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Intelligent Observer and Control Design for Nonlinear Systems
Springer-Verlag Berlin Heidelberg GmbH
Dierk Schräder (Ed.)
Intelligent Observer and Control Design for Nonlinear Systems
With 178 Figures
Springer
Editor: Prof. Dr.-Ing. Dr.-Ing. h. c. Dierk Schröder Technical University ofMunich Institute for Electrical Drive Systems Arcisstrasse 21
D-80333 München Germany
Contributors: Prof. Dr.-Ing. Dr.-Ing. h. c. Dierk Schröder Dr.-Ing. UlrichLenz Dipl.-Ing. Michael Beuschel Dipl.-Ing. FranzD. Hangl Dr.-Ing. ThomasFrenz Dr.-Ing. Dieter Strobl Dr.-Ing. Stephan Straub Dr.-Ing. Kurt Fischle Dipl.-Ing.MartinRau Dipl.-Ing. Anne Angermann
Library of Congress Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - Cip-Einheitsaufnahme Intelligent ob server and control design for nonlinear systems! Dierk Schröder (ed.). -Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore ; Tokyo : Springer, 2000
ISBN 978-3-642-08346-4 ISBN 978-3-662-04117-8 (eBook) DOI 10.1007/978-3-662-04117-8
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© Springer-Verlag Berlin Heidelberg 2000 Originally published by Springer-Verlag Berlin Heidelberg New York in 2000. Softcover reprint of the hardcover 1 st edition 2000
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Preface
Research is a continuous effort. Engineers and research groups are creating new strategies and solutions by using results of other scientists who have been working on the same topic for a long time, thus aquiring a deeper understanding. Indeed we are dependent upon one another and should support each other.
This book is a contribution in this continuous line of scientific efforts and is a result of the research by PhD-students at my institute. We would like to present our ideas and results and we hope very much to provide support for other scientists interested in this area.
The starting point of our considerations is: We are engineers, and therefore we have basic knowledge of the system under consideration. But often there is a lack of precise information for a sufficiently accurate model, due to structured or unstructured uncertainties or, more severe, nonlinearities. How can we get this desired information? The idea is to identify unknown parts of the plant by a learning procedure. An idea which was already proposed by others, but we think we are able to contribute some new aspects and extensions to this area.
One aspect of our research is to assume in the first step that we know the linear part of the non linear plant, but we do not know the type and the parameters of the nonlinearities. In real life these nonlinearities are not smooth in general, typical nonlinearities in motion control are e.g. friction and backlash. So we concentrated on this topic, the identification of type and parameters of the nonlinearities. This led to dynamic learning structures, providing exact information ab out existing nonlinearities. With this information we achieved a much more precise model of the nonlinear plant. The next steps are nonlinear observers and the controller design. One of the major guidelines in our work is that the learning process is mathematically proven stable and convergent. Therefore these intelligent strategies could be used off-line and on-line.
A second fascinating idea is to learn the optimal controller even though one has only a very limited amount of knowledge of the non linear plant. There have been proposals for such a scenario, but up to now there have been very important restrictions. We reduced these restrictions to some extent, but there is additional research necessary; for example to reduce the learning time or to separate the effects of unknown disturbing inputs to the plant during learning. A combination
VI
of the first and second approach leads to a possible design of a nonlinear state space controller, where existing nonlinearities are taken into account already during the procedure of controller design.
We noticed that these methods are applicable in different areas of motion control, e.g. electrical drives, machine tools, processing machines with continuous moving webs (rolling mills, printing machines), or even identification and control in combustion engines. Therefore we decided to gather our results up to now in this book and thus provide an easy access for other researchers. We also hope to get information from their experiences and new results and to start a fruitful discussion. Thank you in advance.
München, October 1999 Dierk Schröder
Contents
1
1.1
1.2
1.3
1.4
Introduction - Control Aspects
Dierk Schröder
Linear Plants .
Linear Plants with Uncertain Parameters
Linear Plants and Nonlinear Controllers
Nonlinear Plants . . . . . . . . . . . . .
1
4
5
8
9
1.5 Our Conceptions of Nonlinear Control Strategies and Observation 13
1.6
2
2.1
2.1.1
2.1.2
2.1.3
2.1.3.1
2.1.3.2
2.1.4
2.2
2.3
2.4
2.5
References. . . . .
Motion Control
Dierk Schröder
Control of Electromechanical Systems
Introduction
Cascaded Control .
State-Space Control
Proportional State-Space Controller
State-Space Control with Integrating Contribution
Generalized Considerations for Electromechanical Systems
Actuator, Mechanical System and Process .....
Objectives of this book (Example Motion Control)
Conclusions
References .
15
19
19
19
20
27
27
34
40
46
59
63
64
VIII
3
3.1
3.2
3.3
3.3.1
3.3.2
3.3.3
3.4
3.4.1
3.4.2
3.4.2.1
3.5
3.5.1
3.5.2
3.5.3
3.6
3.7
4
4.1
4.1.1
4.1.2
4.1.3
4.2
4.2.1
4.2.2
4.2.2.1
4.2.2.2
4.2.2.3
4.2.3
4.3
Learning in Control Engineering
Ulrich Lenz
Intelligent Control as Artificial Intelligence .
Artificial Intelligence Realized by a Non-Biologie Structure .
Basic Structures for Control
Open-Loop Control .
Closed-Loop Control .
"Conditional Feedback" Control Structure
Scopes for Intelligent Control .
Methods of Intelligent Control .
Application of Learning in Control Engineering
Example: Direct and Indirect Approach
Requirements for Adaptive Methods
Stability ............... .
Improving the Controller's Performance
Expandable Knowledge ........ .
Classification due to System's Structure or Restrietions .
References . . . . . . . . . . . . . . . . .
Nonlinear Function Approximators
Michael Beuschel
Nonlinear Function Approximation
Concepts of Function Approximation
Basis Functions for Function Approximation.
Universal and Convergent Function Approximation
Neural Nctworks as Function Approximators .
Radial Basis Function (RBF) Network ....
General Regression Neural Network (GRNN)
GRNN at Multidimensional Input Space
Polynomial Activation (DANN)
Restricted Update Area . . . .
Other Neural Network Approaches
Neuro-Fuzzy Systems as Function Approximators
Contents
67
67
68
69
69
70
71
72
73
73
74
75
76
78
78
78
80
83
83
84
85
86
88
88
89
91
92
92
93
94
Contents
4.3.1
4.3.2
4.4
4.5
4.6
5
Principles of Neuro-Fuzzy Systems ......... .
Neuro-Fuzzy Example: Tuning of Output Fuzzy Sets
Example ..
Conclusion
References .
Systematic Intelligent Observer Design
Ulrich Lenz
IX
94
96
98
101
102
105
5.1 Definitions: Dynamic Systems Containing an Isolated Nonlinearity 107
5.1.1
5.1.2
5.2
5.3
5.3.1
5.3.2
5.3.2.1
5.3.2.2
5.3.2.3
5.3.2.4
5.3.2.5
5.3.2.6
5.4
5.4.1
5.4.2
5.4.2.1
5.4.2.2
5.5
5.6
6
6.1
6.2
Dynamic System with an Isolated Nonlinearity
Approximation of a Static Nonlinearity .....
Hybrid Notation of Signals in the Time and Frequency Domain
Systematic Observer Design
Conditions ........ .
Observer Design for Identification .
Observer Approach ........ .
Dimensioning the Observer Feedback Matrix L
Specification of the Error Transfer Function H (s)
Deriving a Stable Adaptation Law Using Known Error Models.
Reflections on Parameter Convergence
Simplifying the Observer Design
Intelligent Observer Design Following the Luenberger Approach
Prerequisites ....... .
Systematic Observer Design
Deriving the Error Transfer Function
Adaptation Law
Summary.
References .
Identification of Separable Nonlinearities
Franz Hangl
Plants with Separable Nonlinearities
Nonlinear Observer Approach ....
108
110
111
111
111
113
113
113
114
115
121
124
124
124
126
126
127
131
132
135
135
136
x Contents
6.2.1 Identification with Accessible States ............ 136
6.2.1.1 Adaptive Observer According to the Luenberger Observer 136
6.2.2 Identification of Nonlinearities in Plants with Unknown Internal States . . . . . . . . . . . . 139
6.2.2.1 Neural Observer Approach. . 139
6.2.3 The Error Decoupling Filter. 141
6.2.3.1 The Adaptive Law . . . . . . 143
6.3 Implementation of A-Priori Knowledge . 144
6.3.1 Additive A-Priori Knowledge . . . 145
6.3.2 Multiplicative A-Priori Knowledge 146
6.4 References . . . . . . . . . . . . . . 148
7 Identification and Compensation of Friction 149
Thomas Frenz
7.1
7.2
7.3
7.4
7.5
7.6
8
8.1
Introduction
Design of Hardware
Implementation: Learning of Friction Characteristic .
Application: Compensation of Friction Influence .
Conclusion
References .
Detection and Identification of Backlash
Dieter Strobl
Introduction
149
155
157
160
163
165
167
167
8.2 Example System for Backlash Identification 168
8.2.1 Model of an Elastic Two-Mass System . . . 168
8.2.2 Identifiability of the Backlash Characteristic 169
8.2.3 State Space Description of the Nonlinear System 170
8.3 Identification of Backlash ... . . . . . . . . . . 172
8.3.1 Representation of Backlash for the Identification with a Neural Network . . . . . . . . . . . . . . . . 172
8.3.2 Load-Side Backlash Observer (LBO) 174
8.3.2.1 State Space Representation . . . . . 174
Contents
8.3.2.2
8.3.3
8.3.3.1
8.3.3.2
8.4
8.5
8.5.1
8.5.2
8.6
8.7
9
9.1
9.2
9.2.1
9.2.2
9.2.3
9.3
9.3.1
9.3.2
9.3.3
9.4
9.5
10
10.1
10.2
10.3
10.4
10.5
Observer Design and Error Model ...
Motor-Side Backlash Observer (MBO)
State Space Representation . . . .
Observer Design and Error Model.
Simulation Examples ..
Experimental Validation
Experimental Set-Up and Parameters
Results of Online Backlash Identification .
Conclusion
References .
Identification of Isolated Nonlinearities in Rolling Mills
Stephan Straub
Introduction
Neural Networks in Rolling Mills
Plant Description . . . . . . . . .
Compensation of Winder Eccentricities .
Identification of the Roll Bite
Experimental Results .
Plant Description . . .
Identification Results .
Compensation Results
Conclusion
References .
Input-Output Linearization: an Introduction
Kurt Fischle
A U seful Canonical Form for Nonlinear Systems .
Basic Concept of Input-Output Linearization
Simplified Ideal Control Law
Short Summary .
References. . . .
XI
174
178
178
179
180
183
183
184
187
188
189
189
189
189
191
198
204
204
207
209
214
215
217
217
223
228
232
233
XII
11
11.1
Stable Model Reference Neurocontrol
Kurt Fischle
Introduction
11.2 Description of the Concept
11.3 Application Example: Nonlinear Second-Order System
11.3.1 Simulation Results . .
11.3.2 Experimental Results .
11.4 Modifications .....
11.4.1 Modification for Plants with Control Saturation
11.4.2 Modification for Plants with LgLT1 h(J;.) cf: const.
11.4.3 Method with Differentiation of y ........ .
11.4.4 Modifications for Reduction of the Learning Times
11.5 Short Summary.
11.6 References ....
12 Dynamic Neural Network Compositions
Stephan Straub
12.1 Introduction
Contents
235
235
237
240
240
244
247
249
249
250
250
251
253
255
255
12.2 Classification of Identification Methods . 256
12.2.1 Motivation....... 256
12.2.2 Different Net Structures 257
12.2.3 Nonlinear Observer Structures and Dynamic Identificators 260
12.3 Identification of Systems with Unknown Structure Using a Dy-namic Identificator . . . . . . . . . . . 265
12.3.1 Motivation and Theoretical Approach 266
12.3.2 Design of a Dynamic Identificator . 269
12.3.3 Possible Control Concepts 272
12.3.4 Simulation 1: Example . . 274
12.3.5 Simulation 2: Two-Mass System 275
12.3.6 Simulation 3: Inverse Control 278
12.4 Conclusion 280
12.5 References . 281
Contents XIII
13 Further Strategies for Nonlinear Control with Neural Net-works 283
Martin Rau, Anne Angermann
13.1 Introduction 283
13.2 Compensation and State-Space Control Strategies for a Class of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . .. 284
13.2.1 Systems with Isolated Nonlinearities and Nonlinear Observer . 285
13.2.2 Exact Compensation of Isolated Nonlinearities. . . . . 285
13.2.2.1 Transfer Function Description of the Nonlinear System
13.2.2.2 Compensation Algorithm .......... .
13.2.2.3 Realization of the Compensation Filter K(s) .
13.2.3 State-Space Control of the Compensated System
13.2.3.1 Simulation Example
286
287
288
291
293
13.2.4 Conclusions..... 297
13.2.5 Alternate Compensation and Control Design. 297
13.3 Nonlinear Control with a Controllable Canonical Form 302
13.4 Nonlinear Control Design with Neural Networks and Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . .. 304
13.4.1 Model Reference Neuro Control for Systems with Isolated Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 305
13.4.2 Considerations on Numerical Optimization in Nonlinear Control 307
13.5 Time-Optimal Tension Control of Continuous Moving Webs Sys-tems .....
13.5.1 Introduction
13.5.2 Controller Design.
13.5.3 Experimental Validation
13.5.4 Conclusion
13.6 References .
List of Figures
Index
308
308
313
321
323
326
336
337
