TP Model Transformation Based Control Design Frameworks

by Peter Baranyi

TP Model Transformation Based Control Design Frameworks This book covers new aspects and frameworks of control design and optimization based on the TP model transformation and its various extensions The author outlines the three main steps of polytopic and LMI based control design 1 development of the qLPV state space model 2 generation of the polytopic model and 3 application of LMI to derive controller and observer He goes on to describe why literature has extensively studied LMI design but has not focused much on the second step in

Publisher : Springer International Publishing

Author : Peter Baranyi

ISBN : 9783319196046

Year : 2016

Language: en

File Size : 8.73 MB

Category : Engineering Transportation

Péter Baranyi

TP-Model
TransformationBasedControl Design
Frameworks

TP-Model Transformation-Based-Control Design
Frameworks

Péter Baranyi

TP-Model Transformation
-Based-Control Design
Frameworks

123

Péter Baranyi
Technology and Economics
Szecheny Istvan University
and Budapest Univerity of Technology
and 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 2016
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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 our
experiences in control design. I hope the frameworks introduced in the book will
help to radically decrease the amount of analytical work that is performed, often
unnecessarily, by researchers and engineers working in the field of control design
optimization. If our experience can serve as any basis for generalization, many
existing analytical approaches can be substituted by more flexible and effective
numerical methods.
The TP model transformation-based frameworks provide a simple, generic, and
flexible way to interface between identification stages and, primarily, linear matrix
inequality-based control design theories. Further, they support stability verification
purposes in general, even in cases where identification and design are based on very
different representations. Finally, the presented frameworks lay the foundations for
convex hull manipulation-based control design optimization.
I would like to express my appreciation to my friends Prof. Yeung Yam and
Prof. Péter Várlaki for their strong support and for their help in shaping, through
many discussions, a broader scientific and conceptual view behind the TP model
transformation. I am indebted to the work of young researchers Dr. Béla Takarics,
Dr. Péter Galambos, Dr. Ádám Csapó, Patricia Gróf, József Kuti, and Szöllösi
Alexandra, who have helped in preparing a large number of experimental case
studies and in extending the TP-tool MATLAB toolbox. I am grateful to Anna
Szemereki for her help in managing all the related research work and projects
that made it possible for the research group to focus on the research behind this
book. Finally, I would like to thank our collaborators and graduate students, past
and present, for their inputs and contributions to research on this subject.
Budapest, Hungary
January 2016

Péter Baranyi

v

Contents

Acronyms and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
The Key Messages of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
Part I Generalized TP Model Transformation
1

2

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
TP Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
TP Model of qLPV Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
TP Model: TS Fuzzy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
HOSVD and Quasi-HOSVD Based Canonical Form
of TP Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
4
5
6
8
10

Algorithms of the TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Original TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Bi-Linear TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Enriched TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Convex TP Model Transformation: Convex Hull Manipulation . . .
2.4.1
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Pseudo TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
Partial TPC Model Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
Multi TP Model Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11
11
14
17
22
24
25
25
28
35
43
44
48
49
vii

viii

Contents

2.8
2.9

Generalized TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interpolation of the Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.1
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Unifying the Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Operations Between TP Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Towards Approximation in Case of Non-TP Functions . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52
54
55
59
60
61
62

Part II TP Model Transformation Based Control Design and
Optimalization Frameworks
3

TP Model Transformation is a Gateway Between
Identification and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65
66

4

TP Model Transformation Based Control Design Structure . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
71

5

General Stability Verification and Control Design. . . . . . . . . . . . . . . . . . . . . .
5.1
Key Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Decoupling the Design, Optimization, and Stability
Verification: Generalized Design Frameworks. . . . . . . . . . . . . . . . . . . . . .
5.3.1
Multi-Way Convex Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2
Main and Independent TP Model Component
Analysis via the HOSVD Based Canonical Form. . . . . . . . .
5.3.3
Convex Hull Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4
LMI Based System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.5
Exact System Reconstruction: Unified TP
Model Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.6
LMI Based Stability Verification . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73
73
74
77
79
82
82
83
84
86
86

6

TPI Model Transformation for the Class of Non-qLPV Models . . . . . .
6.1
Key Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
TPI Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Example of Re-identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87
87
88
89
89

7

TP Model Transformation for Systems Including Time Delay . . . . . . .
7.1
TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Example of the TP Model Transformation. . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91
91
92
93

Part III Analysis of the TP Model Based Design Frameworks
via a Complex Example
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Contents

ix

8

qLPV Model of the 3DoF Prototypical Aeroelastic Wing Section. . . . . 97
8.1
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2
Including Stribeck Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

9

TP Model Based Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1
Exact and Convex TP Model of the 3DoF Aeroelastic
Wing Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Control Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3
Selecting LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4
Results of the Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1
Controller 1: Asymptotic Stabilization and
Decay Rate Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2
Controller 2: Constraint on the Control Value . . . . . . . . . . . .
9.4.3
Controller 3: State Feedback Control
Including Stribeck Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.4
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.5
Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103
103
104
106
107
107
107
108
108
109
115

10

Convex Hull Manipulation Based Optimization . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Convex Hull Manipulation Based Design Framework . . . . . . . . . . . . .
10.1.1 Key Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.2 Step 1: Convex TP Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.3 Step 2: Convex TP Model Interpolation . . . . . . . . . . . . . . . . . . .
10.1.4 Step 3: LMI Based Design and Stability Verification . . . . .
10.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Determination of the Feasibility Region. . . . . . . . . . . . . . . . . . .
10.2.2 Results of the Numerical Simulations . . . . . . . . . . . . . . . . . . . . .

117
117
118
118
118
120
120
120
121

11

Complexity Manipulation Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 The Control Design Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.1 Main TP Model Component Analysis:
HOSVD Based Canonical Form of the Model . . . . . . . . . . . .
11.1.2 LMI Based System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.3 Exact System Reconstruction: Unified
Weightings in the Polytopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.4 LMI Based Stability Verification . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.5 Maximizing Omega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Evaluation of the Benefits of the Proposed Control Design . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131
131

12

132
133
137
137
137
138
144

TP Model Manipulation Influences the Control
Performance and the Feasibility of LMI Based Design. . . . . . . . . . . . . . . . . 145
12.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
12.1.1 Initialization of the Numerical Analysis . . . . . . . . . . . . . . . . . . . 145

x

Contents

12.1.2

Results of the 2D Analysis: Feasibility and
Convex Hull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.3 Results of the 3D Analysis: Feasibility,
Convex Hull, and Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.4 Results of the 4D Analysis: Feasibility,
Convex Hull, Complexity, and Parameter Space . . . . . . . . . .
12.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Control Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 Control Performance Results of the Numerical
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.2 Evaluation and Comparison of the Derived
Cases and the Best Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146
148
148
154
154
154
156
160

Part IV TP Model Based Control Design of the Dual-Excenter
Vibration Actuator
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
13

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

14

Convex TP Model of the Dual Excenter Vibration System . . . . . . . . . . . . 171
14.1 The Quasi-HOSVD Based Canonical Form:
Approximation and Complexity Trade-Off . . . . . . . . . . . . . . . . . . . . . . . . . 171
14.2 The Convex TP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

15

Derivation of the Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 LMI Based Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179
179
182
184

Part V Control of the Impedance Model Including Varying
Time Delay via TP Model Transformation
16

Impedance Control for Force Reflecting Telemanipulation. . . . . . . . . . . .
16.1 Impedance Control with Feedback Delay . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Control Structure for Stability Preservation. . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187
188
190
193

17

Impedance Model with Varying Feedback Delay in TP
Model Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 The Quasi-HOSVD Based Canonical Form . . . . . . . . . . . . . . . . . . . . . . . .
17.1.1 Exact Quasi-HOSVD Based Canonical Form . . . . . . . . . . . . .
17.1.2 Executing Trade-off by TP Model Transformation . . . . . .
17.2 Manipulation of the Convex Hull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.1 The Vertices of the Exact TP Model . . . . . . . . . . . . . . . . . . . . . . .
17.2.2 The 5 Vertices of the Reduced TP Model . . . . . . . . . . . . . . . . .

195
195
195
198
199
204
208

Contents

18

xi

17.2.3 The 4 Vertices of the Reduced TP Model . . . . . . . . . . . . . . . . .
17.2.4 The 3 Vertices of the Reduced TP Model . . . . . . . . . . . . . . . . .
17.3 Validation of the Convex TP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.1 Constant Time-Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.2 Varying Time-Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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211
211
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215

TP Transformation Based Control Design for Impedance
Controlled Robot Gripper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1 The Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Execution of the TP Model Transformation . . . . . . . . . . . . . . . . . . . . . . .
18.3 LMI-Based Multi-Objective Controller and Observer Design . . . . .
18.4 Resulting Controller and Observer Gains . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.1 Controller-Observer 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.2 Controller-Observer 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.3 Controller-Observer 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.5 Evaluation and Validation of the Control Design . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217
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218
218
219
220
220
221
221
230

Acronyms and Abbreviations

CHOSVD
CNO
DoF
HOOI
HOSVD
INO
IRNO
LMI
LMIs
LPV
LTI
NN
NO
PDC
qLPV
qNN
qSN
RHOSVD
RNO
SN
SVD
TP
TP model
TPC

Compact HOSVD
Close-to-normality
Degree of freedom
Higher-order orthogonal iteration
High-order singular value decomposition
Inverse normality
INO and RNO
Linear matrix inequality
Linear matrix inequalities
Linear parameter-varying
Linear time-invariant
Nonnegativeness
Normality
Parallel distributed compensation
quasi-LPV
quasi-NN
quasi-SN
Reduced HOSVD
Relaxed normality
Sum normalization
Singular value decomposition
Tensor product
Finite element TP-type polytopic model
Pseudo TP model transformation

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