Operator Theory - Masaryk University Operator Theory and Indeï¬پ nite Inner Product Spaces Presented

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Text of Operator Theory - Masaryk University Operator Theory and Indeï¬پ nite Inner Product Spaces...

  • Birkhäuser Verlag Basel . Boston . Berlin

    Operator Theory and Indefi nite Inner Product Spaces Presented on the occasion of the retirement of Heinz Langer in the Colloquium on Operator Theory, Vienna, March 2004

    Matthias Langer Annemarie Luger Harald Woracek Editors

    Introduction to Mathematical Systems Theory Linear Systems, Identification and Control

    Christiaan Heij André Ran Freek van Schagen

  • Authors:

    Christiaan Heij Econometric Institute Faculty of Economics Erasmus University Rotterdam P.O. Box 1738 3000 DR Rotterdam The Netherlands e-mail: heij@few.eur.nl

    André Ran Department of Mathematics Faculty of Exact Sciences Vrije Universiteit Amserdam De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: ran@few.vu.nl

    Bibliographic information published by Die Deutsche Bibliothek

    available in the Internet at .

    be obtained.

    Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF Printed in Germany

    Freek van Schagen Department of Mathematics Faculty of Exact Sciences Vrije Universiteit Amserdam De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: f.van.schagen@few.vu.nl

  • Contents

    Preface ix

    1 Dynamical Systems 1

    1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Systems and Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 State Representations . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2 Input-Output Systems 11

    2.1 Inputs and Outputs in the Time Domain . . . . . . . . . . . . . . . 11 2.2 Frequency Domain and Transfer Functions . . . . . . . . . . . . . . 14 2.3 State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Equivalent and Minimal Realizations . . . . . . . . . . . . . . . . . 20

    3 State Space Models 25

    3.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Structure Theory of Realizations . . . . . . . . . . . . . . . . . . . 31 3.4 An Algorithm for Minimal Realizations . . . . . . . . . . . . . . . 34

    4 Stability 39

    4.1 Internal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Input-Output Stability . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Stabilization by State Feedback . . . . . . . . . . . . . . . . . . . . 45 4.4 Stabilization by Output Feedback . . . . . . . . . . . . . . . . . . 49

    5 Optimal Control 53

    5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Linear Quadratic Control . . . . . . . . . . . . . . . . . . . . . . . 59

  • vi Contents

    6 Stochastic Systems 67 6.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.3 ARMA Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.4 State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.5 Spectra and the Frequency Domain . . . . . . . . . . . . . . . . . . 79 6.6 Stochastic Input-Output Systems . . . . . . . . . . . . . . . . . . . 81

    7 Filtering and Prediction 83 7.1 The Filtering Problem . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.2 Spectral Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.3 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.4 The Steady State Filter . . . . . . . . . . . . . . . . . . . . . . . . 96

    8 Stochastic Control 101 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.2 Stochastic Dynamic Programming . . . . . . . . . . . . . . . . . . 102 8.3 LQG Control with State Feedback . . . . . . . . . . . . . . . . . . 105 8.4 LQG Control with Output Feedback . . . . . . . . . . . . . . . . . 108

    9 System Identification 115 9.1 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.2 Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9.3 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.4 Estimation of Autoregressive Models . . . . . . . . . . . . . . . . . 121 9.5 Estimation of ARMAX Models . . . . . . . . . . . . . . . . . . . . 124 9.6 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    9.6.1 Lag Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 9.6.2 Residual Tests . . . . . . . . . . . . . . . . . . . . . . . . . 129 9.6.3 Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . 130 9.6.4 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . 132

    10 Cycles and Trends 133 10.1 The Periodogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.2 Spectral Identification . . . . . . . . . . . . . . . . . . . . . . . . . 138 10.3 Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 10.4 Seasonality and Nonlinearities . . . . . . . . . . . . . . . . . . . . . 146

    11 Further Developments 151 11.1 Continuous Time Systems . . . . . . . . . . . . . . . . . . . . . . . 151 11.2 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 11.3 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 11.4 Infinite Dimensional Systems . . . . . . . . . . . . . . . . . . . . . 155 11.5 Robust and Adaptive Control . . . . . . . . . . . . . . . . . . . . . 156

  • Contents vii

    11.6 Stochastic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 11.7 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 159

    Bibliography 161

    Index 165

  • Preface

    This book has grown out of more than ten years of teaching an introductory course in system theory, control and identification for students in the areas of Business Mathematics and Computer Science, Econometrics and Mathematics at the ‘Vrije Universiteit’ in Amsterdam. The interests and mathematical background of our students motivated our choice to focus on systems in discrete time only, because the topics can then be studied and understood without preliminary knowledge of (deterministic and stochastic) differential equations. This book requires some preliminary knowledge of calculus, linear algebra, probability and statistics, and some parts use elementary results on Fourier series.

    The book treats the standard topics of introductory courses in linear systems and control theory. Deterministic systems are discussed in the first five chapters, with the following main topics: realization theory, observability and controllabil- ity, stability and stabilization by feedback, and linear-quadratic optimal control. Stochastic systems are treated in Chapters six to eight, with main topics: realiza- tion, filtering and prediction (including the Kalman filter), and linear-quadratic Gaussian optimal control. Chapters nine and ten discuss system identification and modelling from data, and Chapter eleven concludes with a brief overview of further topics.

    Exercises form an essential ingredient of any successful course in this area. The exercises are not printed in the book and are instead incorporated on the accompanying CD-Rom. The exercises are of two types, i.e., theory exercises to train mathematical skills in system theory and practical exercises applying system and control methods to data sets that are also included on the CD-Rom. Many exercises require the use of Matlab or a similar software package.

    We benefitted greatly from comments of many colleagues who, over the years, participated in teaching from this book. In particular, we mention the contribu- tions of (in alphabetical order) Sanne ter Horst, Rien Kaashoek, Derk Pik and Alistair Vardy. We thank them for their comments, which have improved the text considerably. In addition, many students helped us in improving the text by asking questions and pointing out misprints.

  • Chapter 1

    Dynamical Systems

    1.1 Introduction

    Many phenomena investigated in such diverse areas as physics, biology, engineer- ing, and economics show a dynamical evolution over time. Examples are thermo- dynamics and electromagnetism in physics, chemical processes and adaptation in biology, control systems in engineering, and decision making in macro economics, finance, and business economics. The main questions analysed in this book are the following.

    • What type of mathematical models can be used to study such dynamical processes?

    • Once a model class is selected and we know the parameters in the model, how can we achieve specific objectives such as stability, uncertainty reduction and optimal decision making?

    • If we do not know the parameters in the model exactly, how can we estimate them from available data and how reliable is the obtained model?

    The first question is the topic of Chapters 2, 3 and 6, the second one of Chapters 4, 5, 7, 8 and 9, and the third one of Chapters 9 and 10. The answers to these questions will in general depend on accidental particularities of the problem at hand. However, there are important common characteristics of these problems which can be expressed in terms of mathematical models. We first give some examples to illustrate the main ideas in modelling, estimation, forecasting and control.

    Example 1.1. Suppose that for a certain good the market functions as follows. The quantity currently pr