Modeling and Identification of Linear Parameter-Varying Systems.pdf

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Cover

Modeling and Identification of Linear Parameter-Varying Systems

Lecture Notes in Control and Information Sciences 403

ISBN 364213811X

Preface

Contents

Acronyms

List of Symbols

Chapter 1 Introduction

New Challenges for System Identification

The Birth of LPV Systems

The Present State of LPV Identification

The Identification Cycle

General Picture of LPV Identification

LPV-IO Representation Based Methods

LPV-SS Representation Based Methods

Similarity to Other System Classes

Challenges and Open Problems

Perspectives of Orthonormal Basis Function Models

The Gain-Scheduling Perspective

The Global Identification Perspective

Approximation via OBF Structures

The Goal of the Book

Overview of Contents

Chapter 2 LTI System Identification and the Role of OBFs

The Concept of Orthonormal Basis Functions

Signal Spaces and Inner Functions

General Class of Orthonormal Basis Functions

Takenaka-Malmquist Basis

Hambo Basis

Kautz Basis

Laguerre Basis

Pulse Basis

Orthonormal Basis Functions of MIMO Systems

Basis Functions in Continuous-Time

Modeling and Identification of LTI Systems

The Identification Setting

Model Structures

Properties

Linear Regression

Identification with OBFs

Pole Uncertainty of Model Estimates

Validation in the Prediction-Error Setting

The Kolmogorov n-Width Theory

Conclusions

Chapter 3 LPV Systems and Representations

General Class of LPV Systems

Parameter Varying Dynamical Systems

Representations of Continuous-Time LPV Systems

Representations of Discrete-Time LPV Systems

Equivalence Classes and Relations

Equivalent Kernel Representations

Equivalent IO Representations

Equivalent State-space Representations

Properties of LPV Systems and Representations

State-Observability and Reachability

Stability of LPV Systems

Gramians of LPV State-Space Representations

Conclusions

Chapter 4 LPV Equivalence Transformations

State-Space Canonical Forms

The Observability Canonical Form

Reachability Canonical Form

Companion Canonical Forms

Transpose of SS Representations

LTI vs. LPV State Transformation

From State-Space to the Input-Output Domain

From the Input-Output to the State-Space Domain

The Idea of Recursive State-Construction

Cut-and-Shift in Continuous-Time

Cut-and-Shift in Discrete-Time

State-Maps and Polynomial Modules

State-Maps Based on Kernel Representations

State-Maps Based on Image-Representations

State-Construction in the MIMO Case

Conclusions

Chapter 5 LPV Series-Expansion Representations

Relevance of Series-Expansion Representations

Impulse Response Representation of LPV Systems

Filter Form of LPV-IO Representations

Series Expansion in the Pulse Basis

The Impulse Response Representation

LPV Series Expansion by OBFs

The OBF Expansion Representation

Series Expansions and Gain-Scheduling

The Role of Gain-Scheduling

Optimality of the Basis in the Frozen Sense

Optimality of the Basis in the Global Sense

Conclusions

Chapter 6 Discretization of LPV Systems

The Importance of Discretization

Discretization of LPV System Representations

Discretization of State-Space Representations

Complete Method

Approximative State-Space Discretization Methods

Discretization Errors and Performance Criteria

Local Discretization Errors

Global Convergence and Preservation of Stability

Guaranteeing a Desired Level of Discretization Error

Switching Effects

Properties of the Discretization Approaches

Discretization and Dynamic Dependence

Numerical Example

Conclusions

Chapter 7 LPV Modeling of Physical Systems

Towards Model Structure Selection

General Questions of LPV Modeling

Modeling of Nonlinear Systems in the LPV Framework

First Principle Models

Linearization Based Approximation Methods

Multiple Model Design Procedures

Substitution Based Transformation Methods

Automated Model Transformation

Summary of Existing Techniques

Translation of First Principle Models to LPV Systems

Problem Statement

The Transformation Algorithm

Handling Non-Factorizable Terms

Properties of the Transformation Procedure

Conclusions

Chapter 8 Optimal Selection of OBFs

Perspectives of OBFs Selection

Kolmogorov n-Width Optimality in the Frozen Sense

The Fuzzy-Kolmogorov c-Max Clustering Approach

The Pole Clustering Algorithm

Properties of the FKcM

Simulation Example

Robust Extension of the FKcM Approach

Questions of Robustness

Basic Concepts of Hyperbolic Geometry

Pole Uncertainty Regions as Hyperbolic Objects

The Robust Pole Clustering Algorithm

Properties of the Robust FKcM

Simulation Example

Conclusions

Chapter 9 LPV Identification via OBFs

Aim and Motivation of an Alternative Approach

OBFs Based LPV Model Structures

The LPV Prediction-Error Framework

The Wiener and the Hammerstein OBF Models

Properties of Wiener and Hammerstein OBF Models

OBF Models vs. Other Model Structures

Identification of W-LPV and H-LPV OBF Models

Identification with Static Dependence

The Identification Setting

LPV Identification with Fixed OBFs

Local Approach

Global Approach

Properties

Examples

Approximation of Dynamic Dependence

Feedback-Based OBF Model Structures

Properties of Wiener and Hammerstein Feedback Models

Identification by Dynamic Dependence Approximation

Properties

Example

Extension towards MIMO Systems

Scalar Basis Functions

Multivariable Basis Functions

Multivariable Basis Functions in the Feedback Case

General Remarks on the MIMO Extension

Conclusions

Appendix A Proofs

References

Index

Lecture Notes in Control and Information Sciences 403 Editors: M. Thoma, F. Allgöwer, M. Morari

Roland Tóth Modeling and Identiﬁcation of Linear Parameter-Varying Systems ABC

Series Advisory Board P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis Author Dr. Roland Tóth Delft University of Technology Faculty of Mechanical, Maritime and Materials Engineering Delft Center for Systems and Control Mekelweg 2 2628 CD, Delft The Netherlands Co-Editors Dr. Peter S.C. Heuberger Delft University of Technology Faculty of Mechanical, Maritime and Materials Engineering Delft Center for Systems and Control Mekelweg 2 2628 CD, Delft The Netherlands Prof. Dr. Paul M.J. Van den Hof Delft University of Technology Faculty of Mechanical, Maritime and Materials Engineering Delft Center for Systems and Control Mekelweg 2 2628 CD, Delft The Netherlands ISBN 978-3-642-13811-9 e-ISBN 978-3-642-13812-6 DOI 10.1007/978-3-642-13812-6 Lecture Notes in Control and Information Sciences ISSN 0170-8643 Library of Congress Control Number: 2010928272 c 2010 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientiﬁc Publishing Services Pvt. Ltd., Chennai, India. Printed in acid-free paper 5 4 3 2 1 0 springer.com

I dedicate this book to my beloved wife Andrea, my “great” son S´andor and my little daughter Lujza.

Preface Through the past 20 years, the framework of Linear Parameter-Varying (LPV) systems has become a promising system theoretical approach to han- dle the control of mildly nonlinear and especially position dependent systems which are common in mechatronic applications and in the process indus- try. The birth of this system class was initiated by the need of engineers to achieve better performance for nonlinear and time-varying dynamics, com- mon in many industrial applications, than what the classical framework of Linear Time-Invariant (LTI) control can provide. However, it was also a pri- mary goal to preserve simplicity and “re-use” the powerful LTI results by extending them to the LPV case. The progress continued according to this philosophy and LPV control has become a well established ﬁeld with many promising applications. Unfortunately, modeling of LPV systems, especially based on measured data (which is called system identiﬁcation) has seen a limited development since the birth of the framework. Currently this bottleneck of the LPV frame- work is halting the transfer of the LPV theory into industrial use. Without good models that fulﬁll the expectations of the users and without the under- standing how these models correspond to the dynamics of the application, it is diﬃcult to design high performance LPV control solutions. This book aims to bridge the gap between modeling and control by investigating the fundamental questions of LPV modeling and identiﬁcation. It explores the missing details of the LPV system theory that have hindered the formula- tion of a well established identiﬁcation framework. By proposing an uniﬁed LPV system theory that is based on a behavioral approach, the concepts of representations, equivalence transformations, and means to compare model structures are re-established, giving a solid basis for an identiﬁcation theory. It is also explored when and how ﬁrst-principle nonlinear models can be eﬃ- ciently converted to LPV descriptions and what are the pitfalls that must be avoided. Building on well founded system theoretical concepts, the classical LTI prediction-error framework is extended to the LPV case via the use of series-expansion representations.

VIII Preface Beside completing the system theoretical aspects and founding of an LPV prediction-error framework, the book proposes a novel identiﬁcation approach based on orthonormal basis functions, which provides an eﬃcient and easy to use approach of LPV identiﬁcation. It has been shown in the LTI case that decomposing dynamical systems in terms of orthogonal expansions enables the accurate approximation of the system with a ﬁnite length expansion. By tuning the basis functions to the underlying system characteristics, the rate of convergence can be drastically decreased. This leads to highly accurate models (small bias) being represented by a few parameters (small variance), which in fact can be estimated eﬃciently via simple linear regression. This philosophy gives the basic concept of the proposed identiﬁcation approach, for which the applicability in practical scenarios is investigated and powerful algorithms are provided and analyzed. The work presented here is the result of 5 years of research at the Delft University of Technology under the supervision of Prof. Paul M.J. Van den Hof and Peter S.C. Heuberger. Starting with the initial intention to apply simplicity of orthonormal basis function models to overcome the challenges of LPV system identiﬁcation, a long road has led to the theory which is presented in this book. Walking on this road, many excellent people have contributed to this work. Especially Prof. Jan C. Willems, at the Catholic University of Leuven, whose advice and vision about mathematical modeling helped me to ﬁnd the right track that lead to the LPV behavioral theory, that forms the basis of many concepts of this book. Prof. Carsten Scherer, at the Delft University of Technology, whose advice and extensive knowledge on LPV systems and control has always helped me to ﬁnd the missing wheel. Federico Felici, whose earth-moving questions and discussions made me aware of missing key points of the LPV system theory. But the main thanks goes to Prof. Paul M.J. Van den Hof and Peter S.C. Heuberger whose excellent guid- ance led me through these years, culminating in my Ph.D. thesis [188], which served as the basis of this book. As co-editors, they also had a signiﬁcant role in establishing this book in its current from. The book is written as a research monograph with a broad scope, trying to cover the key issues from system theory to modeling and identiﬁcation. It is meant to be interesting for both researchers and engineers but also for graduate students in systems and control who would like to learn about the LPV framework. It also oﬀers an easy to use guide for engineers about the oﬀ-shelf solutions in LPV modeling and identiﬁcation. I hope that you as a reader will enjoy reading it as much as it was fun to write it. Delft, December 2009 Roland T´oth

Contents 1 2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 New Challenges for System Identiﬁcation . . . . . . . . . . . . . . . . . 1.2 The Birth of LPV Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Present State of LPV Identiﬁcation . . . . . . . . . . . . . . . . . . 1.3.1 The Identiﬁcation Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 General Picture of LPV Identiﬁcation . . . . . . . . . . . . . . 1.3.3 LPV-IO Representation Based Methods . . . . . . . . . . . . 1.3.4 LPV-SS Representation Based Methods . . . . . . . . . . . . 1.3.5 Similarity to Other System Classes . . . . . . . . . . . . . . . . 1.4 Challenges and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Perspectives of Orthonormal Basis Function Models. . . . . . . . 1.5.1 The Gain-Scheduling Perspective . . . . . . . . . . . . . . . . . . 1.5.2 The Global Identiﬁcation Perspective . . . . . . . . . . . . . . 1.5.3 Approximation via OBF Structures . . . . . . . . . . . . . . . . 1.6 The Goal of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Overview of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LTI System Identiﬁcation and the Role of OBFs . . . . . . . . . 2.1 The Concept of Orthonormal Basis Functions . . . . . . . . . . . . . 2.2 Signal Spaces and Inner Functions . . . . . . . . . . . . . . . . . . . . . . . 2.3 General Class of Orthonormal Basis Functions . . . . . . . . . . . . 2.3.1 Takenaka-Malmquist Basis . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Hambo Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Kautz Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Laguerre Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Pulse Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Orthonormal Basis Functions of MIMO Systems . . . . . 2.3.7 Basis Functions in Continuous-Time . . . . . . . . . . . . . . . 1 1 3 4 4 6 9 11 14 15 17 17 18 18 19 20 21 21 22 24 24 25 26 26 26 26 27

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