Robust control system design by mapping specifications ...Robust Control System Design by Mapping...

Robust Control System Design

by Mapping Specifications

into Parameter Spaces

Vom Promotionsausschuss

der

Fakultat fur Elektrotechnik und Informationstechnik

der

Ruhr-Universitat Bochum

zur

Erlangung des akademischen Grades

Doktor-Ingenieur

genehmigte

D I S S E R T A T I O N

von

Michael Ludwig Muhler

aus Creglingen

Bochum, 2007

iii

Dissertation eingereicht am: 20. Oktober 2006

Tag der mundlichen Prufung: 20. Marz 2007

1. Berichter: Prof. Dr.-Ing. Jan Lunze

2. Berichter: Prof. Dr.-Ing. Jurgen Ackermann

iv

Fur meine Eltern, Ute und Maria

v

Acknowledgments

I am indebted to those who have given me the opportunity, support, and time to write

this doctoral thesis.

It is a pleasure to thank my advisor Professor Jurgen Ackermann for his encouragement

and advice during my studies. He has always been ready to give his time generously to

discuss ideas and approaches, while giving me the freedom to choose the direction of my

work. His insights and enthusiasm will have a long-lasting effect on me.

I would also like to thank my supervisor Professor Jan Lunze at the Ruhr-University

Bochum for his interest in my work. His support and willingness to work with me across

the miles and the years is greatly appreciated.

I am greatly indebted to my former office-mate, Paul Blue, for creating a friendly and

stimulating work atmosphere, and for many discussions, and also to my other colleagues

at DLR Oberpfaffenhofen, especially, Dr. Tilman Bunte, Dr. Dirk Odenthal, Dr. Dieter

Kaesbauer, Dr. Naim Bajcinca and Gertjan Looye.

Special thanks to Professor Bob Barmish, who initially encouraged me to pursue post-

graduate studies.

I would like to express my gratitude to Airbus for financial support during a three year

grant. Thanks to Dr. Michael Kordt, my contact at Airbus in Hamburg. Financial aid

from the DAAD for the conference presentations of parts of this thesis is gratefully ac-

knowledged.

The final write-up of this thesis would not have been possible without the support of my

supervisor at Robert Bosch GmbH, Dr. Hans-Martin Streib.

Finally, special thanks to my parents and to my wife Ute for their continuous encourage-

ment, patience and support.

Korntal-Munchingen, March 2007 Michael Muhler

vii

Contents

Nomenclature xi

Abstract xiii

Zusammenfassung xiv

1 Introduction 1

1.1 The Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background and Previous Research . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Goal of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Control Specifications and Uncertainty 6

2.1 Parametric MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 MIMO Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 MIMO Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Symbolic State-Space Descriptions . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Transfer Function to State-Space Algorithm . . . . . . . . . . . . . 10

2.2.2 Minimal Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Uncertainty Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Real Parametric Uncertainties . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Multi-Model Descriptions . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.3 Dynamic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 MIMO Specifications in Control Theory . . . . . . . . . . . . . . . . . . . 22

2.4.1 H∞ Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Passivity and Dissipativity . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.3 Connections between H∞ Norm and Passivity . . . . . . . . . . . . 29

2.4.4 Popov and Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.5 Complex Structured Stability Radius . . . . . . . . . . . . . . . . . 33

2.4.6 H2 Norm Performance . . . . . . . . . . . . . . . . . . . . . . . . . 34

viii

2.4.7 Generalized H2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.8 LQR Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.9 Hankel Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Integral Quadratic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.1 IQCs and Other Specifications . . . . . . . . . . . . . . . . . . . . . 40

2.5.2 Mixed Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.3 Multiple IQCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Mapping Equations 42

3.1 Eigenvalue Mapping Equations . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Algebraic Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.1 Continuous and Analytic Dependence . . . . . . . . . . . . . . . . . 46

3.3 Mapping Specifications into Parameter Space . . . . . . . . . . . . . . . . . 48

3.3.1 ARE Based Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2 H∞ Norm Mapping Equations . . . . . . . . . . . . . . . . . . . . . 51

3.3.3 Passivity Mapping Equations . . . . . . . . . . . . . . . . . . . . . 53

3.3.4 Lyapunov Based Mapping . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.5 Maximal Eigenvalue Based Mapping . . . . . . . . . . . . . . . . . 56

3.4 IQC Parameter Space Mapping . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.1 Uncertain Parameter Systems . . . . . . . . . . . . . . . . . . . . . 57

3.4.2 Kalman-Yakubovich-Popov Lemma . . . . . . . . . . . . . . . . . . 57

3.4.3 IQC Mapping Equations . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.4 Frequency-Dependent Multipliers . . . . . . . . . . . . . . . . . . . 60

3.4.5 LMI Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5.1 ARE Mapping Equations . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5.2 Lyapunov Mapping Equations . . . . . . . . . . . . . . . . . . . . . 68

3.5.3 IQC Mapping Equations . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Further Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.7 Comparison and Alternative Derivations . . . . . . . . . . . . . . . . . . . 70

3.8 Direct Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Algorithms and Visualization 73

4.1 Aspects of Symbolic Computations . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.1 Asymptotes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.2 Parametrization of Curves . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.3 Topology of Real Algebraic Curves . . . . . . . . . . . . . . . . . . 78

4.3 Algorithm for Plane Algebraic Curves . . . . . . . . . . . . . . . . . . . . . 80

ix

4.3.1 Extended Topological Graph . . . . . . . . . . . . . . . . . . . . . . 81

4.3.2 Bezier Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Path Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4.1 Common Problems of Path Following . . . . . . . . . . . . . . . . . 84

4.4.2 Homotopy Based Algorithm . . . . . . . . . . . . . . . . . . . . . . 84

4.4.3 Predictor-Corrector Continuation . . . . . . . . . . . . . . . . . . . 86

4.5 Surface Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.6 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6.1 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.7 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7.1 Color Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7.2 Visualization for Multiple Representatives . . . . . . . . . . . . . . 91

5 Examples 93

5.1 MIMO Design Using SISO Methods . . . . . . . . . . . . . . . . . . . . . . 93

5.2 MIMO Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.1 H2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.2 H∞ Norm: Robust Stability . . . . . . . . . . . . . . . . . . . . . . 98

5.2.3 Passivity Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3 Example: Track-Guided Bus . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.1 Design Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.2 Robust Design for Extreme Operating Conditions . . . . . . . . . . 105

5.3.3 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4 IQC Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5 Four Tank MIMO Example . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Summary and Outlook 113

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

A Mathematics 115

A.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.2 Algebraic Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 116

References 121

xi

Nomenclature

Acronyms

ARE algebraic Riccati equation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22

CRB complex root boundary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 44

IQC integral quadratic constraint, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5

IRB infinite root boundary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 44

KYP Kalman-Yakubovich-Popov, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 57

LFR linear fractional representation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

LHP left half plane, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 43

LMI linear matrix inequality, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 28

LQG linear quadratic Gaussian, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 36

LQR linear quadratic regulator, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 36

LTI linear time-invariant, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 6

MFD matrix fraction description, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

MIMO multi-input multi-output, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 6

PSA parameter space approach, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 2

RHP right half plane, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 8

RRB real root boundary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 44

SISO single-input single-output, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 8

xii

Symbols

∗ conjugate transpose A(s)∗ = A(−s)T , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 24

ζ damping factor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 43

∼= equivalent state-space realization, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 7

:= definition, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22

den denominator, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

diag diagonal matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 11

Im image or range space of a matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 117

Im imaginary part of imaginary number, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 43

⊗ Kronecker product, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 115

⊕ Kronecker sum, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 54

vec column stacking operator, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 54

lcm least common multiple, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

Lm2 [0,∞) space of square summable functions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 39

Re real part of imaginary number, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 24

σ largest singular value, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22

σ singular value, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22

Λ eigenvalue spectrum, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 33

trace trace of a matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 34

Variables

G(s) general transfer matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 7

q uncertain parameters, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 3

xiii

Abstract

Robust controller design explicitly considers plant uncertainties to determine the con-

troller structure and parameters. Thereby, the given specifications for the control system

are fulfilled even under perturbations and disturbances.

The parameter space approach is an established methodology for systems with uncertain

physical parameters. Control specifications, for example formulated as eigenvalue criteria,

are hereby mapped into a parameter space. The graphical presentation of admissible

parameter regions leads to easily interpretable results and allows intuitive parametrization

and analysis of robust controllers.

The goal of this thesis is to extend the parameter space approach by new specifications

and to broaden the applicable system class. A uniform concept for mapping specifications

into parameter spaces is presented for this purpose. This enables the generalized deriva-

tion of mapping equations and the identical software implementation for the mapping.

Moreover, it allows to extend the parameter space approach by additional specifications

which can be mapped. Furthermore, the applicable system class can be broadened. All

relevant specifications for linear multivariable systems including the H2 and H∞ norm are

covered by this approach. Beyond that, specifications for nonlinear systems can be used

in conjunction with the parameter space approach. In particular, the mapping of integral

quadratic constraints is introduced.

A brief outline of specifications for multivariable systems introduces into the parameter

space mapping. All specifications are established using a similar mathematical description

that forms the basis for the generalized mapping equations. The mapping equations are

then obtained by converting the generalized algebraic specification description into a

specialized eigenvalue problem.

A symbolic-numerical algorithm is developed to realize the specification mapping. Various

graphical means to visualize the results in a parameter plane are explored. This is moti-

vated by specifications which yield a performance index. Various examples demonstrate

the extension of the parameter space approach and the new possibilities of the concept.

xiv

Zusammenfassung

Beim Entwurf von robusten Reglern werden Unsicherheiten der Regelstrecke explizit

berucksichtigt, um die Struktur und Parametrierung des Reglers so festzulegen, daß die

gestellten Anforderungen an das regelungstechnische System trotz auftretender Storungen

und Streckenveranderungen erfullt werden. Hierzu steht mit dem Parameterraumver-

fahren eine anerkannte Methodik fur Systeme mit unsicheren physikalischen Parame-

tern zur Verfugung. Hierbei werden regelungstechnische Spezifikationen, die zum Beispiel

als Eigenwertkriterien formuliert sind, in einen Parameterraum abgebildet. Die grafische

Darstellung von zulassigen Gebieten in einer Parameterebene fuhrt zu einfach interpretier-

baren Resultaten und ermoglicht die intuitive Parametrierung und Analyse von robusten

Reglern.

Ziel der Arbeit ist die Erweiterung des Parameterraumverfahrens um Spezifikationen sowie

die Vergroßerung der anwendbaren Systemklasse. Hierzu wird ein einheitliches Konzept

zur Abbildung von Spezifikationen in Parameterraume vorgestellt. Dieses erlaubt die ve-

rallgemeinerte Herleitung von Abbildungsgleichungen und die identische softwaretech-

nische Realisierung der Abbildung. Neben allen relevanten Spezifikationen fur lineare

Mehrgroßensysteme, wie die H2 und H∞ Norm, erlaubt das vorgestellte Konzept die

Anwendung des Parameterraumverfahrens auf nichtlineare Systeme. Insbesondere wird

die Abbildung von integral-quadratischen Bedingungen aufgezeigt.

Ein kurzer Abriß der Spezifikationen fur Mehrgroßensysteme fuhrt in die Abbildung in den

Parameterraum ein. Alle Spezifikationen werden in einer gleichartigen mathematischen

Formulierung dargestellt, die die Basis fur die verallgemeinerten Abbildungsgleichungen

bildet. Die Abbildungsgleichungen beruhen auf der Uberfuhrung der allgemeinen alge-

braischen Darstellung fur die Spezifikationen in ein spezielles Eigenwertproblem.

Um die Anwendung des hier vorgestellten Konzeptes zu ermoglichen, wird ein symbolisch-

numerischer Algorithmus zur Durchfuhrung der Abbildung von Spezifikationen entwick-

elt. Verschiedene Moglichkeiten zur grafischen Darstellung der Resultate in einer Pa-

rameterebene werden vorgestellt, insbesondere fur Spezifikationen die Gutewerte liefern.

Mehrere Beispiele stellen die Erweiterung des Parameterraumverfahrens und die neuen

Moglichkeiten des Konzeptes dar.

1

1 Introduction

1.1 The Control Problem

Why should we use feedback at all? The pure dynamics of a stable plant can be simply

modified to the desired dynamics using feedforward control.

In the real world every plant is subject to external disturbances. If we want to alter the

systems response to disturbances or signal uncertainty we have to use feedback.

Another fundamental reason for feedback control arises from instability. An unstable

plant cannot be stabilized by any feedforward system. Feedback control is mandatory for

these plants, even without signal and model uncertainty.

The third fundamental reason for using feedback control is the just mentioned model un-

certainty. The term model uncertainty here includes discrepancy between the true system

and the model used to design the controller. Reasons for deviations are model imperfec-

tions. For example, the modeling of an electric wire as a resistor is known to be perfect up

to a certain frequency range. More elaborate models derived from first principles might

include a resistor-capacitor chain. But even this model is only valid in a certain frequency

range because eventually the encountered physical phenomena reach atomic scale. Thus

model uncertainty is not just present in models obtained from measurements and iden-

tification. Every model, even a model derived from physical modeling, is only valid to a

certain extent.

Further model uncertainties can be design imposed, such as limitations on the complexity

of the design model or certain model types, for example linear models.

Classical control aims at stabilizing a system in the presence of signal uncertainty. Ro-

bust control extends this goal by designing control systems that not only tolerate model

uncertainties, but also retain system performance under plant variations.

While the goal that a feedback control system should maintain overall system performance

despite changes in the plant has been around since the early days of control theory, this

property is nowadays explicitly called robustness.

2 Introduction

1.2 Background and Previous Research

As a reaction to the poor robustness of controllers based on optimization and estimation

theory the field of robust control theory emerged, where plant variations play a key role.

Several different approaches to deal with plant variations mainly influenced by the uncer-

tainty characterization have evolved [Ackermann 1980, Doyle 1982, Lunze 1988, Safonov

1982]. Central topics in robust control theory common to all approaches are

• Uncertainty characterization.

• Robustness analysis.

• Robust controller synthesis.

For systems with real parametric uncertainty, e.g., an unknown or varying system param-

eter, the parameter space approach (PSA)1 is a well established method for robustness

analysis and robust controller design [Ackermann et al. 2002]. The basic idea of the pa-

rameter space approach is to map a condition of specification for a system into a plane of

parameters, i.e., the set of all parameters for which the specification holds is determined.

Initially the PSA considered eigenvalue specifications for linear systems. Its roots can be

traced back to the 19th century, where mathematicians such as [Hermite 1856, Maxwell

1866, Routh 1877], inspired by the first mechanical control systems, studied the basic

question related to stability of whether a given polynomial

p(s) = ansn + . . . + a1s + a0 = 0, (1.1)

has only roots with negative real parts. Interestingly these early accounts of stability ana-

lysis tried to find a solution that can be expressed using the coefficients of the polynomial,

thereby avoiding the explicit computation of roots.

Vishnegradsky was the first to visualize the stability condition in a coefficient parameter

plane in [1876], analyzing the stability of a third order polynomial p(s) = s3+a2s2+a1s+1

with respect to varying a1 and a2. This idea became the building block of the parameter

space approach.

Based on Hermite’s work, [Hurwitz 1895] reported an algebraic condition in terms of

determinants. This stability condition has been extensively used in control theory and

extended to robustness analysis.

Initially the parameter space method considered stability of a linear system described by

the characteristic equation. By mapping the stability condition it originally allowed to

analyze robustness with respect to two specific coefficients.

1Sometimes also referred to as parameter plane method

1.2 Background and Previous Research 3

The parameter space method was then extended to robust root clustering or Γ-stability,

by specifying an eigenvalue region [Ackermann et al. 1991, Mitrovic 1958]. This allows to

indirectly incorporate time-domain specifications and thereby robust performance.

The coefficients ai of (1.1) do not directly relate to plant or controller parameters, and

therefore hamper the application to control problems. Therefore robust control theory

considered polynomials with coefficients that depend on a parameter vector q ∈ Rp:

p(s, q) = an(q)sn + . . . + a1(q)s + a0(q) = 0. (1.2)

If q consists of only one parameter q, then robust stability can be evaluated by plotting

a generalized root locus [Evans 1948], where q takes the role of the usual linear gain k.

Robust, multimodal eigenvalue based parameter space approach is state-of-the-art. The

underlying theory is thoroughly understood for linear time-invariant systems with uncer-

tain parameters.

In general, the parameter space approach maps a given specification, e.g. a permissible

eigenvalue region, into a space of uncertain parameters q ∈ Rp, see Figure 1.1. Usually

the specification is mapped into a parameter plane because this leads to understandable

and powerful graphical results. Moreover, since we can map several specifications con-

secutively, this approach actually allows multiobjective analysis and synthesis of control

systems.

Recently, this approach was extended to the frequency domain for Bode specifications

[Besson and Shenton 1997, Hara et al. 1991, Odenthal and Blue 2000] and Nyquist dia-

grams [Bunte 2000]. Static nonlinearities were considered in the parameter space approach

in [Ackermann and Bunte 1999]. Finally [Muhler 2002] derived mapping equations for

multi-input multi-output systems, including H2,H∞ norms and passivity specifications.

During the 1990s there has been considerable interest in design methods such as H∞

and µ-analysis that require only control specifications and yield a controller including

the structure and parameters. While this seems to be attractive from the point that the

design engineer has not to waste time on thinking of a reasonable control structure and

possibly try several different structures. All of these design methods have the disadvantage

that they lead to very high controller orders. The direct order reduction of the resulting

controllers is a nontrivial task, and often destroys some of the required or desired features

of the initial high-order controllers.

Using the parameter space approach as a design tool we have to specify a controller

structure, e.g., a PID controller, and the parameters of the controller are iteratively tuned

until all design specifications are fulfilled. Thus the parameter space approach falls into

the category of fixed control structure design methods. Other approaches are given by

classical design methods or parameter optimization [Joos et al. 1999].

4 Introduction

σ q1

jω q2

p(s, q)

Figure 1.1: Mapping stability condition into parameter plane

The clear advantage of fixed structure methods is that the control engineer has full control

about the resulting complexity of the control system. This allows to handle implementa-

tion issues directly during the design process.

We will not consider special feedback structures in this thesis. This approach is backed

by the fact that all two degree of freedom configurations have basically the same prop-

erties and potentials, although some structures are especially suitable for some design

algorithms.

1.3 Goal of the Thesis

The main objective of this thesis is to extend the parameter space approach by new

specifications and to broaden the applicable system class, e.g., multivariable or nonlinear.

The basic idea of the parameter space approach (PSA) is to map control specifications

for a given system into the space of defined varying parameters. The boundaries of

parameter sets that fulfill the specifications are hereto determined. Usually we consider

two parameters at a time and control specifications are mapped into a parameter plane.

This allows intuitive interpretation of the graphical results.

By mapping we actually mean the identification of parameter regions (or subspaces) for

which the specifications are fulfilled. In other words we are interested in the set of all

parameters Pgood that fulfill a given specification. The boundary of this good set is

characterized by the equality case of the specification. Mathematically this good set is

given by a mapping equation.

Mapping equations form the mathematical core of the PSA. They combine the control

specific system description with specifications that a control design requires to hold for

the system.

1.4 Outline 5

This thesis presents a unified approach to consider various control specifications for multi-

variable systems in the parameter space approach. It is shown how various specifications

can be formulated using the same mathematical framework.

Since there is no straightforward way to solve the resulting mapping equations a second,

but not less important goal is to find and explore computational methods to solve the

mapping problem.

The results in this thesis can be transferred and applied to time-discrete systems. The

required methodology can be directly taken from [Ackermann et al. 2002]. Hence we do

not extensively cover the application of the time-continuous results in this thesis to the

time-discrete case.

1.4 Outline

Chapter 2 serves multiple purposes. We start with some control theoretic background.

Subsequently the various specifications are presented. Besides the introduction and some

information, the focus lies on a uniform mathematical description of the criteria. This

allows uniform treatment and development of mapping equations and finally mapping

algorithms.

Chapter 3 then presents the mapping equations used to map the specifications into pa-

rameter space. Beyond the mapping equations for specific specifications introduced in

Section 2.4, we consider mapping equations for general integral quadratic constraint (IQC)

specifications [Muhler and Ackermann 2004].

The remaining part of the thesis deals with the practical application of the presented con-

trol theory to practical problems. To this end, we take a closer look at algorithms suitable

for the mapping equations arising from the various control specifications in Chapter 4.

And we explore graphical means to visualize the results in a parameter plane. This is mo-

tivated by specifications that can be related to performance. Here not just the fulfillment

of a condition, for example stability, is crucial, but we are interested in optimizing the

attainable performance level. Therefore contour-like plots with color-coded performance

levels in a parameter plane reveal additional insight.

The application of the derived mapping equations and the mapping algorithms is demon-

strated on various examples in Chapter 5. Concluding remarks and perspectives for

further work are given in Chapter 6.

For the convenience of the reader, we summarize some mathematical background material

and elaborate proofs in Appendix A.

6 Control Specifications and Uncertainty


This chapter introduces the system class considered in this thesis, namely multivariable

parametric systems. We are mainly concerned with linear time-invariant (LTI) systems

throughout the thesis. Nevertheless, some results for nonlinear systems are given that fit

into the used framework.

After presenting the general multi-input multi-output (MIMO) model, we give a brief

overview of important properties of MIMO systems and their limitations. Section 2.3

considers uncertainty structures used to model control systems.

The main part of this chapter is Section 2.4, which presents various control system spec-

ifications used for MIMO systems. This section presents some arguments why it might

be useful to extend the classical eigenvalue based parameter space approach by MIMO

specifications mainly derived from the frequency domain. The main goal of Section 2.4 is

to present all specifications in a way so that they fit into the same mathematical frame-

work. This formulation makes it possible to derive mapping equations in Chapter 3 and

incorporate them into the parameter space approach.

2.1 Parametric MIMO Systems

There has been an enormous interest in the design of multivariable control systems in

the last decades, especially frequency domain approaches [Doyle and Stein 1981],[Francis

1987],[Maciejowski 1989]. We do not intend to give a comprehensive treatment of all

the aspects of multivariable feedback design, and refer the reader to the cited literature.

Thus, the scope of this section is limited to the presentation of the basic concepts and

some examples.

We consider uncertain, LTI systems with parametric state-space realization

x(t) = A(q)x(t) + B(q)u(t), y(t) = C(q)x(t) + D(q)u(t) (2.1)

or transfer matrix representation G(s, q), i.e.,

y(s) = G(s, q)u(s) = (C(q)(sI − A(q))−1B(q) + D(q))u(s), (2.2)

where u ∈ Rm and y ∈ Rp are vectors of signals.

2.1 Parametric MIMO Systems 7

The short-hand notation

G(s, q) ∼=

A(q) B(q)

C(q) D(q)

(2.3)

will be used to represent a state-space realization

x(t)

y(t)

=

A(q) B(q)

C(q) D(q)

x(t)

u(t)

,

for a given transfer matrix G(s, q).

The parameters q ∈ Rp are unknown but constant with known bounds. The set of all

possible parameters is denoted as Q. If not stated otherwise, we assume upper and lower

bounds qi = [q−i ; q+i ] for each dimension and the operating domain q ∈ Q is also referred

to as the Q-box (see Figure 2.1). Since the parameter space approach does not favor

controller over plant uncertainties, we will not discriminate these in general equations.

Thus, usually q is used for both controller and plant uncertainties. If controller parameters

are explicitly mentioned they are also denoted by ki.

The mapping plane can be a plane of uncertain parameters for robustness analysis, or

a plane of controller parameters in a control design step. Also a mix of both parameter

types is useful for the design of gain scheduling controllers.

q1q−

1 q+

1

q2

q−

2

q+

2

Q

Figure 2.1: Box-like operating domain Q

We will use the symbol G(s) for general transfer matrices arising in the considered control

problem. A specific plant will be denoted P (s), and transfer matrices for controllers are

denoted K(s). Thus, G(s) includes arbitrary transfer matrices, general plant descriptions

including performance criteria, or even open or closed loop transfer matrices. We use the

standard notation for specific transfer matrices such as the sensitivity function S(s) and

the complementary sensitivity function T (s).


2.1.1 MIMO Specifications

The main objective of the parameter space approach is to map specifications relevant

for dynamic systems (2.1) and (2.2) into the parameter space or into a parameter plane.

Apart from stability, the most important objective of a control system is to achieve certain

performance specifications. One way to describe these performance specifications is to use

the size of certain signals of interest. For example, the performance of a regulator could

be measured by the size of the error between the reference and measured signals. The size

of signals can be defined mathematically using norms. Common norms are the Euclidean

vector norms

||x||1 :=n∑

i=1

|xi|, ||x||2 :=

√√√√

n∑

i=1

|xi|2, ||x||∞ := max1≤i≤n

|xi|.

The performance of a control system with input and output signals measured by one

of the above norms, not necessarily the same, can be evaluated by the induced matrix

norms. The most prominent matrix norms used in control theory are the H∞ and H2

norms, which will be considered in Section 2.4.1 and Section 2.4.6, respectively.

2.1.2 MIMO Properties

MIMO systems exhibit some properties not known for single-input single-output (SISO)

systems. These differences make it difficult to apply standard SISO design guidelines to

MIMO systems, e.g., for eigenvalues or loop shapes. At least care has to be taken when

simply using these rules.

While for a SISO system the behavior can be characterized by the gain and phase for

a single channel, these entities depend on the direction of the input for MIMO systems.

The same applies to eigenvalues. While eigenvalues can be used to describe the behav-

ior of a SISO system effectively, for MIMO systems the directionality of the associated

eigenvectors becomes important. This can be also seen from a design point of view. The

eigenvalues of a controllable system with available state information can be moved to

any desired location using Ackermann’s formula [Ackermann 1972]. For multivariable

systems there are additional degrees of freedom that can be used to shape the closed-loop

eigenvectors or other design specifications.

It is a well known fact that right half plane (RHP) zeros impose fundamental limitations

on control of SISO systems. While these zeros can be found by inspection of the numerator

of the transfer function of a SISO system, this does not hold for MIMO systems. Although

all elements of a transfer matrix G(s) are minimum-phase, RHP zeros may exist for the

overall multivariable system.

2.2 Symbolic State-Space Descriptions 9

The role of RHP zeros is further emphasized by some design methods, e.g., successive

loop closure, where zeros can arise during intermediate steps. Nevertheless sometimes we

can take advantage of the additional degree of freedom found in MIMO systems to move

most of the deteriorating effect of an RHP zero to a particular output channel.

2.2 Symbolic State-Space Descriptions

All methods and algorithms presented in this thesis require a symbolic state-space de-

scription. In particular the mapping equations for control system specifications presented

in Chapter 3 are based on a parametric, linear state-space description of the considered

system as in (2.1). The purpose of this section is to present an algorithm that calculates

a symbolic state-space description from a given symbolic transfer function, because this

is essential for the methods developed in this thesis.

Such a system description can be obtained by first-principle modeling such as Lagrange

functions or balance equations, where it might be necessary to symbolically linearize

the equations. Note that this linearization preserves the parametric dependency on the

uncertain parameters q. The references Otter [1999], Tiller [2001] give a good introduction

to object oriented modeling where the used software symbolically transforms and modifies

the system description.

While a parametric transfer matrix is easily obtained from a state-space description by

evaluating the symbolic expression G(s, q) = C(q)(sI−A(q))−1B(q)+D(q), the opposite

is much more involved.

For SISO systems or systems with either a single input or output, a canonical form

provides a minimal realization. Particular variants are the controllable and observable

canonical form [Chen 1984, Kailath 1980]. These canonical forms can be easily obtained

in symbolic form from the coefficients of a transfer function.

Consider a multivariable transfer matrix G(s). One way to obtain a state-space description

is to form canonical forms of all transfer functions gij(s) and combine them to get a model

with input-output relation equivalent to the considered transfer matrix G(s). This model

will be nonminimal, i.e., it contains spurious states, which are non-controllable or non-

observable, or both.

For MIMO systems the dimension of a minimal realization is exactly the McMillan de-

gree [Chen 1984]. There exist standard methods to determine a minimal realization for

a numerical state-space model. Unfortunately these algorithms are not transferable to

symbolic transfer matrix descriptions.


2.2.1 Transfer Function to State-Space Algorithm

Besides system representations in state-space and transfer function form, a matrix frac-

tion description (MFD) is another useful way of representing a system. Actually these

models are the keystone of all linear fractional representation (LFR) based robust control

methods, where the idea is to isolate the uncertainty from the system inside a single block.

The aim here is to present a symbolic algorithm. It will be formulated using right coprime

factorization. A dual left coprime version is possible, but does not provide any advantages

over the presented one.

Any transfer matrix G(s) can be written as a right or left matrix fraction of two polynomial

matrices,

G(s) = Nr(s)Mr(s)−1, (2.4a)

G(s) = Ml(s)−1Nl(s). (2.4b)

The numerator matrices Nr(s) and Nl(s) have the same dimension as the transfer ma-

trix G(s), whereas the denominator matrices Mr(s) and Ml(s) are square matrices of

matching dimension. Special variants are coprime factorizations, which will be discussed

later in Section 2.2.2.

A right (left) MFD for a given transfer matrix G(s) ∈ Rl,m is easily obtained as follows.

Determine the polynomial denominator matrix M(s) as a diagonal matrix, where the

entries mii are the least common multiple of all denominator polynomials in the i-th

column (row) of G(s), i.e.,

mii = lcm(den g1i, den g2i, . . . , den gli), i = 1, . . . , m, (2.5a)

and for left MFDs we use

mii = lcm(den gi1, den gi2, . . . , den gil), i = 1, . . . , l. (2.5b)

The fraction-free numerator matrices Nr(s) and Nl(s) are then determined by simply

evaluating

Nr(s) = G(s)Mr(s), (2.6a)

Nl(s) = Ml(s)G(s). (2.6b)

Having found a column-reduced MFD a state-space realization can be determined using

the so-called controller-form [Kailath 1980]. The algorithm presented here will work for

proper and strictly proper systems.


Given a right MFD G(s) = Nr(s)Mr(s)−1, the input-output relation y(s) = G(s)u(s) can

be rewritten as

Mr(s)ξ(s) = u(s), (2.7a)

y(s) = Nr(s)ξ(s), (2.7b)

where ξ(s) is the so-called partial state.

The polynomial matrices Nr(s) and Mr(s) are now decomposed as

Mr(s) = MhcS(s) + MlcΨ(s), (2.8a)

Nr(s) = NlcΨ(s) + NftMr(s). (2.8b)

The decomposition matrices are computed as follows. Let the highest degree of all poly-

nomials in the i-th column of Mr(s) be denoted as ki. The matrix S(s) is diagonal

with S(s) = diag[sk1 , . . . , skm]. Then the matrix Mhc is the highest-column-degree coeffi-

cient matrix of Mr(s).

The term MlcΨ(s) contains the lower-column-degree terms of Mr(s), where Mlc is a coef-

ficient matrix and Ψ(s) a block diagonal matrix:

Ψ(s)T =

1 s · · · sk1−1 0 · · · 0

0 1 s · · · sk2−1 · · · 0

0 0 · · · 0

0 0 · · · 1 s · · · skm−1

.

Output equation (2.8b) is obtained by first computing the direct feedthrough matrix as

Nft = lims→∞

G(s) = lims→∞

Nr(s)Mr(s)−1.

The remaining task is to compute the trailing coefficient matrix Nlc by columnwise coef-

ficient evaluation of NlcΨ(s) = Nr(s) − NftMr(s) for orders of s up to degree ki for the

i-th column.

Having found the decomposition (2.8), a state-space description is now easily obtained by

assembling m integrator chains with ki integrators in the i-th chain. The total order nt

of the system will be given by

nt =

m∑

i=1

ki.


A basic state-space realization of S(s) is given by

A0 = block diag[Ak1 , . . . , Akm], (2.9a)

B0 = block diag[Bk1 , . . . , Bkm], (2.9b)

C0 = Int. (2.9c)

where An is an n × n Jordan block matrix with corresponding input matrix Bn,

An =

0 1

· · ·· · ·0 1

0

, An ∈ Rn,n,

BTn =

[

0 · · · 0 1]

, Bn ∈ Rn, 1.

The correct input-output behavior is then achieved by closing a feedback loop around the

core integrator chains. The final state-space realization is given by

A = A0 − B0M−1hc Mlc, (2.10a)

B = B0M−1hc , (2.10b)

C = Nlc, (2.10c)

D = Nft. (2.10d)

2.2.2 Minimal Realization

A state-space realization is minimal, if it is controllable and observable, and thus contains

no subsystems that are not controllable or observable, or both.

State-space representations are in general not unique. Nevertheless, minimal state-space

realizations are unique up to a change of the state-space basis. More important, the

number of states is constant and minimal. This minimality is especially important for the

symbolic parameter space approach methods presented in Chapter 3, since they minimize

the computational burden in handling and solving the symbolic equations.

Since the 1960s the minimal realization problem has attracted a lot of attention and

a wide variety of algorithms have emerged, e.g., Gilbert’s approach based on partial-

fraction expansions [Gilbert 1963] or Kalman’s method, which is based on controllability

and observability and reduces a nonminimal realization until it is minimal [Kalman 1963].

Note that an input-output description reveals only the controllable and observable part

of a dynamical system.


Rosenbrock [Rosenbrock 1970] developed an algorithm, which uses similarity transforma-

tions (elementary row or column operations), to extract the controllable and observable,

and therefore minimal subpart of a state-space realization. Variants of this algorithm are

now implemented in Matlab and Slicot.

We will use a similar approach based on MFDs, which directly fits into the results pre-

sented in Section 2.2.1. Consider a right MFD,

G(s) = Nr(s)Mr(s)−1,

with polynomial matrices Nr(s) and Mr(s). We now determine the greatest common right

divisor Rgcd(s) of Nr(s) and Mr(s), such that

Nr(s) = Nr(s)Rgcd(s)−1, (2.11)

Mr(s) = Mr(s)Rgcd(s)−1, (2.12)

and we obtain the right coprime factorization

G(s) = Nr(s)Mr(s)−1. (2.13)

The greatest common right divisor of two polynomial matrices can be found by consec-

utive row operations, or left-multiplication with unimodular matrices, until the stacked

matrix [Mr Nr]T is column reduced. Since all steps in finding Rgcd are either multiplica-

tion or addition of polynomials the algorithm is fraction free and can be easily applied

to parametric matrices. Note that we are not interested in special coprime factorization,

e.g., stable factorization over RH∞. So we can symbolically compute Rgcd(s), e.g., using

Maple.

2.2.3 Example

The above algorithm is illustrated by a small MIMO transfer matrix. Consider the follow-

ing plant [Doyle 1986], which is an approximate model of a symmetric spinning body with

constant angular velocity for a principal axis, and two torque inputs for the remaining

two axes,

G(s) =1

s2 + a2

s − a2 a(s + 1)

−a(s + 1) s − a2

. (2.14)

For this example a right MFD is easily obtained by inspection or using (2.5a) and (2.6a),

Mr(s) =

s2 + a2

s2 + a2

, Nr(s) =

s − a2 a(s + 1)

−a(s + 1) s − a2

. (2.15)


It is obvious that if by simply following the algorithm given in Section 2.2.1, we would

end up with a state-space description of order four. To minimize the order, we use the

minimal realization procedure of Section 2.2.2.

It turns out that Nr(s) is actually a greatest common right divisor of Mr(s) and Nr(s),

such that Rgcd(s) = Nr(s), and using (2.11), we immediately determine a right coprime

factorization with

Mr(s) =−1

1 + a2Nr(s), Nr(s) = I. (2.16)

We can then proceed with the decomposition (2.8) and using (2.9) and (2.10) we finally

obtain a minimal state-space realization of order two,

G(s) ∼=

0 a

−a 0

1 a

−a 1

1 0

0 1

0 0

0 0

.

2.3 Uncertainty Structures

The main aim of control is to cope with uncertainty.

System models should express this uncertainty,

but a very precise model of uncertainty is an oxymoron.

George Zames

This section contains material on the description of uncertainty for models used in control

system design. While in general the material can be applied to models from different

application areas, e.g., chemical reactors or power plants, the exposition is especially

suited for models of mechanical systems arising in vehicle dynamics.

Control engineers, unless only experiments are used, are working with mathematical mod-

els of the system to be controlled. These models should describe all system responses vital

for the considered system performance. That is, other system properties are neglectable.

Apart from matching the responses of the true plant as close as possible, models should

be simple enough to facilitate design.

By the term uncertainty we will summarize

1. all known variations of the model,

2. differences or errors between models and reality.

2.3 Uncertainty Structures 15

The first type of uncertainty refers to variations of the model due to parameter changes.

For parametric LTI systems a mathematical description is given in (2.1) and (2.2). As

mentioned earlier these system descriptions are easily obtained from first principle model-

ing.

The parameters q are unknown but constant. That is, we assume that the parameters

do not change during a regular operation of the system; or the change is so slow that

they can be treated as quasi-stationary. For parameters which change rapidly or whose

rate of change lies well within the system dynamics special care has to be taken. In this

case the results obtained by treating the parameters as constant or quasi-constant might

be false or misleading. Representing the changing parameters through an unstructured

uncertainty description might be a remedy. Another approach is to utilize the mapping

equations derived in Section 3.4. The stability of a system with an arbitrary fast varying

parameter is analyzed in Example 3.1.

The uncertain parameters might be states of a full-scale model describing the plant be-

havior with respect to input and output variables. For example, the mass of an airplane

can be considered as a fixed parameter for directional control while it is actually a state

of a model and decreasing with fuel consumption used for full flight evaluation.

From a frequency point of view the system knowledge generically decreases with frequency.

While there are system models which are accurate in a specific frequency range, e.g., for

flexible mechanical systems, at sufficiently high frequencies precise modeling is impossible.

This is a consequence of dynamic properties which occur in physical systems.

We propose the following modeling philosophy:

Use the physical knowledge about the plant to include parametric uncer-

tainties as real perturbations with known bounds. Additional uncertainties are

modeled by (unstructured) dynamic uncertainties. Any information, e.g., di-

rectionality, although not expressible as parametric uncertainties about these

uncertainties should be incorporated in analysis and design.

Thus the overall model which includes an uncertainty description is denoted as

G = G(q, ∆), (2.17)

where ∆ represents unstructured uncertainty considered in detail in Section 2.3.3. The

term unstructured is not to be interpreted literally. The uncertainty description might

actually contain some degree of structure. But here it is used as uncertainty which

cannot be described as real parameter variation. The uncertainty description used here

is an extension of pure parametric models used in the classical parameter space approach

to include unstructured uncertainty.


The frequency domain approaches to robust control, i.e., the popular H∞ and µ control

paradigms, use a different representation of uncertainty. For H∞ robust controllers all

uncertainties have to be captured by a norm-bounded uncertainty description. The model

is described by a nominal plant and H∞ norm-bounded stable perturbations. Thus even

parametric uncertainty, i.e., a detailed parametric model, has to be approximated by a

norm-bounded uncertainty description. Furthermore all uncertainty has to be lumped in

a single uncertainty transfer function or uncertainty block, see Figure 2.2.

∆

G z

u v

w

Figure 2.2: General lumped uncertainty description

While the structured singular value or µ approach tries to alleviate problems associ-

ated with the unstructured uncertainty description by using structured and unstructured

uncertainties, the single uncertainty block ∆ remains. Parametric uncertainties can be

rendered in this single block uncertainty, although some conservativeness is associated

with this approach. See Section 2.3.1 for some comments regarding the transformation of

parametric models into the single block form of Figure 2.2.

Robustness of a control system is not only affected by the plant uncertainty. There are

many other aspects which have to be taken care of, when it comes to successful and robust

operation of a control system. This includes failure of sensor and actuators, fragility of

the implemented controller, physical constraints. Furthermore the opening and closing

of individual loops for a multivariable system can be crucial, especially during manual

start-up or tuning. Nevertheless, robustness refers to robustness with respect to model

uncertainty in this thesis. We will usually try to find a fixed linear controller that robustly

satisfies all design specifications. Apart from robust stability, for real control systems some

performance specification has to be achieved robustly.

2.3.1 Real Parametric Uncertainties

Real parametric uncertainties are lumped into a single vector q ∈ Rp. The general

models for parametric LTI systems were given in (2.1) and (2.2). Since the concept

of real parametric uncertainties is pretty straightforward we will consider some special


variants and the transformation of a parametric model into a model where all parametric

uncertainty is lumped into a single block.

State perturbations

State perturbations are important in the analysis of robust stability. The most general

state perturbation model is given by the following state-space description

x(t) = (A + Aq(q)) x(t), (2.18)

where Aq(q) is a matrix whose entries are polynomial fractions and Aq(0) = 0 , i.e., A is

the nominal state matrix for q = 0 . Usually only pure polynomial matrices are considered,

since a fractional matrix can be transformed into a polynomial matrix by multiplication

with the least common multiple of all denominators.

Of special interest are representations where all perturbations are combined in a single

block, such as in the general lumped uncertainty description of Figure 2.2. Actually, for

polynomial state perturbations we can find a representation of the form

x(t) =(A + U∆(I − W∆)−1V

)x(t), (2.19)

where ∆ is a diagonal matrix of the form ∆ = diag[q1Im1 , q2Im2 , . . . , qpImp]. The inte-

gers mi are the multiplicities with which the i-th parameter appears in ∆. Figure 2.3

shows a block diagram for state perturbation (2.19).

∆v

Vu

U

W

∫

A

x

Figure 2.3: State perturbation block diagram

If no product terms of parameters are present in (2.18) we can write the system as

x(t) =

(

A +

p∑

i=1

Ai(qi)

)

x(t), (2.20)

where Ai(qi) is the perturbation matrix depending on the parameter qi.


Affine state perturbations

A further specialization is obtained, if the perturbations are affine in the unknown pa-

rameters, i.e.,

Ai(qi) = Aiqi, i = 1, . . . , p.

Real structured perturbations

An even more special variant of affine state perturbations are the so-called real structured

perturbations.

Thus (2.20) can be written as

x(t) = (A + U∆V ) x(t), (2.21)

where ∆ is a diagonal matrix with real uncertain parameters, ∆ = diag[q1, . . . , qp] ∈ Rp, p.

Lumped real parametric uncertainty

In this section we revisit the general state perturbation representation (2.18)

x(t) = (A + Aq(q)) x(t),

and we investigate how to obtain a lumped real parametric uncertainty description (2.19),

x(t) = (A + U∆(I − W∆)−1V ) x(t),

where all perturbations are inside a single block ∆.

For fractional, polynomial parameter dependency such a representation can be found by

extracting all non-reducible factors and representing them using a diagonal uncertainty

matrix with the individual factors as elements, e.g., using a tree decomposition [Barmish

et al. 1990a]. Another technique uses Horner factorization [Varga and Looye 1999]. See

[Magni 2001] for an overview of realization and order reduction algorithms.

Representation (2.19) is shown as a block diagram in Figure 2.3. From this block diagram

it becomes obvious that the uncertainty block ∆ can be pulled out and the system can

be put into the lumped uncertainty form of Figure 2.2. This is shown in Figure 2.4. The

transfer function for the uncertainty block from output u(s) to input v(s) is

Guv(s) = (I − W∆)−1V (sI − A)−1U. (2.22)


∆

W

∫

A

Ux

V

vu

Figure 2.4: Lumped state perturbation

Example 2.1 Consider the following system with state perturbation:

x(t) =

0 1

−1 −2

+

−q1 + q1q2 q1q2

q2 q1 − q2

x(t) (2.23)

This representation can be reduced to the following minimal form:

∆ =

q1

q1

q2

q2

, W =

0

0

−1 1 0

0

,

U =

1 0 1 0

0 1 0 1

, V T =

−1 0 0 1

0 1 0 −1

�

2.3.2 Multi-Model Descriptions

A multi-model description consists of a finite number of fixed model descriptions of the

form

x(t) = Aix(t) + Biu(t), y(t) = Cix(t) + Diu(t), i ∈ 1, . . . , p, (2.24)


where p is the number of individual models. Thus a multi-model description usually does

not contain any parameters. Multi-model descriptions are easily treated in the parameter

space approach by consecutively mapping a specification for each model.

2.3.3 Dynamic Uncertainty

The term dynamic uncertainty might be a bit misleading in the sense that all other

uncertainty structures presented in Section 2.3 are describing uncertainties of dynamic

systems. By dynamic uncertainty we refer to uncertainties whose underlying dynamics

are not precisely known and are possibly varying within known bounds.

Dynamic uncertainty operators are often associated with modeling errors that are not

efficiently described by parametric uncertainty. Unmodeled dynamics and inaccurate

mode shapes of aero-elastic models [Lind and Brenner 1998] are examples of modeling

errors that can be described with less conservatism by dynamic uncertainties than with

parametric uncertainties. These dynamic uncertainties are typically complex in order to

represent errors in both magnitude and phase of signals.

The set of unstructured uncertainties ∆ is given as all stable transfer functions (rational

or irrational) of appropriate dimension that are norm bounded:

∆ := {∆ ∈ RH∞, ||∆|| < l(ω)}. (2.25)

We will use the H∞ norm throughout the thesis (see Section 2.4.1 for a review of the H∞

norm) and usually the following normalization condition holds: ||∆||∞ ≤ 1. This normal-

ization can always be enforced by using suitable weighting functions.

There are several possibilities how to describe plant perturbations using unstructured

uncertainties ∆. The most prominent and most intuitive are the (output) multiplicative

and additive perturbations:

Gp(s) = G(s) + Wa(s)∆(s), (2.26)

Gp(s) = (I + ∆(s)Wo(s)) G(s), (2.27)

where Wa(s) and Wo(s) are weights such that ||∆(s)||∞ ≤ 1. See Figure 2.5.

Similar plant perturbations are inverse additive uncertainty, inverse multiplicative output

and (inverse) multiplicative input uncertainty. Another common form is coprime factor

uncertainty:

Gp(s) = (Ml + ∆M)−1(Nl + ∆N), (2.28)

where Ml, Nl is a left coprime factorization of the nominal plant model. This uncertainty

description, suggested by McFarlane and Glover [1990], is mainly used in an H∞ norm


Wa ∆

G G

Wo ∆

Figure 2.5: Additive and multiplicative output uncertainty

loop-shaping procedure, where the open-loop shapes are shaped by weights and the ro-

bustness of the resulting plant to this type of uncertainty is maximized. Usually, no

problem-dependent uncertainty modeling is used in this approach, see [Skogestad and

Postlethwaite 1996] for a thorough treatment.

For plants with different physically motivated perturbations, e.g., input and output mul-

tiplicative uncertainty, it is possible to lump all uncertainties into a single perturbation,

see Figure 2.2. Unfortunately even for unstructured perturbations the resulting overall

uncertainty ∆ is block-diagonal and therefore structured. A straightforward application

of the small-gain theorem (see Theorem 2.2 in Section 2.4.1) will be obviously conserva-

tive, because the system is checked for a much larger set of uncertainties which actually

cannot appear in the real system.

Back in 1982, Doyle and Safonov introduced simultaneously equivalent entities to measure

the robustness margin of a system with respect to structured uncertainties. For a general

system as in Figure 2.2 the so-called structured singular value µ is defined as

µ∆(G(s)) :=1

min{σ(∆) | det[I − G(s)∆] = 0} ,

where σ is the maximal singular value (see Section 2.4.1). The value µ∆(G(s)) is a simple

scalar measure of the smallest structured uncertainty which destabilizes the feedback

system.

The µ approach has been extended to the synthesis of robust controllers and there are

several related toolboxes. Yet, the exact computation of µ is not possible except for

special cases. Thus all available software tries to compute meaningful bounds for µ. This

emphasizes the goal of this thesis to treat real parameter variations directly and only

represent them into a lumped uncertainty ∆, if the parameters are changing fast or their

number is large and we want to refrain from gridding.

Another rational of this approach comes from the fact that µ with respect to pure real

uncertain parameters is discontinuous in the problem data [Barmish et al. 1990b]. That

is, for small changes of the nominal system, maybe due to a neglected parameter, the

stability margin might be subject to large, discontinuous changes. Put in other words,


for a specific model the real µ is not a good indicator of robustness because it might

deteriorate for an infinitesimal perturbation of the considered plant.

2.4 MIMO Specifications in Control Theory

This section reviews the various specifications and objectives relevant for design and ana-

lysis of multivariable control systems. All specifications will be formulated using algebraic

Riccati equations (AREs) or Lyapunov equations. See Section 3.2 for an introduction to

AREs. While there will be no special notation for parametric dependencies, the considered

systems might depend on several real parameters q ∈ Rp.

The specifications are briefly motivated from a general control theoretic point of view.

Special attention is given to reasons why it might be advantageous to include these specifi-

cations into the parameter space approach. For motivation of the specifications presented

in this section see for example [Boyd et al. 1994] or [Scherer et al. 1997].

Apart from the introduction of the specifications, the main aim of this section is to

present them in a mathematical formulation, that is tractable for the mapping equations

developed in Chapter 3.

2.4.1 H∞ Norm

Probably the most prominent norm used in control theory to date is the H∞ norm.

The H∞ norm of a transfer function G(s) is defined as the peak of the maximum singular

value of the frequency response

||G(s)||∞ := supω

σ(G(jω)), (2.29)

where σ is the largest singular value or maximal principal gain of an asymptotically stable

transfer matrix G(s). Note that (2.29) defines the L∞ norm, if the stability requirement

is dropped.

There are several interpretations of the H∞ norm. A signal related interpretation is given

by

||G||∞ = supw 6=0

||Gw||2||w||2

.

Consider a scalar transfer function G(s), then the infinity norm can be interpreted as

the maximal distance of the Nyquist plot of G(s) from the origin or as the peak value

of the Bode magnitude plot of |G(jω)|. In that sense, frequency response magnitude

specifications [Odenthal and Blue 2000] can be recast as scalar H∞ norm problems.

2.4 MIMO Specifications in Control Theory 23

For SISO systems the H∞ norm is simply the maximum gain of the transfer function,

whereas for MIMO systems it is the maximum gain over all directions. Thus the H∞

norm takes the directionality of MIMO systems into account.

For MIMO systems, the H∞ norm describes the maximum amplitude of the steady state

response for all possible unit amplitude sinusoidal input signals. In the context of stochas-

tic input signals, the H∞ norm can be interpreted as the square root of the maximal energy

amplification for all input signals with finite energy.

Note that unlike the induced matrix norms ||A||p, which are related to vector norms ||x||p,the norms used for matrix functions are not directly related to the namesake signal norms.

For example, the L1 norm is another norm, frequently used for LTI systems,

||G||1 := supw 6=0

||Gw||∞||w||∞

.

The H∞ norm can be used to evaluate nominal stability of a system without uncertainty.

By evaluating the H∞ norm of special transfer functions, e.g., a weighted sensitivity

function, performance and robustness of a control system can be assessed. We will show

how to incorporate the latter feature into the PSA in the next chapter.

Based on the control theoretic useful mathematical properties the so-called H∞ prob-

lem was defined by Zames [1981]. Using the general control configuration of Figure 2.6,

the standard H∞ optimal control problem is to find all stabilizing controllers1 K which

minimize

||Fl(P, K)||∞, (2.30)

where Fl(P, K) := P11 +P12K(I −P22K)−1P21 is a lower linear fractional transformation.

Often one is content with a suboptimal controller which is close to the optimal. Then

the H∞ control problem becomes: given a γ > γmin, where γmin is the minimal, achievable

value, find all stabilizing controllers K such that

||Fl(P, K)||∞ < γ. (2.31)

Following [Zames 1981] a number of different formulations and solutions were developed.

One successful approach to solve H∞ control problems involves AREs, see [Doyle et al.

1989, Petersen 1987]. The ARE based algorithm of Doyle et al. [1989] is summarized in

[Skogestad and Postlethwaite 1996, p. 367]. The formulation based on AREs will become

important in Chapter 3 when we derive mapping equations for H∞ norm specifications.

1We use K for controllers to avoid ambiguity with the output-state matrix C


P

K

z

u v

w

Figure 2.6: General control configuration

Hence we do not pursue the automatic solution of (2.31), since we are trying to incor-

porate H∞ criteria into the PSA. Nevertheless the achievable level γ is of interest when

mapping an H∞ specification.

The following theorem, which is known as the bounded real lemma [Boyd et al. 1994],

provides an important link between H∞ control problems and AREs and will therefore

become important in Chapter 3. This theorem, besides its theoretical significance, is often

used as a preparation for the solution of the H∞ problem.

Theorem 2.1 Bounded real lemma

Consider a linear system with transfer function G(s) and corresponding minimal

state-space realization G(s) = C(sI − A)−1B + D. Then the following statements

are equivalent:

(i) G(s) is bounded-real, i.e., G(s)∗G(s) ≤ I, ∀ Re s > 0;

(ii) G(s) is non-expansive, i.e.,∫ τ

0

y(t)T y(t)dt ≤∫ τ

0

u(t)T u(t)dt, τ ≥ 0;

(iii) the H∞ norm of G(s) with A being stable, σ(D) < γ, and γ = 1 satisfies

||G(s)||∞ ≤ γ;

(iv) the algebraic Riccati equation

γXBS1

r B∗X +γC∗S1

l C−X(A−BS1

r D∗C)− (A−BS1

r D∗C)∗X = 0 , (2.32)

with γ = 1 has a Hermitian solution X0 such that all eigenvalues of the ma-

trix A − BB∗X0 lie in the open left half-plane, where Sr = (D∗D − γ2I)

and Sl = (DD∗ − γ2I).

�


Note: The equivalence of (iii) and (iv) in Theorem 2.1 was stated using the parameter γ

such that we can map different performance levels γ for an H∞ norm specification into

parameter space in Chapter 3.

All robust stability conditions for uncertain systems using the H∞ norm can be based on

the following rather general result [Zhou et al. 1996].

Theorem 2.2 Small gain theorem

Consider the feedback system of Figure 2.2, with stable G(s). Then the closed-loop

system is stable for all ∆ ∈ RH∞ with

||∆||∞ ≤ 1

γif and only if ||G||∞ < γ.

�

Small gain theorems have a long history in control theory, starting with [Sandberg 1964].

The above printed version is a norm based gain version [Zhou et al. 1996]. There are even

more general versions for nonlinear functionals.

Note that the small gain theorem can be very conservative. For example, unity feedback

of stable first-order systems with gain greater than one is not covered.

Owen and Zames [1992] make the following observation which is quoted:

The design of feedback controllers in the presence of non-parametric and un-

structured uncertainty . . . is the raison d’etre for H∞ feedback optimization,

for if disturbances and plant models are clearly parametrized then H∞ meth-

ods seem to offer no clear advantages over more conventional state-space and

parametric methods.

Next, consider an SISO control specification, which can be formulated using the H∞ norm.

Nyquist stability margin

An important measure of robustness for SISO transfer functions is the so-called Nyquist

stability margin. The Nyquist stability margin is defined as the minimal distance of the

Nyquist curve from the critical point (−1, 0),

ρ := minω

|1 + G0(jω)|, (2.33)

where G0(s) is the open-loop transfer function.

Observe that the Nyquist stability margin is related to the sensitivity function S(s) by

ρ =1

||S(s)||∞, (2.34)

where S(s) = 1/(1 + G0(s)).


2.4.2 Passivity and Dissipativity

The roots of passivity as a control concept can be traced back to the 1940’s, when re-

searchers in the Soviet Union applied Lyapunov’s methods to stability of control systems

with a nonlinearity. But it took up to 1971, when Willems [1971] formulated the notion

of passivity in a system theoretic framework.

The most striking feature of passivity is that any interconnection of passive systems is

passive. Figure 2.7 illustrates some connections of passive subsystems which comprise a

passive system.

Passive

System 1

Passive

System 2 y2

y

y1

u

w1

u1 Passive

System 1

Passive

System 2 u2

w2

y1

y2

Figure 2.7: Interconnection of passive systems

This fact can be used to design robust controllers by subdividing the complete control

system into passive subsystems and designing a passive controller. If the plant is not

passive, a suitable approach is to fix a controller which leads to a passive controlled sub-

system. On top of this, additional performance enhancing controllers can be determined

which preserve passivity. This approach is similar to the classical feedback - feedforward

filter design steps of many control design approaches.

Since passivity is also defined for nonlinear systems this concept can be applied to control

systems with either nonlinear plant or controller. This approach is easily extended to

parametric robustness by checking or guaranteeing that a system is passive under all

parameter variations.

We will consider quadratic MIMO systems, i.e., the dimension of the input equals the

output dimension. This is a mandatory assumption for passivity. For dissipativity, which

can be seen as the generalization of passivity, this is not necessary, see Definition 2.1 on

page 28. Nevertheless, commonly used dissipativity definitions, e.g., (2.40), assume that

the system is quadratic.

For a linear system passivity is equivalent to the transfer matrix G(s) being positive-real,

which means that

G(s) + G(s)∗ ≥ 0 ∀ Re s > 0. (2.35)


In the time-domain a system is said to be passive if∫ τ

0

u(t)T y(t) dt ≥ 0, ∀ τ ≥ 0, x(0) = 0 . (2.36)

Passivity can be interpreted for physical systems, if the term u(t)T y(t) is a power, e.g.,

current and voltage for electrical and co-located force and velocity for mechanical systems.

Equation (2.36) then says that the difference between supplied and withdrawn energy is

positive.

For SISO transfer functions passivity can be checked graphically by plotting the Nyquist

diagram. If the resulting curve lies in the right half plane then the system is passive.

The following lemma [Anderson 1967] translates the frequency-domain condition (2.35)

into a matrix condition which will lead to mapping equations.

Lemma 2.3 Positive Real Lemma

Consider a linear, time-invariant system G(s) = C(sI − A)−1B + D, with (A, B)

stabilizable, (A, C) observable and D + D∗ nonsingular. Then G(s) is positive real

or passive, if there are matrices L, W , and X = X∗ > 0, such that

A∗X + XA = −L∗L, (2.37a)

XB − C∗ = −L∗W, (2.37b)

D + D∗ = W ∗W. (2.37c)

�

Using elementary matrix operations the unknown matrices L and W can be eliminated

to give the ARE

A∗X + XA + (XB − C∗)(D + D∗)−1(XB − C∗)∗ = 0 . (2.38)

Condition (2.35) is equivalent to the following statement: There exists X = X ∗ satisfying

the ARE (2.38). This equivalence can be found in [Willems 1971]. Using Theorem 3.1

the equivalence between (2.35) and the non-existence of pure imaginary eigenvalues of a

Hermitian matrix can be established. This was first done in [Lancaster and Rodman 1980].

Equation (2.38) will be used to obtain mapping equations for passivity in Chapter 3.

In the preceding formulations a non-zero feedthrough D with D + D∗ nonsingular was

assumed. It turns out that for strictly-proper systems with D + D∗ singular matters

become more difficult. Equation (2.38) disintegrates into:

A∗X + XA < 0, (2.39a)

XB − C∗ = 0 . (2.39b)


The linear matrix inequality (LMI) (2.39a) has to hold, while the constraint (2.39b) is

satisfied. Because this system of equations and inequalities does not fit into the ARE

framework we will use the more general dissipativity in Chapter 3 in order to develop

mapping equations for passivity.

A system is said to have dissipation η if∫ τ

0

(u(t)T y(t) − ηu(t)T u(t)) dt ≥ 0, ∀ τ ≥ 0, x(0) = 0 . (2.40)

Thus passivity corresponds to non-negative dissipation. A system has dissipativity η, if

the following ARE has a Hermitian solution

A∗X + XA + (XB − C∗)(D + D∗ − 2ηI)−1(XB − C∗)∗ = 0 . (2.41)

The AREs (2.38) and (2.41) will be used later to derive algebraic mapping equations

for LTI systems. Thus these AREs allow to incorporate passivity and dissipativity for LTI

systems into the parameter space approach.

As mentioned earlier, the concept of dissipativity carries over to nonlinear systems. In

order to do this we need a more general definition of dissipativity [Willems 1971].

Definition 2.1 Let x(t) = f(x(t), u(t)), y(t) = h(x(t), u(t)) be a system, with state x(t),

input u(t) and output y(t). This system is dissipative with respect to a supply rate S, if

there is a (storing) function V (x), such that

V (x(τ)) − V (x(0)) ≤∫ τ

0

S(u(t), y(t)) dt, (2.42)

is satisfied. �

Note that the latter definition is formulated in terms of the internal state x(t), while the

formulation in (2.40) is related to the input-output behavior of a system. For quadratic

systems, where the input and output dimensions are equal, this definition can now be

specialized using particular supply rates [Sepulchre et al. 1997].

Definition 2.2 Let x(t) = f(x(t), u(t)), y(t) = h(x(t), u(t)) be a system, with state x(t),

input u(t) and output y(t). This system is α−input dissipative, β−output dissipative,

if (2.42) is satisfied with S(u, y) = uT y − α||u||2, S(u, y) = uT y − β||y||2, respectively.

�

Definition 2.2 defines the general α-input and β-output dissipativity. There is no as-

sumption on the values of these parameters. In comparing Definition 2.2 with (2.40)

we conclude that (2.40) corresponds to η-input passivity. After (2.40) we noted that

passivity corresponds to non-negative dissipation η. While the interconnection properties

of passive systems are well-known, we are now interested in feedback interconnections of

dissipative systems used in Definition 2.2.


Theorem 2.4 [Sepulchre et al. 1997]

Let two systems be feedback interconnected as in the right part of Figure 2.7. The

closed loop system is asymptotically stable, if System 1 is α-input dissipative, Sys-

tem 2 is β-output dissipative, and α + β > 0 holds.

�

This theorem assures that the feedback interconnection of two purely passive systems,

i.e., α = β = 0, is stable. We will use Theorem 2.4 and Definition 2.2 to apply the

passivity results to famous results for nonlinear control systems in Section 2.4.4.

Remark 2.1 Quadratic Constraints

Both H∞ norm and dissipativity specifications fit into the more general framework

of quadratic constraints of the form

∫ τ

0

y(t)

u(t)

T

Q S

ST R

y(t)

u(t)

dt ≤ 0, ∀ τ ≥ 0, x(0) = 0 . (2.43)

�

We will consider an even more general set of specifications, which includes (2.43) in

Section 2.5.

2.4.3 Connections between H∞ Norm and Passivity

For SISO systems both the H∞ norm and the passivity condition resemble classical gain

and phase conditions. They are even more related and can be even translated into each

other.

Using the bilinear Cayley transformation

G(s) = (I − G(s))(I + G(s))−1, with det [I + G(s)] 6= 0, (2.44)

the following equivalence holds [Anderson and Vongpanitlerd 1973, Haddad and Bernstein

1991].

Theorem 2.5

(i) G(s) is positive-real, i.e., G(s) + G(s)∗ ≥ 0 ∀ Re s > 0;

(ii) ||G(s)||∞ < 1.

�


By Theorem 2.5 a positive realness condition can be recast as an H∞ norm condition

and vice versa. The application of this theorem in the time-domain is facilitated by the

following conversion.

Let G(s) have the state-space realization

G(s) ∼=

A B

C D

,

then a realization for G(s) is given by

G(s) ∼=

A − B(I + D)−1C B(I + D)−1

−2(I + D)−1C (I − D)(I + D)−1

. (2.45)

If we want to apply the results in the opposite direction, we can exchange the sym-

bols G and G, since the Cayley transformation is bilinear, i.e., the converse of transfor-

mation (2.44) is simply G(s) = (I − G(s))(I + G(s))−1 with det[I + G(s)] 6= 0.

For systems with a static feedthrough matrix D which satisfies det[I + D] = 0 the con-

version based on (2.45) fails. In this case, the transformation from a positive-real to

an H∞ norm condition can be done by using a positive realness preserving congruence

transformation G(s) → G(s) = V ∗G(s)V :

G(s) ∼=

A BV

V ∗C V ∗DV

,

where V is a nonsingular matrix such that det[I +V ∗DV ] 6= 0. For the converse transfor-

mation from an H∞ norm to a positive-real condition we can use a sign matrix transfor-

mation G(s) → G(s) = G(s)S, where S is a sign matrix such that det[I + DS] 6= 0 [Boyd

and Yang 1989].

2.4.4 Popov and Circle Criterion

Although this thesis is mainly concerned with specifications for LTI systems, the presented

theory can be applied to criteria for nonlinear systems. Absolute stability theory allows

to analyze the stability of an LTI system in the feedforward path interconnected with

a static nonlinearity in the feedback path. There are two well-known variants, namely

the Popov and Circle criterion, which can be formulated as the feedback of two passive

systems [Khalil 1992, Kugi and Schlacher 2002].

Both criteria assume a sector nonlinearity. Figure 2.8 shows the general feedback structure

for absolute stability, where Ψ(y) represents the nonlinearity.


Circle criterion

Consider the feedback structure given in Figure 2.8 where the multivariable static non-

linearity Ψ(y) satisfies the sector condition

(Ψ(y) − K1y)T (Ψ(y) − K2y) ≤ 0, (2.46)

with K1, K2 matrices which satisfy K2 − K1 > 0. Thus the nonlinearity is contained in

the sector K1, K2. Note that apart from the positive definiteness of K2 −K1 there are no

further assumptions such as SISO.

[

A B

C 0

]

Ψ(y)

u y

Figure 2.8: Feedback structure for absolute stability

Using an equivalence transformation the feedback loop in Figure 2.8 can be put in the

form Figure 2.9. It can be shown that the nonlinear System 2 in Figure 2.9 is β-output

dissipative, with β = 1. From Theorem 2.4 then follows the Circle criterion, see [Khalil

1992, Vidyasagar 1978]:

Theorem 2.6 Circle Criterion

Consider the feedback structure in Figure 2.8 with a sector nonlinearity Ψ(y) which

satisfies (2.46). The closed loop is absolutely stable, if the LTI system

x(t) = (A − BK1C)x(t) + Bu(t)

y(t) = (K2 − K1)Cx(t)(2.47)

is α-input dissipative with α > −1.

�


[

A B

C 0

]

K

K−1Ψ(y)

K1

K1

System 2

System 1

Figure 2.9: Equivalent feedback loop for circle criterion

Theorem 2.6 can now be formulated as an ARE using the µ-input dissipativity ARE (2.41)

with µ = −1,

XBB∗X + X(2A − BKsC) + (2A − BKsC)∗X + C∗K∗dKdC = 0 , (2.48)

where Ks = K1 + K2, and Kd = K2 − K1.

The condition for absolute stability given in Theorem 2.6 is sufficient. Because this does

not imply necessity there might be considerable conservativeness. If we sacrifice the

generality of the multivariable sector condition (2.46), we can derive sufficient conditions

which can be much tighter. This leads to the Popov criterion considered next.

Popov criterion

For the Popov criterion the considered nonlinearity Ψ(y) is a simple decentralized func-

tion Ψj = Ψj(yj) with

0 ≤ kj ≤ Ψj(yj), j = 1, . . . , m, (2.49)

i.e., K = diag(ki) ∈ Rm, m.


Using an equivalence transformation similar to the one considered for the circle criterion

the nonlinearity Ψ(y) can be embedded into a dissipative subsystem. The decentralized

constraint on the nonlinearity offers some freedom, which can be used to include the

factor (µs + 1)−1 in the nonlinear subsystem while still maintaining dissipativity. This

additional degree of freedom µ can be chosen arbitrarily.

Theorem 2.7 Popov criterion

Consider the feedback structure in Figure 2.8 with a stationary, sector nonlinear-

ity Ψ(y) which satisfies (2.49). The closed loop is absolutely stable, if there is a µ ∈ R

such that the LTI system

x(t) = Ax(t) + Bu(t),

y(t) = KC((I + µA)x(t) + µBu(t)),(2.50)

is α-input dissipative with α > −1.

�

Let C = KC(I + µA), then the ARE related to Theorem 2.7 is given by

A∗X + XA + (XB − C∗)(2I + µKCB + µB∗C∗K∗)−1(XB − C∗)∗ = 0 . (2.51)

2.4.5 Complex Structured Stability Radius

The complex structured stability radius of the system

x(t) = (A + U∆V )x(t) (2.52)

is defined by

rC = inf{

||∆|| : Λ(A + U∆V ) ∩ C+ 6= ∅

}

, (2.53)

where ∆ is a complex matrix of appropriate dimension, C+ denotes the closed right half

plane, and ||∆|| is the spectral norm of ∆.

Lemma 2.8

Let G(s) = V (sI − A)−1U . Then

rC(A, U, V ) = ||G||−1∞ . (2.54)

�


A proof can be found in [Hinrichsen and Pritchard 1986]. Thus the determination of the

complex structured stability radius is equivalent to the computation of the H∞ norm of

a related transfer function.

Note that not all possible perturbations are expressible with a single block perturba-

tion (2.52), e.g.,

x(t) =

−1 0

2 −2

+

0 δ1

δ2 δ3

x(t).

2.4.6 H2 Norm Performance

The H2 norm is a widely used performance measure that allows to incorporate time-

domain specifications into control design. The H2 norm of a stable transfer matrix G(s)

is defined as

||G(s)||2 :=

(

trace1

2π

∫ ∞

−∞

G(jω)∗G(jω)dω

)1/2

.

The above norm definition can be used for generic square integrable functions on the

imaginary axis and is then called L2 norm. Thus, strictly speaking without the stability

condition, we are mapping the L2 norm instead of the H2 norm.

The H2 norm arises, for example, in the following physically meaningful situation. Let

the system input be zero-mean stationary white noise of unit covariance. Then, at steady

state, the variance of the output is given by the square of the H2 norm. This can be seen

from the general definition of the root mean square response norm for systems driven by

a particular noise input with power spectral density matrix Sw:

||G||rms,s :=

(

trace1

2π

∫ ∞

−∞

G(jω)∗Sw(jω)G(jω)dω

)1/2

.

The H2 norm is only finite, if the transfer matrix G(s) is strictly proper, i.e., the direct

feedthrough matrix D = 0 (or D(q) = 0 ). Hence we assume D = 0 in this subsection,

which is a valid assumption for almost any real physical system.

By Parseval’s theorem, the H2 norm can be expressed as

||G||2 =

(

trace

∫ ∞

0

H(t)T H(t)dt

)1/2

, (2.55)

with H(t) = CeAtB being the impulse matrix of G(s) = C(sI − A)−1B.


From this follows

||G||22 = trace [BT

∫ ∞

0

eAT tCT CeAtdt B]

= trace [BT WobsB], (2.56)

where

Wobs :=

∫ ∞

0

eAT tCT CeAtdt

is the observability Gramian of the realization, which can be computed by solving the

Lyapunov equation

AT Wobs + WobsA + CT C = 0 . (2.57)

Alternatively a dual output-controllable formulation exits for ||G||22, which involves the

controllability Gramian Wcon,

||G||22 = trace [CWconCT ], (2.58)

where Wcon can be determined from

AWcon + WconAT + BBT = 0 . (2.59)

The H2 norm is different from the specifications presented so far in that a specifica-

tion ||G||2 ≤ γ cannot be expressed by an ARE. In that sense the H2 norm does not

really fit into the ARE framework. But this specification can be formulated by means of

the more special Lyapunov equation, which is affine in the unknown Wobs.

2.4.7 Generalized H2 Norm

In the scalar case, the H2 norm can be interpreted as the system gain, when the in-

put are L2 functions and the output bounded L∞ time functions. Thus the scalar H2

norm is a measure of the peak output amplitude for energy bounded input signals. Low

values for this quantity are especially desirable if we want to avoid saturation in the sys-

tem. Unfortunately this interpretation does not hold for the H2 norm in the vector case.

Following [Wilson 1989], the so-called generalized H2 norm [Rotea 1993] is defined by

||G||2,gen = λ1/2max

∫ ∞

0

G(t)T G(t)dt, if ||y||∞ = sup0≤t≤∞

||y(t)||2,

or

||G||2,gen = d1/2max

∫ ∞

0

G(t)T G(t)dt, if ||y||∞ = sup0≤t≤∞

||y(t)||∞,


depending on the type of L∞ norm chosen for the vector valued output y. Here, λmax

and dmax denote the maximum eigenvalue and maximum diagonal entry of a non-negative

matrix, respectively.

The generalized H2 norm can also be expressed as

||G||2,gen = λ1/2max [BT WobsB],

or

||G||2,gen = d1/2max [BT WobsB],

depending on the L∞ norm chosen, where Wobs is the observability Gramian.

2.4.8 LQR Specifications

The classical linear quadratic regulator (LQR) problem, which aims to minimize the

objective function

J =1

2

∫ ∞

0

(x(t)T Qx(t) + u(t)T Ru(t))dt (2.60)

for a state-feedback controller u(t) = −Kx(t), was introduced back in 1960 [Kalman and

Bucy].

A core advantage is the easily interpretable time-domain specification, which allows a

transparent tradeoff between disturbance rejection and control effort utilization. Another

feature is the fact that the easily solvable optimization problem produces gains that

coordinate the multiple controls, i.e., all loops are closed simultaneously.

The classical LQR problem is to find an optimal input signal u(t) which drives a given

system x(t) = Ax(t) + Bu(t) by minimizing the performance index (2.60). The optimal

solution for the constant gain matrix K is given by

K = R−1BT X,

where X is the unique positive semidefinite solution of the ARE

AT X + XA − XBR−1BT X + Q = 0 . (2.61)

This ARE is another example for the wide-spread use of AREs in control theory.

The signal-oriented formulation of LQR is the linear quadratic Gaussian (LQG) control

problem. In LQG, input signals are considered stochastic and the expected value of the

output variance is minimized. Mathematically this is equal to the 2-norm of the stochastic

output.


If frequency dependent weights on the signals are included, we arrive at the so-called

Wiener-Hopf design method, which is nothing else than the H2 norm problem considered

in Section 2.4.6.

The classical LQR problem can be cast as an H2 norm problem [Boyd and Barratt 1991].

Consider a linear time-invariant system described by the state equations

x(t) = Ax(t) + Bu(t) + w(t) , (2.62a)

z(t) =

R

12 0

0 Q12

u(t)

x(t)

, (2.62b)

where u(t) is the control input, w(t) is unit intensity white noise, and z(t) is the output

signal of interest.

The square root W12 of a square matrix W is defined as any matrix V = W

12 which

satisfies W = V T V or W = V V T . The matrix square root exists for symmetric, positive

definite matrices. One possible algorithm to obtain the matrix square root is to use lower

or upper triangular Cholesky decompositions.

The LQR problem is then to design a state-feedback controller u(t) = −Kx(t), which

minimizes the H2 norm between w(t) and z(t). From this follows that the performance

index J is given as

J = ||Gw→z(s)||22,

where Gw→z(s) is the transfer function of (2.62) from w(t) to z(t), with state-space

realization

Gw→z

∼=

A − BK I

−R12 K 0

Q12 0

.

The parametric LQR control design allows us to explicitly incorporate control effort spec-

ifications into a robust controller design, which is not possible with pure eigenvalues

specifications. It also applies to parametric SISO systems.

The LQG problem provides another example why it can be advantageous to combine clas-

sical, non-robust methods and the PSA. Back in 1978 [Doyle] showed that there are LQG

controllers with arbitrary small gain margins.

The PSA based LQR and H2 norm design described in this thesis allows to combine the

transparent robustness of PSA with the easy tunability of LQR. The author considers the


combination of the invariance plane based pole movement [Ackermann and Turk 1982]

and LQR performance evaluation in the PSA as a very promising method to design robust

state-space controllers.

2.4.9 Hankel Norm

The Hankel norm of a system is a measure of the effect of the past system input on the

future output. It is known [Glover 1984] that the Hankel norm is given by

||G||hankel = λ1/2max[WobsWcon],

where the controllability Gramian Wcon is the solution of

AWcon + WconAT + BBT = 0 .

The Gramian Wobs measures the energy that can appear in the output and Wcon measures

the amount of energy that can be stored in the system state using an excitation with a

given energy.

The Hankel norm is extensively used in connection with model reduction. For example,

reduced state-space models with minimal deviations in the input-output behavior can be

achieved by neglecting modes with the smallest Hankel singular values, which are defined

as the positive square roots of the eigenvalues of the product of both Gramians

σi =√

λi[WobsWcon]. (2.63)

2.5 Integral Quadratic Constraints

This section presents a recently developed, unified approach to robustness analysis of gen-

eral control systems. Namely we consider integral quadratic constraints (IQCs) introduced

by Megretski and Rantzer [1997].

In general, IQCs provide a characterization of the structure of a given operator and

the relations between signals of a system component. An IQC is a quadratic constraint

imposed on all possible input-output pairs in a system.

The IQC framework combines results from three major control theories, namely input-

output, absolute stability, and robust control. Using IQCs specifications from all these

research fields can be formulated by the same mathematical language. Actually it has

2.5 Integral Quadratic Constraints 39

been shown that some conditions from different theories lead to identical IQCs and are

therefore equivalent.

Since all specifications expressible as IQCs share the same mathematical formulation we

can use the same computational methods to map them into parameter space.

In a system theoretical context the following general IQC is widely used. Two bounded

signals w ∈ Lm2 [0,∞) and v ∈ Ll

2[0,∞) satisfy the IQC defined by the self-adjoint

multiplier Π(jω) = Π(jω)∗, if

∫ ∞

−∞

v(jω)

w(jω)

∗

Π(jω)

v(jω)

w(jω)

dω ≥ 0, (2.64)

holds for the Fourier transforms of the signals. Consider the bounded and causal opera-

tor ∆ defined on the extended space of square integrable functions on finite intervals. If

the signal w is the output of ∆, i.e., w = ∆(v), then the operator ∆ is said to satisfy

the IQC defined by Π, if (2.64) holds for all signals v ∈ Ll2[0,∞). We use the shorthand

notation ∆ ∈ IQC(Π) for operator-multiplier pairs (∆, Π) for which this property holds.

Thus the multiplier Π gives a characterization of the operator ∆. The operator ∆ rep-

resents the nonlinear, time-varying, uncertain or delayed components of a system. For

example, let ∆ be a saturation w = sat(v) then the multiplier

Π =

1 0

0 −1

,

defines an IQC which holds for this nonlinear operator. Note, that this multiplier is

not necessarily unique. Actually there might be an infinite set of valid multipliers.

See [Megretski and Rantzer 1997] for a summarizing list of important IQCs, and [Jonsson

2001] for a detailed treatment.

We consider the general configuration of a causal and bounded linear time-invariant (LTI)

transfer function G(s), and a bounded and causal operator ∆ which are interconnected

in a feedback manner

v = Gw + e2,

w = ∆(v) + e1,

where e1 and e2 are exogenous inputs. See Figure 2.10.

The stability of this system can be verified using the following theorem.


G

∆

we1

v e2

Figure 2.10: General IQC feedback structure

Theorem 2.9 [Megretski and Rantzer 1997]

Let G(s) ∈ RHl×m∞ , and let ∆ be a bounded causal operator. Assume that

(i) for τ ∈ [0; 1], the interconnection (G, τ∆) is well-posed,

(ii) for τ ∈ [0; 1], the IQC defined by Π is satisfied by τ∆,

(iii) there exists ε > 0 such that

G(jω)

I

∗

Π(jω)

G(jω)

I

≤ −εI, ∀ω ∈ R. (2.65)

Then, the feedback interconnection of G(s) and ∆ is stable.

�

Note that the considered feedback interconnection uses positive feedback only.

2.5.1 IQCs and Other Specifications

Many specifications presented in Section 2.4 can be cast as IQCs. In particular, we can

find a multiplier for all specifications which can be formulated as AREs. For example, for

the prominent condition that H∞ norm is less than one (small gain theorem), the IQC

multiplier is given by

Π =

I 0

0 −I

,

and

Π =

0 I

I 0

,

defines a valid multiplier for passivity.

2.5 Integral Quadratic Constraints 41

2.5.2 Mixed Uncertainties

For multiple, mixed uncertainties with different descriptions, e.g., LTI and time-varying,

the individual multipliers Πk can be combined into a single multiplier. The IQC can be

then used to verify stability with respect to both uncertainties occurring simultaneously.

A typical example for a system with mixed uncertainties is a system which has a satura-

tion actuator nonlinearity and where the plant is modeled using a multiplicative output

uncertainty.

Let a mixed uncertainty ∆ be given as,

∆ =

∆1 0

0 ∆2

, (2.66)

with associated IQC multipliers Π1 and Π2, which characterize the uncertainties ∆1

and ∆2. Then the overall IQC multiplier Π is given as the chess board like block (transfer)

matrix

Π =

Π1(1,1)0 Π1(1,2)

0

0 Π2(1,1)0 Π2(1,2)

Π∗1(1,2)

0 Π1(2,2)0

0 Π∗2(1,2)

0 Π2(2,2)

. (2.67)

2.5.3 Multiple IQCs

As mentioned previously, IQC multipliers are not unique. In fact, sometimes there are

not just IQCs with different parameters, but there are IQCs with fundamental differences.

In order to reduce the conservatism associated with the IQC uncertainty description, con-

vex parametrizations have been proposed [Jonsson 2001]. If ∆ ∈ IQC(Πk), k = 1, . . . , n,

then the convex combination of multipliers satisfies ∆ ∈ IQC(∑n

k=1 λkΠk), where λk ≥ 0.

In the PSA a dual and most times more practical approach is to utilize different IQCs by

iteratively mapping the individual IQCs. A good approximation of the uncertainty set,

e.g., using numerical LMI optimization, alleviates the increased conservativeness which

results from the reduced optimization variables used in the IQC stability test.

42 Mapping Equations

3 Mapping Equations

This chapter presents the mapping equations for control specifications considered in Chap-

ter 2. In particular we will present mapping equations based on algebraic Riccati and

Lyapunov equations. This approach is then extended in Section 3.4 to the uniform IQC

framework which allows to cover a large set of specifications and to broaden the system

classes and uncertainty descriptions.

Before we consider the new results for mapping additional specifications, we briefly present

some known results about eigenvalue based mapping equations.

3.1 Eigenvalue Mapping Equations

The roots of the parameter space approach stem from robust stability and have been

extended to eigenvalue specifications [Ackermann 1980]. A review of eigenvalue based

mapping equations is given in the following section.

Many important properties of control systems are characterized by eigenvalue specifica-

tions. This section briefly reviews the associated mapping equations. This serves two

purposes. First the mapping equations presented in this chapter will be compared to the

well-known equations for eigenvalue specifications. Second, as discussed in Section 2.4.7

and 2.4.9, some norm specifications lead to eigenvalue problems. Furthermore, it will be

shown that mapping several specifications considered in Section 2.4 involves a stability

condition, for example the H∞ norm. Generally, the PSA allows to separately map differ-

ent specifications. It seems to be always advantageous to map eigenvalue specifications,

especially since this can be done very efficiently. In that sense the results in this thesis

do not replace but extend the mapping of eigenvalue specifications.

Consider the characteristic polynomial pc(s) of a control system (SISO or MIMO). Math-

ematically pc(s) is calculated as the determinant of the matrix [sI − A], where A is the

closed loop system matrix, so that the roots of pc(s) coincide with the eigenvalues of A.

Without loss of generality, let the characteristic equation of a parametric system with n

states and coefficients ai be given as

pc(s, q) =

n∑

i=0

ai(q)si. (3.1)

3.1 Eigenvalue Mapping Equations 43

As shown in Ackermann et al. [1991], many control system relevant specifications can be

expressed by the condition, that the eigenvalues lie within an eigenvalue region Γ. Any

root s = sr of pc(s) satisfies pc(sr) = 0. Thus, if we are interested in roots lying on the

boundary ∂Γ of a region Γ, e.g., the left half plane (LHP) for asymptotic stability, we

have to check if pc(s) becomes zero for any s lying on ∂Γ. A simple condition is that

both the real and imaginary part of pc(s) equal zero. Therefore mapping equations for

eigenvalue specifications are given by

e1(q, α) = Re pc(s = s(α), q) = 0, (3.2a)

e2(q, α) = Im pc(s = s(α), q) = 0, (3.2b)

where s(α) is an explicit parametrization of the boundary ∂Γ with the running parame-

ter α. Some parametrizations for common specifications are given in Table 3.1.

The parametrization of a Γ boundary influences the order of mapping equations for an

actual system. In turn, this order determines the required complexity when these map-

ping equations are solved. Thus we will discuss the order of mapping equations for Γ

specifications.

All of the parametrizations given in Table 3.1 are either affine in α or contain fractional

terms that are quadratic in α. The resulting mapping equations with s = s(α) are

algebraic equations in α of order n for affine dependency, and 2n for parametrizations

with quadratic terms. Both, the number of states present in the control system and the

parametrization of the given specification determine the degree of the polynomial pc(s) in

the variable α.

Table 3.1: Parametrization of Γ boundaries

Hurwitz stability: Re s < 0 s = jα

Real part limitation: Re s < σ s = σ + jα

Damping: ζ s = α + j ζ1−ζ2 α

Absolute value (circle): |s| < r s = r( 2α1+α2 + j1−α2

1+α2 )

Parabola: Re s + aIm2 s = 0 s = −aα2 + jα

Ellipses: 1a2 Re

2s − 1b2

Im2s = 1 s = −a 1−α2

1+α2 + jb 2α1+α2

Hyperbola: 1a2 Re

2s + 1b2

Im2s = 1 s = −a 1+α2

1−α2 + jb 2α1−α2

The preceding paragraph discussed the complexity of the mapping equations with respect

to the parametrization variable α. We will now focus on the parameters q. The parametric

dependence of e1(q, α) and e2(q, α) on the parameters q only depends on the way those

parameters enter into the coefficients ai(q) of pc(s). Thus, if these coefficients are affine


in q, the resulting mapping equations will be affine in q. This affine dependence is of great

impact, since for this case the mapping equations can be easily solved for two parameters.

For the synthesis of robust controllers the affine dependence of pc(s) on two controller

parameters can be enforced by choosing an appropriate controller structure for SISO sys-

tems. However, for MIMO systems additional conditions have to hold. If two parameters

are affinely present only in a single row or column of A, we cannot get any terms resulting

from products of parameters present in A, since pc(s) is calculated as det[sI − A]. For

MIMO systems with static-gain feedback, this is the case, if two parameters related to

the gains of the same input or output are considered.

The mapping equation (3.2) determines critical parameters for general complex eigenval-

ues s. The critical parameters obtained by (3.2) are thus called complex root bound-

ary (CRB). Furthermore two special cases exist. These are the so-called real root bound-

ary (RRB) and infinite root boundary (IRB). The first corresponds to roots being purely

real which can be mapped by using (3.2a) solely. The latter is characterized by roots

going through infinity. The characteristic polynomial pc(s) has a degree drop for an IRB,

i.e., one or more leading coefficients vanish. The RRB and IRB conditions will mathe-

matically reappear for MIMO specifications, although their interpretation is less intuitive

there.

The mathematical theory required for the mapping equations based on Riccati equations is

more involved. We will present the relevant properties in the next section. See Appendix A

for more details.

3.2 Algebraic Riccati Equations

This section gives an overview of algebraic Riccati equations (AREs). The general ARE

for the unknown matrix X is given by

XRX − XP − P ∗X − Q = 0 , (3.3)

where P , R, and Q are given quadratic complex matrices of dimension n with Q and R

Hermitian, i.e., Q = Q∗, R = R∗. Although in most applications in control theory P , R,

and Q will be real, the results will be given for complex matrices where possible.

An ARE is a matrix equation that is quadratic in an unknown Hermitian matrix X. It can

be seen as a matrix extension to the well-known scalar quadratic equation ax2 +bx+c = 0

that obviously has two, not necessarily real solutions.

An ARE has in general many solutions. Real symmetric solutions, and especially the

maximal solution, play a crucial role in the classical continuous time quadratic control

3.2 Algebraic Riccati Equations 45

problems [Kwakernaak and Sivan 1972]. Numerical algorithms for finding real symmetric

solutions of the ARE have been developed (see, e.g., [Laub 1979]). Many important

problems in dynamics and control of systems can be formulated as AREs [Boyd et al.

1994], [Zhou et al. 1996]. The importance of Riccati equations and the connection between

frequency domain inequalities, e.g., ||G(s)||∞ ≤ 1, has been pointed out by Willems [1971].

Later on it was shown, that the famous and long sought solution to the state-space H∞

problem can be found using AREs [Petersen 1987].

In general, the symbolic solution of AREs is not possible due to the rich solution struc-

ture. Despite that, a successive symbolic elimination of variables using Grobner bases is

considered in Forsman and Eriksson [1993]. Although this symbolic elimination might

reveal some insight into the structure and parameter dependency of the solution set, the

explicit solution is obtainable only for degenerate cases. The authors of the latter report

also conclude, that the computational complexity of the required symbolic computations

can be quite large.

For R = 0 the general ARE reduces to an affine matrix equation in X. These so-called

Lyapunov equations have proven to be very useful in analyzing stability and controllabil-

ity [Gajic and Qureshi 1995], while the design of control systems usually involves AREs.

Since an ARE is in general not explicitly solvable, there is no direct way to obtain map-

ping equations from AREs. These mapping equations will be derived using some special

properties of AREs.

Associated with the general ARE (3.3) is a 2n × 2n Hamiltonian matrix:

H :=

−P R

Q P ∗

. (3.4)

The matrix H in (3.4) can be used to obtain the solutions to the equation (3.3), see

Theorem A.1 for a constructive method. For the PSA these solutions are not relevant.

This is equivalent to the fact that the PSA does not compute actual eigenvalues when

mapping Γ specifications. Nevertheless we will use a particular property of (3.4). Namely

the set of all eigenvalues of H is symmetric about the imaginary axis. To see that,

introduce

J :=

0 −I

I 0

.

It follows that J−1HJ = −JHJ = −H∗. Thus H and −H∗ are similar and λ is an

eigenvalue of H if and only if −λ is.

The following well-known theorem [Zhou et al. 1996] provides an important link between

solutions of AREs and the Hamiltonian matrix H.


Theorem 3.1 Stabilizing solutions

Suppose that R ≥ 0, Q = Q∗, (P, R) is stabilizable, and there is a Hermitian

solution of (3.3). Then for the maximal Hermitian solution X+ of (3.3), P −RX+

is stable, if and only if the Hamiltonian matrix H from (3.4) has no eigenvalues on

the imaginary axis.

�

Note: A Hermitian solution X+ (resp. X−) of (3.3) is called maximal (resp. minimal)

if X− ≤ X ≤ X+ for all X satisfying (3.3), where X1 ≤ X2 means that X2 − X1 is

non-negative definite.

A proof of Theorem 3.1 is given in the Appendix in Section A.2. Theorem 3.1 shows that

the non-existence of pure imaginary eigenvalues is a necessary and sufficient condition

that the ARE (3.3) has a maximal, stabilizing, Hermitian solution. Since we have seen in

Chapter 2 that the adherence of many control specifications is equivalent to the existence

of a maximal, stabilizing, Hermitian solution of an ARE, we can test this adherence by

checking if the associated Hamiltonian matrix H has no pure imaginary eigenvalues.

The PSA deals with uncertain parameters q. The purpose of the next subsection is to

extend the previous results for invariant matrices to matrices with uncertain parameters.

3.2.1 Continuous and Analytic Dependence

So far we considered AREs with constant matrices. Suppose now that the matrices P, Q,

and R are analytic functions of a real parameter q ∈ R, i.e., P = P (q), Q = Q(q),

and R = R(q). We are thus concerned with the parametric ARE

X(q)R(q)X(q) − X(q)P (q) − P (q)∗X(q) − Q(q) = 0 , (3.5)

and associated Hamiltonian matrix H(q) defined analogously to (3.4).

Before we turn to the analytic dependence of maximal, stabilizing solutions and the

important equivalence of the eigenvalue properties of the Hamiltonian matrix H similar

to Theorem 3.1, we study the continuity of all maximal solutions of (3.5) with respect

to q.

If an ARE has Hermitian solutions, then there is a maximal Hermitian solution X+ for

which Λ(P −RX+) lies in the closed left half-plane. See Theorem A.2 for a rigorous state-

ment of this fact. The behavior of X+ as a function of P, Q, and R will be characterized

subsequently.

3.2 Algebraic Riccati Equations 47

Lemma 3.2 (Lancaster,Rodman)

The maximal Hermitian solution X+ of (3.5) is a continuous function of the ma-

trices (P, Q, R) ∈ Cn,n.

�

Although Lemma 3.2 is concerned with the continuity of maximal Hermitian solutions,

these solutions are in general not differentiable. That is, the maximal parametric solu-

tion X+ = X+(q) is not necessarily an analytic function of the real parameter q. The

analyticity of maximal Hermitian solutions is ensured by the invariance of the number of

pure imaginary eigenvalues of H.

The following theorem provides the link between control system specifications and the

Hamiltonian matrix H(q). This theorem forms the corner stone of ARE based mapping

equations. It appeared partly in a mathematical context in [Ran and Rodman 1988].

See [Lancaster and Rodman 1995] for a detailed exposition.

Theorem 3.3 (Lancaster,Rodman)

Let P = P (q), Q = Q(q), and R = R(q) be analytic, complex n×n matrix functions

of q on a real interval [q−; q+], with R(q) positive semidefinite Hermitian, Q(q)

Hermitian, and (P (q), R(q)) stabilizable for every q ∈ [q−; q+]. Assume that for

all q ∈ [q−; q+], the Riccati equation (3.5) has a Hermitian solution. Further assume

that the number of pure imaginary or zero eigenvalues of

H(q) :=

−P (q) R(q)

Q(q) P (q)∗

(3.6)

is constant. Then the maximal solution X+(q) of (3.5) is an analytic function of the

parameter q ∈ [q−; q+]. Conversely, if X+(q) is an analytic function of q ∈ [q−; q+],

then the number of pure imaginary or zero eigenvalues of H(q) is constant.

�

Proof:

The proof of this theorem is rather involved and provides no insight for the successful

application, i.e., the derivation of the mapping equations. It is therefore omitted

for brevity. See [Lancaster and Rodman 1995] for a sketch of the proof.

�


Theorem 3.3 is applicable to AREs, where the matrices P, Q and R are real. We get

necessarily real maximal solutions X+ for this control theory relevant case.

The previous results can be generalized to the case when P (q), Q(q) and R(q) are analytic

functions of several real variables q = (q1, . . . , qp) ∈ Q, where Q is an open connected set

in Rp.

3.3 Mapping Specifications into Parameter Space

In this section we present the mapping equations for the specifications given in Section 2.4

for systems with uncertain parameters.

For eigenvalue specifications the boundary of the desired region in the eigenvalue plane

is mapped into a parameter plane by the characteristic polynomial. Using the real and

imaginary part of the characteristic polynomial, two mapping equations are obtained

which depend on a generalized frequency α and the uncertain parameters q. The mapping

equations presented in this section will have a similar structure.

Actually, all mapping equations presented in this thesis will consist of pe individual equa-

tions with uncertain parameters q ∈ Rp and pe − 1 auxiliary variables. Usually pe is

either 1 or 2.

Thus if the vector of uncertain parameters q is of dimension p = 1, we get either a single

equation or a regular system of equations that can be solved for q. This allows to explicitly

determine the critical parameter values of q for which the specification is marginally ful-

filled. Related to this case is the dual problem of direct performance evaluation considered

in Section 3.8.

For p > 1 we get an underdetermined set of equations. The case p = 2 is not only

important for the easily visualized plots, but also because it admits tractable solution

algorithms considered in Chapter 4. For p > 2 gridding of p − 2 parameters is necessary

to determine the resulting solution sets.

3.3.1 ARE Based Mapping

While we provided the definition of H∞ norm, dissipativity and complex stability radius

specifications, we pointed out that all of these specifications are equivalent to the existence

of a maximal, Hermitian solution of an ARE. Using Theorem 3.1 we can in turn formulate

the adherence of the given specifications as the non-existence of pure imaginary eigenvalues

of an associated Hamiltonian matrix. This is a well-known fact used in standard numerical

algorithms [Boyd et al. 1989].

3.3 Mapping Specifications into Parameter Space 49

−3 −2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ℜ{λ}

ℑ{λ}

Figure 3.1: Appearance of pure imaginary eigenvalues

Consider now the uncertain parameter case. Using Theorem 3.3 we can extend this

equivalence to systems with analytic dependence on uncertain parameters. Given a spe-

cific parameter q∗ ∈ Rp for which a maximal, Hermitian solution X+(q∗) exists, we know

from Theorem 3.1 that the Hamiltonian matrix (3.6) has no pure imaginary eigenvalues.

Using Theorem 3.3 we can extend this property as long as the number of eigenvalues on

the imaginary axis is constant. In other words, having found a parameter for which a

specification described by an ARE holds, the same specification holds as long as the num-

ber of imaginary eigenvalues of the associated Hamiltonian matrix (3.6) is zero. Hence,

the boundary of the region for which the desired specification holds is formed by param-

eters for which the number of pure imaginary eigenvalues of (3.6) changes. A new pair

of imaginary eigenvalues of (3.6) arises, if either two complex eigenvalue pairs become

a double eigenvalue pair on the imaginary axis, or if a double real pair becomes a pure

imaginary pair. Note: Another possibility is a drop in the rank of H, which corresponds

to eigenvalues going through infinity. The appearance of pure imaginary eigenvalues is

depicted in Figure 3.1.

Let us first discuss the appearance of pure imaginary eigenvalues through a double pair

on the imaginary axis. The matrix H(q) has an eigenvalue at λ = jω if

det [jωI − H(q)] = 0. (3.7a)

In general a polynomial f(ω) has a double root not only if f(ω) = 0, but the derivative

of f(ω) with respect to the argument ω has to vanish also

∂f(ω)

∂ω= 0.


Since the characteristic polynomial of H(q) (3.7a) is a polynomial, we obtain

∂

∂ωdet [jωI − H(q)] = 0, (3.7b)

as the second condition for a double eigenvalue at λ = jω. Equations (3.7a) and (3.7b)

define two polynomial equations that can be used to map a given specification into pa-

rameter space.

A necessary condition for a real eigenvalue pair to become a pure imaginary pair through

parameter changes is

det [jωI − H(q)] ω=0 = det H(q) = 0. (3.8)

Additionally the opposite end of the imaginary axis has to be considered

det [jωI − H(q)] ω=∞. (3.9)

Equation (3.9) is just the coefficient of the term with the highest degree in ω of the

determinant det[jωI − H(q)].

Equation (3.8) is not sufficient, since it determines all parameters for which (3.6) has a

pair of eigenvalues at the origin. This includes real pairs that are just interchanging on

the real axis. To get sufficiency we have to check all parameters satisfying (3.8), if there

are only real eigenvalues. Figure 3.2 shows the two possible eigenvalue paths that are

determined by (3.8). The same check has to be performed for solutions of (3.9), because

here eigenvalues can interchange at infinity and (3.9) is not sufficient as well.

−3 −2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ℜ{λ}

ℑ{λ}

−3 −2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ℜ{λ}

ℑ{λ}

Figure 3.2: Possible eigenvalue paths for real root condition

From the eigenvalue spectrum of a Hamiltonian matrix follows that (3.7a) is purely real

and does not contain any imaginary terms. This fact is interesting in the following com-

parison and we will revisit this property in Section 3.5, where we evaluate the complexity

of the mapping equations.


Furthermore, using this eigenvalue property the following simplifications can be made for

(3.7a) and (3.7b). Equation (3.7b) contains the factor ω, which can be neglected since

the solution ω = 0 is independently mapped using (3.8). After dropping this factor, both

equations contain only terms with even powers of ω.

The mapping equations (3.7), (3.8) and (3.9) have a similar structure like the well-known

equations for pole location specifications presented in Section 3.1, where (3.7) can be

interpreted as the condition for the CRB, (3.8) as the RRB equivalent, and (3.9) as

the IRB condition. Actually, using the above approach and Lyapunov’s famous matrix

equation for Hurwitz stability of an autonomous system x(t) = Px(t) [Boyd et al. 1994],

P TX + XP = −Q, Q = QT > 0,

leads to mapping equations for the CRB, RRB and IRB, which have the same solution set

as the equations (3.2) derived from the characteristic polynomial. The only difference is

that the ARE based mapping equations for Hurwitz stability are factorizable containing

the same term squared. This can be easily seen from the Hamiltonian matrix H (3.4),

where P appears twice on the diagonal with R = 0 .

3.3.2 H∞ Norm Mapping Equations

We will use the H∞ norm as an example for ARE based mapping equations. Many

important properties for this particular example are equivalent for other specifications

discussed in Chapter 2, for example passivity.

Using Theorem 3.1 and the ARE (2.32) of the bounded real lemma, we get the following

well-known theorem, see [Boyd et al. 1989], where this result is used to derive a numerical

bisection algorithm to compute the H∞ norm.

Theorem 3.4

Let A be stable, γ > σ(D) and define the matrices Sr = (D∗D − γ2I) and Sl =

(DD∗ − γ2I). Then ||G||∞ ≥ γ if and only if

Hγ =

A − BS−1

r DT C −γBS−1r BT

γCT S−1l C −AT + CT DS−1

r BT

(3.10)

has pure imaginary eigenvalues, i.e., at least one.

�


This theorem provides the Hamiltonian matrix needed in the general mapping equa-

tions (3.7), (3.8) and (3.9). In order to get the mapping equations, we have to compute

the determinant of the partitioned matrix det[jωI − H]. For a 2 × 2 block matrix

M =

M11 M12

M21 M22

the determinant is

det M = det M11 det[M22 − M21M

−111 M12

],

with M11 nonsingular. For ease of presentation let D = 0 , then det[jωI − Hγ] can be

written as

det [jωI − Hγ] = det [jωI − A] det[jωI + AT + γ2CT C(jωI − A)−1BBT

]. (3.11)

The first factor of the latter equation is just the Hurwitz stability condition. This is in line

with the fact that the H∞ norm requires the transfer function G(s) being stable. When

we directly compute det[jωI − Hγ] the stability related factor det[jωI − A] is canceled.

Hence, stability has to be mapped separately. Without this additional condition we

are actually mapping an L∞ norm condition. Concluding, apart from the H∞ norm

condition det[jωI − Hγ ], the Hurwitz stability condition has to be mapped additionally

using the mapping equations for Hurwitz stability described in Section 3.1.

The next point to notice about the H∞ norm mapping is that the static feedthrough

condition σ(D) < γ in Theorem 3.4 is implicitly mapped by (3.8) and (3.9). This can be

shown using a frequency domain formulation. See Section 3.7, which considers alternative

derivation of mapping equations for some specifications.

Note that the mapping equations for the H∞ norm are not just characterizing the pa-

rameters for which ||G||∞ = γ. Actually, all parameters for which G(s, q) has a singular

value σ(G) = γ are determined. Thus, we might get boundaries in the parameter space

for which the i-th biggest singular value σi has the specified value γ. This is similar to

eigenvalue specifications, where we get boundaries for each eigenvalue crossing of the Γ-

boundary. Analogously to the Γ-case, the resulting regions in the parameter space have

to be checked, if the specifications are fulfilled or violated (possibly multiple times).

If the critical gain condition on the static feedthrough matrix σ(D) = γ is fulfilled,

e.g., if D = γI, then Sl and Sr are singular. A remedy is to either consider a dif-

ferent performance level γ or to use a high-low frequency transformation and map the

condition ||G(s)||∞ = γ, where G(s) = G(s−1), since ||G||∞ = ||G||∞. A state-space

representation of G(s), which retains stability if A is stable, is given as

G(s) ∼=

A−1 −A−1B

CA−1 D − CA−1B

. (3.12)


To make the presented theory more clear we will consider a very simple exemplary system

in Example 3.2 on page 71.

3.3.3 Passivity Mapping Equations

The Hamiltonian matrix for dissipativity follows from the ARE (2.41) as

Hη =

A + BSC −BSB∗

C∗SC −(A + BSC)∗

, (3.13)

where S = (2ηI − D − D∗)−1. The specific mapping equations are then easily formed

using (3.7), (3.8), and (3.9).

In Section 2.4.2 the ARE for passivity was only valid for D 6= 0 . The following derivation

therefore relies on the more general dissipativity to obtain mapping equations for the not

uncommon case D = 0 .

For D = 0 , S reduces to S = 12η

I, and the Hamiltonian matrix Hη becomes

Hη =

A + 1

2ηBC − 1

2ηBB∗

12η

C∗C −(A + 12η

BC)∗

, (3.14)

The mapping equations for passivity can be obtained by evaluating the limit of the general

equations with H = Hη where η goes to zero, after extracting factors which only depend

on η. Since in general, det[jωI − Hη] is either fractionless with respect to η or a fraction

with denominator η, we get the relevant factor by simply evaluating the numerator.

More specifically the two mapping equations for the CRB condition are given as,

limη→0

num det [jωI − Hη] = 0, (3.15)

and

∂

∂ωlimη→0

num det [jωI − Hη] = 0. (3.16)

The RRB condition is given by

limη→0

num det Hη = 0, (3.17)

and the IRB follows as the leading coefficient of (3.15) with respect to ω

lcω limη→0

num det [jωI − Hη] = 0. (3.18)


For all four mapping equations (3.15)–(3.18) the limit has to be taken after the determi-

nant is calculated. A simple passivity example is presented in Example 5.6.

A dual approach to obtain mapping equations for passivity would be to use the results

from Section 2.4.3 and transform the passivity or positive realness condition into an H∞

norm problem.

Not surprisingly, we run into exactly the same singularity for D = 0 , which arises if we use

the passivity ARE (2.38) directly. Using Theorem 2.5 and the transformation (2.44) the

condition that G(s) = C(sI − A)−1B is positive real can be formulated as ||G(s)||∞ < 1,

where

G(s) ∼=

A − BC 2B

−C I

.

The resulting matrices Sr and Sl for the associated Hamiltonian matrix Hγ in Theorem 3.4

thus become zero.

3.3.4 Lyapunov Based Mapping

We now turn to special variants of AREs. If the quadratic term R of the general ARE (3.3)

vanishes, a so-called Lyapunov equation is obtained:

P ∗X + XP + Q = 0 , (3.19)

where Q = Q∗. Apply the Kronecker stacking operator (see Section A.1) to get

(P ∗ ⊕ P ) vec(X) = − vec(Q). (3.20)

This equation has a unique solution X if and only if λi(P ) + λj(P ) 6= 0, ∀ i, j.

In control theory, equation (3.19) is commonly used to test stability, controllability, and

observability. While stability is not of interest here, because it can be easily handled

by the classical PSA, Chapter 2 presented several specifications, e.g., the H2 norm, that

were formulated with observability and controllability Gramians Wobs and Wcon, and the

associated Lyapunov equations

A∗Wobs + WobsA + C∗C = 0 , (3.21a)

AWcon + WconA∗ + BB∗ = 0 . (3.21b)

We shall first consider the parametric solution of a Lyapunov equation. Equation (3.19)

is an affine equation in the unknown matrix elements xij of X = XT . Thus (3.19) consti-

tutes a system of 12n(n + 1) linear equations, where n is the dimension of the quadratic


matrix P ∈ Cn,n. This system can be readily solved for X(q) using a computer alge-

bra system for any parametric dependency of P (q) and Q(q). Note that the symmetry

of (3.19) should be used to minimize the computational effort, instead of using the full

system (3.20). In comparing (3.19) and (3.21), we see that the symbolic observability and

controllability Gramians Wobs(q) and Wcon(q) are obtained as the solution of a system

of 12n(n + 1) linear equations.

We will now present the mapping equation for the H2 norm. Using equation (2.56), a

specification on the H2 norm like ||G(s, q)||2 ≤ γ can be mapped into the parameter

space. In order to use (2.56) as a mapping equation, we need the parameter dependent

observability Gramian Wobs(q). This matrix can be obtained as described above.

Substituting this parametric solution Wobs(q) into (2.56), the parametric mapping equa-

tion is obtained as

||G(s)||22 = trace[B(q)T Wobs(q)B(q)

]= γ2, (3.22)

where γ2 specifies the desired performance level.

The dual output controllable formulation is given by

||G(s)||22 = trace [C(q)Wcon(q)C(q)] = γ2. (3.23)

Equation (3.22) is an implicit equation in the uncertain parameters q. Since the desired

or achievable performance level is not known a priori, it is recommended to vary γ and

determine the set of parameters P2,good for which ||G(s)||22 = γ2 holds for multiple values γ.

Also, a gray-tone or color coding of the different sets P2,good(γ) is useful. The visualization

of parameter space results will be considered in greater detail in Section 4.7.

The H2 norm mapping equation (3.22) is a single equation, that depends only on the

system parameters q. This is in line with the fact that the general definition of the H2

norm includes an integral over all frequencies. Thus there is no auxiliary variable α or

frequency ω in the mapping equations.

This fact makes (3.22) useful on its own, as opposed to being solely used for mapping H2

norm specifications into a parameter plane, especially to analyze and design control sys-

tems for more than two parameters. Furthermore (3.22) will be used to directly evalu-

ate H2 norms in Section 3.8.

If the parameters q enter in a polynomial fashion into A(q), B(q), C(q), the mapping

equation (3.22) is a polynomial equation. There may be some special cases, when (3.22)

is affine in one or more parameters, but in general this equation is polynomial in q, even

if A(q), B(q), and C(q) are affine in q.


In general there is no difference in using either (3.22) or (3.23). Nevertheless depending

on the complexity of B(q) and C(q) one or the other mapping equation might be easier

to solve in isolated cases.

3.3.5 Maximal Eigenvalue Based Mapping

We conclude this section with the remaining specifications presented in Chapter 2 for

which no mapping equations have been derived yet. Both the Hankel and the general-

ized H2 norm can be expressed as a function of a parametric matrix. These associated

matrices can be computed using the solution of parametric Lyapunov equations.

To get mapping equations for the Hankel and generalized H2 norms, we apply standard

results for mapping eigenvalue specifications. Namely a condition λmax(M) = γ, where M

is a non-negative matrix leads to the mapping equation

det [γI − M ] = 0.

Accordingly the condition dmax(M) = γ, M ≥ 0 leads to the system of mapping equations

mii = γ, i = 1, . . . , n.

3.4 IQC Parameter Space Mapping

Our goal is to use the unifying framework of IQCs to find mapping equations which allow

to incorporate an even larger set of specifications. We will show that mapping IQCs ex-

tends the parameter space beyond the ARE mapping equations presented in the previous

section.

Having found the mapping equations for general IQC specifications enables us to con-

sider specifications from the input-output theory, absolute stability theory and the robust

control field. Specifications from all these research fields can be used in conjunction

with the parameter space approach. Using the same mathematical formulation the same

computational methods can be used for these different specifications.

IQCs have been introduced in Chapter 2. A brief treatment of IQCs was given in Sec-

tion 2.5, where the basic stability condition (iii) of Theorem 2.9 is given as

G(jω)

I

∗

Π(jω)

G(jω)

I

≤ −εI, ∀ω ∈ R. (3.24)

In the current section we state the main result, the mapping equations for IQCs. We

will first consider fixed, frequency-independent multipliers. Section 3.4.4 will then show

3.4 IQC Parameter Space Mapping 57

how to map specifications based on IQCs with frequency-dependent multipliers Π(jω) into

parameter space. As an example a nonlinear system is analyzed in Section 3.4.4 using a

frequency-dependent multiplier.

For many uncertainties not just a single multiplier but sets of multipliers exist to char-

acterize the uncertainty structure. Often these sets can be described by parametrized

multipliers. We will show how to utilize these additional degrees of freedom in order

to minimize the conservativeness inherent in mapping only a single multiplier in Sec-

tion 3.4.5. An exemplary problem that utilizes LMI optimization is given in Example 3.1,

Section 3.4.5.

3.4.1 Uncertain Parameter Systems

Consider an uncertain LTI system G(s, q) ∈ RHl×m∞ interconnected to a bounded causal

operator ∆ as shown in Figure 2.10. The parameters q ∈ Rp are uncertain but constant

parameters with possibly known range. The operator ∆ might represent various types of

uncertainties, including constant uncertain parameters not contained in q. Let Π be a

constant multiplier that characterizes the uncertainty ∆ with partition

Π =

Π11 Π12

Π∗12 Π22

. (3.25)

Conditions (i) and (ii) of Theorem 2.9 are parameter-independent. Hence, the parameter-

dependent stability condition (3.24), which can be written as

G(jω)∗Π11G(jω) + Π22 + Π∗12G(jω) + G(jω)∗Π12 ≤ −εI, ∀ω ∈ R, (3.26)

has to be fulfilled by all parameters in Pgood.

The avenue to determine mapping equations will be the application of the Kalman-

Yakubovich-Popov (KYP) lemma. Previously known results on how to map specifications

expressible as AREs are then easily applied to IQC specifications.

3.4.2 Kalman-Yakubovich-Popov Lemma

In this section, the well known Kalman-Yakubovich-Popov (KYP) lemma is discussed.

The KYP lemma relates very different mathematical descriptions of control theoretical

properties to each other. In particular, it shows close connections between frequency-

dependent inequalities, AREs and LMIs. Nowadays a very popular application of the

KYP lemma is to derive LMIs for frequency domain inequalities, since efficient numerical

algorithms for the solution of LMI problems exist.


Theorem 3.5 (Kalman, Yakubovich, Popov)

Let (A, B) be a given pair of matrices that is stabilizable and A has no eigenvalues

on the imaginary axis. Then the following statements are equivalent.

(i) R > 0 and the ARE

Q + XA + AT X − (XB + S)R−1(XB + S)T = 0 (3.27)

has a stabilizing solution X = XT ,

(ii) the LMI with unknown X

XA + AT X XB

BT X 0

+

Q S

ST R

> 0 (3.28)

has a solution X = XT ,

(iii) for a spectral factorization the condition

(jωI − A)−1B

I

∗

Q S

ST R

(jωI − A)−1B

I

> 0 (3.29)

holds ∀ω ∈ [0;∞),

(iv) there is a solution to the general LQR problem

minu

J =

∫ ∞

0

(x(t)T Qx(t) + 2x(t)T Su(t) + u(t)T Ru(t)) dt (3.30)

with x(t) = Ax(t) + Bu(t), x(0) = x0 and limt→∞ x(t) = 0.

(v) R > 0 and the Hamiltonian matrix

H =

A − BR−1ST BR−1BT

Q − SR−1ST −AT + SR−1BT

(3.31)

has no eigenvalues on the imaginary axis.

Proof. See, for example, [Rantzer 1996] or [Willems 1971].

�

In order to apply the KYP lemma we will need the following remark.


Remark 3.1

The spectral factorization condition (3.29) of the KYP lemma can be extended to

the case where a transfer function G(s) = C(sI − A)−1B + D appears in the outer

factors similar to the IQC condition (2.65):

G(jω)

I

∗

Π11 Π12

Π∗12 Π22

G(jω)

I

=

(jωI − A)−1B

I

∗

M

(jωI − A)−1B

I

, (3.32)

where

M =

Q S

ST R

=

C D

0 I

T

Π11 Π12

Π∗12 Π22

C D

0 I

.

�

Remark 3.2

The minimum of J in (3.30) equals x0X+x0, where X+ is the largest symmetric

solution of the ARE (3.27). The optimal input u is then given by

u(t) = −Kx(t), where K = R−1(XB + S).

�

3.4.3 IQC Mapping Equations

We are now ready to derive the main result, the mapping equations for IQC specifications.

Let G(s, q) have a state-space realization A(q), B(q), C(q), D(q), i.e.,

G(s, q) = C(q)(sI − A(q))−1B(q) + D(q).

We will not express the parametric dependence of matrices in the remainder for notational

convenience.

Using Remark 3.1 the basic IQC condition (2.65) in Theorem 2.9 for a constant multi-

plier Π with partition (3.25) can be transformed into the condition

(jωI − A)−1B

I

∗

M

(jωI − A)−1B

I

≤ −εI, (3.33)


where the multiplier is transformed into

M =

CT Π11C CT (Π12+Π11D)

(DT Π11+Π∗12)C DT Π11D+Π∗

12D+DT Π12+Π22

.

Since we are interested in mapping equations describing the boundaries of a parameter

set Pgood, we consider marginal satisfaction of (3.33), i.e., ε = 0.

Now, use statements (3.29) and (3.31) of the KYP lemma to get the equivalent condition

that the Hamiltonian matrix

H =

A 0

CT Π11C −AT

−

B

CT (Π12 + Π11D)

Π−122

CT (Π12 + Π11D)

−B

∗

, (3.34)

with Π22 = Π22 + DT Π12 + Π∗12D + DTΠ11D has no eigenvalues on the imaginary axis.

We have now formulated the adherence of a given IQC specification as the non-existence

of pure imaginary eigenvalues of an associated Hamiltonian matrix. Using Theorem 3.3

and the same arguments and methods as in Section 3.3.1, we can substitute (3.34) into

the ARE based mapping equations (3.7a), (3.8) and (3.9) to get mapping equations for

IQC based specifications. Thus the KYP lemma established the important relationship

which allows to derive mapping equations for IQC conditions. The consequence is that

the statements about the properties of ARE based mapping equations can be directly

transferred to the IQC mapping equations, e.g., the non-sufficiency of (3.8).

We extend the previous IQC results by considering frequency-dependent multipliers in

the next section.

3.4.4 Frequency-Dependent Multipliers

Consider the case when the multiplier Π is frequency-dependent, i.e., Π = Π(jω).

The common frequency domain criterion used in conjunction with IQCs for frequency-

dependent multipliers is

G(jω)

I

∗

Π(jω)

G(jω)

I

< 0, ∀ω ∈ [0;∞), (3.35)

where G(jω) ∈ RHl×m∞ and Π(jω) ∈ RHl+m×l+m

∞ .

A particular example of a frequency-dependent multiplier is the strong result by Zames

and Falb [1968] for SISO nonlinearities1. Put into the IQC framework, an odd nonlinear

1The results in [Zames and Falb 1968] can be only applied to MIMO nonlinearities using additional

restrictions, see [Safonov and Kulkarni 2000].


operator, e.g., saturation, satisfies the IQC defined by

Π(jω) =

0 1 + L(jω)

1 + L(jω)∗ −2 − L(jω) − L(jω)∗

,

where L(s) has an impulse response with L1 norm less than one.

Following [Megretski and Rantzer 1997], any bounded rational multiplier Π(jω) can be

factorized as

Π(jω) = Ψ(jω)∗ΠsΨ(jω), (3.36)

where Ψ(jω) absorbs all dynamics of Π(jω) and Πs is a static matrix. Actually factor-

ization (3.36) is known as a J-spectral factorization, which plays an important role on

its own in H∞ and H2 control theory [Green et al. 1990], or as a canonical Wiener-Hopf

factorization in operator theory [Bart et al. 1986].

We will now derive mapping equations for frequency-dependent multipliers by simple

transformations and equivalence relations. Rewrite (3.36) as

Π(jω) =

Ψ(jω)

I

∗

Πs 0

0 0

Ψ(jω)

I

, (3.37)

and apply a transformation similar to Remark 3.1, where Ψ(jω) has state-space represen-

tation Ψ(jω) = Cπ(jωI − Aπ)−1Bπ + Dπ to get

Π(jω) =

(jωI − Aπ)−1Bπ

I

∗

CT

π ΠsCπ CTπ ΠsDπ

DTπ ΠsCπ DT

π ΠsDπ

︸︷︷︸

Mπ

(jωI − Aπ)−1Bπ

I

. (3.38)

Next, partition the input matrix Bπ of Ψ(jω) according to the signals in the general IQC

feedback loop (see Figure 2.10) as Bπ = [Bπ,v Bπ,w], which was suggested in the deriva-

tion of a linear quadratic optimal control formulation of (3.35) in [Jonsson 2001], and

use (3.38) to write condition (3.35) as

(jωI − Aπ)−1(Bπ,vG(jω) + Bπ,w)

G(jω)

I

∗

Mπ


G(jω)

I

< 0, (3.39)

for all ω ∈ [0;∞).


Let G(jω) = C(jωI − A)−1B + D, then the following state-space representation for the

factor to the right of Mπ in (3.39) can be deduced:


G(jω)

I

=

Aπ Bπ,vC Bπ,vD + Bπ,w

0 A B

I 0 0

0 C D

0 0 I

(3.40)

=

A B

C D

. (3.41)

Using this state-space representation, condition (3.39) becomes

(C(jωI − A)−1B + D)∗Mπ(C(jωI − A)−1B + D) < 0, ∀ω ∈ [0;∞), (3.42)

or

(jωI − A)−1B

I

∗

CT

DT

Mπ

[

C D]

︸︷︷︸

M

(jωI − A)−1B

I

< 0, ∀ω ∈ [0;∞). (3.43)

As a result, we have obtained an IQC condition with a frequency-independent multi-

plier M , where

M =

CT

DT

Mπ

[

C D]

=

Q S

ST R

, (3.44)

and where (A, B) represents an augmented system composed of both the LTI system

dynamics and the multiplier dynamics. The multiplier matrices can be computed as,

Q =

CT

π ΠsCπ CTπ ΠsDπ,vC

CT DTπ,vΠsCπ CT DT

π,vΠsDΠ,vC

, (3.45)

S =

CT

π Πs(Dπ,vD + Dπ,w)

CT DTπ,vΠs(Dπ,vD + Dπ,w)

, (3.46)

R =[

(Dπ,vD + Dπ,w)T Πs(Dπ,vD + Dπ,w)]

. (3.47)

As in the frequency-independent case discussed in the previous section, we can now apply

the KYP lemma in order to obtain the Hamiltonian matrix which will lead to the map-

ping equations. Namely, use statements (3.29) and (3.31) of the KYP lemma to get the


equivalent condition that the Hamiltonian matrix

H =

A − BR−1ST BR−1BT

Q − SR−1ST −AT + SR−1BT

(3.48)

has no eigenvalues on the imaginary axis.

Hence frequency-dependent bounded rational multipliers Π(jω) can be mapped into pa-

rameter space using basic matrix transformations and the results from the previous sec-

tion.

3.4.5 LMI Optimization

For a system with fixed parameters, all multipliers considered so far led to a simple

stability test, which could be evaluated by computing the eigenvalues of a Hamiltonian

matrix (3.31). For systems with uncertain parameters q ∈ Rp, we showed how to map

an IQC specification into a parameter plane. But in general, the uniqueness of a multiplier

is not given.

While the main idea behind the IQC framework is to find a suitable multiplier for an

uncertainty, for many uncertainties a set of possible multipliers exists. Especially for

nonlinearities and time-delay systems there is an enormous list of publications involving

different multipliers. See [Megretski and Rantzer 1997] for some references. Depending on

the considered LTI system one or the other multiplier might be advantageous and yield

less conservative results.

For example, consider the following multiplier from [Jonsson 1999] for a system

x(t) = (A + U∆V )x(t), (3.49)

with slowly time-varying uncertainty ∆ and known rate bounds

Π(jω) =

Z Y T − jωW T

Y + jωW −X

. (3.50)

Jonsson [1999] derives a set of LMI conditions to check stability involving real matri-

ces W, X = XT , Y and Z = ZT . These matrices can be easily obtained solving a convex

optimization problem. The result of the optimization is not only a binary stability check,

but also an optimal multiplier Π(jω).

There are two different possibilities to exploit the degrees of freedom in the multiplier

formulation during the mapping process.


One approach would be to use a limited set of parameter points (q1, q2) for which we

obtain optimal multipliers and subsequently determine the set of good parameters Pgood

for each individual multiplier. The actual overall set of uncertain parameters which fulfill

the specification is then given as the union of all individual good sets.

The second approach could be denoted as adaptive multiplier mapping. Hereby we ob-

tain successive multipliers as we actually determine the boundary by moving along the

boundary of the set Pgood. Thus we adaptively correct the optimal multiplier on the way

as we generate the boundary by solving an underlying optimization problem.

While the first approach needs to solve a limited and predefined number of optimization

problems, the adaptive multiplier mapping requires a possibly large number of optimiza-

tion, which is not known a priori. Nevertheless the second approach gives the actual

set Pgood directly and there is no need to determine the union of individual sets. Fur-

thermore, when the actual mapping is expensive, it might be favorable to use a single

adaptive mapping run.

Example 3.1 The following example shows how the IQC results of the current section

can be used to extend the parameter space approach to stability checks with respect to

varying parameters.

Consider the following parametric system, which fits into the setup of problem (3.49),

x(t) =

q1 + q2 1 + q2

−3 −12− q1

+

0 2

5

25

0

∆

x(t), (3.51)

where ∆ is a diagonal matrix containing an arbitrarily fast varying parameter δ. The

parameter δ is assumed to be bounded in the interval [0; 1].

Note that if we treat δ as a constant parameter, then the set of good parameters is given

by δ = 0 since all stability sets for δ > 0 contain the set for δ = 0. So from the perspective

of constant parameters we do not lose anything, if we allow δ to deviate from δ = 0 to

values in the interval [0; 1].

We assume arbitrary fast variations of the parameter δ, therefore the matrix W , which

appears in (3.50), equals zero. Following [Jonsson 1999], system (3.49) is stable with

respect to ∆, if a feasible solution for the following LMI problem exists.


−3 −2 −1 0 1 2−3

−2

−1

0

1

2

q1

q2

Figure 3.3: Stability regions using adaptive LMI optimization

Find X = XT , Y, Z = ZT , P = P T such that,

X ≥ 0, (3.52)

AT P + PA + V T ZV PU + V T Y T

UT P + Y V −X

< 0, (3.53)

Z > 0, (3.54)

Z + Y + Y T − X > 0. (3.55)

We determine the stability boundaries in the q1, q2 parameter plane. In the first step we

calculate the maximal stability boundary using adaptive multiplier mapping. In each step

optimal values for the X, Y, and Z matrices, which determine the multiplier (3.50), are

computed. The resulting stability boundaries are shown in Figure 3.3 as a solid curve.

The figure also shows the common real root boundary, assuming uncertain, but constant δ

values, as a dotted line, and the complex root boundaries for δ = 0, and δ = 1 as dashed

lines.

Figure 3.4 shows the stability regions depicted as solid curves, which are obtained using

optimal multipliers for the points (0, 0), (1, 0), and (−1,−0.5). For comparison the

stability region obtained using adaptive multiplier mapping is shown as a dash-dotted


−3 −2 −1 0 1 2−3

−2

−1

0

1

2

q1

q2

Figure 3.4: Stability regions using multiple fixed IQC multipliers

curve. The figure shows that a small number of reference points might suffice to get a

rather accurate approximation of the true stability region. It should be noted that the LMI

problem (3.52) is only a feasible problem. Although the resulting multiplier guarantees

stability for the reference point, i.e. A(q1, q2), the resulting size of the stability region in

the (q1, q2) parameter plane is not necessarily maximal. Thus treating the parameters q1

and q2 as additional uncertainty to the ∆ uncertainty and augmenting the LMI problem

might lead to even larger stability regions.

For q1 = 0 and q2 > 0 the true stability limit is q2 = 0.23018, and instability occurs

near the origin for a switching behavior of δ. Whereas the optimal multiplier yields a

guaranteed stability for q2 = 0.067, which exemplifies the conservativeness of this IQC

condition. Nevertheless Figure 3.3 and Figure 3.4 show that the IQC condition gives

very accurate results for stability regions, where the varying parameter does not affect

the stability region. And the simplicity of the used multiplier (frequency-independent)

suggests that a more complicated multiplier might even improve the shown results. �

3.5 Complexity 67

3.5 Complexity

This section discusses the complexity of the mapping equations derived above. In par-

ticular we will look at the order of the equations as a function of the number of states

in the system and as a function of the input and output dimensions. Furthermore the

complexity with respect to the uncertain parameters q is considered.

3.5.1 ARE Mapping Equations

The complexity of ARE based mapping equations is determined by the Hamiltonian ma-

trix H defined in (3.4). The corresponding Hamiltonian matrix for all ARE expressible

specifications in this thesis, i.e., H∞ norm (2.32), passivity (2.38), dissipativity (2.41), cir-

cle criterion (2.48), Popov criterion (2.51) and complex structured stability radius (2.54),

can be written in the following form:

H =

A + BS1C −BS2B

T

CT S3C −(A + BS1C)T

, (3.56)

where Si, i = 1, . . . , 3, are specification dependent, nonsingular matrices, which depend

on D. For uncertain parameters q in the plant description, all matrices A, B, C and Si

can be parameter dependent.

The dimension of H depends only on the size of A ∈ Rn,n, where n equals the number

of states in the system, H ∈ R2n, 2n. The dimension of H does not increase with the

number of inputs and outputs. Thus complexity does not increase when we consider

MIMO instead of SISO systems.

The order of det[jωI −H] with respect to ω is 2n and corresponds directly to the number

of states. Note that (3.7a) has only terms with even powers of ω. Thus after we eliminate

the factor ω, the second, double root condition (3.7b) has order 2(n − 1) with respect

to ω.

The order of the mapping equations with respect to a single parameter q or qi can be

determined by studying a general determinant. The determinant of an arbitrary ma-

trix M ∈ Rn,n can be calculated as the algebraic sum of all signed elementary products

from the matrix. An elementary product is given by multiplying n entries of an n × n

matrix, all of which are from different rows and different columns:

det M =∑

π∈Sπ

sgn(π)m1π1m2π2 · · ·mnπn, (3.57)

where the index π in the above sum varies over all possible permutations Sπ of {1, . . . , n}.The total number of possible permutations is n!, which therefore equals the number of

terms in the defining sum of determinant.


For n = 3 the position of the individual factors for the 3! elementary products is shown

below:

∗∗

∗

∗∗

∗

∗∗

∗

∗∗

∗

∗∗

∗

∗∗

∗

.

The matrix A appears twice on the diagonal of H. Because an entry aij appears both at hij

and hn+j,n+i, there is always an elementary product that contains both hij and hn+j,n+i.

Thus the general determinant det[jωI − H], depends quadratically on the entries of A.

Hence, even if the entries of A depend affinely on qi we will get quadratic mapping equa-

tions in qi. This holds for special canonical representations, which yield a characteristic

polynomial with affine parameter dependency. The quadratic dependency of the map-

ping equations makes it clear that, even for really simple parameter dependence, we have

to deal with mapping equations that are more complicated than the mapping equations

obtained for eigenvalue specifications.

For parametric entries of the input and output matrices B and C it is obvious that we

get quadratic terms for each entry, since the Hamiltonian matrix H already has quadratic

terms due to BS2BT and CTS3C.

The appearance of entries from the D matrix in the final mapping equations depends

on the considered specification. Thus, there is no obvious way to show the dependence

of the mapping equations with respect to parameters in D. Nevertheless, evaluating the

dependence for each specification expressible as an ARE or by looking at the general IQC

Hamiltonian matrix (3.34), with Π22 = Π22 + DT Π12 + Π∗12D + DT Π11D, we can deduce

that entries of D will reappear as quadratic terms in the mapping equations.

3.5.2 Lyapunov Mapping Equations

Lyapunov based mapping equations can be obtained by evaluating one of the two equa-

tions given in (3.21). In order to determine the mapping equations we first have to solve

a system of linear equations, where the parametric coefficient matrix has size 12n(n + 1),

where n is the number of states of the considered transfer matrix. The resulting Gramian

has the same dimensions as A.

In the final step, we have to compute a parametric matrix product including the just

obtained Gramian, and determine the trace. Thus the main obstacle to compute Lyapunov

mapping equations is the symbolic solution of a linear system.

3.6 Further Specifications 69

3.5.3 IQC Mapping Equations

The order of the IQC mapping Hamiltonian matrix is given by the number of plant states

plus the number of states required to realize the multiplier. Thus, only for frequency-

dependent multipliers there is a multiplier state-space augmentation, and the complexity

increases. Hence, the accuracy gained by using a higher order multiplier has to be paid

for by an increased Hamiltonian matrix complexity and therefore increased computational

requirements.

3.6 Further Specifications

So far we covered a list of specifications that includes almost the entire list of specifi-

cations commonly used in control engineering. Nevertheless some candidates of unequal

importance are missing: the structured stability radius µ, the entropy, and the L1 norm.

The structured stability radius µ has been introduced to cope with parametric uncer-

tainties in the H∞ norm framework. To date there are only approximate methods to

estimate its value. So, there is no way to map µ specifications into a parameter space.

Since the structured stability radius tries to deal with mixed uncertainties it considers a

similar problem as the methods presented in this thesis. Having the definition of µ in

mind, we do not expect advantageous insights of mapping a specification for parametric

uncertainties into a parameter plane.

Another specification considered in the control literature is the entropy of a system. This

entity is actually not a norm, but it is closely related to the H2 and H∞ norm.

The γ-entropy [Mustafa and Glover 1990] of a transfer matrix G(s) is defined as

Iγ(G) :=

−γ2

2π

∫ ∞

−∞

ln det[I − γ2G(jω)G(jω)∗

]dω, if ||G||∞ < γ,

∞, otherwise.

(3.58)

The γ-entropy has been used as a performance index in the context of so-called risk

sensitive LQG stochastic control problems, and reappeared in the H∞ norm framework

in [Doyle et al. 1989] as the so-called central controller, where the selected controller

satisfying an H∞ norm condition minimizes the γ-entropy.

The γ-entropy of G(s), when it is finite, is given by

Iγ(G) = trace [BT XB], (3.59)

where X is the positive definite solution of the ARE

AT X + XA + CT C +1

γ2XBBT X = 0 . (3.60)


Although the familiar ARE might suggest that mapping equations for this specifications

can be derived, this is not the case. The existence of a solution to (3.60) can be tested with

the familiar Hamiltonian matrix (3.6). But in order to compute and map Iγ we actually

need this solution. Therefore the entropy does not fit into the presented framework and

no algebraic mapping equations can be derived.

Another example of a specification, that cannot be easily mapped, is the L1 or peak gain

norm:

||G(s)||1 := sup||w||∞ 6=0

||Gw||∞||w||∞

. (3.61)

For a given system, the total variation of the step response is the peak gain of its transfer

function, see [Lunze 1988]. This specification is not computable by algebraic equations.

Boyd and Barratt [1991] suggest to use numerical integration to calculate it. A remedy

might be to use lower and upper bounds given in [Boyd and Barratt 1991], and map H∞

and Hankel norm specifications. Namely, if G(s) has n poles, then the L1 norm can be

bounded by the Hankel and H∞ norm:

||G||∞ ≤ ||G||1 ≤ (2n + 1)||G||hankel ≤ (2n + 1)||G||∞. (3.62)

3.7 Comparison and Alternative Derivations

A particular strength of the approach pursued in this thesis is the generality. All spec-

ifications expressible as an ARE or Lyapunov equation can be considered. Nevertheless

there are alternative derivations of mapping equations for some specifications.

Most authors to date have considered either eigenvalue or frequency domain specifications

for SISO systems [Besson and Shenton 1999, Bunte 2000, Hara et al. 1991, Odenthal

and Blue 2000]. Reasons for this might be the elegant derivation of eigenvalue mapping

equations, which are computationally very attractive for practically relevant problems.

For frequency domain specifications the extension to MIMO systems is nontrivial and the

complexity associated with SISO mapping equations might have hindered the application

to MIMO systems.

3.8 Direct Performance Evaluation

The PSA maps specifications into a parameter plane. We are neither interested in the

direct, numerical evaluation of a specification, e.g., ||G(s, q∗)||∞, where q∗ is a fixed pa-

rameter vector, nor in the solution of an optimal control problem traditionally considered

3.8 Direct Performance Evaluation 71

in the H2 and H∞ literature. Nevertheless the presented mapping equations can be used

for the direct performance evaluation for some of the presented specifications.

Although in a different mathematical framework, Schmid [1993] devoted a large portion

of his work to the direct evaluation of the H∞ norm, see also [Kabamba and Boyd 1988].

The mapping equations establish conditions, that allow the direct computation of the H∞

norm. Thus, instead of using the numerically attractive bisection algorithm widely used

in control software, the H∞ norm can be computed by solving the two algebraic equa-

tions (3.7a) and (3.7b) in the two unknowns ω and γ. These positive solutions provide

candidate values γ for the H∞ norm. Additionally, the solutions of (3.8) and (3.9) are

computed and the H∞ norm of the system is given by the maximal value over all candi-

date solutions. As a byproduct, we get the frequency ω, for which the maximal singular

value occurs.

For the H2 norm the direct evaluation is simply possible by solving the linear equa-

tion (3.21b) and substituting the solution into the mapping equation (3.22).

Example 3.2 We use the mapping equations to directly compute the H∞ norm for the

open-loop transfer function

G(s) =1

(s + 1)(s + 2)

2 −2s

s 3s + 2

. (3.63)

A state-space representation for G(s) is given by

G(s) ∼=

−2 0

0 −1

1 2

1 1

−2 2

2 −1

0 0

0 0

. (3.64)

The mapping equations (3.7a) and (3.7b) become

e1 = γ4ω4 + (5γ4 − 14γ2)ω2 + 4γ4 − 8γ2 + 4,

e2 = ω(4γ4ω2 + 10γ4 − 28γ2).

This polynomial system of equations has only one single relevant solution ω = 1, γ =√

2

with ω, γ > 0.

The DC-gain condition (3.8) is

det Hγ = γ4 − 2γ2 + 1,


which has the positive solution γ = 1. Note that this can be easily observed by the fact

that G(j0) = I, and thus the singular values are given by σ1 = σ2 = 1.

There is no solution for (3.9) and we can conclude that the maximal singular value of G(s)

is given for ω = 1 and ||G(s)||∞ =√

2. �

3.9 Summary

The main contribution of this chapter is the presentation of a uniform framework that

allows to derive mapping equations for parametric control system specifications expressible

by AREs.

Besides ARE expressible specifications, previously unknown mapping equations for IQC

based stability tests are determined. Using the results in this section we can draw from the

vast number of available IQCs and incorporate them into the parameter space approach.

Furthermore, application of standard parameter space methods allows to include an even

larger list of specifications into control system analysis and design.

The resulting equations are similar to well-known Γ-stability mapping equations. This

allows similar computational methods for the mapping, although the complexity is in

general higher, due to the quadratic nature of the specifications.

73

4 Algorithms and Visualization

The purpose of computing is insight, not numbers.

Richard Hamming

The main contribution of this chapter is the presentation of algorithms which solve the

mapping problem, i.e., they plot the critical parameters in a parameter plane. Thus these

algorithms are a prerequisite to the successful application of the presented MIMO control

specifications in the parameter space context.

Geometrically, the mapping is either a curve plotting or surface-surface intersection prob-

lem. These can be approached with a variety of different techniques, including numerical,

analytic, geometric, and algebraic methods, and using various methods such as subdivi-

sion, tracing, and discretization. While classic algorithms used a single method, recently

algorithms which combine multiple methods or so-called hybrid algorithms appeared.

We will follow this line and combine previous known results such that the new combination

of basic building blocks forms an efficient algorithm for the mapping problem.

The robustness of the algorithm has to be considered. It is easy to generate a specialized,

fast algorithm for special curve intersections. This is actually the topic of solid geom-

etry, vision or computer-aided design, where the complexity of the geometrical objects

considered is known beforehand.

For practical experimentation the algorithms were implemented using Maple and Mat-

lab.

The Lyapunov based mapping equations in Section 3.3.4, e.g., for the H2 norm, directly

lead to an implicit polynomial equation

f(x, y) =

n∑

i,j=0

aijxiyj = 0, (4.1)

where x and y represent the two parameters of the parameter plane. In order to plot the

parameters, which satisfy a control system specification, we have to determine the curve Cof real solutions satisfying (4.1). General properties of algebraic curves will be presented

in Section 4.2.


For ARE based mapping equations we get two polynomial equations

f1(x, y, ω) = 0,

f2(x, y, ω) = 0,(4.2)

and we are interested in the plane curve of real solutions (x, y) ∈ R2 which can be

obtained for all positive ω. Mathematically (4.2) defines a spatial curve in R3. Thus we

are interested in the plane curve C given as the projection of the spatial curve (4.2) onto

the plane ω = 0. Since (4.2) is a generalization of (4.1) we will first treat plane parameter

curves, and consider the extension to (4.2) in Section 4.5.

4.1 Aspects of Symbolic Computations

This section considers general aspects of symbolic computation and compares them to nu-

merical computations. This gives rationals for designing hybrid algorithms which employ

both symbolic and numerical computations, such that the overall results are obtained fast

and robust.

Symbolic computation is considerably different from numerics, as new aspects such as

internal data representation, memory usage, and coefficient growth arise, while difficult

aspects of numerical algorithms, e.g., rounding errors and numerical instability, disap-

pear [Beckermann and Labahn 2000]. The main advantage of symbolic computations is

the fact that general equations with unknown variables can be solved. Furthermore the

exact arithmetic avoids common problems of numerical algorithms introduced by finite

precision.

A common symbolic algebra package such as Maple can handle 216−1 individual symbolic

expressions. Compare this to the determinant of a matrix A with unknown entries aij.

The determinant of a matrix with size n has n! expressions. Thus even the modest size

of n = 9 exceeds the capacities of available software. And since the factorial grows much

faster than polynomial, even with more memory and computer power, the memory usage

will be always a limiting factor in the application of symbolic algorithms to real world

problems.

These limitations impose some restrictions on the application of the presented control

specifications. Although the mapping equations given in Chapter 3 can be computed

symbolically, treating all parameters as variable complicates the computations drastically.

A remedy is to substitute all parameters not part of the parameter plane into which the

specifications are mapped with their numerical values. For example, when we are mapping

the H2 norm specification into a q1, q2 parameter plane, all remaining parameters in the

state space model should be replaced by their numerical values prior to the mapping

4.2 Algebraic Curves 75

equation computation. Thus, if we are gridding a parameter q3, we will generate the

mapping equations for each grid point. This is much more attractive than computing

universal mapping equations, where q3 appears as a free parameter.

The computational complexity of symbolic algorithms cannot be measured by operational

complexity. The main reason is the varying cost associated with operations such as

addition or multiplication. The cost here depends on the size of the components and the

size of the result. This is related to the way computer algebra systems such as Maple

store expressions. Here all expressions are stored in an expression tree or more precise a

direct acyclic graph. The following example shows the possible explosion on the number

of expressions for a very small problem:

p1(s) = s2 + q1s + q2,

p2(s) = s2 + (q1 − q2)s + q1,

p1(s)p2(s) = s4 + (2q1 − q2)s3 + (q2

1 − q1q2 + q1 + q2)s2 + (q2

1 − q22 + q1q2)s + q1q2.

So in order to evaluate the computational complexity for symbolic algorithms we need bit

complexity. The cost of intermediate operations might be especially large if the coefficients

of expressions are in the field of quotients Q. Here the cost might grow exponentially.

Thus fraction-free algorithms which avoid expressions lying in Q are very attractive from

a computational cost view.

4.2 Algebraic Curves

This section presents some basic facts about algebraic curves which will be important

for the algorithms presented in the subsequent sections. The content presented here is

rather self contained, since in general algebraic geometry and curves are not covered

in basic engineering courses. While algebraic curves can be generally represented in

explicit, implicit and parameter form, we will only consider curves defined in implicit

form f(x, y) = 0 as in (4.1), because this is the natural description of the mapping

equations.

For each summand aijxiyj of (4.1), the degree is defined as the sum of the individual

powers dij = i + j. The total degree of the polynomial is then given as the maxi-

mum deg f = max dij. For systems of polynomials as in (4.2), the complexity is measured

by its total degree, which is the product of the total degrees of all individual polynomi-

als deg f =∏

deg fi.

We will distinguish two different classes of points or solutions of f(x, y) = 0, which arise

in the following definition.


Definition 4.1 We call (x0, y0) ∈ C a singular point of curve (4.1), if both partial deriva-

tives of f vanish at that point:

fx(x0, y0) = fy(x0, y0) = 0,

If a point is not singular, then it is called regular. A curve of degree d has at most 12d(d−1)

singular points. A curve C is regular, if every point of C is regular. �

The singular points are not only important for the topology of a curve, e.g. number of self

intersecting points, but are also vital to numerical algorithms. In most cases numerical

algorithms will become inaccurate or show slow convergence in the vicinity of singular

points. Furthermore, phenomena such as the birth of new branches (or more formal

bifurcations) and branch switching can happen at singular points.

A remedy for singular points is to develop algorithms which either can handle singular

points or compute and detect them and switch to specialized methods to handle singular

points.

If point (x0, y0) is a regular point of C then C has a well-defined tangent direction

at (x0, y0), with tangent line equation

(x − x0)fx(x0, y0) + (y − y0)fy(x0, y0) = 0. (4.3)

For singular points the tangent lines (possibly multiple) have to be computed using the

higher derivatives. A singular point is said to have multiplicity m, if all partial derivatives

up to degree m vanish.

The fundamental theorem of algebra tells that a polynomial of degree n has n roots

in C. The number of possible intersection points of two curves, defined implicitly by two

polynomials, is bounded by Bezout’s theorem. Applied to curves, it states that in general

two irreducible curves of degree m and n have exactly mn common points in the complex

plane C2, i.e., they intersect in at most mn real points [Coolidge 2004].

Besides the singular points, there are other special points of curves. First consider so-

called critical points with horizontal or vertical tangent, i.e., a partial derivative fx(x0, y0)

or fy(x0, y0) equals zero.

Other important points are inflection points, or flexes. A regular point of a plane curve

is an inflection point, if the tangent is parallel to the curve, i.e., the curvature is zero. A

regular point (x0, y0) of f(x, y) = 0 is an inflection point, if and only if

det

fxx fxy fx

fxy fyy fy

fx fy 0

= 0. (4.4)


4.2.1 Asymptotes of Curves

Another property of curves are asymptotes. These are lines which are asymptotically

approached. In general, points on an asymptote do not belong to the solution of (4.1).

An easy and mathematical complete description of asymptotes can be obtained by homog-

enization of the polynomial. A homogeneous polynomial contains only irreducible com-

ponents with the same degree n. Any algebraic curve is homogenized by introducing the

auxiliary variable z and multiplying each term in fi with zm such that the new term is of or-

der n. The obtained polynomial p(x, z) contains the original problem since p(x, 1) = f(x).

Furthermore for z = 0 we get the solutions at infinity. The asymptotes of the homogenized

polynomial are thus given by the solution of p(x, 0).

The actual asymptotes of a curve might possess an offset

cxx + cyy + c0 = 0,

which is easily computed by substituting this equation into the curve equation. If an

offset c0 6= 0 exists, the resulting equation has to be zero for all values of x, y respectively.

Thus, we can for example solve the coefficient f(c0) of the highest degree of x.

Example 4.1 Consider the curve y3−y2−yx2+2xy+x2+1. The homogenized polynomial

is p(x, y, z) = y3 − y2z − yx2 + 2xyz + x2z + z3. Setting z = 0 the terms relevant for the

asymptotes are y3 − yx2. This polynomial can be easily factored into (y − x)(y + x)y.

The actual asymptotes are then computed as y − x + 1 = 0, y + x− 1 = 0 and y = 1. �

4.2.2 Parametrization of Curves

Parametric curves of the form

[x(α), y(α)], α ∈ R, (4.5)

where x(α) and y(α) are rational polynomials in α, are very easily plotted by evaluating

the polynomials for sufficiently many values of parameter α. This form can be always

transformed into an implicit curve (4.1) by eliminating α, e.g., using resultants. The

opposite transformation, called parametrization, is not necessarily possible. The theory

of algebraic curves states that an implicit curve can be parametrized if and only if the

curve is rational. Rationality of a curve is given, if the so-called genus of a curve equals

zero [Walker 1978]. The genus is characterized by the degree and properties of the singular

points. Let C be an irreducible curve of degree d which has δ double points and κ cusps.

Then

genus(C) =1

2(d − 1)(d − 2) − (δ + κ). (4.6)


Note that the curve is reducible if genus(C) < 0. As a consequence of (4.6), linear

and quadratic curves can be always parametrized. For cubic curves (d = 3), at least

one singularity (double point or cusp) has to be present. So in general for curves with

degree d > 2, it is not possible to derive a rational parametrization. See [van Hoeij 1997]

for a method to compute parametrizations for rational curves.

4.2.3 Topology of Real Algebraic Curves

The set of real zeros Cf = {(x, y) ∈ R2 | f(x, y) = 0} of a bivariate rational polynomial

is usually referred to as a real algebraic curve. Such a curve might have special points or

singularities, where the tangent is not well defined, e.g., isolated points, self-intersections,

or cusps. Figure 4.1 depicts the quartic curve

f(x, y) = (x2 + 4y2)2 − 12x3 + 96xy2 + 48x2 − 12y2 − 64x, (4.7)

which has three singularities, two self-intersections and a cusp.

−3 −2 −1 0 1 2 3 4 5−3

−2

−1

0

1

2

3

x

y

Figure 4.1: A quartic curve with exactly three singularities

Numerous papers, e.g., [Arnon and McCallum 1988, Sakkalis 1991], consider the problem

of determining the topology of real algebraic curves defined by a polynomial f(x, y) = 0.

The topology is approached by means of an associated graph, that has the same critical

points as vertices, and where an edge of the graph represents a curve segment connecting

two vertices. Figure 4.2 shows the topological graph of curve (4.7).


The common steps to compute the topological graph of f(x, y) are (see e.g., [Gonzalez-

Vega and Necula 2002]):

1. Determine the x-coordinates of the critical points by computing the discriminant

of f(x, y) with respect to y, and determine its real roots x1, . . . , xm. Each vertical

line x = xi contains at least a critical point of the curve.

2. Compute the vertices of the graph, by computing the y-coordinates yij, i.e., deter-

mine the real roots of f(xi, y) = 0.

3. For each vertex (xi, yij) compute the number of branches emanating to the left and

right.

4. Construct the graph by appropriately connecting the vertices. This can be simply

done by ordering the vertices in terms of the coordinate y. Note that the connected

graph is uniquely determined, since any incorrect branch between two vertices leads

to at least one intersection of two edges at a non-critical point.

Some published algorithms precede this scheme by an initial step involving a linear change

of coordinates, which ensures that there are no vertical lines that contain two critical

points.

In step 1, the discriminant is used to reduce the system of equations

f(x, y) = fy(x, y) = 0,

to a univariate polynomial. Numerically calculating the real roots of a univariate polyno-

mial is standard, and software packages such as Matlab and Maple have no difficulty

to determine accurate solutions efficiently. Other approaches to solve a system of n poly-

nomials in n unknowns are for example interval analysis, homotopy methods [Allgower

and Georg 1990, Morgan 1987], elimination theory and Grobner bases.

Note that there can be multiple branches between two vertices, see for example the con-

nection between vertices V1 and V3 in Figure 4.2. The usual approach to perform step 3

is to compute additional solutions of f(x, y) in the vicinity of critical vertical points and

singular points or in between them.

We propose to evaluate a Taylor series expansion of f(x, y) for a critical vertical point.

With f(x0, y0) = fy(x0, y0) = 0, we get the quadratic approximation

fx(x0, y0)∆x +1

2fxx(x0, y0)∆x2 + fxy(x0, y0)∆x∆y +

1

2fyy(x0, y0)∆y2 = 0. (4.8)

For a singular point with multiplicity m, we can obtain the direction of tangents by solvingm∑

i=1

(m

i

)

fxiym−i(x0, y0)∆xi∆ym−i = 0, (4.9)

evaluated at (x0, y0).


Figure 4.2: Topology of a quartic curve with exactly three singularities

4.3 Algorithm for Plane Algebraic Curves

A scheme for a general algorithm which determines the plane algebraic curve of an implicit

bivariate polynomial f(x, y) is:

1. Preprocessing

2. Determine the topological graph

3. Approximate all curve segments

Phase 1 usually involves factorization and possible coordinate changes which alleviate the

subsequent computations.

There are numerous approaches to perform phase 3, e.g., path-following, Bezier curve

approximation and piecewise-linear approximations.

Using an extended topological graph, we aim to get a very robust and efficient algorithm,

which allows to use predictor-corrector based path following for individual regular curve

segments. The extended topological graph divides the curve into a number of easily

traceable curve segments. From the extended topological graph we already know the

behavior of a curve in the vicinity of a singular point. This allows to determine the curve

close to a singular point, avoiding numerical problems.

Before the final curve is plotted, a Bezier approximation can be displayed. This allows

fast response times to user inputs and gives the user a first impression of the final results.

4.3 Algorithm for Plane Algebraic Curves 81

4.3.1 Extended Topological Graph

The basic topological graph, described in Section 4.2.3 is now extended, such that each

segment is convex inside a triangle. This allows to get a robust algorithm for plotting the

overall curve.

As a first step, we divide segments with critical vertical points on both vertices. The

tangents of both vertices never cross because they are parallel.

Second, we include all inflection points. These can be computed by solving f(x, y) = 0,

and (4.4) using a resultant method. The extended topological graph, has the property

that all curve segments are convex and lie inside a triangle formed by the vertices and

the intersection of the tangents at both vertices.

V1

V2

T12

Figure 4.3: Convex segment of rational curve

Consider a curve segment with vertices V1 and V2. Since there are no singular, critical

vertical or inflection points, the gradient of the tangent monotonically changes from the

angle at V1 to the angle at V2. Thus, the curve segment is convex and it has to lie in the

triangle formed by V1, V2, and the intersection of both tangents labeled T12. See Figure 4.3

which shows a convex segment inside the bounding triangle.

Thus, we not only have a topological graph with convex curve segments, but we know

the triangular area in which all points on a curve segment have to lie. The individual

triangles can intersect each other, although this happens only for very degenerate curves.

The intersection of triangles can be eliminated by introducing additional points into the

extended topological graph until no intersection occurs. Figure 4.4 shows the extended

topological graph of the curve defined by

f(x, y) = x4 + y4 − x3 − y3 − x2y + xy2, (4.10)

and the actual dotted curve. This curve has a triple multiplicity at V6 = (0, 0) and two

flexes at V2 and V3.


V1

V2

V3

V4

V5

V6

Figure 4.4: Extended topological graph

4.3.2 Bezier Approximation

The Bezier curve is a parametric curve important in computer graphics [Farin 2001]. A

quadratic Bezier curve is the path traced by the function

p(α) = (1 − α)2p0 + 2α(1 − α)p1 + α2p2, α ∈ [0; 1]. (4.11)

The curve passes through the end points p0 and p2 with tangent vectors p1−p0 and p2−p1.

See Figure 4.5 for a simple quadratic Bezier curve.

The functions (1−α)2, 2α(1−α), and α2 are degree two Bernstein polynomials that serve

as blending functions for the control points p0, p1, and p2. The Bernstein polynomials

are non-negative and add to one. Thus p(α) is an affine combination of the points p0, p1,

and p2 contained in the triangle p0 p1 p2. Geometrically quadratic Bezier curves are

parabolas.

p0

p1

p2

Figure 4.5: Quadratic Bezier

We will use information from the extended topological graph to sketch the curve. Using

simple quadratic Bezier curves for each curve segment, a good approximation of the

4.4 Path Following 83

true curve with bounded error can be sketched. The highest computational burden is

associated with computing the support point for a Bezier spline involving a singular

point. The branch tangents of the singular point are hereto calculated. See Figure 4.6

for a Bezier based approximation of curve (4.10). Note the small deviation of the Bezier

approximation from the true dotted curve.

p2

p1

p0

Figure 4.6: Bezier approximation of quadratic curve

4.4 Path Following

In this section we will consider the approximation of a single continuous curve segment us-

ing path following or curve tracing. While there are several approaches to path following,

we particularly treat predictor-corrector continuation methods [Allgower and Georg 1990].

In virtue of the polynomial equation defining a curve, the required numerical calculations

can be performed with high precision, and thus this approach is very suitable.

Before we present high-fidelity predictor-corrector algorithms, we will first consider com-

mon problems of gradient based path following algorithms in Section 4.4.1. We then

present a very easily implementable formulation of the path following problem, using ho-

motopy in Section 4.4.2, before we extend this to a full-scale predictor-corrector algorithm

in Section 4.4.3.


(x0, λ0)

(x1, λ1)(x2, λ2)

Figure 4.7: Branch skipping

4.4.1 Common Problems of Path Following

A common problem is branch skipping, i.e., while the path of a single branch is followed,

the algorithm misleadingly converges to a point which is on a separate branch not con-

tinuously connected to the branch currently followed. See Figure 4.7 for an example,

where (x2, y2) is wrongly connected to (x1, y1). Branch skipping might lead to missed

curve segments, or incorrect segment connection.

In extreme cases consecutive branch skipping might lead to branch looping, where the

algorithm enters an infinite loop, while connecting points from different branches, see

Figure 4.8.

4.4.2 Homotopy Based Algorithm

The earliest account of a continuation method can be found in [Poincare 1892]. The idea

of using a differential equation to solve a system on nonlinear equations was first explicitly

reported in [Davidenko 1953]. Davidenko’s approach is a subset of homotopic methods,

which can be used to the curve segment approximation problem.

Two functions y = f(x) and y = g(x) are homotopic, if one function can be continuously

deformed into the other, in other words, if there is a homotopy between them: a continuous

function y = h(α, x), with h(0, x) = f(x) and h(1, x) = g(x). The easiest homotopy is

given by the affine interpolation

h(α, x) = (1 − α)f(x) + αg(x). (4.12)

The solution of parametrized nonlinear equations and algebraic mapping equations in

particular can be formulated as a homotopy. To this end, one variable of f(x, y) = 0 is

4.4 Path Following 85

(x1, λ1)

(x2, λ2)

(x3, λ3)

(x4, λ4)

(x5, λ5)

Figure 4.8: Branch looping

used as a homotopic parameter. For example, let x be the homotopic parameter. We then

try to obtain an explicit solution of y as a function of x. Hereto, the total differential

of f(x, y) is determined as

fxdx + fydy = 0. (4.13)

Now write this equation to get the ordinary differential equation

dy

dx= −fx

fy, (4.14)

known as Davidenko’s equation.

We can now merely use a numerical initial value problem solver to solve (4.14) . While this

method has been successfully used by some authors, the exploitation of the contractive

properties of the curve by Newton type predictors is preferable.

The main difference in using (4.14) for path following and the solution of nonlinear equa-

tions by homotopic methods is the intermediate accuracy. A homotopy (4.12) has to

simply track all paths approximately and the ultimate requirement is only the solution

for t = 1. Whereas the curve segment approximation for the parameter space approach

requires an acceptable solution for intermediate steps too.


4.4.3 Predictor-Corrector Continuation

An efficient and robust predictor-corrector method possesses the following features [All-

gower and Georg 1992]:

1. efficient higher order predictor,

2. fast converging corrector,

3. effective step size adaptation,

4. detection and handling of special points such as bifurcation or turning points.

These properties are important if we want to successfully approximate complicated or

difficult curves, e.g., arising from H∞ norm specifications for systems with high-order or

polynomial parameter dependency, or both. Yet, the precision should be high enough in

order to make sure that the build up of numerical error does not lead to an increase in

the number of required iterations or prevents convergence of the corrector step at all. If

necessary, e.g., in the vicinity of singularities, some software packages have the ability to

explicitly change the precision.

Note: Although predictor-corrector methods are commonly used to integrate ordinary

differential equations, these methods are considerable different than the methods described

in this section. While we can use the contractive properties of the solution set following a

solution path, this is not the case for initial value problem solvers. Actually the corrector

converges in the limit only to an approximate point for differential equations.

Predictors

During the predictor step a point close to the curve with some distance from the current

point (xk, yk) is determined. Very commonly and sufficient for the parameter space curve

approximation is an Euler predictor, where the predictor step uses the tangent to the

curve,

xk+1 = xk + hkt(xk), (4.15)

with current step length hk > 0, and tangent vector t(xk) at the curve point xk = (xk, yk).

An even more simple predictor step can be performed using a secant predictor, which uses

two previous points to approximate the current direction

xk+1 = xk + hk(xk − xk−1). (4.16)

4.5 Surface Intersections 87

Correctors

A straightforward corrector is given by a Newton type iteration,

xk+1,n+1 = xk+1,n + ∆xk+1,n, n = 0, 1, . . . (4.17)

where ∆xk+1,n is the solution of the linear equation

[fx(xk+1,n) fy(xk+1,n)] ∆xk+1,n = −f(xk+1,n). (4.18)

The n-th iteration of the corrected point (xk+1, yk+1) is denoted xk+1,n. Due to the convex

nature of the curve segments, low iteration numbers n are sufficient to get close to the

real curve.

Step length control

Any efficient path following algorithm has to incorporate a step length control mechanism

because the local properties of the followed curves vary largely on the curve. Of course,

any step length adaption will depend on the desired curve tracing accuracy. Furthermore,

a robust step length control will prevent path skipping illustrated in subsection 4.4.1.

Due to the extended topological graph and the subdivision into convex curve segments,

we can employ a very simple step length control, e.g., by using the function value at the

final point (xk, yk) of the corrector step as an error model.

Prevention of path skipping

Using the bounding triangle of a convex segment, the path skipping can be avoided

robustly by checking that the predicted point, and points computed during the correction

phase are inside this triangle.

4.5 Surface Intersections

This section describes the algorithms proposed for solving the implicit surface-surface

intersection problem which is essential to the parameter space approach.

Both polynomial equations in (4.2) define a surface in R3. The intersection of these two

surfaces forms a spatial curve in R3. If a parametrization exists, which is in general not

true, this curve can be written as

[x(α), y(α), ω(α)]. (4.19)


Since we are only interested in the parameters x and y which satisfy (4.2), the generalized

frequency ω can be treated as an auxiliary variable. Thus, the critical boundaries in the

parameter plane are obtained as the projection of the space curve onto the plane ω = 0.

The projection of the intersection onto this plane, is mathematically described by the

resultant of both polynomials

r(x, y) = res(f1(x, y, ω), f2(x, y, ω), ω). (4.20)

Actually, for all mapping boundaries presented in Section 3.3.1 and Section 3.4, we have

the additional constraint, that f2 is the derivative of f1 with respect to ω,

f2(x, y, ω) =∂f1(x, y, ω)

∂ω.

Thus, r(x, y) in (4.20) can be written as

r(x, y) = res(f1(x, y, ω),∂f1(x, y, ω)

∂ω, ω),

which is the discriminant of f1 with respect to ω. And we obtain

r(x, y) = disc(f1(x, y, ω), ω). (4.21)

Finally, we can show that (4.21) contains not just the CRB condition (3.7a), but this

polynomial has additional factors which are equivalent to the RRB (3.8) and IRB condi-

tion (3.9).

Utilizing this property, the critical boundaries can be determined by evaluation the dis-

criminant (4.21), and factorizing this polynomial, with additionally eliminating double

factors. Subsequently, all critical boundaries can be plotted by consecutively plotting the

resulting curves in the (x, y) plane using the algorithm developed in Section 4.3.

As an alternative we can evaluate the CRB projection separately, after eliminating the

factor ω, using the resultant equation (4.20), while the RRB and IRB conditions are

already implicit equations. For this approach, (4.20) will contain a term simply squared.

Thus, it suffices to consider only the argument of the square.

4.6 Preprocessing

The computational burden of generating the solution curves of algebraic equations can

be alleviated in many cases by symbolic preprocessing. By preprocessing we mean any

transformation of the mapping equation system which preserves the solution structure and

alleviates the determination of the actual solution set. For some systems this symbolic

preprocessing is actually mandatory.

An intuitive preprocessing step is to scale the equations. Furthermore, using a computer

algebra system, e.g., Maple, we can use factorization.

4.6 Preprocessing 89

4.6.1 Factorization

If a polynomial f(x, y) is factorizable into individual polynomial factors, then the solution

set can be determined by treating the individual factors separately. A polynomial can be

possibly symbolically factorized, if the coefficients are integers. We therefore assume that

the coefficients are integers, prior to factorization. This can be done by normalizing all

rational coefficients. In a separate step, the integer coefficients can be factored such that

the polynomial is primitive over the integers.

All algorithms presented for plotting the curve of an implicit polynomial are faster when

factors are considered consecutively. Therefore, assume that the polynomial is irreducible.

4.6.2 Scaling

Scaling transforms the equations such that the coefficients are not extremal. The purpose

of scaling is to make a problem involving equations numerically more tractable. This is a

pretty vague goal and it should be clear that depending on the algorithm which is used

for the problem there is no theoretical best scaling. Thus, we have to use common sense

in choosing a scaling. In general any algorithm will benefit from a problem which has

coefficients with absolute values close to one and only small variations within equations.

There are two types of scaling. The multiplication of the equation by common factor is

called equation scaling, while the transformation of a variable of the form x = constant · xis referred to as variable scaling.

The scaled form of a univariate polynomial equation anxn + . . . + a1x + a0 = 0 with

equation scaling factor 10c1 and variable scaling x = 10c2x is given by

10c1+nc2+log10 an xn + . . . + 10c1+c2+log10 a1 x + 10c1+log10 a0 = 0. (4.22)

The coefficients can now be centered about unity and the variation of coefficients mini-

mized by solving linear equations. See Chapter 5 of [Morgan 1987] for an implementable

algorithm.

4.6.3 Symmetry

The successful exploitation of symmetry can not only lead to a much more efficient algo-

rithm but also to a more robust one. Having identified the axis of symmetry we have to

follow the path on one side only. The path on the opposite side can be mirrored. Thus,

symmetry will immediately lead to reduction of the computational cost by a factor of 1/2.


4.7 Visualization

Visualization is concerned with exploring data and information graphically - as a means

of gaining understanding and insight into the data [Earnshaw and Wiseman 1992].

Nowadays it is common understanding, that a robust control toolbox should provide a

user-friendly, possibly graphical interface, as suggested in [Boyd 1994]. While all compu-

tations are done in the background and only the results are finally visualized. See [Muhler

et al. 2001, Sienel et al. 1996] for a toolbox which enables plant descriptions via block

diagrams and allows to graphically specify Γ-stability eigenvalue specifications. As pro-

posed in [Muhler and Odenthal 2001] this approach can be extended to frequency domain

specifications.

We will explore several ways how to visualize specifications in a parameter plane. A

straightforward approach is to simply plot the critical boundaries of parameter region,

which fulfill a specification. This is equivalent to plots generated for Γ-stability specifi-

cation, see Figure 3.3 for an IQC stability example. Varying line styles might be used to

distinguish different specifications, e.g., Γ-stability and H∞ norm.

Further improvements of the visualization can be achieved by using the following methods:

• Overlay

• Complementary Colormaps

• Slave Cursor

4.7.1 Color Coding

Most frequency domain specifications, e.g., Nyquist stability margin, yield a scalar value

for fixed parameters. Thus, it is possible to determine regions with performance in a

specific range. Color coding these regions according to their performance level allows

immediate assessment of performance satisfaction.

Note that critical boundary lines resulting from eigenvalue specifications can be over-

laid on top of color coded contour plots. Hence, multiple objectives can be represented

simultaneously in a parameter plane.

The Nyquist stability margin is used to explain the color coding scheme. Annuli in the

Nyquist plane are color coded according to their distance from the critical point (-1,0).

In order to make the color coded plots intuitively evident, we propose a traffic light color

coding scheme. This color map resembles the colors of a traffic light ranging from green

to red. We use colors close to red to visualize regions with poor performance and colors

4.7 Visualization 91

close to green for good performance. Thus, the plots are readily understandable by the

control designer. Alternatively gray-scale color coding could be employed with black for

poor and white for good performance, if the use of colors is not possible. Figure 4.9 shows

the used color coding scheme for the Nyquist stability margin.

Stability margin ρm

Im(G

(jω))

Re(G(jω))−2.5 −2 −1.5 −1 −0.5 0 0.5

−1

−0.5

0

0.5

1

0

1.0

0.75

0.5

0.25

Figure 4.9: Color coding for the Nyquist stability margin

Nyquist performance can now be visualized in the parameter plane by determining pa-

rameter sets which lead to performance in a specific range. These sets are color coded

according to their performance level. Thus, we seek to determine the set of parameters

Ki with Nyquist stability margin in a specific range

Ki := {k1, k2 : ρ−i ≤ ρ(k1, k2) ≤ ρ+

i }.

Using different values of ρ−i , ρ+

i , we can exactly determine the sets Ki. The boundary lines

obtained for each value of ρ represent the contour lines, which are used to color code the

sets in the parameter space according to their performance level.

4.7.2 Visualization for Multiple Representatives

In this section we consider the visualization of admissible sets and frequency domain

performance for multiple representatives.

Eigenvalue specifications

Similar to the nominal design case, the eigenvalue specifications are mapped into the

controller parameter plane for each representative. Intersection of sets leads to a set KΓ of


admissible controller parameters, for which the Γ-specifications are simultaneously fulfilled

for all representatives [Ackermann et al. 1993], see Figure 4.10 where superscripts (1)

and (2) denote two different representatives.

��

��

��

��

��

�

�

��

��

� �

� � �"!

� � � !

# � � !$

# � �%!$

# $

Figure 4.10: Mapping of ∂Γ for multiple representatives

Appropriate visualization enables the designer not only to identify the admissible set, but

to identify specifications and operating points which constrain the admissible set.

Color coding for multiple representatives

For multiple representatives a scalar function which is to be maximized can be visualized

by worst case color coding. For several representatives only the lowest value is relevant.

Therefore only the minimal value over all representatives is color coded and visualized.

Visualizing the Nyquist stability margin for several representatives in the controller pa-

rameter plane can be done by plotting the lowest value obtained for all representatives.

Figure 4.11 shows a simple example for two representatives.

for Rep 1+2Rep 1 worst caseRep 2

Figure 4.11: Worst case color coding example

93

5 Examples

This chapter presents applications of the derived mapping equations. Various practical

examples demonstrate the usability for robust control system design.

The model description is given as a parametric state-space model as in (2.1) or as a

transfer matrix representation (2.2).

For ease of presentation, we consider only systems with the same number of inputs and

outputs, i.e., m = p. Nevertheless, all results are valid for non-square systems with m 6= p.

5.1 MIMO Design Using SISO Methods

For SISO control systems, classical gain and phase margins are good measures of robust-

ness. Furthermore, loop-shaping techniques provide a systematic way to attain good ro-

bustness margins and desired closed-loop performance. The methods introduced in [Ack-

ermann et al. 2002, Sections 5.1−5.4] facilitate such a design for the PSA. However, the

classical gain and phase margins are not reliable measures of robustness for multivariable

systems.

The simplest approach to multivariable design is to ignore its multivariable nature and

just look at one pair of input and output variables at a time. Sometimes this approach is

backed up by decoupling, although robust decoupling is in general difficult to achieve.

A classical design procedure using this idea for multivariable systems is the sequential loop

closing method, where a SISO controller is designed for a single loop. After this design

has been done successfully, that loop is closed and another SISO controller is designed for

a second pair of variables, and so on.

Example 5.1 Consider the following plant

G(s) =1

(s + 1)(s + 2)

2 −2s

s 3s + 2

. (5.1)

In the first step, we design a constant gain controller k11(s) = k1. The transfer function

seen by this controller is g11(s) = 2/((s + 1)(s + 2)). Setting k1 = 1 leads to a stable

94 Examples

transfer function. After closing this loop, the transfer function seen by a controller from

output 2 to input 2 is

g22(s) = g22(s) −g12(s)k11(s)g21(s)

1 + k11(s)g11(s)=

3s + 4

s2 + 3s + 4.

A stabilizing controller for this transfer function is k22(s) = 1. The resulting decentralized

controller is thus given by the identity matrix K(s) = I. We will come back to this

example with a better solution in Example 5.3 and Example 5.4. �

This method has a number of weaknesses. During the controller design, the resulting scalar

transfer function for the i-th step might be nonminimum-phase, although all members

of G(s) are minimum-phase transfer functions. This might pose a severe constraint for

the control design, since nonminimum-phase transfer functions limit the maximal usable

gain.

5.2 MIMO Specifications

5.2.1 H2 Norm

Example 5.2 Consider the attitude control of a satellite for one axis. The transfer

function is given by

Y (s) =1

Izs2U(s),

where Iz is the moment of inertia for the z-axis. We are now designing a state-feedback

controller u(t) = −(k1x1(t) + k2x2(t)) which minimizes the objective

J =1

2

∫ ∞

0

(x1(t)2 + x2(t)

2 + u(t)2) dt.

The resulting matrices for this problem are

x(t) =

0 1

−k1

Iz−k2

Iz

x(t) + w(t),

z(t) =

−k1 −k2

1 0

0 1

x(t).

5.2 MIMO Specifications 95

Using (3.21a) the observability Gramian becomes

Wobs =1

2k1k2

k1(k

21 + 1)Iz + k2

1 + k22 Iz(k

21 + 1)k2

Iz(k21 + 1)k2 Iz(k

21 + 1)Iz + k1k

22 + k1

.

And we get

J2 =(k2

1 + 1)I2z + (k1 + k2)(k

21 + k1k2 + 2)Iz + k2

1 + k22

2k1k2

as the mapping equation which is quadratic in k2.

Figure 5.1 shows the resulting parameter sets Q2 for J2 = 12 and two different moments of

inertia Iz = 1 and Iz = 2. It can be seen that the operating point with the highest moment

of inertia is the limiting case for the LQR specification. As an additional specification the

set QΓ for which the damping ζ of the closed-loop system is at least ζ = 0.9 is determined

in Figure 5.1, and shown by a dotted line. The figure shows that a robust controller can

be determined from the set of parameters which satisfy both specifications, marked as the

light shaded area.

0 1 2 3 4 5 60

2

4

6

8

10

12

14

Iz = 2

Iz = 1

k1

k2

Figure 5.1: LQR and Γ-stability boundaries for attitude control example

�

Example 5.3 Revisit Example 5.1. We will design a decoupled constant-gain output-

feedback controller u(t) = −Ky(t) with

K =

k1 0

0 k2

, (5.2)

96 Examples

which minimizes the following LQR-like performance index

J =1

2

∫ ∞

0

(y(t)T y(t) + α u(t)T u(t)) dt. (5.3)

This performance index treats both outputs equally, which is reasonable since the open-

loop plant has similar gains for these outputs. The parameter α provides an adjustable

design knob, which allows an intuitive tradeoff between the integral error of the com-

manded output and the actuator effort. For this specific example, we assume α = 1.

The open-loop plant has pure real eigenvalues at {−1,−2}. Thus, we additionally require

the rather stringent specification that all closed-loop eigenvalues should have at least a

minimum damping of ζ = 1.0.

We will solve this problem by mapping the design requirements into the k1, k2 controller

parameter plane. To this end, we formulate the LQR output problem (5.3) in the H2

norm framework by employing the results of Section 2.4.8.

Using the fact that y(t) = Cx(t), the LQR weight matrices in (2.62b) for this problem

are given by

Q = CT C, R = I.

In order to apply the algebraic mapping equations (3.23), we need a state-space description

of the system. A minimal realization of the system (5.1) is given by

G(s) ∼=

−2 0 −2 −4

0 −1 −2 −2

1 −1 0 0

−1 1/2 0 0

. (5.4)

We incorporate the controller u(t) = −Ky(t) in parametric form into (2.62a) and (2.62b)

to get the state-space system G(s) defined in the H2 norm mapping equation (3.23). For

the particular problem considered in this example, these equations are given by

x(t) = (A − BKC)x(t) + w(t), (5.5)

z(t) =

−K

I

Cx(t). (5.6)


And the parametric transfer function G(s)w→z becomes

G(s)w→z

∼=

2(k1 − 2k2 − 1) −2(k1 − k2) 1 0

2(k1 − k2) −2k1 + k2 − 1 0 1

−k1 k1 0 0

k2 −12k2 0 0

1 −1 0 0

−1 1/2 0 0

.

The controllability Gramian for this problem is obtained from (3.21b) as

Wcon(k1, k2) =1

6(k1 + 1)(k2 + 1)2�

4k2

1 − 5k1k2 + 5/2k22 + 3k1 + 3/2 (k1 − k2)(4k1 − 5k2 − 1)

(k1 − k2)(4k1 − 5k2 − 1) 4k21 − 11k1k2 + 10k2

2 − 3k1 + 9k2 + 3

.

Finally, the resulting performance index can be computed as

J =(4k1 + 5k2 + 9)(2k2

1k2 + 2k21 + k1k

22 + k2

2 + k1 + 2k2 + 3)

24(k1 + 1)(k2 + 1)2. (5.7)

For a given performance level J = J∗, (5.7) provides an implicit mapping equation in the

unknowns k1 and k2. The minimal achievable performance level J for the decentralized

output feedback (5.2) is bounded from below by the performance level Jfull obtainable

with a dense static-gain feedback controller. Using classical LQR theory, Jfull is easily

calculated as Jfull = 0.824. In general J > Jfull, therefore, we will map J = 1 into the

parameter plane. Figure 5.2 shows the resulting parameter set Q2 for J = 1. The region

satisfying both LQR and Γ-stability requirements is shaded in the figure. The figure

actually shows that we can set k1 = 0, while still obtaining reasonable controllers. Note

that without the damping specification on the closed-loop eigenvalues, we would still need

to map the Hurwitz-stability requirement to get the correct LQR set. For this example,

stability is assured if

k1 > −1 ∧ k2 > −1.

The actual boundary values k1 = k2 = −1 appear in the denominator of (5.7).

The minimal obtainable performance level J for the decentralized controller can be com-

puted using the algebraic equation (5.7). For k1 = 0.2074, k2 = 0.9329, we get J = 0.8421,

which is only slightly higher than Jfull. Compare this to J = 1.125 for the controller de-

signed in Example 5.1.

98 Examples

k2

k1

J < 1

–3

–2

–1

0

1

2

3

–3 –2 –1 1 2 3

Figure 5.2: LQR boundaries (solid) and Γ-stability (dashed) for MIMO control example

The numerical values of J change with the state-space representation considered. Thus,

the actual values of J should be only considered to measure the relative performance of

a controller compared to a reference controller, e.g., the dense optimal controller or zero-

gain controller (open-loop system). This becomes apparent when one is computing J for

the state-space representations (5.4) and (3.64), which both lead to the same input-output

behavior but different quantitative values of J . �

5.2.2 H∞ Norm: Robust Stability

We use robust stabilization as a classical control problem that fits into the H∞ framework

to motivate the mapping of H∞ norm specifications. Different from the traditional liter-

ature about H∞ control theory [Zhou et al. 1996],[Francis 1987] we will treat structured

(parametric) and unstructured uncertainties.

The well-known small-gain theorem (see Theorem 2.2 in Section 2.4.1 or [Zhou et al.

1996]) states that a feedback system composed of stable operators will remain stable, if

the H∞ norm of the product of all operators is smaller than unity.

As an example, consider a plant G(s) with multiplicative, unknown uncertainty ∆(s) at

the output as in Figure 2.5 and associated weighting function W0. The block diagram for

the closed feedback loop with controller K(s) is shown in Figure 5.3.


K G

Wo ∆

Figure 5.3: Feedback system including plant with multiplicative uncertainty

The problem is, how large can ||∆||∞ be, so that internal stability is preserved? Us-

ing simple loop transformations, we can isolate the uncertainty ∆, which can be seen

in Figure 5.4.

W0(I + GK)−1GK

∆

Figure 5.4: Feedback system including plant with isolated multiplicative uncertainty

Using the small-gain theorem (Theorem 2.2), we get the following sufficient condition for

internal stability with respect to unstructured multiplicative output uncertainty:

||∆||∞ ≤ 1

||W0(I − GK)−1GK||∞. (5.8)

Example 5.4 We analyze the robust stability of the plant given in Example 5.1 for de-

centralized static-gain controllers (5.2) with respect to unstructured multiplicative output

uncertainty. Consider the weighting function

Wo(s) =3s + 1

2

s + 3I.

This implies a moderate relative uncertainty of up to 16% for low frequencies, which

increases to higher frequency reaching 100% at about 1 [rad/sec] and finally going to 300%

in the high frequency range.

Figure 5.5 shows the gray-tone coded sets of parameters that correspond to different

tolerable uncertainty sizes. Dark areas correspond to poor robustness, whereas areas

100 Examples

−0.5 0 0.5 1 1.5 2

−0.5

0

0.5

1

1.5

2Robust stability for multiplicative uncertainty

k1

k2

Figure 5.5: Stability with respect to unstructured multiplicative uncertainty

with lighter colors indicate good robustness. The controller designed in Example 5.3

with k1 = 0.2074, k2 = 0.9329 yields better robustness than the initial controller from

Example 5.1 with k1 = k2 = 1.

Comparing the results in Figure 5.2 and Figure 5.5, we see that by varying k2 there is a

tradeoff between robustness and performance. �

5.2.3 Passivity Examples

Example 5.5 Consider a general strictly proper second order system in pole-zero factor-

ized form

G(s) =s − z1

(s − p1)(s − p2). (5.9)

Using a controllable canonical form state-space representation we obtain the Hamilto-

nian Hη as

Hη =

0 1 0 0

−p1p2 − z1

2ηp1 + p2 + 1

2η0 − 1

2η

z1

2η− z1

2η0 p1p2 + z1

2η

− z1

2η12η

−1 −p1 − p2 − 12η

.


The resulting CRB

(p1 + p2 − z1)ω2 + p1p2z1 = 0,

2(p1 + p2 − z1)ω = 0,

has no valid solution for ω 6= 0 and dismantles into the RRB and IRB conditions:

RRB: p1p2z1 = 0,

IRB: p1 + p2 − z1 = 0.

Evaluating eigenvalues of Hη with z1 = p1 + p2 for η → 0 we can deduce the simple rule

that a general second order transfer function (5.9) is passive, if

z1 > p1 + p2, ∀p1, p2 < 0. (5.10)

�

Example 5.6 Consider the following robust passivity problem. Let

G(s) =s2 + a1s + a0

s3 + 8s2 + 17s + 10. (5.11)

Then a controllable state-space realization in canonical form is given by

G(s) ∼=

0 1 0

0 0 1

−10 −17 −8

0

0

1

a0 a1 1 0

.

The passivity boundaries in the (a0, a1) parameter plane and the resulting good parameter

set Pgood are shown in Figure 5.6. Obviously the parameter set Pgood is contained in the

first quadrant which corresponds to minimum phase systems. Furthermore it can be easily

seen from this plot that there is no weakly minimum phase passive system, i.e., a1 = 0.

To illustrate the robust passivity, the Nyquist plots for three exemplary passive systems

are shown in Figure 5.7, which correspond to the three circular markers on the edges

of Pgood in Figure 5.6.

�

102 Examples

−2 0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

PSfrag replacements

Pgood

a0

a1

Figure 5.6: Robust passivity

−0.2 0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Real

Imag

ω=−∞

ω=+∞ ω=0 ω=0 ω=0

Figure 5.7: Nyquist curves for passive systems

5.3 Example: Track-Guided Bus 103

5.3 Example: Track-Guided Bus

We demonstrate the application of the presented methods by designing a controller for

a track-guided bus introduced in [Ackermann et al. 1993, Ackermann and Sienel 1990],

see also [Muhler and Ackermann 2000]. The task is to minimize the distance of the bus

from a guideline. We investigate automatic steering based on feedback of the lateral

displacement. An actuator which commands the front wheel steering angle δf is used.

The lateral displacement y is measured by a sensor at the front bumper. Figure 5.8 shows

a sketch of the bus with front wheel angle as input and the displacement sensor.

fδ

Guideline

CG

y

Front Sensor

Figure 5.8: Track-guided bus

After an appropriate controller has been found for the nominal operating point the next

step is to design a controller which simultaneously satisfies the specifications for several

operating conditions. Thus, we repeat the previous design for the four vertices of the

operating domain Q and try to find a simultaneous solution and choose the controller

parameters accordingly.

A controller which stabilizes the four vertex conditions is likely to successfully stabilize

the whole operating domain. But satisfaction of the specifications for the four vertices is

not sufficient for all parameters q ∈ Q. Hence, as a final step we have to do a robustness

analysis in the (q1, q2)-plane to check for satisfaction of performance specifications in the

entire operating domain Q. Note that we understand the analysis for the whole operating

domain Q as an essential step in the robust controller design process. If initially the

performance criteria are not satisfied for the entire operating domain, then the design is

repeated with further representatives of Q in addition to the vertices.

The transfer function of the bus with uncertain parameters mass m (normalized by the

friction coefficient between tire and road) and velocity v is given by

G(s) =a0v

2 + a1vs + a2mv2s2

s3(b0 + b‘0mv2 + b1mvs + m2v2s2)

.

104 Examples

The operating domain is given by m ∈ [9.95; 32]t and v ∈ [3; 20]m/s, which represents

all possible operating conditions relevant for a city bus. Figure 5.9 shows the operating

domain for the bus with the four vertex points.

Figure 5.9: Operating domain for bus

The following controller structure was used in [Ackermann et al. 1993]:

K(s) =k1 + k2s + k3s

2

d0 + d1s + d2s2 + d3s3.

The coefficients k1, d0 . . . d3 are fixed. A root locus plot shows that the controller pa-

rameters k2, k3, which determine the zeros of K(s), are most crucial for the design step.

Therefore we design the controller in the k2, k3-plane.

5.3.1 Design Specifications

The specifications for the closed loop system can be expressed through Γ-stability. All

roots should lie left to the hyperbola

( σ

0.35

)2

−( ω

1.75

)2

= 1. (5.12)

This guarantees a maximal settling time T = 2.9 s and a minimal damping ζ = 0.196.

A suitable controller which simultaneously stabilizes the four vertices of the operating

domain for these specifications was determined in [Ackermann et al. 1993] as

K(s) =253(0.15s2 + 0.7s + 0.6)

(s + 25)(s2 + 25s + 625).

We are are extending these specifications by trying to maximize the Nyquist stability

margin.

5.3 Example: Track-Guided Bus 105

5.3.2 Robust Design for Extreme Operating Conditions

We design a simultaneous controller for the four vertex operating conditions. The task is

to tune k2, k3 such that the roots lie left of (5.12) and the worst Nyquist stability margin

is maximized for the four representatives.

By mapping the Nyquist stability margin using (2.34) and the Γ-stability boundaries for

the four vertices Figure 5.10 is generated. The plots are arranged as in Figure 5.9 with

minimal v and m in the lower left corner.

Figure 5.10: Color coded ρ and Γ-stability boundaries

To make the joint analysis for all four vertices easier we generate the worst-case overlay

by determining the set of simultaneous Γ-stabilizing controllers through intersection and

plot the worst-case Nyquist stability margin for the four vertices. Figure 5.11 shows the

worst case overlay for four representatives in the (k2, k3)-plane. From this plot we can

choose k2, k3 values from the admissible set with maximal worst-case ρ. For a controller

with values k2 = 0.7, k3 = 0.15 Γ-stability is guaranteed for vertex conditions, but we

could only achieve a poor Nyquist stability margin.

5.3.3 Robustness Analysis

The controller resulting from the previous design process satisfies the given specifications

at least for the extremal operating conditions. As a final step we verify the Γ-stability

and Nyquist stability margin specifications for the whole range of operating conditions.

We therefore map the eigenvalue and Nyquist stability margin specifications into the v, m-

plane. Figure 5.12 shows the Γ-stability boundaries and the color coded Nyquist stability

106 Examples

k20.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.05

0.1

0.15

0.2

0.25

controllerselected

k3 = 0.15

k2 = 0.7

Figure 5.11: Worst-Case Overlay

margin in the (v, m) parameter plane. The Γ-stability boundaries do not intersect the

operating domain, depicted by a rectangle. In addition the color coded Nyquist stability

margin in Figure 5.12 is sufficient for the entire operating domain. Hence, the designed

controller guarantees robust satisfaction of the given specifications.

Instead of using Figure 5.12 as a mere robust stability check, we get valuable information

about the performance for different operating conditions. The most critical operating con-

dition regarding Nyquist margin occurs for maximal velocity and minimal mass, whereas

for the Γ-stability margin the worst case is maximal velocity and maximal mass.

5.4 IQC Examples

Example 5.7 Consider the following nonlinear control example depicted in Figure 5.13

with a PI controller, a dead zone which models the actuator, and a linear plant G(s). The

transfer function of the controller is given by KPI(s) = k1 + k2

s. The plant is given by

G(s) =qs + 1

s2 + s + 1,

where q ∈ R is an uncertain parameter.

We aim at analyzing the robustness of the system with respect to variations in q. Fur-

thermore we want to tune the controller such that robustness to parameter variations is

achieved.

5.4 IQC Examples 107

v

m

Γ−stability and Frequency−Domain Analysis

0 5 10 15 20 25 30

5

10

15

20

25

30

35

40

45

50

Figure 5.12: Analysis in the operating domain

The given feedback interconnection is called critical since the worst case linearization is

at best neutrally stable. Note that the transfer function KPI(s)G(s) is unbounded which

prevents the application of standard stability criteria for nonlinear systems which require

bounded operators.

rPI G(s)

y

Figure 5.13: Dead zone PI controller example

We use the Zames-Falb IQC derived in [Jonsson and Megretski 1997], where it was shown

that an integrator and a sector bounded nonlinearity can be encapsulated in a bounded

operator that satisfies the following IQC

Π(jω) =

0 1 − H(jω)∗

1 − H(jω) − 2kRe(1 − H(jω)− kF (jω))

, (5.13)

where

F (s) =H(s) − H(0)

s,

where H(s) is a stable transfer function with L1 norm less than one, and the parameter k

equals the static gain of the open loop linear part k = k2G(0). This IQC corresponds to

the Zames-Falb IQC for slope restricted nonlinearities [Zames and Falb 1968].

108 Examples

Let the integral gain k2 = 2/5 and H(s) = 1/(s + 1). For our particular example the

parameter k equals the proportional gain k = k2. We map the stability condition into

the (k1, q) parameter plane. This allows to evaluate robustness with respect to q, while

we can select the controller gain k1 to maximize the robustness.

Since the multiplier Π(jω) in (5.13) is frequency-dependent, we use the method described

in Section 3.4.4 to reformulate the IQC stability problem with a constant multiplier.

The multiplier evaluates to

Π(jω) =

0jω

jω − 1

jω

jω + 1−5ω2 + 2

ω2 + 1

. (5.14)

This multiplier is not positive definite, so that most algorithms for spectral factorization

fail. A remedy is to use a constant offset matrix Π0 which makes the remainder positive

definite:

Π(jω) = Π0 + Πp(jω).

The transfer matrix Πp(jω) can now be factorized. So we alter (3.36) from Π(jω) =

Ψ(jω)∗ΠsΨ(jω), into Π(jω) = Ψ(jω)∗ΠsΨ(jω) + Π0, to get the spectral factorization

Π(jω) =

Ψ(jω)

I

∗

Πs 0

0 Π0

Ψ(jω)

I

,

which again has the form (3.36).

For this particular example the augmented system (A, B) in (3.43) is of forth order:

A =

−1 0 1 − q2 − 52k1 1 − 5

2k1q2

0 −1 0 0

0 0 0 1

0 0 −1 −1

, B =

0

1

0

1

,

and the submatrices Q, S and R of the multiplier M relevant for the mapping equations

are given as:

5.4 IQC Examples 109

M =

Q S

ST R

=

4 −2 −2 + 2q2 + 5k1 −2 + 5k1q2 0

−2 3 1 − q2 − 52k1 1 − 5

2k1q2 0

−2 + 2q2 + 5k1 1 − q2 − 52k1 0 0 1 − q2 − 5

2k1

−2 + 5k1q2 1 − 52k1q2 0 0 1 − 5

2k1q2

0 0 1 − q2 − 52k1 1 − 5

2k1q2 −5

.

The corresponding mapping equations are of eighth order containing k1 and q2 as param-

eters.

The resulting stability boundaries are shown in Figure 5.14. The set of stable parame-

ters Pgood contains the origin.

−2 0 2 4 6 8 10−2

0

2

4

6

8

10

PSfrag replacements

Pgood

k1

q

Figure 5.14: Stability boundaries

To evaluate the conservativeness of the results numerical simulations were performed using

the nonlinear system. The simulations showed that the upper line shown in Figure 5.14 is

far from the real boundary, while the lower boundary is very close to the actual boundary.

Figure 5.15 not only shows the nonlinear boundaries (solid) but also the stability bound-

aries for a linear system (dashed) which lacks the nonlinear dead zone actuator. The

results show that the nonlinear stability region is only slightly smaller than the linear

counterpart. Although the mathematical description of the curves is different, however.

�

110 Examples

−1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

PSfrag replacements

Pgood

k1

q

Figure 5.15: Comparison of linear and nonlinear system

5.5 Four Tank MIMO Example

A multivariable laboratory process is considered to show a practical application of the

robust control analysis and synthesis methods presented in this thesis. The process is

the level control of a system of four interconnected water tanks. The process has been

described in [Johansson 2000]. The system is shown in Figure 5.16. The system not only

shows considerable cross-couplings, but has an adjustable multivariable zero. Furthermore

it has static and dynamic nonlinearities.

The task is to control the water levels of the first and second tank by varying the flows

generated by the two pumps. The inputs are the control voltages for the pumps v1 and v2

and the outputs are level measurement voltages y1 and y2.

The nonlinear system model can be derived from mass balances and energy conservation

in the form of Bernoulli’s law for flows of incompressible, non-viscous fluids

h1 = − a1

A1

√

2gh1 +a3

A1

√

2gh3 +γ1k1

A1

v1

h2 = − a2

A2

√

2gh2 +a4

A2

√

2gh4 +γ2k2

A2

v2

h3 = − a3

A3

√

2gh3 +(1 − γ2)k2

A3

v2

h4 = − a4

A4

√

2gh4 +(1 − γ1)k1

A4

v1,

(5.15)

where Ai is the cross-section of tank i, ai the cross-section of outlet i, and hi the water

level of the i-th tank. The output signals for the measured levels are proportional to the

water level y1 = kch1 and y2 = kch2.

5.5 Four Tank MIMO Example 111

Tank 2

Tank 4Tank 3

γ2v2γ1v1

Tank 1

Pump 1 Pump 2

v1 v2

(1− γ1)v1 (1− γ2)v2

Figure 5.16: Schematic diagram of the four-tank process

The transfer matrix linearized for a given static operating point is

G(s) =

γ1c11

(T1s + 1)

(1 − γ2)c12

(T1s + 1)(T3s + 1)

(1 − γ1)c21

(T2s + 1)(T4s + 1)

γ2c22

(T2s + 1)

. (5.16)

Let the ratio of water diverted to tank one rather than tank three be γ1 = 0.33 and

the corresponding ratio of pump two is set to γ2 = 0.167. All other nominal parameter

values are given in Table 5.1. The transfer matrix (5.16) then has two multivariable

zeros at s = −0.197 and s = 0.118. The multivariable RHP zero limits the achievable

performance for this system.

Table 5.1: Parameter values for four tank example

A1, A2, A3, A4 [cm2] [30.0, 30.0, 20.0, 20.0]

a1, a2, a3, a4 [cm2] [0.1, 0.067, 0.067, 0.1]

k1, k2 [cm3/Vs] 3.0

kc [V/cm] 1.0

112 Examples

We consider decentralized PI control with input-output pairing y1 − u2 and y2 − u1 sug-

gested by relative gain array analysis. Here, we link both Kp and Ki parameters of the

individual loops, i.e. Kp = Kp1 = Kp2 and Ki = Ki1 = Ki2 and map a real part limitation

of Re s < 0.004 and a nominal performance ||WpS||∞ specification, where S is the sensi-

tivity and Wp = (250s+10)/(1000s+1) a performance weighting function. The resulting

plot is shown in Figure 5.17 where good performance is represented by light colors and

the eigenvalue specification is depicted by dashed lines.

Taking an optimal value from Figure 5.17 the resulting multivariable controller can be

fine-tuned, for example by considering the individual loops or by evaluating robustness

with respect to changes in the nominal parameter values.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Four tank PI controller design

Kp

Ki

Figure 5.17: PI controller design for four tank example

113

6 Summary and Outlook

We conclude with a brief summary of the work presented, and remarks on future directions

for related research.

6.1 Summary

Robust control of systems with real parameter uncertainties is a topic of practical impor-

tance in control engineering. In this thesis we considered the mapping of new specifica-

tions. Although well-known and often used, some of these specifications have not been

used in the parameter space context. Furthermore, the considered specifications can be

mapped for multivariable systems. Special criteria for nonlinear systems are presented.

To this end, not only standard specifications like the Popov criterion are considered, but

the mapping of versatile integral quadratic constraints is introduced.

Starting point of the thesis is the observation that many practical specifications, for exam-

ple H∞ norm and passivity, can be formulated using the same mathematical framework.

Namely algebraic Riccati equations (AREs). An important link forms the KYP lemma,

which translates different mathematical formulations of specifications into each other.

A corresponding set of specifications can be formulated using Lyapunov equations, a spe-

cial case of an ARE. Important representatives are the H2 norm and LQR specifications.

These allow to express performance specifications in parameter space. These specifica-

tions directly lead to a single implicit mapping equation, which is in general at least

quadratic in the uncertain parameters.

Mathematical results on the analytic dependence of solutions for parametric AREs led to

the derivation of mapping equations. Hereby control specifications formulated as AREs

can be converted into a specific eigenvalue problem using properties of associated Hamil-

tonian matrices.

The introduction of mapping equations for IQCs broadens the applicable system class even

further by enabling to consider specifications for input-output theory, absolute stability

theory, and the robust control field. The exploitation of additional degrees of freedom

provided by variable uncertainty characterization is shown.

114 Summary and Outlook

Mapping equations for ARE and Lyapunov equation based specifications are similar in

structure to root locus specifications. Nevertheless, due to the quadratic nature of the

specifications, which shows in the AREs, the mapping equations are in general more

complex and nontrivial to solve.

Practical aspects of the thesis therefore include the presentation of a hybrid symbolic-

numerical algorithm. This algorithm exploits the properties of the algebraic mapping

equations to determine characteristic points on the resulting curves in a parameter plane.

The topology of the curve then allows to either numerically connect the individual curve

points by curve following or to approximate the curve using a Bezier approximation. The

uniform mathematical description of the mapping equations allows to employ a single

mapping algorithm for all specifications.

The mapping equations are based on a parametric state-space realization. A symbolic

algorithm is hereto presented, which calculates a state-space realization for a given transfer

function.

This thesis shows that classical eigenvalue criteria and modern norm and nonlinear spec-

ifications can be combined in the parameter space approach to yield an efficient control

engineering tool that takes all practical aspects like stability, performance and robustness

into account.

6.2 Outlook

We note related future research topics.

In this thesis, a comprehensive methodology to map control specifications into parameter

spaces is presented. The successful and wide-spread use of these methods necessitates

an efficient and robust software implementation of the given algorithms. Furthermore, a

user-friendly front-end facilitates the practical application of the thesis results. It can be

used to specify the considered system and corresponding specifications and to evaluate the

graphical results. One possible way is a computer toolbox with graphical user interaction.

While the new possibilities have been demonstrated on some practical examples, one line

of research is to apply the methods to a large-scale real-world problem and to investigate

the limits and computational burdens.

There are numerous ways to characterize uncertainty. While the approach pursuit in this

thesis is to deal explicitly with real uncertain parameters, one research topic is to evaluate

different uncertainty characterizations.

115

A Mathematics

This appendix reviews some mathematical topics used in the thesis, that are not neces-

sarily treated in graduate level engineering courses. Particularly all theorems needed to

prove the fundamental theorems in Chapter 3 are stated here.

We shall always work with finite dimensional Euclidean spaces, where

Rn = {(x1, . . . , xn) : x1, . . . , xn ∈ R},Cn = {(x1, . . . , xn) : x1, . . . , xn ∈ C}.

A point in Rn is denoted by x = (x1, . . . , xn) and the coordinate or natural basis vectors

are written as ei :

e1 =

1

0...

0

, e2 =

0

1...

0

, . . . , en =

0

0...

1

.

All arguments of functions are either given in parenthesis or omitted for brevity.

A.1 Algebra

This section reviews basic facts from tensor algebra. The use of tensors facilitates the

notation of matrix derivatives used for some algorithms presented in Chapter 4. Fur-

thermore tensors are supported by symbolic and numerical software packages such as

Matlab and Maple and therefore allow an easy implementation of the presented algo-

rithms. See [Graham 1981] for a good reference on tensor algebra.

Given two matrices A ∈ CnA,mA , B ∈ CnB ,mB , the Kronecker product matrix of A and B,

denoted by A ⊗ B ∈ CnA∗nB ,mA∗mB , is defined by the partitioned matrix

A ⊗ B :=

a11B a12B · · · a1mAB

a21B a22B · · · a2mAB

...... · · ·

...

anA1B anA2B · · · anAmAB

.

116 Mathematics

The Kronecker power of a matrix is defined similar to the standard matrix power as

X⊗,2 = X ⊗ X, X⊗,3 = X ⊗ X⊗,2, X⊗,i = X ⊗ X⊗,i−1, (A.1)

where the superscript ⊗, i denotes the i-th Kronecker power.

The Kronecker sum of two matrices A ∈ CnA, nA B ∈ CnB , nB is defined as

A ⊕ B := A ⊗ InB+ InA

⊗ B,

where In is the identity matrix of order n.

Furthermore, let vec(X) denote the vector that is formed by stacking the columns of X

into a single column vector:

vec(X) :=[

x11 x21 . . . xm1 x1n x2n . . . xmn

]T .

Using this stacking operator the following properties hold for complex1 matrices with

matching dimensions:

vec(AXB) = (BT ⊗ A) vec(X),

and

vec(AX + XB) = (BT ⊕ A) vec(X).

A.2 Algebraic Riccati Equations

This section will review important facts about algebraic Riccati equations (AREs). In

order to make this section rather self-contained we will state theorems presenting basic

properties of AREs.

The general algebraic matrix Riccati equation is given by

XRX − XP − P ∗X − Q = 0 , (A.2)

where R, P and Q are given n×n complex matrices with R and Q Hermitian, i.e., R = R∗

and Q = Q∗.

Together with (A.2) we will consider the matrix function

R(X) = XRX − XP − P ∗X − Q. (A.3)

1Note that T denotes the transpose, while ∗ is used for the complex conjugate transpose

A.2 Algebraic Riccati Equations 117

Associated with (A.2) is a 2n × 2n Hamiltonian matrix:

H :=

−P R

Q P ∗

. (A.4)

The following theorem [Zhou et al. 1996] gives a constructive description of all solutions

to (A.2).

Theorem A.1 ARE solutions

Let V ⊂ C2n be an n-dimensional invariant subspace of H, and let X1, X2 ∈ Cn×n

be two complex matrices such that

V = Im

X1

X2

.

If X1 is invertible, then X = X2X−11 is a solution to the Riccati equation (A.2)

and Λ(P − RX) = Λ(H|V), where H|V denotes the restriction of H to V.

�

Proof:

We only show the first part of the theorem that proofs the construction of solutions.

The following equation holds because V is an M invariant subspace:

−P R

Q P ∗

X1

X2

=

X1

X2

Λ.

Premultiply the above equation by [X −I ] and then postmultiply with X−11 to get

[

X −I]

−P R

Q P ∗

I

X

= 0 ,

XRX − XP − P ∗X − Q = 0 .

This shows that X = X2X−11 is actually a solution of (A.2).

�

118 Mathematics

Theorem A.1 shows that we can determine all solutions of (A.2) by constructing bases for

those invariant subspaces of H. For example, the invariant subspaces can be found by com-

puting the eigenvectors vi and corresponding generalized eigenvectors vi+1, . . . , vi+ki−1

related to eigenvalues λi of H with multiplicity ki. Taking all combinations of these vec-

tors, which have at least one actual eigenvector vi and are therefore H invariant, we can

calculate the solutions X = X2X−11 from

X1

X2

=[

vi vj

]

, i 6= j.

Before we turn to the important stabilizing solutions of an ARE, we consider maximal

solutions. The following theorem will be used in the proof of Theorem A.3 also.

Theorem A.2 Maximal solutions

Suppose that R = R∗ ≥ 0, Q = Q∗, (P, R) is stabilizable, and there is a Hermitian

solution of the inequality R(X) ≤ 0. Then R(X) = 0 has a maximal Hermitian

solution X+ for which X+ ≥ X for every Hermitian solution X of R(X) ≤ 0 holds.

Furthermore the maximal solution X+ guarantees that all eigenvalues of P − RX+

lie in the closed left half-plane.

�

See [Kleinman 1968], [Lancaster and Rodman 1995, p. 232] or [Zhou et al. 1996] for a

proof. The proof is constructive, i.e., it not only proofs the preceding statements, but

it also gives an iterative procedure to determine the maximal solution X+. A Newton

procedure can be derived to solve the equation R(X) = 0.

Decompose R into R = BB∗. Since the pair (P, R) is stabilizable, it can be seen from

the controllability matrix that the pair (P, B) is also stabilizable and there is a stabilizing

feedback matrix F0 for which P0 = P − BF0 is stable.

Then we determine X0 as the unique, Hermitian solution of the Lyapunov equation

X0P0 + P ∗0 X0 + F ∗

0 F0 + Q = 0.

In order to apply the Newton procedure to the matrix function R(X), we need the first

Frechet derivative [Ortega and Rheinboldt 1970]:

dRX(H) = −(H(P − RX) + (P − RX)∗H).

The Newton procedure is than given as

dRX(Xk+1 − Xk) = −R(Xk), k = 0, 1, 2, . . . (A.5)

A.2 Algebraic Riccati Equations 119

This can be written as the following Lyapunov equation,

Xk+1(P − RXk) + (P − RXk)∗Xk+1 = −XkRXk − Q, k = 0, 1, 2, . . . (A.6)

We can now finally turn to the important Theorem 3.1 which forms the basis for the

mapping equations, because it converts conditions for an ARE into an eigenvalue problem.

The theorem is restated here, see [Lancaster and Rodman 1995, p. 196]. The actual

mapping equations can be derived using the analytic extension given by Lancaster and

Rodman in 1995.

Theorem A.3 Stabilizing solutions

Suppose that R ≥ 0, Q = Q∗, (P, R) is stabilizable, and there is a Hermitian solution

of (A.2). Then for the maximal Hermitian solution X+ of (A.2), P −RX+ is stable,

if and only if the Hamiltonian matrix H defined in (A.4) has no eigenvalues on the

imaginary axis.

�

Proof:

Since the pair (P, R) is only required to be stabilizable, we start with a decomposi-

tion into a controllable and stable part, such that the matrices P , R, Q, and X can

be written as

P =

P11 P12

0 P22

, R =

R11 0

0 0

, Q =

Q11 Q12

Q∗12 Q22

, X =

X11 X12

X∗12 X22

, (A.7)

where the pair (P11, R11) is controllable and P22 is stable. This decomposition leads

to a Riccati equation in standard form for the controllable pair (P11, R11)

X11R11X11 − X11P11 − P ∗11X11 − Q11 = 0 , (A.8)

a Sylvester equation in X12, and a Lyapunov equation in X22. Furthermore the

term P − RX can be written as

P − RX =

P11 − R11X11 P12 − R11X12

0 P22

, (A.9)

and the ARE (A.8) has the associated Hamiltonian H11.

We are now ready to proof that stability of P − RX+ follows from H having not a

single pure imaginary eigenvalue. To this end, suppose that H has no eigenvalues

on the imaginary axis. The decomposition above leads to

Λ(H) = Λ(H11) ∪ Λ(−P22) ∪ Λ(P ∗22), (A.10)

120 Mathematics

and any solution X11 of (A.8) has the property Λ(P11 − R11X11) ⊆ Λ(H11). Using

Theorem A.2 it follows that (A.8) has a maximal solution X+11, and Λ(P11−R11X

+11)

is stable. From the triangular decomposition and in particular (A.9) then follows

that P − RX+ holds for the corresponding solution X+ of (A.2).

To proof the converse, assume that X+ is a maximal solution of (A.2) and P −RX+

is stable. Let X+ be decomposed similar to X in (A.7). From (A.9) then follows

that P11 − R11X+11 is stable and that X+

11 is a maximal solution of (A.8). There-

fore H11 has no pure imaginary eigenvalues and the proof is concluded, since (A.10)

implies that the same holds for H.

�

References 121

References

Ackermann, J., Der Entwurf linearer Regelungssysteme im Zustandsraum, Regelungstech-

nik, 1972, vol. 20, pp. 297–300.

Ackermann, J., Parameter space design of robust control systems, IEEE Trans. on Auto-

matic Control, 1980, vol. 25, pp. 1058–1072.

Ackermann, J., A. Bartlett, D. Kaesbauer, W. Sienel, and R. Steinhauser, Robust Control,

Springer-Verlag, 1993.

Ackermann, J., P. Blue, T. Bunte, L. Guvenc, D. Kaesbauer, M. Kordt, M. Muhler,

and D. Odenthal, Robust Control: The Parameter Space Approach, Springer-Verlag,

London, 2002.

Ackermann, J. and T. Bunte, Robust Prevention of Limit Cycles for Robustly Decoupled

Car Steering Dynamics, Kybernetika, 1999, vol. 35, no. 1, pp. 105–116.

Ackermann, J., D. Kaesbauer, and R. Munch, Robust Γ-stability analysis in a plant pa-

rameter space, Automatica, 1991, vol. 27, pp. 75–85.

Ackermann, J. and W. Sienel, Robust control for automatic steering, in Proc. American

Control Conf., San Diego, 1990, pp. 795–800.

Ackermann, J. and S. Turk, A common controller for a family of plant models, in Proc.

IEEE Conf. on Decision and Control, Orlando, 1982, pp. 240–244.

Allgower, E. L. and K. Georg, Numerical Continuation Methods, Springer-Verlag, 1990.

Allgower, E. L. and K. Georg, Continuation and Path Following, Acta Numerica, 1992,

vol. 2, pp. 1–64.

Anderson, B. D. O., A system theory criterion for positive real matrices, SIAM Journal

on Control, 1967, vol. 5, pp. 171–182.

Anderson, B. D. O. and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern

Systems Theory Approach, Prentice-Hall, 1973.

122 References

Arnon, D. S. and S. McCallum, A polynomial time algorithm for the topological type of a

real algebraic curve, Journal of Symbolic Computation, 1988, vol. 5, pp. 213–236.

Barmish, B. R., J. Ackermann, and H. Z. Hu, The tree structured decomposition: a

new approach to robust stability analysis, in Proc. Conf. on Information Sciences and

Systems, Princeton, 1990a, pp. 133–139.

Barmish, B. R., P. P. Khargonekar, Z. C. Shi, and R. Tempo, Robustness margin need

not be a continuous function of the problem data, Systems and Control Letters, 1990b,

vol. 15, pp. 91–98.

Bart, H., I. Gohberg, and M. A. Kaashoek, Constructive Methods of Wiener-Hopf Fac-

torization, Birkhauser, 1986.

Beckermann, B. and G. Labahn, Numeric and Symbolic Computation of problems defined

by Structured Linear Systems, Reliable Computing, 2000, vol. 6, pp. 365–390.

Besson, V. and A. T. Shenton, Interactive Control System Design by a mixed H∞ Pa-

rameter Space Method, IEEE Trans. on Automatic Control, 1997, vol. 42, no. 7, pp.

946–955.

Besson, V. and A. T. Shenton, An Interactive Parameter Space Method For Robust Per-

formance in Mixed Sensitivity Problems, IEEE Trans. on Automatic Control, 1999,

vol. 44, no. 6, pp. 1272–1276.

Boyd, S., Robust Control Tools: Graphical User-Interfaces and LMI Algorithms, Systems,

Control and Information, 1994, vol. 38, no. 3, pp. 111–117, special issue on Numerical

Approaches in Control Theory.

Boyd, S., V. Balakrishnan, and P. Kabamba, A bisection method for computing the H∞

norm of a transfer matrix and related problems, Mathematics of Control, Signals and

Systems, 1989, vol. 2, no. 3, pp. 207–219.

Boyd, S. and C. Barratt, Linear Controller Design: Limits of Performance, Prentice-Hall,

1991.

Boyd, S., L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in

System and Control Theory, SIAM studies in applied mathematics, 1994.

Boyd, S. and Q. Yang, Structured and Simultaneous Lyapunov Functions for System

Stability Problems, Int. Journal of Control, 1989, vol. 49, no. 6, pp. 2215–2240.

Bunte, T., Mapping of Nyquist/Popov Theta-stability margins into parameter space, in

Proc. 3rd IFAC Symp. on Robust Control Design, Prague, Czech Republic, 2000.

References 123

Chen, C. T., Linear System Theory and Design, Hold, Rinehart and Winston, New York,

1984.

Coolidge, J. L., A Treatise on Algebraic Plane Curves, Dover Publications, 2004.

Davidenko, D., On a new method of numerical solution of systems of nonlinear equations

(in Russian), Dokl. Akad. Nauk USSR, 1953, vol. 88, pp. 601–602.

Doyle, J. C., Guaranteed margins for LQG regulators, IEEE Trans. on Automatic Control,

1978, vol. 23, no. 4, pp. 756–757.

Doyle, J. C., Analysis of feedback systems with structured uncertainties, IEE Proc., 1982.

Doyle, J. C., Redondo Beach lecture notes, Internal Report, Caltech, Pasadena, 1986.

Doyle, J. C., K. Glover, P. Khargonekar, and B. Francis, State-space solution to standard

H2 and H∞ control problems, IEEE Trans. on Automatic Control, 1989, vol. 34, no. 8,

pp. 831–847.

Doyle, J. C. and G. Stein, Multivariable feedback design: Concepts for a classical/modern

synthesis, IEEE Trans. on Automatic Control, 1981, vol. 26, no. 1, pp. 4–16.

Earnshaw, R. A. and N. Wiseman, An Introductory Guide to Scientific Visualization,

Springer, 1992.

Evans, W. R., Graphical analysis of control systems, Trans. AIEE, 1948, vol. 67, no. 2,

pp. 547–551.

Farin, G., Curves and Surfaces for Computer Aided Geometric Design, Morgan Kauf-

mann, 2001.

Forsman, K. and J. Eriksson, Solving the ARE symbolically, Technical Report LiTH-ISY-

R-1456, Dept. of Electrical Engineering, Linkoping University, 1993.

Francis, B., A course in H∞ control theory, Springer, New-York, 1987.

Gajic, Z. and M. Qureshi, Lyapunov Matrix Equation in System Stability and Control,

Mathematics in Science and Engineering Series, Academic Press, 1995.

Gilbert, E. G., Controllability and observability in multi-variable control systems, Int.

Journal of Control, 1963, vol. 1, no. 2, pp. 128–151.

Glover, K., All optimal Hankel-norm approximation of linear multivarible systems and

their L∞- error bounds, Int. Journal of Control, 1984, vol. 39, pp. 1115–1193.

124 References

Gonzalez-Vega, L. and I. Necula, Efficient topology determination of implicitly defined

algebraic plane curves, Computer Aided Geometric Design, 2002, vol. 19, no. 9, pp.

719–743.

Graham, A., Kronecker Products and Matrix Calculus with Applications, J. Wiley & Sons,

1981.

Green, M., K. Glover, D. Limebeer, and J. C. Doyle, A J-spectral approach to H∞ control,

SIAM Journal on Control and Optimization, 1990, vol. 28, pp. 1350–1371.

Haddad, W. M. and D. S. Bernstein, Robust stabilization with positive real uncertainty:

Beyond the small gain theorem, Systems and Control Letters, 1991, vol. 17, pp. 191–208.

Hara, S., T. Kimura, and R. Kondo, H∞ control system design by a parameter space

approach, in Proc. 10th Int. Symp. on Mathematical Theory of Networks and Systems,

Kobe, Japan, 1991, pp. 287–292.

Hermite, C., Sur le nombre des racines d’une equation algebrique comprise entre des

limites donnees, J. Reine Angewandte Mathematik, 1856, vol. 52, pp. 39–51, English

translation Int. Journal of Control 1977.

Hinrichsen, D. and A. J. Pritchard, Stability radius for structured perturbations and the

algebraic Riccati equation, Systems and Control Letters, 1986, vol. 8, pp. 105–113.

Hurwitz, A., Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit

negativen reellen Theilen besitzt, Mathematische Annalen, 1895, vol. 46, pp. 273–284.

Johansson, K. H., The Quadruple-Tank Process: A Multivariable Laboratory Process with

an Adjustable Zero, IEEE Trans. on Control Systems Technology, 2000, vol. 8, no. 3,

pp. 456–465.

Jonsson, U., A Popov Criterion for Systems with Slowly Time-Varying Parameters, IEEE

Trans. on Automatic Control, 1999, vol. 44, no. 4, pp. 844–846.

Jonsson, U., Lectures on Input-Output Stability and Integral Quadratic Constraints, Lec-

ture Notes, Royal Institute of Technology, Stockholm, Sweden, 2001.

Jonsson, U. and A. Megretski, The Zames-Falb IQC for Critically Stable Systems, Tech-

nical report, MIT, 1997.

Joos, H.-D., A. Varga, and R. Finsterwalder, Multi-Objective Design Assessment, in IEEE

Symp. on Computer-Aided Control System Design, Kohala, Hawaii, USA, 1999.

Kabamba, P. T. and S. P. Boyd, On parametric H∞ optimization, in Proc. IEEE Conf.

on Decision and Control, Austin, Texas, 1988, vol. 4, pp. 1354–1355.

References 125

Kailath, T., Linear Systems, Prentice-Hall, 1980.

Kalman, R. E., Mathematical description of linear dynamical systems, SIAM Journal on

Control, 1963.

Kalman, R. E. and R. S. Bucy, New results in linear filtering and prediction theory, ASME

Trans. Series D: J. Basic Engineering, 1960, vol. 83, pp. 95–108.

Khalil, H. K., Nonlinear Systems, Macmillan, 1992.

Kleinman, D. L., On an iterative technique for Riccati equation computation, IEEE Trans.

on Automatic Control, 1968, vol. 13, pp. 114–115.

Kugi, A. and K. Schlacher, Analysis and Synthesis of Non-linear Dissipative Systems:

An Overview (Part 2) (in German), Automatisierungstechnik, 2002, vol. 50, no. 3, pp.

103–111.

Kwakernaak, M. A. and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience,

1972.

Lancaster, P. and L. Rodman, Existence and uniqueness theorem for the algebraic Riccati

equation, Int. Journal of Control, 1980, vol. 32, no. 2, pp. 285–309.

Lancaster, P. and L. Rodman, Algebraic Riccati equations, Oxford Science Publications,

1995.

Laub, A. J., A Schur method for solving the algebraic Riccati equations, IEEE Trans. on

Automatic Control, 1979, vol. 24, pp. 913–925.

Lind, R. and M. J. Brenner, Robust Flutter Margin Analysis that Incorporates Flight Data,

Technical report, NASA, 1998.

Lunze, J., Robust Multivariable Feedback Control, Prentice-Hall, London, 1988.

Maciejowski, J. M., Multivariable Feedback Design, Electronic Systems Engineering Series,

Addison-Wesley, 1989.

Magni, J. F., Linear Fractional Representations with a Toolbox for Use with MATLAB,

Toolbox Manual, ONERA, Technical Report TR 240/01 DCSD, 2001.

Maxwell, J. C., On Governors, Proc. Royal Soc. London, 1866, vol. 16, pp. 270–283.

McFarlane, D. and K. Glover, Robust Controller Design Using Normalized Coprime Factor

Plant Descriptions, vol. 138 of Lecture Notes in Control and Information Sciences,

Springer-Verlag, 1990.

126 References

Megretski, A. and A. Rantzer, System analysis via integral quadratic constraints, IEEE

Trans. on Automatic Control, 1997, vol. 42, no. 6, pp. 819–830.

Mitrovic, D., Graphical analysis and synthesis of feedback control systems, Trans. AIEE,

1958, vol. 77, no. 2, pp. 476–503.

Morgan, A., Solving polynomial systems using continuation for engineering and scientific

problems, Prentice-Hall, 1987.

Muhler, M., Mapping MIMO Control Specifications into Parameter Space, in Proc. IEEE

Conf. on Decision and Control, Las Vegas, NV, 2002, pp. 4527–4532.

Muhler, M. and J. Ackermann, Representing multiple objectives in parameter space us-

ing color coding, in Proc. 3rd IFAC Symp. on Robust Control Design, Prague, Czech

Republic, 2000.

Muhler, M. and J. Ackermann, Mapping Integral Quadratic Constraints into Parameter

Space, in Proc. American Control Conf., Boston, MA, 2004, pp. 3279–3284.

Muhler, M. and D. Odenthal, Paradise: Ein Werkzeug fur Entwurf und Analyse robuster

Regelungssysteme im Parameterraum, in 3. VDI/VDE-GMA Aussprachetag, Rechn-

ergestutzter Entwurf von Regelungssystemen, Dresden, 2001.

Muhler, M., D. Odenthal, and W. Sienel, Paradise User’s Manual, Deutsches Zentrum

fur Luft und Raumfahrt e. V., Oberpfaffenhofen, 2001.

Mustafa, D. and K. Glover, Minimum Entropy H∞ Control, Lecture Notes in Control and

Information Sciences, Springer-Verlag, 1990.

Odenthal, D. and P. Blue, Mapping of frequency response magnitude specifications into

parameter space, in Proc. 3rd IFAC Symp. on Robust Control Design, Prague, Czech

Republic, 2000.

Ortega, J. M. and W. C. Rheinboldt, Iterative solution of nonlinear equations in several

variables, Academic Press, New York, 1970.

Otter, M., Objectoriented Modeling of Physical Systems (in German), Automatisierung-

stechnik, 1999, vol. 47/48, series ”Theorie fur den Anwender”, 17 parts from 1/1999 to

12/2000.

Owen, J. G. and G. Zames, Robust H∞ disturbance minimization by duality, Systems and

Control Letters, 1992, vol. 19, no. 4, pp. 255–263.

Petersen, I. R., Disturbance attenuation and H∞-optimization: A design method based on

the algebraic Riccati equations, IEEE Trans. on Automatic Control, 1987, vol. 32, pp.

427–429.

References 127

Poincare, H., Les Methodes Nouvelles de la Mecanique Celeste, Gauthier-Villars, Paris,

1892.

Ran, A. C. M. and L. Rodman, On parameter dependence of solutions of algebraic Riccati

equations, Mathematics of Control, Signals and Systems, 1988, vol. 1, pp. 269–284.

Rantzer, A., On the Kalman-Yakubovich-Popov lemma, Systems and Control Letters,

1996, vol. 28, pp. 7–10.

Rosenbrock, H. H., State-space and multivariable theory, Nelson, London, 1970.

Rotea, M. A., The generalized H2 control problem, Automatica, 1993, vol. 29, no. 2, pp.

373–385.

Routh, E. J., A Treatise on the Stability of a Given State of Motion, Macmillan, London,

1877.

Safonov, M. G., Stability margins of diagonally perturbed multivariable feedback systems,

IEE Proc., 1982.

Safonov, M. G. and V. V. Kulkarni, Zames-Falb Multipliers for MIMO Nonlinearities,

Int. Journal of Nonlinear and Robust Control, 2000.

Sakkalis, T., The topological configuration of a real algebraic curve, Bulletin of the Aus-

tralian Mathematical Society, 1991, vol. 43, no. 1, pp. 37–50.

Sandberg, I. W., On the L2-boundedness of solutions of nonlinear functional equations,

Bell Syst. Tech. J., 1964.

Scherer, C., P. Gahinet, and M. Chilali, Multiobjective output-feedback control via LMI

optimization, IEEE Trans. on Automatic Control, 1997, vol. 42, no. 7, pp. 896–911.

Schmid, C., Zur direkten Berechnung von Stabilitatsrandern, Wurzelortskurven und H∞-

Normen, Habilitationsschrift, Fortschritt-Berichte, Reihe 8, Nr. 367, VDI-Verlag, 1993,

(In German).

Sepulchre, R., M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control, Springer-

Verlag, London, 1997.

Sienel, W., T. Bunte, and J. Ackermann, Paradise - A Matlab-based robust control

toolbox, in Proc. IEEE Symp. on Computer-Aided Control System Design, Dearborn,

MI, 1996, pp. 380–385.

Skogestad, S. and I. Postlethwaite, Multivariable Feedback Control, John Wiley & Sons,

1996.

128

Tiller, M., Introduction to Physical Modeling with Modelica, Kluwer Academic Publishers,

Boston, 2001.

van Hoeij, M., Rational Parametrizations of Algebraic Curves using a Canonical Divisor,

Journal of Symbolic Computation, 1997, vol. 23, pp. 209–227.

Varga, A. and G. Looye, Symbolic and numerical software tools for LFT-based low order

uncertainty modeling, in IEEE Int. Symp. on Computer-Aided Control System Design,

Hawaii, 1999, pp. 1–6.

Vidyasagar, M., Nonlinear System Analysis, Prentice-Hall, 1978.

Vishnegradsky, I. A., Sur la theorie generale des regulateurs, Compt. Rend. Acad. Sci.,

1876, vol. 83, pp. 318–321.

Walker, R. J., Algebraic Curves, Springer, 1978.

Willems, J. C., Least squares stationary optimal control and the algebraic Riccati equation,

IEEE Trans. on Automatic Control, 1971, vol. 16, no. 6, pp. 621–634.

Wilson, D. A., Convolution and Hankel operator norms for linear systems, IEEE Trans.

on Automatic Control, 1989, vol. 34, no. 1, pp. 94–97.

Zames, G., Feedback and optimal sensitivity: model reference transformations, multiplica-

tive seminorms, and approximate inverse, IEEE Trans. on Automatic Control, 1981,

vol. 26, pp. 301–320.

Zames, G. and P. L. Falb, Stability conditions for systems with monotone and slope-

restricted nonlinearities, SIAM Journal on Control, 1968, vol. 6, no. 1, pp. 89–108.

Zhou, K., J. C. Doyle, and K. Glover, Robust and optimal control, Prentice-Hall, 1996.

129

Lebenslauf

Michael Ludwig Muhler

geboren am 3. Februar 1973 in Creglingen

verheiratet, ein Kind

September 1983 - Mai 1992 Gymnasium Weikersheim

Oktober 1993 - Juli 1997 Studium der Technischen Kybernetik

an der Universitat Stuttgart

August 1997 - September 1998 Graduate Studies

in Chemical and Electrical Engineering

University of Wisconsin, Madison

Oktober 1998 - Januar 1999 Studium der Technischen Kybernetik

an der Universitat Stuttgart

Januar 1999 - August 2002 Wissenschaftlicher Mitarbeiter beim

Deutschen Zentrum fur Luft- und Raumfahrt

Oberpfaffenhofen

Institut fur Robotik und Mechatronik

seit September 2002 Mitarbeiter der Robert Bosch GmbH

Stuttgart

Korntal-Munchingen, im Marz 2007

Robust control system design by mapping specifications ...Robust Control System Design by Mapping...

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