments, the value of �.t/ was varied as a sine function of time�.t/ D 0:03 C sin.t�/0:025. The input signal was a square wave with a frequencyof 2 Hz and amplitude of 1 N. Figure 17.17 shows the results of the simulation.

Reference 215

Fig. 17.17 Comparison under varying delay

It can be observed that the two different systems produce different behaviorsaround the two terminal positions. The reason behind this difference is that thesquare wave and the sine function in this example has the same period, and thus,the outer terminate position is encountered at larger momentary delays.

Reference

1. E. Biglieri, K. Yao, Some properties of singular value decomposition and their applications todigital signal processing. Signal Process. 18(3), 277–289 (1989)

Chapter 18TP� Transformation Based Control Designfor Impedance Controlled Robot Gripper

Abstract The goal of this chapter is to design the controller to the model. Firstwe select the CNO type reduced TP model derived in the previous chapter then weexecute the LMI based design. We derive three controllers with different conditions.Finally we check and analyze the resulting controllers via numerical simulations.

Keywords Impedance control • Control design • Time delay

18.1 The Control Problem

We begin this section by recalling the equation of motion of the impedance modelthat is embedded in the control structure proposed in Sect. 16.2.

Rx.t/ DFh.t/

mC

Fc.t/

m�

b

mPx.t/ �

k

mx.t � �.t//: (18.1)

Our goal when designing stability-preserving controllers is to provide a controlsignal Fc.t/ that fulfills a set of criteria relevant to the performance of the impedancemodel during teleoperated grasping. For example, such criteria might include thefollowing:

1. The supervised impedance model must be stable, so Px.t/ ! 0 as t ! 1, ifFh.t/ ! const.

2. Stability must be guaranteed through a control signal that is constrained as(kFc.t/k2 < ).

3. The conservativeness of the solution should be relaxed.

Accordingly, we are looking for a pair of state observer and state feedback gainswhich match the criteria listed above.


217

218 18 TP� Transformation Based Control Design for Impedance Controlled Robot. . .

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

feedback delay [s]

wei

ghts


18.2 Execution of the TP� Model Transformation

Taking as a basis all of the TP models found earlier, we first check whether ornot they lead to feasible LMIs. As a full treatment of this step would extend past thescope of this book, we only recall the reduced CNO-type TP model developed earlierin Chap. 17, which consists of 3 vertices and which resulted in the best controlperformance. In the remainder of this chapter, this type of TP model is applied tothe synthesis of controllers and observers (Fig. 18.1).

18.3 LMI-Based Multi-Objective Controllerand Observer Design

The convex TP type polytopic model described in the previous section can be welladapted to LMI based controller and observer design approaches. As a result of themany and rapid advances we are witnessing in relevant fields, it is now possible tofind the LMIs that are appropriate to such models in terms of the desired controlrequirements. In this section, we use the same structure when designing both ourobserver and controller, and apply a set of well-known LMI theorems, including onethat ensures constraints relevant to the control value, as detailed in earlier sections.

18.4 Resulting Controller and Observer Gains 219

However, at the same time we also consider the disturbance rejection property of thesystem. To do this, we first assume that disturbances appear in the model as follows:

Px.t/ D A.p/x.t/ C B.p/u.t/ C E.p/v.t/; (18.2)

such that v.t/ is the disturbance. In the specific case of the impedance model, thedisturbance affects the system states through the same input gains as the controlsignal (i.e., through u.t/), as a result of which E.p/ D B.p/. Disturbance rejectioncan be realized by minimizing � subject to:

supkv.t/k2¤0

ky.t/k2

kv.t/k2

� �: (18.3)

The following LMI theorem, derived in Chap. 3 of Tanaka and Wang [2], meetsthe above criteria:

Theorem 18.1 (Disturbance Rejection). The feedback gains Fr that stabilizesthe system and minimize � in (18.3) can be obtained by solving the followingminimization problem based on LMIs.

minimizeX;M1;:::;Mr

�2

2

6664

� 1

2fXAT

i � MTj BT

i C AiX � BiMj

CXATj � MT

i BTj C AjX � BjMig

!

� 12.Ei C Ej/

12X.Ci C Cj/

T

� 12.Ei C Ej/

T �2I 012.Ci C Cj/X 0 I

3

7775

> 0

(18.4)

for r < s � R, except pairs .r; s/ such that 8p.t/ W wr.p.t//ws.p.t// D 0, and whereMr D FrX.

Remark 18.1. Note that if the LMIs of Theorems 9.1 and 18.1 are solved simulta-neously, X in Eq. (18.4) becomes identical to P1 in Theorem 9.1.

18.4 Resulting Controller and Observer Gains

To complete this numerical example, this section summarizes the controller andobserver gains obtained based on the LMIs in Theorems 9.1, 9.3, and 18.1. Inorder to satisfy the relevant performance requirements, all the LMIs of the appliedtheorems must be solved simultaneously. This section describes the solutions ofsuch LMI combinations:


• Controller 1 is obtained from Theorem 9.1, and provides asymptotically stableobserver and controller performance.

• Controller 2 is obtained from Theorems 9.1 and 9.3, and provides asymptoticallystable observer and controller performance with a constrained control signal( < 200).

• Controller 3 is obtained from Theorems 9.1, 9.3, and 18.1, and provides asymp-totically stable observer and controller performance with constrained controlsignal and with adherence to the predefined disturbance rejection performance( < 200 and E D B ).

18.4.1 Controller-Observer 1

Fcno31 D

�4:693366697674861 � 101

�7:568178739709092 � 101

�

Fcno32 D

9:305971161607453 � 102

8:498685679030033 � 102

�

Fcno33 D

�1:473244940427044 � 102

�1:406297436375449 � 102

�(18.5)

Kcno31 D

�4:038966693651662 � 101

2:773932874373593 � 103

�

Kcno32 D

�4:088474064644633 � 101

2:689376377428563 � 103

�

Kcno33 D

�7:930535944776077 � 101

2:866274487959281 � 103

�: (18.6)


Fcno31 D

�1:083065587303841 � 102

�1:382708449040355 � 102

�

Fcno32 D

1:161938393154183 � 102

8:602599797438528 � 101

�

Fcno33 D

�3:882266623580870 � 102

�4:064172428857575 � 102

�(18.7)

18.5 Evaluation and Validation of the Control Design 221

Kcno31 D

4:202738085358277 � 102

�3:706627491954388 � 102

�

Kcno32 D

4:513238288314137 � 102

�3:980516180904803 � 102

�

Kcno33 D

3:913248140080873 � 102

�3:451262331158127 � 102

�: (18.8)


Fcno31 D

�5:528369213968272 � 101

�7:731154196175747 � 101

�

Fcno32 D

2:347047967210873 � 102

2:129777441712156 � 102

�

Fcno33 D

�2:274116741892123 � 102

�2:405310852966929 � 102

�(18.9)

Kcno31 D

�3:632338488681152 � 101

3:995608638767787 � 102

�

Kcno32 D

1:176614924780354

3:556439446795744 � 102

�

Kcno33 D

�7:536285829290905 � 101

4:457147943312266 � 102

�: (18.10)

18.5 Evaluation and Validation of the Control Design

In the previous parts of this chapter, the entire design process was applied ona concrete delayed impedance model. In this section, the control performanceswith and without the TP type polytopic controller are compared via numericalsimulations. This section also discusses some specific cases which are particularlynoteworthy. Table 18.1 shows the two parameter sets that are used in this section.Stiffness values are chosen based on an existing type of helical spring.


Table 18.1 Parameter sets applied in the validation

Description Parameter SET A SET B Units

Mass m 1 1 kg

viscous damping b 80 120 Ns/m

Stiffness of the environment k 1921 6315 N=m

Delay interval � 0-0:04 0-0:019 s

Within the simulation, a typical grasping process is imitated using a prerecordedoperator input Fh.t/. The mechanical environment of the slave device, consisting inthe simulation of a set of helical springs, is considered to be in physical interactionwith the device during the simulated grasping task. This means that the slave deviceis attached to the environment in such a way that both positive (press) and negative(pull) forces can arise (Fixed contact). If the slave device is not in fixed contact withthe environment (Free space motion), it moves freely at first, before touching theenvironment. The simulations include cases where the time-delay does not change(�.t/ D const:), as well as cases where the time-delay changes randomly during thegrasping task. The latter case emulates real, physical network delays. For anotherexample on the use of such emulation of physical delays, readers are referred to[1]. In the remainder of this section, simulation results are provided that show thefulfillment of the performance conditions:

• Fixed contact

– Constant time-delay (Figs. 18.2 and 18.3)– Varying time-delay (Figs. 18.4 and 18.5)

• Contact after free space motion

– Constant time-delay (Figs. 18.6 and 18.7)– Varying time-delay (Figs. 18.8 and 18.9)

Each figure contains three subplots: The uppermost diagram shows the positionresponse of the impedance model as a function of time with (continuous line) andwithout (dashed line) the TP� controller. The graph in the middle displays the time-delay in the feedback loop (�.t/), while the third diagram shows the intervention ofthe human operator (Fh.t/), the interaction force in the remote environment (Fe.t/),and the control signal (u.t/). In these simulations the actions of the human operator(Fh.t/) were emulated using a predefined force curve.

Figures 18.2 and 18.3 illustrate a case where feedback delay is constant duringthe emulated grasping action. It can be clearly seen that whenever the TP� controlleris switched off, unfavorable oscillations occur around the maximum deformationwith both parameter sets. Switching to our TP� based stability preserving controller,these oscillations can be made to disappear. The same behavior can be observed inFigs. 18.4 and 18.5, which refers to the case where feedback delay varies.


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−

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Fig. 18.2 Simulation with fixed contact under constant time-delay for parameter set A

Figures 18.6, 18.7, 18.8, and 18.9 show a more realistic grasping situation, inwhich the jaw of the gripper is not fastened to the remote object. In this case, thejaw can accelerate through free space motion until it touches the grasped object. Thehigher the velocity (the higher the kinetic energy), the longer the state of the systemoscillates at a higher oscillation amplitude (as long as the TP� controller is switchedoff). With the TP� controller, such oscillatory effects can be curbed down.


Fig. 18.3 Simulation with fixed contact under constant time-delay for parameter set B


Fig. 18.4 Simulation with fixed contact under varying time-delay for parameter set A


Fig. 18.5 Simulation with fixed contact under varying time-delay for parameter set B


Fig. 18.6 Simulation with contact after free space motion under constant time-delay for parameterset A


Fig. 18.7 Simulation with contact after free space motion under constant time-delay for parameterset B


Fig. 18.8 Simulation with contact after free space motion under varying time-delay for parameterset A


Fig. 18.9 Simulation with contact after free space motion under varying time-delay for parameterset B

References

1. S. Hirche, M. Buss, Study of teleoperation using realtime communication network emulation, in2003 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2003. AIM2003. Proceedings, July 2003, pp. 586–591

2. K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix InequalityApproach (Wiley-Interscience, New York, 2001)

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TP-Model Transformation-Based-Control Design Frameworks

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