model order reduction of linear and nonlinear systems in

Model order reduction of linear andnonlinear systems in the Loewner framework

Ion Victor Gosea

a Thesis submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophyin Electrical Engineering

Chair: Prof. Dr. Athanasios C. Antoulas1,2

Second Committee Member: Dr. rer. nat. habil. Mathias Bode1

Third Committee Member: Prof. Dr. Peter Benner 3

1 Jacobs University Bremen2 Rice University Houston3 Max Planck Institute Magdeburg

Date of Defense: January 6th, 2017

Department of Computer Science & Electrical Engineering

Statutory Declaration

Family Name, Given/First Name Ion Victor Gosea

Matriculation number 20327911

What kind of thesis are you submitting:Bachelor-, Master- or PhD-Thesis

PhD-Thesis

English: Declaration of Authorship I hereby declare that the thesis submitted was created and written solely by myself withoutany external support. Any sources, direct or indirect, are marked as such. I am aware of thefact that the contents of the thesis in digital form may be revised with regard to usage ofunauthorized aid as well as whether the whole or parts of it may be identified as plagiarism. Ido agree my work to be entered into a database for it to be compared with existing sources,where it will remain in order to enable further comparisons with future theses. This does notgrant any rights of reproduction and usage, however.

This document was neither presented to any other examination board nor has it beenpublished.

German: Erklärung der Autorenschaft (Urheberschaft) Ich erkläre hiermit, dass die vorliegende Arbeit ohne fremde Hilfe ausschließlich von mirerstellt und geschrieben worden ist. Jedwede verwendeten Quellen, direkter oder indirekterArt, sind als solche kenntlich gemacht worden. Mir ist die Tatsache bewusst, dass der Inhaltder Thesis in digitaler Form geprüft werden kann im Hinblick darauf, ob es sich ganz oder inTeilen um ein Plagiat handelt. Ich bin damit einverstanden, dass meine Arbeit in einerDatenbank eingegeben werden kann, um mit bereits bestehenden Quellen verglichen zuwerden und dort auch verbleibt, um mit zukünftigen Arbeiten verglichen werden zu können.Dies berechtigt jedoch nicht zur Verwendung oder Vervielfältigung.

Diese Arbeit wurde noch keiner anderen Prüfungsbehörde vorgelegt noch wurde sie bisherveröffentlicht.

……………………………………………………………………………………………………….Date, Signature

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AbstractThe Loewner framework is an interpolatory model order reduction technique that uses measuredor computed data, e.g., measurements of the frequency response of a to-be approximated dy-namical system instead of the system matrices, and constructs reduced models based on a rankrevealing factorization of appropriately constructed matrices.

In this thesis, we propose extensions of the classical Loewner framework for reduction oflinear systems to some specific applications such as reducing classes of mildly nonlinear systems.The later includes bilinear, quadratic-bilinear and linear switched systems.

The motivation behind this endeavor is that some of these aforementioned classes of systemscan be viewed as a bridge between linear and nonlinear systems. For example, one can alwayswrite an approximation of a nonlinear system by means of a bilinear system. Moreover, forcertain types of nonlinear systems, we can always find an equivalent quadratic-bilinear modelwithout performing any approximation. Linear switched systems have been extensively studiedin the literature since they offer a valuable addition to the class of linear systems, althoughreduction of such systems is arguably new. They can also be viewed as an intermediate steptowards hybrid systems.

For all the classes of systems that were previously mentioned, the overall strategy for extend-ing the Loewner framework is conceptually similar. After collecting samples of input/outputfrequency domain mappings, e.g., either by means of measuring or by direct computation, onemakes use of a specific arrangement of the data in matrix format. Hence, following some the-oretical considerations, one can build reduced order models directly from the given data. Thereduced systems have similar response to the large-scale original systems. More exactly, the in-put/output mappings for both systems have similar characteristics in the frequency range wherethe samples were considered.

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PublicationsLarge parts of this thesis have been published, are submitted for publication or were presentedat various conferences and workshops. In order of appearance in this document, we mention thefollowing:

1. Chapter 3, Section 3.4 ;

• I.V. Gosea, A.C. Antoulas, Stability preserving post-processing methods applied inthe Loewner framework, IEEE 20th Workshop on Signal and Power Integrity (SPI),2016, Torino, Italy.

2. Chapter 4;

• [10], A.C. Antoulas, I.V. Gosea, A.C. Ionita, Model reduction of bilinear systems inthe Loewner framework, Computational Methods in Science and Engineering, SIAMJournal on Scientific Computing (SISC), Volume 38, Issue 5, pp. B889–B916.

3. Chapter 5;

• I.V. Gosea, A.C. Antoulas, Data-driven model order reduction of quadratic-bilinearsystems, submitted to Numerical Linear Algebra with Applications.• I.V. Gosea, A.C. Antoulas, Model reduction of linear and nonlinear systems in the

Loewner framework: A summary, European Control Conference (ECC), 2015, Linz,Austria.• I.V. Gosea, A.C. Antoulas, Model reduction of qudratic-bilinear systems in the

Loewner framework, 7th Workshop on ”Matrix Equations and Tensor Techniques”(METT VII) 2017, Pisa, Italy.

4. Chapter 6;

• I.V. Gosea, M. Petreczky, A.C. Antoulas, Data-driven model order reduction oflinear switched systems in the Loewner framework, submitted to SIAM Journal onScientific Computing (SISC).• I.V. Gosea, M. Petreczky, A.C. Antoulas, Model order reduction of linear switched

systems (LSS) in the Loewner framework, 20th Conference of the International LinearAlgebra Society (ILAS), 2016, Leuven, Belgium.• I.V. Gosea, A.C. Antoulas, Model reduction of linear switched systems from computed

data, 16th GAMM Workshop on Applied and Numerical Linear Algebra (ANLA),2016, Hamburg, Germany.

5. Appendix A;

• I.V. Gosea, A.C. Antoulas, Approximation of a damped Euler-Bernoulli beam usingthe Loewner framework, technical report, available online at https://arxiv.org/pdf/1712.06031.pdf.

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https://arxiv.org/pdf/1712.06031.pdf


AcknowledgementsI would like to express my sincere gratitude to my adviser Prof. Dr. Athanasios C. Antoulas, whohas been a tremendous mentor for me. I would like to thank him for encouraging my research andfor allowing me to grow as a research scientist. I truly appreciated his advice on both research aswell as on my career in general, his continuous support of my Ph.D. studies and related research,patience, motivation, and immense knowledge. It was an honor and a privilege to have been hisstudent for more than five years.

I would also like to thank the committee members, Dr. Bode and Prof. Dr. Benner foraccepting to take part in the dissertation committee.

Special thanks to my family; words cannot express how grateful I am to my father for all of thesacrifices that he has made on my behalf. Starting in elementary school, twenty years ago, whenI participated in the first mathematics contest, through middle and high school (national andinternational mathematics olympiads) and all the way to the university level, he has supportedme to fulfill my dreams and guided me with his priceless advice.

I would like to express my gratitude to my first math teacher who believed in me and en-couraged me to participate in math competitions.

I would also like to thank all of my friends who helped me in different ways throughout mystudies and for the gift of companionship.

I wish to thank my girlfriend who stayed by my side, supported me and offered me herencouragement in time of need.

I would like to thank Dr. Mihaly Petreczky for suggesting the problem of model reductionof linear switched systems, for the research collaboration and for general useful discussions, inparticular on how to thoroughly write a research paper or a review report.

I wish to thank Dr. Jens Saak for for helpful discussions and for recommending a toolboxwhich was very helpful for implementing some of the numerical experiments.

I would like to thank Pawan Goyal for insightful discussions and for providing helpful softwareused in my simulations.

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To my father Ion.

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Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPublications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Introduction 11.1 Approaches to model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 MOR applied to dynamical linear and nonlinear systems . . . . . . . . . . . . . . 3

2 General Properties of Dynamical Systems 62.1 Approximating and transforming nonlinear systems . . . . . . . . . . . . . . . . . 6

2.1.1 Carleman linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 McCormick relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Variational equation approach . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Special matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Bilinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Quadratic-bilinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Input-output mappings of dynamical systems . . . . . . . . . . . . . . . . . . . . 322.3.1 Volterra series representation (time-domain) . . . . . . . . . . . . . . . . . 322.3.2 Frequency domain mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.3 A new class of transfer functions . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Norms on dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 The Loewner Framework for Linear Systems 463.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Set-up of the Loewner interpolation framework . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.2 Rational interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.3 The Loewner matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Interpolatory reduction methods and the Loewner framework . . . . . . . . . . . . 523.3.1 Sylvester equations for O and R . . . . . . . . . . . . . . . . . . . . . . . . 54

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3.3.2 The Loewner pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.3 Construction of interpolants (models) . . . . . . . . . . . . . . . . . . . . . 553.3.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Enforcing stability in the Loewner framework . . . . . . . . . . . . . . . . . . . . 613.4.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.2 Post processing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 The Loewner Framework for Bilinear Systems 694.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Literature overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Bilinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 The main procedure for extending the Loewner framework from linear to bilinear

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.1 The generalized controllability and observability matrices . . . . . . . . . . 714.3.2 The generalized Loewner pencil . . . . . . . . . . . . . . . . . . . . . . . . 754.3.3 Construction of interpolants . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.4 One-sided interpolation and parametrized reduced models . . . . . . . . . 80

4.4 Volterra series interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.1 One-sided interpolation in the Volterra framework . . . . . . . . . . . . . . 904.4.2 Two-sided mixed interpolation conditions . . . . . . . . . . . . . . . . . . . 91

4.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.5.1 Bilinear controlled heat transfer system . . . . . . . . . . . . . . . . . . . . 924.5.2 Burgers’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.5.3 Chafee-Infante equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 The Loewner Framework for Quadratic-Bilinear Systems 985.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1.1 Literature overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2 Quadratic-bilinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 The main procedure for extending the Loewner framework to QB systems . . . . . 101

5.3.1 Arranging the data in the required format . . . . . . . . . . . . . . . . . . 1015.3.2 The interpolation property . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.3.3 Sylvester-type equations satisfied by the matrices O and R . . . . . . . . . 1035.3.4 The generalized Loewner pencil . . . . . . . . . . . . . . . . . . . . . . . . 1075.3.5 Sylvester equations satisfied by the Loewner matrices . . . . . . . . . . . . 1095.3.6 Construction of reduced order models . . . . . . . . . . . . . . . . . . . . . 110

5.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.1 Burgers’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.2 Nonlinear RLC network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4.3 Chafee-Infante equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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6 The Loewner Framework for Linear Switched Systems 1186.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Linear switched systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.3 The Loewner framework for LSS - the case D=2 . . . . . . . . . . . . . . . . . . . 123

6.3.1 The generalized controllability and observability matrices . . . . . . . . . . 1246.3.2 The generalized Loewner pencil . . . . . . . . . . . . . . . . . . . . . . . . 1326.3.3 Construction of reduced order models . . . . . . . . . . . . . . . . . . . . . 136

6.4 The Loewner framework for linear switched systems - the general case . . . . . . . 1376.4.1 Sylvester equations for Rq and Oq . . . . . . . . . . . . . . . . . . . . . . . 1396.4.2 The Loewner matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.4.3 Construction of reduced order models . . . . . . . . . . . . . . . . . . . . . 142

6.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.5.1 The balanced truncation method proposed in [94] . . . . . . . . . . . . . . 1446.5.2 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.5.3 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.5.4 Third example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7 Conclusion 155

A The Loewner Framework Applied to a Vibrating Beam Model 156A.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

B Various Proofs 164B.0.1 Proof of Proposition 2.2.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

B.1 Proof of Proposition 2.2.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.2 Proof of Proposition 2.2.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166B.3 Proof of Lemma 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Bibliography 170

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List of Tables

3.1 H∞ norm of the error systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 H2 norm of the error systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3 H∞ norm of the error systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 Samples of transfer functions that are matched . . . . . . . . . . . . . . . . . . . . 76

6.1 Relative approximation error for the two modes in the H2 and H∞ norms . . . . . 148

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List of Figures

3.1 RLC circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Singular value decay of the Loewner matrix. . . . . . . . . . . . . . . . . . . . . . 593.3 Frequency response comparison − first frequency band choice. . . . . . . . . . . . 593.4 Frequency response comparison − second frequency band choice. . . . . . . . . . . 593.5 Singular value decay of the Loewner matrix. . . . . . . . . . . . . . . . . . . . . . 603.6 Frequency response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7 Error analysis in frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . 603.8 Singular value decay of the Loewner matrix. . . . . . . . . . . . . . . . . . . . . . 613.9 Frequency response comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.10 Frequency response for each input/output combination. . . . . . . . . . . . . . . . 623.11 Singular values of the Loewner matrix. . . . . . . . . . . . . . . . . . . . . . . . . 663.12 Frequency response for different methods. . . . . . . . . . . . . . . . . . . . . . . 673.13 Variation of the H∞ norm for different parameters γ. . . . . . . . . . . . . . . . . 673.14 Singular values of the Loewner matrix. . . . . . . . . . . . . . . . . . . . . . . . . 683.15 Variation of the number of antistable poles for different k. . . . . . . . . . . . . . 683.16 Frequency response comparison when using the optimal method. . . . . . . . . . . 683.17 Frequency response comparison when using the sub-optimal method. . . . . . . . 68

4.1 Singular values of the Loewner pencil. . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Time domain simulation - output signals. . . . . . . . . . . . . . . . . . . . . . . . 934.3 Time domain simulation - approximation errors. . . . . . . . . . . . . . . . . . . . 944.4 Singular values of the Loewner pencil. . . . . . . . . . . . . . . . . . . . . . . . . . 954.5 Time domain simulation - output signals. . . . . . . . . . . . . . . . . . . . . . . . 954.6 Time domain simulation- approximation errors. . . . . . . . . . . . . . . . . . . . 954.7 Singular values of the Loewner pencil. . . . . . . . . . . . . . . . . . . . . . . . . . 964.8 Time domain simulation - output signals. . . . . . . . . . . . . . . . . . . . . . . . 964.9 Time domain simulation- approximation errors. . . . . . . . . . . . . . . . . . . . 97

5.1 Singular values of the Loewner pencil; (a) bilinear; (b) quadratic-bilinear. . . . . . 1135.2 The poles of the reduced Loewner models. . . . . . . . . . . . . . . . . . . . . . . 1135.3 Time domain simulations − original vs. reduced systems. . . . . . . . . . . . . . . 1145.4 Time-domain approximation error between original and reduced systems. . . . . . 1145.5 Circuit schematic [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.6 Singular value decay of the QB Loewner pencil. . . . . . . . . . . . . . . . . . . . 1155.7 Time-domain simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.8 Relative error between the response of the original system and of the reduced ones.116

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5.9 Singular value decay of the QB Loewner pencil. . . . . . . . . . . . . . . . . . . . 1175.10 Time-domain simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.11 Relative error between the response of the original system and of the reduced ones.117

6.1 Schematic of the evaporator vessel [95]. . . . . . . . . . . . . . . . . . . . . . . . . 1456.2 Frequency response of the original subsystems. . . . . . . . . . . . . . . . . . . . . 1466.3 Decay of the singular values for different matrices. . . . . . . . . . . . . . . . . . . 1476.4 Frequency domain approximation error. . . . . . . . . . . . . . . . . . . . . . . . . 1486.5 Time domain simulation - first choice of input. . . . . . . . . . . . . . . . . . . . . 1496.6 Time domain simulation - second choice of input. . . . . . . . . . . . . . . . . . . 1496.7 Schematic of the tool slide on the guide rails of the stand [83]. . . . . . . . . . . . 1506.8 Frequency response of the original subsystems. . . . . . . . . . . . . . . . . . . . . 1516.9 Decay of the singular values of the different matrices. . . . . . . . . . . . . . . . . 1526.10 The control and switched input signals. . . . . . . . . . . . . . . . . . . . . . . . . 1526.11 Time domain simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.12 Time domain approximation error. . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.1 Original frequency response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.2 Poles and zeros of the original transfer function. . . . . . . . . . . . . . . . . . . . 159A.3 Decay of the Loewner singular values - original samples. . . . . . . . . . . . . . . 160A.4 Frequency response comparison - original vs. Loewner. . . . . . . . . . . . . . . . 160A.5 Poles and zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A.6 Frequency response comparison - original vs. modal. . . . . . . . . . . . . . . . . 161A.7 Error in frequency domain - Loewner vs. modal. . . . . . . . . . . . . . . . . . . . 161A.8 Decay of the Loewner singular values - FE samples. . . . . . . . . . . . . . . . . . 162A.9 Frequency response comparison for various methods. . . . . . . . . . . . . . . . . 162A.10 Frequency response comparison - Loewner vs. Loewner FE. . . . . . . . . . . . . . 163

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Notation

N the set of natural numbersZ the set of integer numbersR the set of real numbersC the set of complex numbersRn×m the set of real valued matrices with n rows and m columnsRe(z) the real part of a complex number zC− the set of complex numbers with negative real partIn or simply I the identity matrix of size n× n0n or simply 0 the null (zero) matrix of size n× nek,n(ek) the kth unit vector of size n1n a column vector of size n with only 1 entriesdiag(x) the diagonal matrix with the vector x on its diagonalblkdiag(. . .) the block diagonal matrix with various blocks xtridiag(x) the tridiagonal matrix with entries xX matrix (in general real valued) of size n×mXT the transpose of a matrix XXk the kth column of a matrix X (i.e. = Xek)Xij the (i, j) entry of a matrix X (i.e. = eTi Xej)tr(X) the trace of a matrix X (i.e. =

∑n

k=1 Xkk)range(X) the range or column space of a matrix X (i.e. =

∑n

k=1 ckXk)ker(X) the kernel or nullspace of a matrix X (i.e. = v ∈ Rn|Xv = 0)eig(X) the set of the eigenvalues of a matrix Xrank(X) the column rank of a matrix Xvec(X) the vectorization operator (i.e. = [X1; X2; . . . ; Xn])vec−1(X) the inverse of the vectorization operator⊗ the Kronecker product operatorJn Jordan block of size n× nL(f) the unilateral Laplace transform of a function fsup(f) the supremum of a function f‖X‖2 the spectral norm of a matrix X‖Σ‖H2 the H2 norm of a dynamical system Σ‖Σ‖H∞ the H∞ norm of a linear system Σ

xv

Chapter 1

Introduction

Dynamical systems are a principal tool in the modeling, prediction, and control of physicalphenomena ranging from heat dissipation in complex microelectronic devices, to vibration sup-pression in large wind turbines, to storm surges before an advancing hurricane. Direct numericalsimulation of these models may be the only possibility for accurate prediction or control of suchcomplex phenomena.

However, an ever-increasing need for improved accuracy requires inclusion of more detail atthe modeling stage, leading inevitably to consideration of larger-scale, more complex dynam-ical systems. Such large-scale systems are often linked to spatial discretization of underlyingtime-dependent coupled partial differential equations (PDEs). Sometimes, additional algebraicconstraints arise which are leading to differential-algebraic equations (DAEs). The simulation ofsuch systems can create considerably large demands on computational resources.

In broad terms, model order reduction (MOR) is used to replace large, complex models of timedependent processes into smaller, simpler models that are still capable of accurately representingthe behavior of the original process under a variety of conditions.

The motivation for MOR stems from the need for accurate modeling of physical phenomenathat often leads to large-scale dynamical systems which require long simulation times and largedata storage. Logically, MOR seems to be a valid candidate for meeting some of the needs thatappear when dealing with large scale engineering problems.

The principal focus is finding a way to accomplish the transformation from the original systemto the reduced system (by means of MOR). This should be done systematically and by use ofsystem-theoretical ideas and by making use of strategies from computational linear algebra.

The goal is an efficient, methodical strategy that yields a dynamical system with a substan-tially lower dimension than that of the original system, hence requiring far less computationalresources for realization, while retaining response characteristics close to the original system.

The reduced order models can be efficiently used as surrogates for the original model, i.e.,by replacing it as a component in large scale simulations. Some general goals for reduced ordermodels are mentioned below,

1. The reduced input-output map should be uniformly “close” to the original.2. Critical system features and structure should be preserved (such as stability, passivity,

Hamiltonian structure, subsystem; interconnectivity, or second-order structure).3. Strategies for computing the reduced system should lead to robust, numerically stable

algorithms and require minimal application-specific tuning.

1

1.1 Approaches to model order reduction

1.1 Approaches to model order reduction.

Model order reduction is commonly used in the simulation and control of complex physicalprocesses. The systems that usually arise in such cases are often too complex to meet theexpediency requirements of interactive design, optimization, or real time control.

MOR has been devised as a means to reduce the dimension of these complex systems to a levelthat is manageable for such requirements. The ensuing methods have been an indispensable toolfor speeding up the simulations arising in various engineering applications involving large-scaledynamical systems.

Generally, large systems arise due to accuracy requirements on the spatial discretization ofpartial differential equations for fluids, structures, or in the context of lumped-circuit approxi-mations of distributed circuit elements. For some applications, see [4, 18].

Model reduction methods can be classified in two broad categories, namely, SVD (singularvalue decomposition)-based and Krylov-based or moment matching methods. Both of thesecategories are included in the broad family of projection based methods for which the statevariable x is approximated by the projected variable x so that Vx = x.

The former category derives its name from the fact that the corresponding reduction methodsare related to the SVD and the associated 2−norm. The most prominent among them is balancedtruncation (BT). This method is based on the computation of controllability and observabilityGramians, or generalized/empirical versions thereof, which leads to the elimination of stateswhich are difficult to reach and observe. The bottleneck in applying them consists in the factthat obtaining the Gramians requires the solution of Lyapunov (or Riccati) matrix equations,which are computationally expensive. However, recent developments allow the approximatesolution, i.e., approximate balancing, of realistic size problems as discussed in [28, 102, 29, 18]and references therein.

The reduction methods in the second category are based on moment matching, that is,matching of the coefficients of power series expansions of the transfer function at selected pointsin the complex plane. These coefficients are the values of the underlying transfer function,together with values of its derivatives. The main underlying problem is rational interpolation(as introduced for MOR purposes in [3, 7]).

The most abundant methods in the literature are closely related to the so-called Krylovsubspace-based iteration, encountered in numerical linear algebra, as well as the Arnoldi orLanczos procedures, and multi-point (rational) versions thereof.

The advantages of balancing reduction methods include preservation of stability and an apriori computable error bound. Krylov-based methods are numerically efficient and have lowercomputational cost, but in general the preservation of other properties is not automatic anddepends on the choice of the expansion points and the form of the projector (e.g. orthogonal oroblique).

For details on the Krylov-based methods as well as other model reduction methods thatwere not particularly emphasized in this thesis such as proper orthogonal decomposition (POD)and reduced basis (RB), we refer the reader to the book [4], the book chapter [8], the surveys[13, 18, 27], and the articles [68, 58, 59].

2

1.2 MOR applied to dynamical linear and nonlinear systems

1.2 MOR applied to dynamical linear and nonlinear sys-tems

Consider a system described by a set of nonlinear differential equations

ΣN :

Ex(t) = f(x(t)) + g(x(t))u(t),y(t) = Cx(t),

(1.1)

where f : Rn → Rn, g : Rn → Rm, E ∈ Rn×n, C ∈ Rp×n, x(t) ∈ Rn, u(t) ∈ Rm. We seekreduced systems of the form

ΣN :

E ˙x(t) = f(x(t)) + g(x(t))u(t),y(t) = Cx(t),

(1.2)

where E ∈ Rr×r, C ∈ Rp×r, x(t) ∈ Rr. The number of inputs and outputs m and p, respectively,remain the same, while r n.

We begin by listing the classes of systems which will be treated in our approach. For simplicity,consider SISO systems (m = p = 1). The first class is the one of linear systems. We make thefollowing choice for the mappings f and g

f(x(t)) = Ax(t), g(x(t)) = B, A ∈ Rn×n, B ∈ Rn.

The dynamics of such system is described by the following set of differential algebraic equations

ΣL :

Ex(t) = Ax(t) + Bu(t) ,y(t) = Cx(t).

(1.3)

For the second class, the one of bilinear systems, we consider

f(x(t)) = Ax(t), g(x(t)) = Nx(t) + B , N ∈ Rn×n,

with the following descriptor space representation of the original system

ΣB :

Ex(t) = Ax(t) + Nx(t)u(t) + Bu(t),y(t) = Cx(t),

(1.4)

and for the reduced system as,

ΣB : E ˙x(t) = Ax(t) + Nx(t)u(t) + Bu(t) , y(t) = Cx(t).

Finally, for the case of quadratic bilinear systems, the functions f(t) and g(t) are chosen as

f(x(t)) = Ax(t) + Q(x(t)⊗ x(t)), g(x(t)) = Nx(t) + B, Q ∈ Rn×n2.

3


The descriptor space representation of the original systems is given, as

ΣQB :

Ex(t) = Ax(t) + Q(x(t)⊗ x(t)

)+ Nx(t)u(t) + Bu(t),

y(t) = Cx(t),(1.5)

while the one corresponding to the reduced-order system, as

ΣQB : E ˙x(t) = Ax(t) + Q(x(t)⊗ x(t)

)+ Nx(t)u(t) + Bu(t) , y(t) = Cx(t).

In the above equations, consider the following dimensions of the reduced matrices

E, A, N ∈ Rr×r, Q ∈ Rr×r2, B ∈ Rr×m, B ∈ Rp×r,

A thorough description of linear systems can be found in [4], while further details concerninganalytic nonlinear systems have been previously discussed in [103, 79, 105].

Throughout this thesis we concentrate on interpolatory model reduction methods. Thesemethods have initially emerged in numerical analysis and linear algebra and are related to rationalinterpolation. We hence seek reduced models whose transfer functions match those of the originalsystems at selected frequencies, or interpolation points.

For the nonlinear case, the methods require the appropriate definition of transfer functions,as it will be shown later for the classes of bilinear and quadratic-bilinear (QB) systems.

As stated before, for linear systems, moment matching methods seek reduced-order modelsthat interpolate the full-order input-output mapping in frequency domain (i.e. the transferfunction H(s) = C(sE−A)−1B) and/or its derivatives, i.e.

H(si) = H(si),djH(s)

dsj∣∣∣∣s=si

= djH(s)dsj

∣∣∣∣s=si

,

at selected frequencies si ∈ C for i > 1, j > 0. For these methods, interpolation typically leadsto approximation of the whole transfer function over a wide interval of frequencies of interest.

In particular, we are going to focus on data driven model order reduction methods andspecifically on the Loewner framework (as was introduced in [91]).

The Loewner framework can be viewed as a step forward from the classical realization problem(see [4]) which can be formulated as follows; given a sequence of scalars/matrices hk ∈ Rp×m, itis required to find a linear system ΣL characterized by (A,B,C) so that hk = CAk−1B. Thesevalues are referred to as Markov parameters and can be viewed as samples of the transfer functionat infinity.

The question arises as to whether such a problem can be solved if information about thetransfer function at different points in the complex plane is instead provided. This is sometimesreferred as the generalized realization problem. This problem can be stated as follows: givendata obtained by sampling the transfer matrix of a linear system, construct a controllable and/orobservable state space model of a system consistent with the data.

The Loewner framework is based on Lagrange rational interpolation, i.e., constructing rationalinterpolants that use a Lagrange basis for the numerator and denominator polynomials. Asshown in [76], this basis choice leads to algorithms that are numerically robust. Also, a rationalfunction is more versatile than a simple polynomial because it has both poles (roots of the

4


denominator) and zeros (roots of the numerator), and can model functions with singularities andhighly oscillatory behavior more easily than polynomials can.

Using rational functions, models that match (interpolate) given data sets of measurementsare computed. In the context of linear systems, we start from data sets that contain frequencyresponse measurements and we seek reduced systems that model these measurements. Thisparticular property will be generalized for the weakly nonlinear systems as mentioned before.

A main ingredient of the Loewner methodology is the Loewner matrix L (originally introducedin [90]). This matrix is a divided difference matrix and can be expressed as a scaled differenceof Cauchy matrices. Its most important property is that it is exclusively written in terms ofthe given measured/computed data. It turns out, that for linear systems, the L matrix can befactored in terms of the E matrix corresponding to the underlying system. The same propertyis going to be kept for all generalizations we propose throughout the thesis.

In the current work, in order to compare the performance of the proposed Loewner methodol-ogy, we will use the so-called IRKA (iterative rational Krylov algorithm) as an alternative MORapproach for most of the cases. This method was originally described in [74] where it was appliedto linear systems. It is an iterative procedure that constructs, upon convergence, reduced ordermodels which are locally optimal in the sense that the H2 norm of the error system is minimal.

In recent years, this procedure had received considerable attention in the MOR community(see for instance [54, 55]). Two generalizations have been proposed for bilinear systems, i.e.,BIRKA (bilinear version of IRKA) in [21], and as well for quadratic-bilinear systems, i.e., TQB-IRKA (truncated quadratic-bilinear version of IRKA) in [26].

5

Chapter 2

General Properties of DynamicalSystems

The purpose of this chapter is to make an inventory of some key features of linear systems andinvestigate how can those be extended to some mildly nonlinear classes of dynamical systems,such as bilinear and quadratic-bilinear.

2.1 Approximating and transforming nonlinear systemsThis section deals with finding simplified or equivalent models for given complex nonlinear prob-lems. Specifically, one requires approximated models written in matrix format (state-space rep-resentation). Then, one could try to asses how good the approximation really is and how theequivalent model captures the properties of the initial nonlinear system.

As the Taylor series expansion is used to approximate infinitely differentiable complex valuedfunctions around a particular point, so do some of the methods presented here for nonlinearsystems. Since we require finite-dimensional models, a truncation is hence in order. That wouldbe the first limitation (since the truncation automatically introduces an error). The other isthat, in some cases, the approximation is usually good only around the expansion point, but notreally exact in other regions. Finally, the dimension of the rewritten system is much higher thanof the original nonlinear system, which is denoted with n, i.e., a polynomial growth in n.

The alternative is equivalently rewriting the dynamics involving analytic nonlinearities byadding variables and/or taking derivatives. In this way no approximation is involved and theincrease in the dimension of the rewritten system is linear in n. The downside is that this appliesonly for special nonlinear functions.

2.1.1 Carleman linearizationWe introduce a powerful numerical tool that allows approximating an original nonlinear systemwith a bilinear system. The technique is commonly known as Carleman linearization (see [103]).

Since the new proposed method is an approximation technique, it follows that the bilinearsystem obtained is not going to be equivalent to the initial one. But, in some cases, an approxi-mation is good enough from a numerical point a view. As it will be covered in this section, one

6

2.1 Approximating and transforming nonlinear systems

of the main properties of this method is that, as the dimension of the bilinear system increases,the original nonlinear system is better and better approximated. Of course, in many cases, onedoes not afford to increase dimensionality after a point, since it results in slow computationaltime and computational memory issues. Consider a nonlinear system described by (1.1),

ΣN :

x(t) = f(x(t)) + g(x(t))u(t),y(t) = Cx(t),

(2.1)

where t > 0, x(0) = x0 and the nonlinear vector-valued functions ,g : Rn → Rn are supposedto be analytic in x. For simplicity, assume that the output depends linearly on the variable x.

Without loss of generality we assume that: x(0) = 0, f(x(0)) = f(0) = 0. If this does nothold, set x(t) = x(t) − x0(t) where x0(t) is the zero input solution. Use the following notationf(x(t)) = f(x(t)+x0(t))− f(x0(t)). Then rewrite the system equations in terms of the new state:

˙x(t) = x(t)− x0(t) = f(x(t)) + g(x(t))u(t)− f(x0(t))= f(x(t) + x0(t))− f(x0(t)) + g(x(t))u(t)= f(x(t)) + g(x(t))u(t),

and also y(t) = C(x(t) + x0(t)). Consequently, it follows thatx(0) = x(0)− x0(0) = 0,

f(x(0))) = f(0) = f(x0)− f(x0) = 0.

Denote the composition of k Kronecker products of the same vector x with itself with

x(k)(t) = x(t)⊗ x(t)⊗ ... ⊗ x(t) ∈ Rnk ,

where x(0) = 1,x(1) = x, x(2) = x ⊗ x and x(j) ⊗ x(k) = x(j+k), ∀j, k > 1. The next step is towrite the following power series expansions in x for the non-linear functions f and g

f(x) =∞∑k=1

Fkx(k), g(x) =∞∑k=0

Gkx(k), (2.2)

where G0 ∈ Rn×1,Fj,Gj ∈ Rnj×nj , j > 1. Here, F1,G1 denote the Jacobian matrices of f andg, respectively. Moreover, F2,G2 denote the matrices containing second derivatives and so on.

It follows that this is just an equivalent way of writing the Taylor series of both functionswhere N represents the truncation index (by keeping only the first N terms in both series).f(x) = ∑N

k=1 Fkx(N) = F1x + F2x(2) + . . .+ FNx(N),

g(x) = ∑N−1k=0 Gkx(k) = G0 + G1x + . . .+ GN−1x(N−1).

(2.3)

By substituting (2.3) into the state equation in (2.1), we write

x(t) =N∑k=1

Fkx(k)(t) +N−1∑k=0

Gkx(k)(t)u(t). (2.4)

7


Notice that the left hand side of the differential equation in (2.4) contains the state variable x,while the right hand side contains various derivations of it (x(2)(t), . . . ,x(N)(t)). We would liketo modify the state variable by adding the terms that are missing (in this way, increasing thedimension of it). As an example, proceed with writing the derivative of x(2)(t) with respect tothe time variable t

x(2)(t) = dx(2)(t)dt

=d(x(t)⊗ x(t)

)dt

= x(t)⊗ x(t) + x(t)⊗ x(t)

=[ N∑k=1

Fkx(k)(t) +N∑k=1

Gkx(k)(t)u(t)]⊗ x(t) + x(t)⊗

[ N∑k=1

Fkx(k)(t) +N−1∑k=0

Gkx(k)(t)u(t)]

=N−1∑k=1

[Fk ⊗ In + In ⊗ Fk]x(k+1)(t) +N−2∑k=0

[Gk ⊗ In + In ⊗Gk]x(k+1)(t)u(t).

Introduce the following summation of j terms, each containing j − 1 Kronecker products, asfollows (for j > 2, k > 1)

Fj,k = Fk(t)⊗ In ⊗ . . .⊗ In + In ⊗ Fk(t)⊗ In ⊗ . . .⊗ In + . . .+ In ⊗ In ⊗ . . .⊗ Fk(t),

where we set F1,k := Fk. In a similar manner define Gj,k for j > 1, k > 0. Now write the timederivative of x(j)(t) (for j > 1) in terms of the new defined matrices

x(j)(t) =N−j+1∑k=1

Fj,kx(k+1)(t) +N−j∑k=0

Gj,kx(k+1)(t)u(t).

Increase the dimension of the original state vector from n to n(N) = n+ n2 + ...+ nN = nN−nn−1 by

introducing a new bilinear state variable (n is the dimension of the nonlinear variable x)

x⊗(t) =[

x(t) x(2)(t) . . . x(N)(t)]T∈ Rn(N)

.

Hence we obtain a bilinear system with the following realizationx⊗(t) = A⊗x⊗(t) + N⊗x⊗(t)u(t) + B⊗u(t),y = C⊗x⊗(t),

(2.5)

where x⊗(0) = 0 and the matrices are introduced as(A⊗,N⊗ ∈ Rn(N)×n(N)

,B⊗,(C⊗

)T∈ Rn(N)

)

A⊗ =

F1,1 F1,2 . . . F1,N

0 F2,1 . . . F2,N−1

0 0. . . F3,N−2

......

......

0 0 . . . FN,1

, N⊗ =

G1,1 G1,2 . . . G1,N−1 0G2,0 G2,1 . . . G2,N−2 0

0 G3,0. . . G3,N−3 0

......

. . ....

...0 0 ... GN,0 0

,

B⊗ =[

G10 0 · · · 0],C⊗ =

[C 0 · · · 0

].

8


This procedure can be viewed as a linearization of the initial nonlinear system since it involvesapproximation techniques (Taylor series expansion and truncation) to yield a bilinear system.The two main challenges with this method are as follows:

1. It is merely an approximation of the original system;2. The dimension of the bilinear system is considerably higher than that of the original system.

Increasing the truncation index will result in better approximation for the bilinear models butthere is a price to pay. The dimension increase might create problems in terms of memory usagefor further reduction applications (starting with n = 10 nonlinear states and by choosing N = 4will yield a dimension higher than 104 for the rewritten bilinear system, i.e., 11110). For generalpractical applications, one would restrict to N = 2.

Example 2.1.1 Consider the following non-linear scalar system characterized by the equations:

ΣN0 :

x(t) = −2x(t) + sin (x(t)) + u(t),y(t) = x(t).

Write the Taylor series around 0 for sin x

sin (x(t)) =∞∑k=1

ckxk(t) = x(t)− x3(t)

3! + x5(t)5! − . . . , ck =

0, k even(−1)(k−1)/2

k! , k odd,

and the state equationx(t) = −x(t)−

∞∑k=3

ckxk(t) + u(t).

We identify the following values:

Fj,1 = −j, Fj,k =

0, k evenj (−1)(k−1)/2

k! , k odd, k > 1,

Gj,0 = j, Gj,k = 0, j, k > 1, C1 = 1, Ch = 0, h > 2.

Decide to truncate at N = 3 and since in this case it holds that x(k)(t) = xk(t), by puttingtogether the new state vector x⊗1 (t) =

[x(t) x2(t) x3(t)

]T∈ R3, write the state and output

equation asx⊗1 (t) = A⊗1 x⊗1 (t) + N⊗1 x⊗1 (t)u(t) + B⊗1 u(t), y(t) = C⊗1 x⊗1 (t),

with the following system matrices are identified

A⊗1 =

−1 0 − 13!

0 −2 00 0 −3

, N⊗1 =

0 0 02 0 00 3 0

, B⊗1 =

100

, C⊗1 =[

1 0 0].

9


Next, we increase the truncation index to N = 5, i.e. x⊗2 (t) =[x(t) x2(t) . . . x5(t)

]T∈ R5.

Hence, write the state and output equation as

x⊗2 (t) = A⊗2 x⊗2 (t) + N⊗2 x⊗2 (t)u(t) + B⊗2 u(t), y(t) = C⊗2 x⊗2 (t),

with the following system matrices:

A⊗2 =

−1 0 − 1

3! 0 15!

0 −2 0 − 23! 0

0 0 −3 0 − 33!

0 0 0 −4 00 0 0 0 −5

, N⊗2 =

0 0 0 0 02 0 0 0 00 3 0 0 00 0 4 0 00 0 0 5 0

, B⊗2 =

10000

, C⊗2 =

[1 0 0 0 0

].

2.1.2 McCormick relaxationIn this section we introduce a transformation which has been known in the literature as theMcCormick relaxation (see [92]). The original use of this methodology was not for MOR sinceit would seem counter-intuitive for the state dimension of a system to be increased at first, inorder to eventually attain reduction of its dimension. Proceed with introducing the definition ofan elementary function that will be used throughout this section.

Definition 2.1.1 The function h : R → R is considered to be elementary if and only if it is apolynomial, rational, exponential, logarithmic, trigonometric, root function, or a composition ofany of these classes of functions.

As mentioned in ([71]), there is a variety of practical applications where these kind of elementaryfunction appear

1. h1(x) = xα11 x

α22 · · ·xαkk , x = [x1 x2 . . . xk]T in chemical rate equations, MOSFET in

saturation mode;

2. h2(x) = xk+x in chemical rate equations and smoothing functions;

3. h3(x) = ex in diodes, bipolar junction transistors and ion-channel model;

4. h4(x) = sin(x) in control systems (where x is the angle to be steered).

Continue with the definition of a polynomial system, as introduced in [71].

Definition 2.1.2 A polynomial system of order tp is described by the following ordinary differ-ential equations (for i ∈ 1, 2, . . . , n)

n∑j=1

ei,jxi =n∑j=1

ai,jxqi,j,11 · · ·xqi,j,nn +

( n∑j=1

bi,jxri,j,11 · · ·xri,j,nn

)u, (2.6)

where qi,j,h, ri,j,h ∈ N. The order of such system is defined as tp = maxi,j(∑n

h=1 qi,j,h,∑nh=1 ri,j,h

).

Also, note that the nonlinear function in the right hand side of (2.6) is linear in the input u.

10


Example 2.1.2 Consider the following polynomial system characterized by the equationsx1 = 2x1x2 + x2

3 − x2x3u,

x2 = 4x2x23 + x1 + 2x1u,

0 = x31x2 − x2

2 + x23u.

In this case, the order of the system is tp = 4 (since the term x31x2 enters in the third equation).

As defined in (1.1), the dynamics of the nonlinear type of systems we analyze is characterizedby the state equation which is a set of ODEs and the state-output equation. The originalassumption is that the output enters linearly in terms of the internal variable, i.e., y = Cx(t).Furthermore, assume for simplicity that the input also enters linearly in the state equation, i.e.,g(x(t)) = Bu(t).The current objective is to show that the set of ordinary differential equations (ODE)

x(t) = f(x(t)) + Bu, (2.7)

can be converted into a polynomial system. Additionaly, one has to show that any polynomialsystem can be converted into quadratic-bilinear system. As introduced in [71], the first procedureis called polynomialization, while the second is known as quadratic-linearization of a polynomialsystem.

Consider that the function f in (2.7) can be split up as f =[

f1(x) f2(x) . . . fn(x)]T

where fi : Rn → R, i ∈ 1, 2, . . . , n.Assume that the original nonlinear function f : Rn → Rn can be written as a linear combina-

tion of elementary functions γ1(x), γ2(x), . . . , γ`(x) with γi(x) : Rn → R

f(x) = aTk x +∑j=1

mk,jγj(x). (2.8)

By plugging in (2.8) into (2.7), it follows that

xk = aTk x +∑j=1

mkjγj(x) + Bku, (2.9)

where ak ∈ Rn and mkj ∈ R for k ∈ 1, 2, . . . , n. Hence rewrite the equations in (2.7) as,

x(t) = Ax(t) + Mγ(x(t)) + Bu(t), (2.10)

where AT = [a1 a2 · · · an] ∈ Rn×n, M = [mkj] ∈ Rn×`, γ(x) = [γ1(x) · · · γ`(x)]T ∈ R` andB = [B1 B2 . . . Bn]T ∈ Rn×m. Since each differential equation contains only some nonlinearfunctions, then the matrix M is typically sparse.

This format of differential equations often appears in various engineering problems. Forexample, in equations that stem from modified nodal analysis (MNA), each γk(x) representsthe current flowing into a node; in chemical rate equations, γk(x) represents the reaction rateaccording to a specific reaction; in mechanical applications, γk(x) is the force to the system.

11


The fact that f is assumed to be written as a linear combination of solely such functions mayseem restrictive. Nevertheless, because of the function composition, such elementary functionscover nonlinear functions that are suitable for a wide range of engineering tasks.

Polynomialization

In a nutshell, this procedure allows rewriting a system described by ODEs with mild nonlinear-ities, i.e., that can be written as linear combination of elementary functions, as a polynomialsystem. As described in [71], there are two main categories for this procedure

1. polynomialization by adding polynomial algebraic equations;2. polynomialization by taking derivatives.

For the first category (which is applicable only in certain circumstances), the first step is tointroduce a new variable zk = γk(x)) which is going to replace γk(x) in the original equations.Then, provided that zk − γk(g) = 0 can be written as a polynomial equation, it is then addedto the others.

Example 2.1.3 Consider a system described by the differential equation (for simplicity of ex-position we sometimes omit to write the time variable, i.e., simply write x instead of x(t))

x = x+ x2 + x3

1− x2 + u.

Hence, by letting z = x3

1−x2 , rewrite this system as:

x = x+ x2 + z + u,

0 = x3 − z + zx2 .

For the second category (which is applicable to a broader set of functions), the first step is, asbefore, to introduce a new variable zk = γk(x). This will replace γk(x) in the original equations.Then, by applying the chain rule, add zk = dγk(x)

dx f(x) as a new polynomial equation .

Example 2.1.4 Start with the nonlinear differential equation x = x+x2+ex+u. By introducingthe new variable z = ex, it follows that z = exx = z(x + x2 + z + u) and hence the two areequivalent

x = x+ x2 + ex + u⇔

x = x+ x2 + z + u,

z = xz + x2z + z2 + zu,

where the latter is indeed a polynomial system (from Definition 2.1.2).

Three important properties of the polynomialization procedure are mentioned in [71]

1. By adding polynomial algebraic or differential equations through variable change, the re-sulting polynomial system is linear in terms of the input u.

2. Starting with system for which its nonlinearity is written in terms of ` elementary functions,the size of the equivalent polynomial system is linear with respect to `.

3. The equivalent polynomial system constructed via this procedure is not unique; there existsa minimum-order polynomial system that corresponds to the original system.

12


Quadratic linearization

This procedure is needed to equivalently transform a polynomial system (as defined in (2.6)),into a quadratic-bilinear system (as defined in (1.5)). As originally introduced in [71], one candistinguish two categories of quadratic linearization (QL)

1. QL performed by adding quadratic algebraic equations;2. QL performed by taking derivatives.

Again, the first procedure is seldom applicable (in most cases one has to combine both ofthem). Nevertheless, introduce new variables so that z = xr1

1 xr22 · · ·xr`n /w where, at each iteration

step, w is either an original variable, or a newly introduced variable up to that iteration. Then,by replacing the product of monomials xr1

1 xr22 · · ·xr`n with w in the original equations, we rewrite

(if possible) the new obtained terms as quadratic algebraic equations.

Example 2.1.5 Start with the nonlinear differential equation x = x + x3 + u. By introducingthe new variable z = x2, it follows that x = x+ xz + u and hence the two are equivalent

x = x+ x3 + u⇔

x = x+ xz + u,

0 = z − x2.

For the second category of QL, the procedure is similar to the polynomialization by means ofiteratively taking derivatives. We present two self-explanatory examples in this direction.

Example 2.1.6 Consider the same nonlinear system as for Example 2.1.1, i.e

ΣN1 :

x = −2x+ sin x+ u,

y = x.

By introducing two additional variables: z = sin x and w = cosx, compute the time derivativesand then substitute the given terms

x = −2x+ z + u,

z = x cosx = −2wx+ wz + wu,

w = −x sin x = 2zx− z2 − zu.

By denoting with x =[x z w

]T, rewrite the original system as a quadratic-bilinear system

with state vector x

˙x =

−2 1 00 0 00 0 0

︸︷︷︸

A1

x +

0 0 0 0 0 0 0 0 00 0 −1 0 0 1

2 −1 12 0

0 1 0 1 −1 0 0 0 0

︸︷︷︸

Q1

(x⊗ x) +

0 0 00 0 10 −1 0

︸︷︷︸

N1

xu+

100

︸︷︷︸

B1

u,

y =[

1 0 0]

︸︷︷︸C1

x.

13


Example 2.1.7 Consider the following nonlinear system with two state variables x1 and x2

ΣN2 :

x1 = e−x2√x2

1 + 1, x2 = −x2 + u,

y = x1 − x2.

By introducing two additional variables:: x3 = e−x2 , x4 =√x2

1 + 1, it follows that:

x1 = x3x4, x2 = −x2 + u, x3 = x3x2 − x3u, x4 = x1x3.

By denoting with x =[x1 x2 x3 x4

]T, rewrite the original system as a quadratic-bilinear

system (with state vector x)

E2 ˙x = A2x + Q2(x⊗ x

)+ N2xu+ B2u, y = C2x,

where

E2 =

1 0 0 00 1 0 00 0 1 00 0 0 1

, A2 =

0 0 0 00 −1 0 00 0 0 00 0 0 0

, N2 =

0 0 0 00 0 0 00 0 −1 00 0 0 0

, B2 =

0100

,

Q2 =

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

, C2 =

1−1

00

.

The main result is that by iteratively taking derivatives and adding algebraic equations, theinitial nonlinear system can be transformed into a quadratic bilinear system.

The advantage of this method is that no approximation is performed: the quadratic-bilinearsystem is equivalent to the initial nonlinear system.

The downside is that the procedure works for certain types of smooth analytic nonliniarities.For application purposes, it works for some cases, but nevertheless, one has to investigate theapplicability of this method to its full extend.

2.1.3 Variational equation approachConsider a nonlinear system ΣN as defined in (1.1). Assume that the nonlinear functions f ,g :Rn → Rn are analytic in x. Without loss of generality we assume that: x(0) = 0, f(x(0)) =f(0) = 0.

Next compute an approximate bilinear model ΣB via Carleman linearization presented inSection 2.1.1 and an exact quadratic bilinear model ΣQB via the reformulation procedure pre-sented in Section 2.1.2. The idea is to decouple these two systems and write them equivalentlyas an infinite linear time-varying representation.

This approach (described in detail in [62, 103]) consists in taking the response to inputs ofthe form u(t) = αu(t) (α is an arbitrary scalar). Since the systems we are analyzing are analyticsystems, it follows that the solution x(t) also exhibits an analytic representation and thus can

14


be written as an infinite power series expansion in the parameter α

x(t) =∞∑k=1

αkxk(t) = αx1(t) + α2x2(t) + α2x3(t) + ... (2.11)

Proceed with the bilinear model obtained via Carleman linearizaton, i.e., as introduced in (1.4)).By substituting (2.11) and the control input u(t) = αu(t) onto the differential equation in (1.4),write

E∞∑k=1

αkxk(t) = A∞∑k=1

αkxk(t) + N( ∞∑k=1

αkxk(t))αu(t) + αBu(t)

=( ∞∑k=1

αkAxk(t))

+( ∞∑k=2

αkNxk−1(t)u(t))

+ αBu(t).

Now, since this differential equation and the equation for the initial state must hold for any valueof α, coefficients of same powers of α can be equated.

By collecting all terms corresponding to powers of α, we get Ex1(t) = Ax1(t) + Bu(t). Ingeneral, collecting all terms corresponding to powers of αk (where k > 1) we obtain Exk(t) =Axk(t) + Nxk−1(t)u(t). Each such equation can be viewed as a linear time-varying equation,Exk(t) = Axk(t) + Bk(t)u(t) by denoting with Bk(t) := Nxk−1(t)u(t), k > 1.

We decouple the bilinear equation into infinite many time-varying linear equations and writethe initial dynamics equivalently as follows

Ex(t) = Ax(t) + Nx(t)u(t)Bu(t) ⇔

Ex1(t) = Ax1(t) + Bu(t),Ex2(t) = Ax2(t) + Nx1(t)u(t),

...

Exk(t) = Axk(t) + Nxk−1(t)u(t),...

Denote with yk(t) = Cxk(t), ∀ k > 1. The initial bilinear system is written equivalently asinfinitely many coupled linear subsystems. Denote with Σ(k)

B the kth subsystem which dynamicsis determined by xk (for k > 2), i.e,

Σ(k)B :

Exk(t) = Axk(t) + Nxk−1(t)u(t),yk = Cxk.

(2.12)

Note that Σ(1)B actually corresponds to the linear system ΣL described in (1.3). We continue

with the quadratic-bilinear model in (1.5) obtained via the proposed relaxation method. Thefollowing holds

Q(x(t)⊗ x(t)

)= Q

( ∞∑k=1

αkxk(t))⊗( ∞∑k=1

αkxk(t))

=∞∑k=2

αk[ k−1∑j=1

Q(xj(t)⊗ xk−j(t)

)].

15

2.2 Special matrices

By substituting (2.11) and the input u(t) = αu(t) onto the differential equation in (1.5), write

E∞∑k=1

αkxk(t) = A∞∑k=1

αkxk(t) + Q( ∞∑k=1

αkxk(t))⊗( ∞∑k=1

αkxk(t))

+ N( ∞∑k=1

αkxk(t))αu(t) + αBu(t)

=∞∑k=1

αkAxk(t) +∞∑k=2

αk[ k−1∑j=1


)]+∞∑k=2

αkNxk−1(t)u(t) + αBu(t).

Now, similarly as for the bilinear case, since this differential equation holds for all α ∈ R,coefficients corresponding to the same powers of α can be equated, and hence doing so for thecoefficients of αk, k > 1 we can write

Exk(t) = Axk(t) +k−1∑j=1


)+ Nxk−1(t)u(t). (2.13)

Decouple the quadratic-bilinear differential equation into infinitely many equations, as

Ex1(t) = Ax1(t) + Bu(t),Ex2(t) = Ax2(t) + Q

(x1(t)⊗ x1(t)

)+ Nx1(t)u(t),

...

Exk(t) = Axk(t) + Q(x1(t)⊗ xk−1(t) + . . .+ xk−1(t)⊗ x1(t)

)+ Nxk−1(t)u(t),

...

The initial quadratic-bilinear system is written equivalently as infinitely many coupled time-varying subsystems. Denote with Σ(k)

QB the kth subsystem (determined by the variable xk)

Σ(k)QB :

Exk(t) = Axk(t) +∑k−1j=1 Q

(xj(t)⊗ xk−j(t)

)+ Nxk−1(t)u(t),

yk = Cxk.(2.14)

The advantage of the variational equation approach is that the solution x(t) can be derived byiteratively solving a series linear systems which are non-linearly coupled. For instance, startingwith the first purely linear subsystem, we directly write the solution as x1(t) =

∫ t0 e

A(t−τ)Bu(τ)dτ(assuming x1(0) = 0). By considering this expression as a pseudo-input for the second subsystem,an expression for x2(t) can be directly derived.

Continuing this iterative procedure, one obtains the so-called Volterra series representationassociated to the system ΣQB. More details on this approach will be available in the upcomingsections.

2.2 Special matricesMost of the ideas and results from this section directly depend on results from classical controltheory. Concepts such as reachability, controllability, observability, minimality or stability are

16


of particular importance for linear model reduction techniques. The objective is to understandhow some of this inherent properties of linear system transcend to bilinear and quadratic-bilinearsystems.

For the time being, we will ignore the E matrix in the differential equations (we focus on ODEsrather than DAEs). Also, if not stated otherwise, the systems will supposed to be single-inputsingle-output (SISO) with m = p = 1.

2.2.1 Linear systemsDefinition 2.2.1 For a linear system ΣL defined in (1.3), introduce the reachability/controllabilitymatrix Vk(A,B) and the observability matrix W`(A,C), as

Vk(A,B) =[

B AB · · · Ak−1B]∈ Rn×k, (2.15)

W`(A,C) =[

CT ATCT · · · (AT )m−1CT]T

= V`(AT ,CT )T ∈ R`×n. (2.16)

Moreover, for k = `, the Hankel matrix Hk ∈ Rk×k is written as the product of the two matrices,as

Hk(A,B,C) =Wk(A,B)Vk(A,C). (2.17)

The matrices in (2.15) and (2.16) satisfy the recurrence relations

Vk(A,B) = [B AVk−1(A,B)], W`(A,C) = [CT ATW`−1(A,C)]T . (2.18)

Note that the Hankel matrix is of particular importance in classical realization theory (see [4]).The linear space Kk(A,B) = range(Vk(A,B)) is also known in the literature as the kth

Krylov subspace generated by matrix A and vector B. Also, the two matrices defined in (2.15)and (2.16) are of particular importance because they offer a quantization of the reachability andobservability concepts.

Definition 2.2.2 The linear system ΣL defined in (1.3) is reachable if, for any T > 0, x(0) ∈ Rn

and x ∈ Rn, there exists a control input u that applied to the system will steer the system fromx(0) to x at time T , i.e., x(T ) = x.

Proposition 2.2.1 The system ΣL is reachable if and only if rank(Vn(A,B)) = n (an equivalentcondition for the system to be reachable is also that rank([sIn −A B]) = n, ∀ s ∈ C).

Definition 2.2.3 The linear system ΣL defined in (1.3) is observable if, the initial conditionx(0) can be uniquely determined from the value of y(t) at any instance of time T > 0.

Proposition 2.2.2 The system ΣL is observable if and only if rank(Wn(A,C)) = n (an equiv-alent condition for the system to be observable is also that rank([sIn −A ; C]) = n, ∀ s ∈ C).

Proposition 2.2.3 A linear system ΣL is minimal if it is simultaneously reachable and observ-able. This condition is equivalent to rank(Hn(A,B,C)) = n.

17


Definition 2.2.4 Introduce the following generalized linear reachability and observability matri-ces associated to the nodes λ1, . . . , λk and µ1, ..., µ`, respectively (where h, i > 2)

Rk =[

Φλ1B Φλ2B · · · ΦλkB]∈ Rn×k, (2.19)

O` =[

ΦTµ1C

T ΦTµ2C

T · · · ΦTµ`

CT]T∈ R`×n, (2.20)

where Φ(s) = (sI−A)−1. The nodes λik1 and µj`1 are chosen from R\eig(A) (i.e differentthan the poles of A).

As a remark, the linear space Kk = range(Rk) is also known in the literature as the kth

rational Krylov subspace generated by matrix A, vector B and nodes λ1, . . . , λk.

Proposition 2.2.4 Let ΣL be a reachable linear system. Then it follows that,

rank(Rk(A,B)) = n ∀ λi ∈ C \ eig(A),

provided that k > n.

Proposition 2.2.5 Let ΣL be an observable linear system. Then it follows that,

rank(O`(A,C)) = n ∀ µj ∈ C \ eig(A),

provided that ` > n.

Definition 2.2.5 Let p,q : R → Rn be two mappings defined as p(t) = eAtB and q(t) =eAT tCT , respectively. Introduce the controllability and the observability infinite Gramians:

P =∫ ∞

0p(t)p(t)T dt =

∫ ∞0

eAtBBT eAT t dt ∈ Rn×n, (2.21)

Q =∫ ∞

0q(t)q(t)T dt =

∫ ∞0

eAT tCTCeAt dt ∈ Rn×n. (2.22)

The two infinite Gramians satisfy the following Lyapunov equations:

AP + PAT + BBT = 0, ATQ+QA + CTC = 0. (2.23)

Proposition 2.2.6 The necessary and sufficient condition for the Gramians to exist is that theA matrix has all poles in the left half plane, i.e., Re(eig(A)) < 0. Moreover, in this case, bothobservability and controllability Gramians are positive definite.

Definition 2.2.6 [4] Let ΣL be a linear system of dimension n with m = p (the number ofinputs and outputs is the same). The system is symmetric if there exists a symmetric matrix Z(with Z = ZT ) so that the following hold

AZ = ZAT , B = ZCT .

Definition 2.2.7 For a symmetric linear system ΣL, introduce the cross Gramian C as

C =∫ ∞

0eAtBCeAt dt ∈ Rn×n, (2.24)

18


Proposition 2.2.7 The cross Gramian in (2.24) satisfies the Sylvester equation AC + CA +BC = 0. Moreover it holds that C2 = PQ.

Definition 2.2.8 Let ΣL be a reachable and observable (hence minimal) continuous-time linearsystem of dimension n. Moreover consider that the poles of the system are all in the left half plane(i.e eig(A) ⊂ C−). The Hankel singular values of are defined as the square roots of the eigenvaluesof the product of the two Gramians defined in (2.21) and (2.22), i.e., σk(ΣL) =

√eig(PQ).

Moreover, if the system ΣL is symmetric, then it follows that σk(ΣL) = eig(C).

2.2.2 Bilinear systemsDefinition 2.2.9 For a bilinear system ΣB defined in (1.4), introduce the reachability and ob-servability matrices, respectively, by means of the following recurrence relations (for k > 2)

Vk(A,N,B) =[

B AVk−1(A,N,B) NVk−1(A,N,B)]∈ Rn×(2k−1), (2.25)

W`(A,N,C) =[

CT ATW`−1(A,N,C)T NTW`−1(A,N,C)T]T∈ R(2`−1)×n, (2.26)

where V1(A,N,B) = B andW1(A,N,C) = C. Note thatW`(A,N,C) = V`(AT ,NT ,CT )T , ∀`.

For example, write the following reachability matrices corresponding to the first three levels asV1(A,N,B) = B, V2(A,N,B) =

[B AB NB

],

V3(A,N,B) =[

B AB A2B ANB NB NAB N2B].

Definition 2.2.10 For a bilinear system ΣB, based on the matrices defined in (2.25) and (2.26),define the augmented reachability and observability matrices, respectively as

Vk(A,N,B) =[V1 V2 · · · Vk

]∈ Rn×

(2k+1−k−2

), (2.27)

W`(A,N,C) =[WT

1 WT2 · · · WT

]T∈ R

(2`+1−`−2

)×n. (2.28)

The following results can be viewed as extensions of the concepts of reachability and observabilityto the class of bilinear systems (and were first introduced in [50] and [36]). For this class ofsystems, we introduce the so-called span-reachability property, which is presented as follows.

Proposition 2.2.8 ([50], Theorem 5) The subset of all states of system ΣB (as introduced in(1.4)) that are reachable from the origin spans a subspace X of Rn that can be expressed asX = range(Vn(A,N,B)).

Proposition 2.2.9 ([50], Theorem 6) The subset of all unobservable states of ΣB can be ex-pressed as Y = ker(Wn(A,N,C)).

Proposition 2.2.10 The system ΣB or order n is reachable if and only if rank(Vn(A,N,B)) =n. Moreover, the system ΣB is observable if and only if rank(Vn(A,N,B)) = n.

19


Definition 2.2.11 As for the linear case, when choosing k = `, one can define an (augmented)Hankel matrix Hk(A,B,C,N) ∈ R(2k+1−k−2)×(2k+1−k−2), as Hk =WkVk.

The next result was originally stated in [50], Theorem 8.

Proposition 2.2.11 A realization A,N,B,C of a bilinearly realizable sequence of kernels isminimal if and only if its state space is both observable and spanned by the states reachable fromthe origin.

Consequently, the bilinear system ΣB of order n is minimal if and only if rank(Hn) = n. Thefollowing matrices were originally defined in [33] and can be viewed as generalizations of thelinear counterparts.

Definition 2.2.12 Introduce the following generalized linear reachability and observability ma-trices associated to the nodes λ1, . . . , λk and µ1, ..., µ`, respectively :

V(h)k (A,N,B) =

[Φλ1NVh−1

k Φλ2NVh−1k . . . ΦλmNVh−1

k

]∈ Rn×kh , (2.29)

W(i)` (A,N,C) =

[ΦTµ1N

T (W i−1` )T ΦT

µ2NT (W i−1

` )T · · · ΦTµ`

NT (W i−1` )T

]T∈ R`i×n, (2.30)

where V(1)k = Vk and W(1)

` =W` Φ(s) = (sI−A)−1. The nodes λik1 and µj`1 are chosen fromR \ eig(A) (i.e different than the poles of A).

The matrices W and V are used as projection matrices for a generalization of the Rational Krylovmethod for reducing the dimension of bilinear systems (see [35]).

V =[V(1)k V(2)

k · · · V(1)k

]∈ Rn×(k+k2+...+kh),

W =[ (W(1)

`

)T (V(2)k

)T· · ·

(W(i)

`

)T ]∈ R(`+`2+...+`i)×n.

Of course, in practice, it is desired to avoid the explicit computation of such matrices, and insteadcompute an orthonormal basis of the column space of much lower dimension.

Definition 2.2.13 Let pk : Rk → Rn and q` : R` → Rn be two mappings defined in a recursiveway as (for k, ` > 2)

pk(t1, . . . , tk) = pk−1(t1, . . . , tk−1)NeAtk ,

q`(t1, . . . , t`) = q`−1(t1, . . . , t`−1)NT eAT t` ,

where p1(t1) = p(t1) = eAt1B and q1(t1) = q(t1) = eAT t1CT , respectively.

Definition 2.2.14 Let Pk be the controllability Gramian associated to the energy functional pk(it follows that this Gramian corresponds to the kth decoupled subsystem of ΣB). Similarly, letQk be the observability Gramian associated to the energy functional qk. Consequently, we write

Pk =∫ ∞

0· · ·

∫ ∞0

pk(t1, . . . , tk)pk(t1, . . . , tk)Tdt1 · · · dtkRn×n, (2.31)

Qk =∫ ∞

0· · ·

∫ ∞0

qk(t1, . . . , tk)qk(t1, . . . , tk)Tdt1 · · · dtk ∈ Rn×n. (2.32)

20


Proposition 2.2.12 The matrices defined in (2.31) and (2.32), respectively, satisfy the followinggeneralized Sylvester equations (for k, ` > 2)

APk + PkAT + NPkNT = 0, ATQ` +Q`A + NTQ`N = 0. (2.33)

Definition 2.2.15 Introduce the controllability and the observability infinite Gramians associ-ated with the bilinear system ΣB, respectively, as

P =∞∑k=1

∫ ∞0· · ·

∫ ∞0

pk(t1, ..., tk)pk(t1, . . . , tk)Tdt1 · · · dtk =∞∑k=1Pk ∈ Rn×n, (2.34)

Q =∞∑`=1

∫ ∞0· · ·

∫ ∞0

qk(t1, . . . , t`)qk(t1, . . . , t`)Tdt1 · · · dt` =∞∑`=1P` ∈ Rn×n. (2.35)

Proposition 2.2.13 The matrices defined in (2.34) and (2.35), respectively, satisfy the followinggeneralized Sylvester equations

AP + PAT + NPNT + BBT = 0, ATQ+QA + NTQN + CTC = 0. (2.36)

For the infinite sums in (2.34) and (2.35) to converge, the following sufficient condition holds (asintroduced in [113]).

Lemma 2.2.1 The controllability Gramian P and the observability Gramian Q as defined in(2.34) and, respectively in (2.35), exist if the matrix A is asymptotically stable (see Definition2.2.34) and the following holds ‖N‖2 <

√2α/β.

In practical applications (time-domain simulations or control related tasks), one would use N =N/γ instead of N (where γ ia a positive scalar so that γ > β‖N‖2/(

√2α)). Then the control

input will be chosen as u(t) = γu(t). In this way, provided that the A matrix is Hurwitz, wesatisfy the necessary conditions for the existence of matrices P and Q as given in Lemma 2.2.1.Finally, introduce the following matrices in a recursive manner (for k > 2),

Ik(A,N) =[

I AIk−1(A,N) NIk−1(A,N)]∈ Rn×(2k−1)n. (2.37)

where I1(A,N) = I. These matrices will be used for defining observability matrices for quadratic-bilinear systems. For example write,

I2(A,N) =[

I A N], I3(A,N) =

[I A A2 AN N NA N2

].

2.2.3 Quadratic-bilinear systemsSimilar results to the ones introduced for the class of bilinear systems in the previous section,i.e., Propositions 2.2.8 to 2.2.11 and also Definitions 2.2.9 to 2.2.12, can be adapted for theclass of quadratic bilinear systems. Before we proceed to explicitly stating these results, we firstintroduce some elements of notation that will be useful in this section and also in Chapter 5.

As pointed out in [33], the Q matrix corresponding to the quadratic-system ΣQB is not uniqueand can be chosen to satisfy certain useful conditions.

21


Proposition 2.2.14 The Q matrix is transformed to Q ∈ Rn×n2 using basic additions

Q(j, (a− 1)n+ b) = Q(j, (b− 1)n+ a) = Q(j, (a− 1)n+ b) + Q(j, (b− 1)n+ a)2 (2.38)

for all j, a, b ∈ 1, 2, . . . , n. This implies thatQ(v⊗w

)= Q

(w⊗ v

), ∀ v,w ∈ Rn,

Q(V⊗ v

)= Q

(v⊗V

), ∀ v ∈ Rn, ∀ V ∈ Rn×n.

(2.39)

Example 2.2.1 Let Q =[

1 3 5 72 4 6 8

]and denote x = [x1 x2]T the state variable. The

term Q(x⊗ x

)that enters the differential equation characterizing the dynamics of ΣQB can be

computed

Q(x⊗ x

)=[

1 3 5 72 4 6 8

] [x2

1 x1x2 x2x1 x22

]T=[

x21 + 8x1x2 + 7x2

22x2

1 + 10x1x2 + 8x22

].

Now, by rewriting the Q matrix as suggested in proposition 2.14, i.e., Q =[

1 4 4 72 5 5 8

], it

follows that the desired entry does not change

Q(x⊗ x

)=[

x21 + 8x1x2 + 7x2

22x2

1 + 10x1x2 + 8x22

]= Q

(x⊗ x

).

Definition 2.2.16 Let Xk be the kth column of the matrix X ∈ Rm×n. The vectorization of Xis represented by the mapping vec : Rm×n → Rmn obtained by including all the columns of Xinto a column vector. It is represented by the mapping vec : Rm×n → Rmn such as vec(X) =[XT

1 . . . XTn ]T . Additionally, introduce the inverse vectorization operation as vec−1

m,n : Rmn →Rm×n so that vec−1

m,n(x) = X, where x = [XT1 . . . XT

n ]T . If m = n, the notation vec−1n (x) is

going to be used instead.

Proposition 2.2.15 Given matrices, X ∈ Rm×n, Y ∈ Rp×q, Z ∈ Rn×r, V ∈ Rq×s andW ∈ Rr×o, the following identities hold:

vec(XZW) =(WT ⊗X

)vec(C), (2.40)(

X⊗Y)(

Z⊗V)

=(XZ

)⊗(YV

), (2.41)(

X⊗Y)V = X⊗

(YV

), for X ∈ Rm×1, (2.42)(

X⊗Y)Z =

(XZ

)⊗Y, for Z ∈ Rn×1. (2.43)

For the proof of the first and second equalities, we refer the readers to [65], Section 12.1.Additionally, note that the third and forth equalities represent just particular cases of the secondone, by choosing n = r = 1 and Z = 1 for (2.42) and, respectively, q = s = 1 and V = 1 for(2.43).

22


The identities in Proposition 2.2.15 will be also used in various proofs throughout Chapter5. In the following we denote with ek,n ∈ Rn the kth unit vector of length n. If the dimension ofsuch vector is implied, we use for simplicity the notation ek instead of ek,n.

Proposition 2.2.16 The Kronecker product of two unit vectors is also an unit vector. Whenmultiplying ej,m ∈ Rm and ek,n ∈ Rn, the product will be an unit vector of size mn, i.e.,

ej,m ⊗ ek,n = e(j−1)n+k,mn. (2.44)

Definition 2.2.17 Let X ∈ Rm×n. For 1 6 i 6 m and 1 6 j 6 n, let X(i,j) represent theelement of matrix X corresponding to the ith row and jth column, i.e. X(i,j) = eTi,mXej,n.

The process of unfolding a tensor into a matrix is sometimes called matricization of a tensor.For a 3-dimensional tensor, there are three different ways to unfold the tensor, depending on themode-µ fibers that are used for the unfolding (see [33]). If the tensor is unfolded using mode-µfibers, it is called the mode-µ matricization of the tensor.

Definition 2.2.18 Let Xtens ∈ Rn×n×n be a three dimensional tensor which is described bythe matrices Xtens

i ∈ Rn×n2, i ∈ 1, 2, . . . , n (the so-called frontal slices). Then, the mode-µ

matricizations (for µ ∈ 1, 2, 3) are given by

X(1) = X =[

Xtens1 Xtens

2 . . . Xtensn

], X(2) =

[ (Xtens

1

)T (Xtens

2

)T. . .

(Xtensn

)T ],

X(3) =[

vec(Xtens1 ) vec(Xtens

2 ) . . . vec(Xtensn )

]T.

Definition 2.2.19 In addition, consider Xtens ∈ Rn×n×n be another three dimensional tensor,so that,

X(1) =[

vec(Xtens1 ) vec(Xtens

2 ) . . . vec(Xtensn )

]T.

Denote with X(−1) to be the mode-1 matricization of the tensor Xtens, defined as

X(−1) =[

Xtens1 Xtens

2 · · · Xtensn

].

Equivalently, we can write the definition of the matrix X(−1) in terms of the matrix X(1) = X,as follows

X(−1) =[

vec−1(XTe1

)vec−1

(XTe2

)· · · vec−1

(XTen

) ]. (2.45)

Example 2.2.2 Let the tensor Qtens ∈ R2×2×2 be described by the following frontal slices

Qtens1 =

[1 32 4

], Qtens

2 =[

5 76 8

].

Then it follows that

Q = Q(1) =[

1 3 5 72 4 6 8

], Q(2) =

[1 2 5 63 4 7 8

],

23


Q(3) =[

1 2 3 45 6 7 8

], Q(−1) =

[1 5 2 63 7 4 8

].

Proposition 2.2.17 Let Qtens ∈ Rn×n×n be a third order tensor with Q,Q(2),Q(3) ∈ Rn×n2 andQ(−1) ∈ Rn×n2 be the matricization matrices introduced in Definitions 2.2.18 and 2.2.19. Then,it follows that

Q(3)(v⊗ In

)=(In ⊗ vT

)QT , ∀ v ∈ Rn, (2.46)

Q(−1)(In ⊗w

)=(wT ⊗ In

)QT , ∀ w ∈ Rn. (2.47)

An interesting result states that, if the matrix Q is written in the format described in Proposi-tion 2.2.14, then it follows that the mode-2 and mode-3 matricizations are equal, i.e., Q(2) = Q(3).

Proposition 2.2.18 Based on the results in Proposition 2.2.17 and assuming that the matrix Qis rewritten as in Proposition 2.2.14 (hence it satisfies the conditions in (2.39)), it follows that

Q(3)(v⊗ In

)= Q(−1)

(In ⊗ v

)=(Q(v⊗ In

))T=(Q(In ⊗ v

))T, ∀ v ∈ Rn, (2.48)

and that the following holds for all V,W ∈ Rn×n

Q(2)(V⊗W

)(Q(2)

)T= Q(3)

(V⊗W

)(Q(3)

)T= Q(−1)

(W⊗V

)(Q(−1)

)T. (2.49)

After presenting a short inventory of some useful notations and properties corresponding toquadratic-bilinear systems, we proceed to extending the quantities mentioned in Definitions 2.2.8to 2.2.12.

Definition 2.2.20 For a quadratic-bilinear system ΣB defined in (1.5), introduce the reachabilityand observability matrices, respectively, by means of the recurrence relations (for k > 2)

Vk =[

B AVk−1 NVk−1 Q(Vk−1 ⊗ V1

). . . Q

(Vk−i ⊗ Vi

). . . Q

(V1 ⊗ Vk−1

) ], (2.50)

W` =[

CT ATWT`−1 NTWT

`−1

(VT`−1 ⊗ IT1

)QTWT

`−1 . . .(VT1 ⊗ IT`−1

)QTWT

`−1

]T, (2.51)

where V1 = B, W1 = C. For simplicity use Vk instead of Vk(A,Q,N,B) and same for W`.

For example, we write the following reachability matrices corresponding to the first levels as

V2 =[

B AB NB Q(B⊗B

) ],

V3 =[

B AB A2B ANB AQ(B⊗B

)NB NAB N2B AQ

(B⊗B

). . . Q

(B⊗B

)Q(AB⊗B

)Q(NB⊗B

)Q(Q(B⊗B

)⊗B

). . .

].

24


Similarly, write the first observability matrices

W2 =[

C ; CA ; CN ; CQ(B⊗ I

) ],

W3 =[

C ; CA ; CA2 ; CNA ; CQ(B⊗A

); CN ; CAN ; CN2 ; CQ

(B⊗N

). . . CQ

(B⊗ I

); CQ

(AB⊗ I

); CQ

(NB⊗ I

); CQ

(Q(B⊗B

)⊗ I

). . .

].

Definition 2.2.21 For a quadratic-bilinear system ΣQB, based on the matrices defined in (2.50)and (2.51), define the augmented reachability and observability matrices, respectively as

Vk =[V1 V2 · · · Vk

], W` =

[WT

1 WT2 · · · WT

`

]T. (2.52)

In the next sections we will try to extend the concepts in Definitions 2.2.13 to 2.2.15 and theresults in Propositions 2.2.12 and 2.2.13, from bilinear systems to quadratic-bilinear systems.We mention the first (as far as the author knows) attempt of introducing infinite Gramians forthis particular class of nonlinear systems.

Quadratic-bilinear Gramians in the literature

The breakthrough for generalizing the infinite Gramian framework to the class of quadratic-bilinear systems was recently made in [25]. There, the authors construct such matrices and showthat they satisfy some generalized Sylvester equations with quadratic terms.

Proposition 2.2.19 Let ΣQB be a quadratic-bilinear system as defined in (1.5). The infinitecontrollability and observability Gramians PQB and QQB, respectively, as defined in [26], satisfythe following equations

APQB + PQBAT + Q(PQB ⊗ PQB

)QT + NPQBNT + BBT = 0, (2.53)

ATQQB +QQBA + Q(2)(PQB ⊗QQB

)(Q(2)

)T+ NTQQBN + CTC = 0. (2.54)

From now on, these matrices will be considered as the true quadratic-bilinear Gramians. One ofthe drawback for using these matrices for further reduction purposes is that the first equationis quadratic in P and hence not only is it challenging to solve for high dimensional systems,but a real and also positive-definite solution may not be always guaranteed. That is why, theso-called truncated Gramians are used instead in [25] (see Corollary 3.4) for practical modelorder reduction tasks (when dealing with large scale systems).

Note also that the second equation in Proposition 2.2.19, i.e., (2.54), can be rewritten byusing the formula in (2.49), so that it is written in terms of the matrix Q(−1), as

ATQQB +QQBA + Q(−1)(QQB ⊗ PQB

)(Q(−1)

)T+ NTQQBN + CTC = 0. (2.55)

A new type of quadratic-bilinear Gramians

As a direct consequence of the drawback mentioned above, we will propose alternative defini-tions of infinite Gramians that lead to equations that can be more easily solved. The proposed

25


equations are linear generalized Sylvester equations while the equation in 2.53, is nonlinear witha quadratic term PQB ⊗ PQB.

Let Υ = N,Q be a set with two elements. Then, for ` > 1, introduce Υ` as the set ofordered tuples of length ` constructed with symbols from Υ.

For example, it follows that Υ2 = (N,N), (N,Q), (Q,N), (Q,Q). Let w be a tupleof length `, i.e., w ∈ Υ` and denote the jth element of w with w(j). Hence, write w =(w(1), w(2), . . . , w(`)), for w(j) ∈ Υ, j ∈ 1, 2, . . . , `.

Let ε be the empty tuple which does not contain any symbol from Υ. Introduce the lengthof a tuple as the mapping | · | : Υ` → N, so that |w| = `⇔ w ∈ Υ`, ` > 1 and |ε| = 0.

Definition 2.2.22 Consider two tuples composed of elements α1, ..., αi, and β1, ..., βj, includedin the set X . Introduce the concatenation of such tuples as the mapping : X i × X j → X i+j

with the following property(α1, α2, . . . , αi

)(β1, β2, . . . , βj

)=(α1, α2, . . . αi, β1, β2, . . . βj

).

Definition 2.2.23 Introduce the function φ : Υ× Rb → Rn×n for b ∈ 1, 2, defined as follows

φ(w,X ) =

eAt1 , if w = N, b = 1, and X = t1,eAt1B⊗ eAt2 , if w = Q, b = 2, and X = t1, t2.

Definition 2.2.24 Let w ∈ Υk be a tuple of length k with w = (w1, w2, . . . , wk), for wj ∈Υ, j ∈ 1, 2, . . . , k. Let p(t) := pε0(t) = eAtB be the mapping from Definition 2.2.5, introducethe following energy functional corresponding to the tuple w, as ( k > 1)

pwk (t1, t2, t3, . . . , th) = eAt1w1φ(w1,X1) · · ·w`φ(wk,Xk)B. (2.56)

For example, when w = (Q,N,Q) ∈ Υ3 is chosen, the energy functional pQ,N,Q3 evaluated at

(t1, t2, . . . , t6) can be written as

pN,Q,N3 (t1, t2, . . . , t6) = eAt1NeAt2

(eAt3B⊗ eAt4

)NeAt5B,

by applying formula (2.56) for w1 = N, w2 = Q, w3 = N, X1 = t2,X2 = t3, t4,X3 = t5.

Definition 2.2.25 Let the linear Gramian be denoted with PL := Pε0 =∫ ∞

0eAtBBT eAT tdt (as

in (2.21)). Introduce the infinite controllability Gramian associated with the tuple w 6= ε asfollows

Pwk =

∫ ∞0· · ·

∫ ∞0

pwk (t1, t2, t3, . . . , th)pw

k (t1, t2, t3, . . . , th)Tdt1dt2 . . . dth, if k > 1. (2.57)

Definition 2.2.26 Consider Υk = w(1)k ,w(2)

k , . . . ,w(2k)k for k > 1. Let the infinite controllabil-

ity Gramian corresponding to level k be defined as the sum of all Gramians Pwk so that |w| = k.

Hence write Pk = ∑2kj=1P

w(j)k

k . For example, write the level two and three Gramians as,

P1 = PN1 + PQ

1 , P2 = PN,N2 + PN,Q

2 + PQ,N2 + PQ,Q

2 .

26


Definition 2.2.27 We introduce the infinite controllability Gramian P corresponding to thequadratic-bilinear system ΣQB as the summation of all level k Gramians, for all k ≥ 0, i.e.,

P = PL + P1 + P2 + . . . =∞∑k=0Pk. (2.58)

Proposition 2.2.20 Let w ∈ Υ and w ∈ Υk, k > 1. Then the recurrence formula holds

AP wwk + P ww

k AT =

−NPwk−1NT , if w = N

−Q(Pwk−1 ⊗ PL

)QT , if w = Q

.

Proposition 2.2.21 The linear Gramian PL satisfies the equation in (2.23); the Gramian cor-responding to level k can be written in terms of the Gramian corresponding to the previous level,as

APk + PkAT + Q(Pk−1 ⊗ PL

)QT + NPk−1NT + BBT = 0, k > 2. (2.59)

Lemma 2.2.2 The controllability Gramian P defined in (2.58), satisfies the following equation:

AP + PAT + Q(P ⊗ PL

)QT + NPNT + BBT = 0. (2.60)

This result directly follows by adding the equations in (2.59) for all levels k > 0

A∞∑k=0Pk + PkAT + Q

( ∞∑k=1Pk−1 ⊗ PL

)QT + N

∞∑k=1Pk−1NT + BBT = 0.

Remark 2.2.1 Under certain conditions, there exists a positive-definite solution to equation(2.60), which can explicitly be written as,

vec(P) = −(In ⊗A + A⊗ In +

(Q⊗Q

)K(vec(PL)⊗ In2

)+ N⊗N

)−1(B⊗B

), (2.61)

where K ∈ Rn4×n4 is a permutation matrix that allows to split up the vectorization of theKronecker product between matrices U,V ∈ Rn in two factors that depend only on U or V, i.e.,

vec(U⊗V) = K(vec(V)⊗ In2

)vec(U).

In practice, it is to be avoided to explicitly use this direct formula to compute the Gramian Psince it involves matrices of size n4 when the original dimension is already large, i.e., n > 103.Instead, one would use an iterative procedure to estimate the infinite summation in (2.58).Next, let us introduce the following energy functionals that are going to be used for constructingan observability Gramian Q that is assigned to the quadratic-bilinear system ΣQB.

Definition 2.2.28 Let w ∈ Υ` be a tuple of length ` with w = (w1, w2, . . . , w`), for wj ∈ Υ, j ∈1, 2, . . . , `. Denoting with q(t) := iε0(t) = CeAt the linear contribution, introduce the followingenergy functional (for ` > 1)

qw` (t1, t2, t3, . . . , th) = CeAt1w1γ(w1,X1) · · ·w`γ(w`,X`). (2.62)

27


For example, choosing w = (Q,N,Q) ∈ Υ3, write the energy functional qQ,N,Q3 evaluated at

(t1, t2, . . . , t6), in the following way

qQ,N,Q3 (t1, t2, . . . , t6) = CeAt1Q

(eAt2B⊗ eAt3

)NeAt4Q

(eAt5B⊗ eAt6

),

by applying formula (2.62) for w1 = Q, w2 = n, w3 = Q, X1 = t2, t3,X2 = t4,X2 = t5, t6.

Definition 2.2.29 Let the linear Gramian be denoted with QL := Qε0 =∫ ∞

0eAtBBT eAT tdt.

Introduce the infinite controllability Gramian associated with the tuple w 6= ε as follows:

Qw` =

∫ ∞0· · ·

∫ ∞0

qw` (t1, t2, t3, . . . , th)qw

` (t1, t2, t3, . . . , th)Tdt1dt2 . . . dth, if ` > 1. (2.63)

Definition 2.2.30 Consider Υ` = w(1)` ,w(2)

` , . . . ,w(2`)` for ` > 1. Let the infinite observability

Gramian corresponding to level ` be defined as the sum of all Gramians Qw` so that |w| = `.

Hence write Q` = ∑2`j=1Q

w(j)`

` . For example, write the first level two and three Gramians as:

Q1 = QN1 +QQ

1 , Q2 = QN,N2 +QN,Q

2 +QQ,N2 +QQ,Q

2 .

Definition 2.2.31 We introduce the infinite observability Gramian Q corresponding to thequadratic-bilinear system ΣQB as the summation of all level ` Gramians, for all ` ≥ 0. Write

Q = QL +Q1 +Q2 + . . . =∞∑`=0Q`. (2.64)

Proposition 2.2.22 Let w ∈ Υ and w ∈ Υ`, ` > 1. Then the recurrence formula holds

ATQww` +Qww

` A =

−NTQwl−1N, if w = N,

−Q(−1)(Qwl−1 ⊗ PL

)(Q(−1)

)T, if w = Q.

Proposition 2.2.23 The linear Gramian QL satisfies equation (2.23); the Gramian Q` corre-sponding to level ` can be written in terms of the Gramian corresponding to the previous level,as

ATQ` +Q`A + Q(−1)(Q`−1 ⊗ PL

)(Q(−1)

)T+ NTQ`−1N + CTC = 0, ` > 1, (2.65)

where the matrix Q(−1) is as in Definition 2.2.19, i.e.,

Q(−1) =[

vec−1(QTe1

)vec−1

(QTe2

)· · · vec−1

(QTen

) ]. (2.66)

Lemma 2.2.3 The observability Gramian Q defined in (2.64), satisfies the following equation:

ATQ+QA + Q(−1)(Q⊗PL

)(Q(−1)

)T+ NTQN + CTC = 0. (2.67)

This result directly follows by adding the equations in (2.65) for all levels ` > 0

AT∞∑`=0Q` +

∞∑`=0Q`A + Q(−1)

( ∞∑`=1Q`−1 ⊗ PL

)(Q(−1)

)T+ NT

∞∑`=1Q`−1N + CTC = 0.

28


Remark 2.2.2 As before, one can explicitly write the solution to (2.67) as,

vec(Q) = −(In⊗AT + AT ⊗ In +

(Q(−1)⊗Q(−1)

)K(vec(PL)⊗ In2

)+ NT ⊗NT

)−1(CT ⊗CT

).

(2.68)

2.2.4 StabilityThe concept of stability is of particular importance when devising model reduction techniques ofdynamical systems. Starting with a stable initial system, one could require the preservation ofthis property for the reduced system as well. So the study of stability is nevertheless necessary.

Linear systems

In the context of linear time invariant systems ΣL ( as defined in (1.3)), the theory behindstability is already thoroughly elaborated (see [4]).

Definition 2.2.32 The system is called stable if all solution trajectories x are bounded for pos-itive time: x(t), t > 0 are bounded. This property can be reformulated in terms of the matrix A.More exactly, all eigenvalues of A must have real parts that are not positive, and, in addition,all pure imaginary eigenvalues have multiplicity one (in this case A is stable).

Definition 2.2.33 The system is called asymptotically stable if all solution trajectories x go tozero as t approaches infinity (x(t)→ 0, t→∞). This property can be reformulated in terms ofthe matrix A, more exactly it means that all eigenvalues of A must have negative real parts (inthis case A is said to be a Hurwitz matrix).

Furthermore, we mention the following results.

Definition 2.2.34 The matrix A is asymptotically stable if and only if there exist real scalarsβ > 0 and 0 < α 6 −maxi(Re(eigi(A))) so that: ‖ eAt ‖6 βe−αt.

Definition 2.2.35 The system is bounded-input, bounded-output (BIBO) stable if any boundedinput u results in a bounded output. This can be reformulated in terms of the impulse responseh; the condition is that h be absolutely integrable, i.e.,

∫∞−∞ |h(t)|dt <∞.

Bilinear systems

Since stability preserving model reduction techniques are of growing interest, the extrapolation ofstability related concepts to bilinear systems is important. There is a close relationship betweenstability and the Volterra series representation (as introduced in [103]).

In contrast to the linear case, bilinear systems include an additional matrix (denoted withN) that influences the evolution of the solution x. Therefore, it is clear that the stability of theA matrix will no longer suffices to guarantee system stability. By factoring x(t), one can rewritethe differential equation characterizing the dynamics of ΣB as

x(t) = Ax(t) + Nx(t)u(t) + B⇔ x(t) = (A + Nu(t))x(t) + Bu(t)⇔ x(t) = A(t)x(t) + Bu(t).

29


This implies the reformulation of the bilinear system as a time varying linear system since thenew defined A matrix is time-dependent, i.e., A(t) = A + Nu(t).

One possibility to simplify this set-up is to consider that the input is constant over time(u(t) = uc, ∀t ∈ R). In this case, it follows that the A is time invariant, i.e, A = A + Nuc.And now, as in the case of linear systems, we impose the following condition for the system tobe asymptotically stable, i.e., Re(eig(A + Nuc)) < 0. But in practical applications, the controlinput is seldom constant and hence one would require other approaches.

Definition 2.2.36 A bilinear system ΣB is bounded-input bounded-output (BIBO) stable if forany bounded input signal u(t), the output signal y(t) is bounded on [0,∞]. Furthermore thisimplies that the corresponding Volterra series of the solution x(t) uniformly converges on [0,∞).

In contrast to the asymptotic stability of a linear system, for bilinear systems one considersbounded-input bounded-output (BIBO) stability instead. The following important statementcan be found in [107].

Lemma 2.2.4 Let ΣB be a bilinear system (as defined in (1.4)) and assume that the A matrixis asymptotically stable. Hence, as stated in Definition 2.2.34, there exist α, β > 0 such that‖eAt| 6 βe−αt, ∀ t > 0. Additionally assume that |u(t)| 6 M uniformly holds on [0,∞] (whereM > 0). The system ΣB is BIBO stable if ‖N‖2 6 α/(Mβ).

The BIBO stability framework already existing for bilinear systems involves only necessaryconditions. One of these is presented here although others that are found in the literature (seefor instance [114] where discrete-time bilinear systems are treated instead).

In summary, the stability of bilinear systems requires the asymptotic stability of A and asufficiently bounded matrix N.

Nonlinear systems

We are going to discuss Lyapunov stability. Consider the autonomous system ΣA described by:x(t) = f(x(t)), f : Rn → Rn, x(t) ∈ Rn (u(t) does not enter the differential equation).

Definition 2.2.37 The function ζ : Rn → R is said to be a Lyapunov function for the systemΣA if, for all solutions x ∈ B, the following holds dζ(x(t))

dt6 0, (where B is some neighborhood

around x = 0).

For linear systems it follows that f(x) = Ax. Choose the quadratic Lyapunov functionas ζ(x) = xTPx with P symmetric (with P = PT ). Then it follows that ζ = xT (ATP +PA)x = xTUx. Usually, the symmetric matrix U is given and thus the problem of constructinga Lyapunov function amounts to solving for P the Lyapunov equation: ATP + PA = U.

Next, find necessary conditions so that ζ(x) is Lyapunov function for the system ΣA. Assumef(x) = ALx + G(x)x where limx→0 G(x) = 0 and AL to be a Hurwitz matrix (here AL is thelinearization matrix and can be computed by evaluating the Jacobian matrix associated with fat 0). Choose a symmetric matrix U and let P be the unique solution of the Lyapunov equationATLP + PAL + U = 0. Hence, write

ζ(x) = xTPf(x) + f(x)TPx = xT (PAL + ATLP)x + 2xTPG(x)x = −xTUx + 2xTPG(x)x

30


⇒ ζ(x) 6 −xTUx + 2‖P‖‖G(x)‖‖x‖2. (2.69)

Now, it is clear that since limx→0 G(x) = 0, then for any β > 0, there exist r > 0 so that‖G(x)‖ < β, ∀ ‖x‖ < r. By substituting this inequality and the well known result xTQx >λmin(U)‖x‖2 into (2.69), it follows that

ζ(x) < −(λmin(U)− 2β‖P‖

)‖x‖2, ∀ ‖x‖ < r.

Finally, by choosing β < λmin(U)/2‖P‖, it follows that ζ(x) is indeed a Lyapunov function forx = f(x), i.e., ζ(x) 6 0, ∀ ‖x‖ < r. By setting c = min‖x‖=r xTPx = λmin(P)r2, it follows thatthe set xTPx < c is a subset (estimate) of the region of attraction.

For quadratic autonomous systems it follows that f(x) = Ax + Q(x ⊗ x

). Assuming A

is Hurwitz, identify AL = A and G(x) = Q(x ⊗ I

). Choose the function ζ(x) = xTPx and

compute its derivative asζ = xTATPx +

(xT ⊗ xT

)QTPx + xTPAx + xTPQ

(x⊗ x

)= −xTUx + 2xTPQ

(x⊗ x

).

Example 2.2.3 Consider the following polynomial autonomous system

ΣA1 :

x1 = −x2,

x2 = x1 + (x21 − 1)x2.

Set x = [x1 x2]T and f(x1, x2) = [−x2 x1 + (x21− 1)x2]T . Compute the linearization matrix and

setting the G function as

AL = ∂f∂x

∣∣∣∣x=0

=[

0 −11 −1

], G(x) =

[0 0

x1x2 0

],

rewrite the dynamics of ΣA1 as x = ALx + G(x)x. Note that AL is indeed Hurwitz (eigenvaluesat −1 ±

√3)/2 and G(x) → 0, x → 0. By introducing an additional state x3 = x2

1 it followsthat x3 = 2x1x1 = −2x1x2. Rewrite the system as

ΣA1 :

x1 = −x2,

x2 = x1 − x2 + x3x2,

x3 = −2x1x2.

Collect the scalar entries xi, i ∈ 1, 2, 3 in to the vector x = [x1 x2 x3]T . Hence write thedynamics of ΣA equivalently as x = Ax + Q

(x⊗ x

)where, the matrices

A =

0 −1 01 −1 00 0 0

, Q =

0T9eT8,9−2eT2,9

∈ R9×9.

Note that the matrix A is not Hurwitz (since it has an eigenvalue at 0). Hence, when applyingthe relaxation method from Section 2.1.2, issues regarding stability might arise.

We modify the dynamics of ΣA1 , by replacing the second equation by x2 = −x1 + (x1− 1)x2.Then this new system (call it ΣA2) is a quadratic-bilinear autonomous system (no need for any

31

2.3 Input-output mappings of dynamical systems

relaxation) with variable x = [x1 x2]T ,

ΣA2 :

x1 = −x2

x2 = x1 + (x1 − 1)x2⇔ x = Ax+Q

(x⊗x

), A =

[0 −11 −1

], Q =

[0 0 0 00 1 0 0

].

Choose U = I2 and obtain P =[

1.5 −0.5−0.5 1

](by solving ATP + PA + U = 0). Hence

ζ(x) = xTPx = 1.5x21 − x1x2 + x2

2 ⇒ ζ(x) = −(x21 + x2

2)− (x31x2 − 2x2

1x22).

By using basic inequalities, such as |x1x2| 6 1/2‖x‖2 and |x1 − 2x2| 6√

5‖x‖2, it follows that

ζ(x) 6 −‖x‖2 + |x1||x1x2||x1 − 2x2| 6 −‖x‖2 +√

52 ‖x‖

4.

It follows that ζ(x) < 0,∀‖x‖ < r, where r =√

2/√

5. Since λmin(P) = (5 −√

5)/4, itfollows that the set Θ(x) < (

√5 − 1)/2 is an estimate of the region of attraction (where

c = λmin(P)r2 = (√

5− 1)/2).


2.3.1 Volterra series representation (time-domain)Definition 2.3.1 [103] Let hn(t1, t2, ..., tn) be a real-valued multivariate function in n variables(hn : Rn → R) so that hn(t1, t2, ..., tn) = 0 if any ti < 0. A system Σhom described by thefollowing input-output relationship

y(t) =∫ ∞−∞

...∫ ∞−∞

hn(σ1, σ2, ..., σn)u(t− σ1)...u(t− σn)dσ1...dσn, (2.70)

is called a nth degree homogeneous system.

In the following we state some properties of systems described by the mapping in (2.70), as

1. The relation in (2.70) is sometimes called the generalized convolution formula.

2. The assumption that hn is one-sided in each variable ti corresponds to causality.

3. The system is not necessary linear (only for n = 1) but it is stationary.

4. Plugging in the input αu(t) will yield the output αny(t) (where α ∈ R and y(t) is theresponse to u(t)).

5. The multi-variable function hn is called the kernel of the system.

In the following, we will be analyzing linear, bilinear and quadratic-bilinear systems. Thefirst class of systems, i.e. linear systems, can be viewed as degree one homogeneous systems.The other two classes consist of an infinite series of homogeneous subsystems.

32


Linear systems

Consider a simplified model of a linear system originally defined in (1.3) with E = In. Addi-tionally, assume x(0) = 0. The zero-input response is found by solving x(t) = Ax(t) whichyields xzi(t) = eAt. Consider the change of variable by introducing z(t) = e−Atx(t). Hencex(t) = eAtz(t); next rewrite the differential equation in the z variable as

z(t) = d

dt

(e−Atx(t)

)= −Ae−Atx(t) + e−Atx(t) = −Ae−Atx(t) + e−At(Ax(t) + Bu(t)) = e−AtBu(t).

Let B(t) = e−AtB, C(t) = eAtC. Using these newly introduced time varying matrices, rewritethe initial time invariant linear system (with variable x) as a time varying linear system (withnew varibale z) in the following way,

ΣtvL :

z(t) = B(t)u(t),y(t) = C(t)z(t), z(0) = 0.

(2.71)

By integrating the differential equation in (2.71) and then substituting z(t) = e−Atx(t), obtain

z(t) =∫ t

0B(τ)u(τ)dτ ⇒ e−Atx(t) =

∫ t

0e−AτBu(τ)dτ → x(t) =

∫ t

0eA(t−τ)Bu(τ)dτ. (2.72)

Since y(t) = Cx(t), write the following input-output mapping,

y(t) =∫ t

0CeA(τ−t)Bu(τ)dτ =

∫ t

0h(t− τ)u(τ)dτ = h(t) ∗ u(t) = u(t) ∗ h(t). (2.73)

where impulse response (or linear kernel) corresponding to the system ΣL is defined as

h(t) = CeAtB. (2.74)

Bilinear systems

Again, as for the linear systems case, consider a simplified model of the bilinear system originallydefined in (1.4) with E = In. Also, assume that x(0) = 0. As before the zero-input response isxzi(t) = eAt. Consider the variable change z(t) = e−Atx(t)⇒ x(t) = eAtz(t) and hence obtain

z(t) = d

dt(e−Atx(t)) = −Ae−Atx(t) + e−Atx(t) = −Ae−Atx(t) + e−At(Ax(t) + Nx(t)u(t) + Bu(t)

)= e−AtNeAtz(t)u(t) + e−AtBu(t).

Let N(t) = e−AtNeAt and as before B(t) = e−AtB, and C(t) = eAtC. Rewrite the initial timeinvariant bilinear system (with variable x) as a time varying bilinear system (with variable z)z(t) = N(t)z(t)u(t) + B(t)u(t),

y(t) = Cz(t), z(0) = 0.(2.75)

33


By integrating the differential equation in (2.75), write

z(t) =∫ t

0N(τ1)z(τ1)u(τ1)dτ1 +

∫ t

0B(τ1)u(τ1)dτ1. (2.76)

By renaming the variables t→ τ1, τ1 → τ2, evaluate z(t) at τ1 using (2.76) and then substitutez(τ1) back into

∫ t0 N(τ1)z(τ1)u(τ1)dτ1. It follows that

z(t) =∫ t

0

∫ τ1

0N(τ1)N(τ2)z(τ2)u(τ2)u(τ1)dτ2dτ1+

∫ t

0

∫ τ1

0N(τ1)B(τ2)u(τ2)u(τ1)dτ2dτ1+

∫ t

0B(τ1)u(τ1)dτ1.

Repeating this procedure ` times, it follows that

z(t) =∫ t

0

∫ τ1

0· · ·

∫ τ`−1

0N(τ1)N(τ2) · · · N(τ`)z(τ`)u(τ`) · · ·u(τ1)dτ` · · · dτ1

+∑k=1

∫ t

0

∫ τ1

0· · ·

∫ t

0

∫ τk−1

0N(τ1) · · · N(τk−1)B(τk)u(τk) · · ·u(τ1)dτk · · · dτ1.

The first term in the expression above vanishes as k →∞. Hence write

z(t) =∞∑k=1

∫ t

0

∫ τ1

0· · ·

∫ τk−1

0N(τ1) · · · N(τk−1)B(τk)u(τk) · · ·u(τ1)dτk · · · dτ1. (2.77)

By replacing N(t) = e−AtNeAt, B(t) = e−AtB in (2.77), it follows that

z(t) =∞∑k=1

∫ t

0

∫ τ1

0· · ·

∫ τk−1

0e−Aτ1NeAτ1 · · · e−Aτk−1NeAτk−1e−AτkBu(τk) · · ·u(τ1)dτk · · · dτ1

=∞∑k=1

∫ t

0

∫ τ1

0· · ·

∫ τk−1

0e−Aτ1NeA(τ1−τ2)N · · ·NeA(τk−1−τk)Bu(τk) · · ·u(τ1)dτk · · · dτ1.

Now, returning to the original variable x(t) and using that x(t) = eAtz(t) we recover it as

x(t) =∞∑k=1

∫ t

0

∫ τ1

0· · ·

∫ τk−1

0eA(t−τ1)NeA(τ1−τ2)N · · ·NeA(τk−1−τk)Bu(τk) · · ·u(τ1)dτk · · · dτ1.

(2.78)The expression in (2.78) is the so-called Volterra series associated with the bilinear system ΣB(it is well established in the context of nonlinear systems, see [103] and [79]). Furthermore, byusing that y(t) = Cx(t), we can write the input-output mapping as follows

y(t) =∞∑k=1

∫ t

0

∫ τ1

0· · ·

∫ τk−1

0CeA(t−τ1)NeA(τ1−τ2)N · · ·NeA(τk−1−τk)Bu(τk) · · ·u(τ1)dτk · · · dτ1.

(2.79)Now, by appropriately considering several changes of variables, obtain different types of kernels.

34


First consider the following change of variabletk = t− τ1,

tj = τk−j − τk−j+1, j ∈ 1, 2, . . . , k − 1⇔ τj = t−

k∑i=k−j+1

ti, ∀j ∈ 1, 2, . . . , k.

By substituting these values into (2.79), one can write the output as,

y(t) =∞∑k=1

∫ t1

0

∫ t2

0· · ·

∫ tk−1

0CeAtkNeAtk−1N · · ·NeAt1Bu(t− tk − ...− t1) · · ·u(t− tk)dtk · · · dt1.

Definition 2.3.2 The kth regular kernel associated to the system ΣB is expressed as

hkreg(t1, . . . , tk) = CeAtkNeAτk−1N · · ·NeAτ1B. (2.80)

Next, consider another change of variable, namely tj = t− τk+1−j j ∈ 1, 2, . . . , k; hence writeτj = t− tk+1−j. By substituting these values into (2.79), write

y(t) =∞∑k=1

∫ t1

0

∫ t2

0· · ·

∫ tk−1

0CeAtkNeA(tk−1−tk)N · · ·NeA(τ1−τ2)Bu(t− t1) · · ·u(t− tk)dtk · · · dt1.

Definition 2.3.3 The kth triangular kernel associated to the system ΣB is expressed as

hktri(t1, . . . , tk) = CeAtkNeA(tk−1−tk)N · · ·NeA(τ1−τ2)B. (2.81)

Note that the triangular kernel is the same as the regular kernel up to a change of variables:

hktri(t1, t2, ..., tk−1, tk) = hkreg(t1 − t2, t2 − t3, ..., tk−1 − tk, tk).

Definition 2.3.4 Introduce the symmetric kernel using the triangular kernel in (2.81), as

hksym(t1, . . . , tk) = 1k!∑π(.)

hktri(tπ(1), . . . , tπ(k)). (2.82)

where π : 1, . . . , n → 1, . . . , n is a permutation.

Quadratic-bilinear systems

We propose a generalization of higher order regular kernels based on Definition 2.3.2 by includingthe matrix Q, which will complicate the structure of the multi-variate time-domain functions.We do not claim that the following characterization is valid for all quadratic-bilinear kernels, butfor a certain class which is of interest.

Definition 2.3.5 Let Υ = N,Q and w ∈ Υk be a tuple of length k with w = (w1, w2, . . . , wk),for wj ∈ Υ, j ∈ 1, 2, . . . , k. Denoting with h(t) := hε0(t) = CeAtB the linear kernel (impulse

35


response), introduce the following kth level generalized kernel together with the tuple w, as

hwk (t1, t2, t3, . . . , th) = CeAt1w1φ(w1,X1) · · ·w`φ(wk,Xk)B, k > 1, (2.83)

where the mapping φ : Υ×Rb → Rn×n was previously introduced in Definition 2.2.23. Moreover,|Xi| ∈ 1, 2, ∀ i ∈ 1, 2, . . . , k and 1 +∑k

i=1 |Xi| = h.

For example, the level 1 kernels can be written as

hN1 (t1, t2) = CeAt1NeAt2B, and hQ

1 (t1, t2, t3) = CeAt1Q(eAt2B⊗ eAt3B

). (2.84)

2.3.2 Frequency domain mappingsThe unilateral (or one-sided) Laplace transform of a function f(t) is defined as,

F(s) = Lf(t) =∫ ∞

0f(t)e−stdt. (2.85)

The generalization of this well known result follows naturally

Definition 2.3.6 Let f : Rn → R be a multivariate function in n variables. The Laplace trans-form of f is defined as,

F(s1, s2, ..., sn) = Lf(t1, t2, ..., tn) =∫ ∞

0...∫ ∞

0f(t1, t2, ..., tn)e−s1t1e−s2t2 ...e−sntndt1...dtn.

For homogeneous systems, apply the Laplace transform of y(t) in (2.70), first for n = 1,

Y(s) = Ly(t) =∫ ∞

0

∫ ∞0

h(τ)u(t− τ)e−stdτdt =∫ ∞

0

∫ ∞0

h(τ)u(t− τ)e−sτe−s(t−τ)dτdt

=∫ ∞

0h(τ)e−sτdτ

∫ ∞0

u(t− τ)e−s(t−τ)dt = H(s)U(s).

Similarly, for a nth degree homogeneous system, by taking the multivariate Laplace transformone can write

Yn(s1, . . . , sn) = H(s1, . . . , sn)U(s1) · · ·U(sn).

Definition 2.3.7 The transfer function H(s) of a linear system ΣL as introduced in (1.3) (withE = In) is defined as the Laplace transfer of the linear kernel h(t) in (2.74). Hence, the transferfunction can be written in terms of the system matrices as follows H(s) = C(sI −A)−1B. Byreplacing the identity matrix with E we write the more general definition

H(s) = C(sE−A)−1B. (2.86)

36


The exact derivation follows

H(s) = Lh(t) =∫ ∞

0h(t)e−stdt =

∫ ∞0

CeAtBe−stdt =∫ ∞

0Ce−(sI−A)tBdt

= C( ∫ ∞

0e−(sI−A)tdt

)B = C(sI−A)−1B.

We make use of the following lemma

Proposition 2.3.1 If the mapping f : Cn → C can be written as a product of two factors thatdo not have any variables ti in common, i.e., f(t1, . . . , tn) = x(t1, . . . , tj)y(tj+1, . . . , tn), then itfollows that applying the multi-variate Laplace transform on both sides, write F(s1, . . . , sn) =X(s1, . . . , sj)Y(sj+1, . . . , sn).

Definition 2.3.8 The kth regular transfer function associated to the system ΣB (as defined in(1.4)) is defined as the Laplace transform of the kth regular kernel in (2.80). It can be expressedin terms of the system matrices as

Hkreg(s1, . . . , sk) = C(skE−A)−1N(sk−1E−A)−1N · · ·N(s1E−A)−1B, (2.87)

where k > 1 and again the identity matrix In was replaced by E.

Similarly, one can write the other two types of transfer function as (∀ k > 1)

Hktri(s1, . . . , sk) = C((s1 + ...+ sk)E−A)−1N((s1 + · · ·+ sk−1)E−A)−1N . . .N(s1E−A)−1B,

Hksym(s1, . . . , sk) = 1n!∑π()

Htri(sπ(1), . . . , sπ(k)).

Example 2.3.1 By choosing k=2, we enumerate the three categories of transfer functions pre-viously defined, as (replace the identity matrix with E)

H2reg(s1, s2) = C(s2E−A)−1N(s1E−A)−1B,H2tri(s1, s2) = C((s1 + s2)E−A)−1N(s1E−A)−1B,

H2sym(s1, s2) = 12[C((s1 + s2)E− E)−1N(s1E−A)−1B + C((s2 + s1)E−A)−1N(s2E−A)−1B].

Note that H2sym(s1, s2) = H2sym(s2, s1) (hence the symmetry of this transfer function).

In model order reduction applications, especially for the ones that involve moment matching,the regular transfer functions are mainly used. For the generalization of the Loewner frameworkto bilinear systems presented in chapter 4, one requires samples belonging to the functions definedin (2.87) in order to build the Loewner matrices.

The growing exponential approach

The properties of growing exponentials can be adapted to construct transfer function descriptionsfor stationary state equations. In this direction, the method introduced in [103] (Section 3.5), isgoing to be adapted to find symmetric transfer functions of quadratic-bilinear systems.

37


First, one applies this procedure for linear systems and then extrapolates it to quadratic bilin-ear systems. The main idea is motivated by the following property of a stable linear continuoustime-invariant system; if the input is a growing exponential, then the output is a scaled versionof the input. Hence by plugging in the input function u(t) = eλt, λ > 0 into (43), write

y(t) =∫ ∞

0h(τ)u(t− τ)dτ =

∫ ∞0

h(τ)eλ(t−τ)dτ =( ∫ ∞

0h(τ)e−λτdτ

)eλt = H(λ)eλt. (2.88)

Moreover, for a linear combination of growing exponentials, i.e., u(t) = ∑ki=1 bie

λit, it followsthat the response is given by y(t) = ∑k

i=1 biH(λi)eλit.Since y(t) = Cx(t), one can consider the variable x to be written in the format x(t) = Geλt

where G ∈ Rn is a constant vector. By plugging in x(t) and u(t) into the differential equationEx(t) = Ax(t) + Bu(t) we get

λEeλt = AGeλt + Beλt ⇒ λE = AG + B⇒ G = (λE−A)−1B.

Since y(t) = Cx(t) = CGeλt, from the relation above write y(t) = C(λE − A)−1Beλt. From(2.88), identify H(λ) = C(λE−A)−1B for an arbitrarily chosen λ > 0. Then the conclusion isthat H(s) = C(sE−A)−1B, ∀s ∈ C as introduced in (2.86).

For homogeneous systems of higher order, assume the input consists in a linear combinationof growing exponentials, as u(t) = ∑k

i=1 eλit and that the solution x(t) is of the form

x(t) =∑m

Gm1,...,mk(λ1, . . . , λk)e(m1λ1+···+mkλk)t, mi ∈ 0, 1.

Substituting into the differential equation and equating coefficients of identical exponentials, wesolve for G1,0,...,0(λ1), G1,1,0,...,0(λ1, λ2), . . . , G1,1,...,1(λ1, . . . , λk) and conclude that

Hisym(s1, . . . , si) = 1i!CG1, 1, . . . , 1︸︷︷︸

i times

,0,...,0(s1, . . . , si), i ∈ 1, 2, . . . , k. (2.89)

Consider the state-space representation of a quadratic-bilinear system ΣQB as introduced in (1.5).As an example, proceed with deriving the first two transfer functions of the system. By choosingk = 2 in (2.89), it follows that

H1sym(s1) = CG1,0(s1), H2sym(s1, s2) = 12CG1,1(s1, s2). (2.90)

First, plug in the following input u(t) = eλ1t + eλ2t, λi > 0, i ∈ 1, 2 and assume the variablex is written as

x(t) = G1,0(λ1)eλ1t + G0,1(λ2)eλ2t + G1,1(λ1, λ2)e(λ1+λ2)t. (2.91)

From this representation also derive

x(t) = λ1G1,0(λ1)eλ1t + λ2G0,1(λ2)eλ2t + (λ1 + λ2)G1,1(λ1, λ2)e(λ1+λ2)t,

x(t)⊗ x(t) =(G1,0(λ1)⊗G0,1(λ2) + G0,1(λ2)⊗G1,0(λ1)

)e(λ1+λ2)t + G1,0(λ1)⊗G1,0(λ1)e2λ1t + . . .

38


Next plug in (2.91) into the differential equation from (1.5). By equating the coefficients of theterms eλit, i = 1, 2 from both sides of the differential equation, it follows that

λ1EG1,0 = AG1,0 + B, λ2EG0,1 = AG0,1 + B⇒

G10 = (λ1E−A)−1B,G0,1 = (λ2E−A)−1B.

(2.92)

Hence, from (2.90) and (2.92) conclude that H1sym(s1) = C(s1E−A)−1B which means that thefirst symmetric transfer function associated to ΣQB is the transfer function corresponding to ΣL.

Furthermore, by equating the coefficients of the term e(λ1+λ2)t from both sides of the differ-ential equation in (1.5), write

(λ1 + λ2)EG11 = AG11 + Q(G10 ⊗G01 + G01 ⊗G10

)+ N(G10 + G01)⇒

G11 =((λ1 + λ2)E−A

)−1(Q(G10 ⊗G01 + G01 ⊗G10) + N(G10 + G01)

). (2.93)

Again applying the result in (2.90), it follows that CG11(s1, s2) = 2H2sym(s1, s2). By substi-tuting the linear contributions G10 and G01 from (2.92) onto the (2.93) and by using the notationΦs = (sE−A)−1, write the second symmetric transfer function as

H2sym(s1, s2) = 12CΦs1+s2Q

(Φs1B⊗Φs2B+Φs2B⊗Φs1B

)+ 1

2CΦs1+s2N(Φs1B+Φs2B). (2.94)

By feeding a linear combination of more growing exponentials as the input u(t), we come upwith higher order transfer functions which can be written as composition of frequency domainfunctionals that depend solely on Q or solely on N and also on both Q and N. Based on (2.94),introduce the following rational functions

HN1 (s1, s2) = CΦ(s1)NΦ(s2)B, HQ

1 (s1, s2, s3) = CΦ(s1)Q(Φ(s2)B⊗Φ(s3)B

), (2.95)

which are of particular importance since they are going to be used in the generalization of theLoewner framework to the class of QB system. More specifically, samples of these sort of functionswill be needed to construct the Loewner matrices. The second symmetric transfer function ofΣQB defined in (2.94) can be written in terms of these rational functions, as

H2sym(s1, s2) = 12

(HQ

1 (s1 + s2, s1, s2) + HQ1 (s1 + s2, s2, s1) + HN

1 (s1 + s2, s1) + HN1 (s1 + s2, s2)

).

2.3.3 A new class of transfer functionsWe proceed to defining a new class of rational functions that will be used for the model orderreduction method proposed in Chapter 5. The advantage is that the construction of thesefunctions only requires the multiplication of certain matrices similar to the case of bilinear systems(mentioned in Chapter 4). Before introducing a more general structure for the desired transferfunctions, one needs to set up a series of notations and definitions.

39


Definition 2.3.9 For Υ = N,Q, b ∈ 1, 2 and S ∈ Cb consider the function Γ : Υ× Cb →Cnb×n, defined as follows:

Γ(w,S) =

Φ(s1), if w = N, S = (s1) and b = 1,Φ(s1)B⊗Φ(s2), if w = Q, S = (s1, s2) and b = 2.

(2.96)

More exactly, in (2.96), the function Γ can be described by Γ(N, s1) = (s1E − A)−1 and byΓ(Q, (s1, s2)) = (s1E−A)−1B⊗ (s2E−A)−1.

Denote with Υ` the set of all ordered tuples of length ` with entries from Υ, i.e., for all ` > 1,write Υ` = (w1, w2, . . . , w`)|wk ∈ Υ, 1 6 k 6 `. Moreover, Υ0 = ε contains only the nullsymbol.

Definition 2.3.10 Let `, h > 0 be two integers so that ` 6 h 6 2` and consider the orderedtuple S = (s0, s1, . . . , sh), where s0, s1, . . . , sh ∈ C. Additionally, let Sk, 1 6 k 6 ` be a tuple of1 or 2 complex numbers from the set s0, s1, . . . , sh so that S1 S2 · · · S` = S. Finally,let w = (w1, w2, . . . , w`) ∈ Υ` be a tuple of length ` > 0. Associate the following level ` rationaltransfer function to the tuple w ∈ Υ`:

Hw` (s0, s1, . . . , sh) =

CΦ(s0)B, ` = 0, w = ε,

CΦ(s0)w1Γ(w1,S1) · · ·w`Γ(w`,S`)B, ` > 1.(2.97)

The rational functions introduced in (2.97) are hence divided in subcategories or levels. Itfollows that a number of 2` functions are associated to level ` > 0. More precisely, the onecorresponding to level 0 is the linear transfer function Hε

0(s) = C Φ(s) B, which can also bedenoted with H(s). Furthermore, the two functions corresponding to level 1 are written as:

HN1 (s0, s1) = C Φ(s0) N Φ(s1)B, HQ

1 (s0, s1, s2) = C Φ(s0) Q(Φ(s1)B⊗Φ(s2)B

). (2.98)

Note that the rational functions in (2.98) can be computed by taking the multivariate Laplacetransform of the time domain kernels in (2.84).

For the second level, let w = (N,Q) and S1 = (s1), S2 = (s2, s3). Then write the functionHw

2 evaluated at the tuple (s0, s1, s2, s3), as follows:

HN,Q2 (s0, s1, s2, s3) = CΦ(s0)w1Γ(w1,S1)w2Γ(w2,S2)B (2.99)

= CΦ(s0)NΦ(s1)Q(Φ(s2)B⊗Φ(s3)

)B. (2.100)

In general, a `th level transfer function defined in (2.97) is a multivariate rational functiondepending on h variables s1, . . . , sh, for `+ 1 6 h 6 2`+ 1.

Note that if w = (N,N, . . . ,N) ∈ Υ`+1, ` > 0 and S` = (s`), then the transfer function Hw`+1

exactly coincides to the bilinear transfer function proposed in [35], i.e.

Hw` (s0, . . . , s`) = CΦ(s0)NΓ(N, s1) · · ·NΓ(N, s`+1)B = CΦ(s0)NΦ(s1) · · ·NΦ(s`)B. (2.101)

40

2.4 Norms on dynamical systems

2.4 Norms on dynamical systemsConsider the following definition of Hq norms, as presented in [4].

Definition 2.4.1 Let F be a complex-valued m× p function (F : C → Cm×p) which is analyticin C+. Then, for q > 1, compute the Hq norm of F, as

‖F‖Hq =

(

supx>0

∫∞−∞ ‖F(x+ jy)‖qq dy

) 1q

, for q ∈ [1,∞)

supz∈C+

‖F(z)‖q, for q =∞(2.102)

where ‖F(z)‖q is the (Schatten) q-norm of the function F evaluated at z.

Also consider the following definition of Lp norms, as presented in [4].

Definition 2.4.2 Let F be a complex-valued m × p function (F : C → Cm×p) that has nosingularities (poles) on the imaginary axis but is not necessarily analytic in either the left or theright half of the complex plane. Then, for q > 1, compute the Lq norm of F, as

‖F‖Lq =

(

supy∈R

∫∞−∞ ‖F(jy)‖qq dy

) 1q

, for q ∈ [1,∞),

supy∈R

σmax(F(jy)), for q =∞.(2.103)

where σmax(F(jy)) is the maximum singular value of the matrix F(jy) evaluated at jy.

Definition 2.4.3 Define the following Hardy spaces of functions which are analytic in C+, i.e.,

RH2 = F ∈ R(s)p×m, iR ⊂ D(F), ‖F‖H2 <∞, (2.104)RH∞ = F ∈ R(s)p×m, iR ⊂ D(F), ‖F‖H∞ <∞. (2.105)

where the H2 and H∞ norms were introduced in Definition 2.4.1.

For linear systems, of particular importance are the cases q ∈ 2,∞. Hence introduce thefollowing norms for measuring the size of the frequency domain input output mapping.

Definition 2.4.4 Consider a linear system ΣL as defined in (1.3) and let H(s) be the transferfunction in (2.86). Then the Hq norms associated with this system are defined as

‖ΣL‖Hq =

(

12π∫∞−∞ tr

(H(jω)∗H(jω)

)dω) 1

2, if q = 2,

supω∈R

σmax(H(jω)), if q =∞.(2.106)

Since the E matrix corresponding to the system ΣL is assumed to be invertible, it followsthat this matrix can be assimilated in the other system matrices, i.e., A and B. Hence, the

41


linear kernel is written as h(t) = CeAtB (as introduced in (2.74)). By applying the well knownresult by Parseval, i.e.,∫ ∞

−∞tr(h(t)Th(t)

)dt = 1

2π

∫ ∞−∞

tr(H(jω)∗H(jω)

)dω,

we can write the H2 norm in time domain in terms of the impulse response (kernel) h(t).

Proposition 2.4.1 Let ΣL be a linear system with E = In. Then the H2 norm of ΣL can bewritten

‖ΣL‖2H2 =

∫ ∞−∞

tr(h(t)Th(t)

)dt. (2.107)

Moreover, assuming ΣL is stable (the matrix A is Hurwitz), then the Gramians P and Qdefined in (2.21) and (2.22), respectively, both exist and are positive definite. Write

∫∞0 tr

(h(t)h(t)T

)dt =

∫∞0 tr

(CeAtBBT eAT tCT

)dt = tr

(CPCT

),∫∞

0 tr(h(t)Th(t)

)dt =

∫∞0 tr

(BT eAT tCTCeAtB

)dt = tr

(BTQB

).

(2.108)

Using that tr(h(t)Th(t)

)= tr

(h(t)h(t)T

)and that h(t) = 0, for t < 0, the following result

naturally follows from (2.107) and (2.108).Proposition 2.4.2 Let ΣL be a stable linear system. Then the H2 norm of ΣL can be writtenas

‖ΣL‖2H2 = tr

(CPCT

)= tr

(BTQB

). (2.109)

An interesting formula that relates the H2 norm of the SISO system ΣL to the poles andresidues of the transfer function H(s) has been proposed in [74]. There, it is assumed that thetransfer function H(s) ∈ C can be written as H(s) = ∑n

i=1ri

s−λi . where λi and ri are the polesand the residues associated to the poles, respectively. Moreover all poles are simple and in theleft half plane, i.e., λi ∈ C−.

Proposition 2.4.3 Let ΣL be a SISO stable linear system; the H2 norm of ΣL can be writtenas

‖ΣL‖2H2 =

n∑i=1

H(−λi)ri. (2.110)

Some of the definitions in this section that apply to linear systems were generalized for bilinearcontrol systems. For example, several derivations corresponding to the H2 norm characterizationof such systems were included in [113], [56] or [53]. As far the author of this thesis is aware of,there is no H∞ norm derivation available in the literature.

Definition 2.4.5 Consider a bilinear system ΣB as defined in (1.4) and let Hk(s1, . . . , sk) bethe kth multivariate regular transfer function associated to this system (as introduced in (2.87)).Then the H2 norm associated with this system is defined as

‖ΣB‖H2 =( ∞∑k=1

supx1>0,...,xk>0

∫ ∞−∞· · ·

∫ ∞−∞‖Hk(x1 + jy1, . . . , xk + jyk)‖2

F dy1 . . . dyk

) 12, (2.111)

where ‖M‖F =√

tr(MM∗) is the Frobenius norm of the matrix M.

42


By applying Parseval’s teorem for several variables, it follows that the H2 norm of the systemΣB in frequency domain is equivalent to its L2 norm in time domain.

Proposition 2.4.4 [56] Consider a linear system ΣB as defined in (1.4) and let hk(t1, . . . , tk)be the kth regular Volterra kernel associated to this system (as introduced in (2.80)).

‖ΣB‖L2 =( ∞∑k=1

∫ ∞−∞· · ·

∫ ∞−∞‖hkreg(t1, . . . , tk)‖2

F dy1 . . . dyk

) 12. (2.112)

Alternatively, as for linear systems, there is a connection the above mentioned norm computed intime-domain and the bilinear infinite Gramians. Assuming that the conditions in Lemma 2.2.1are fulfilled, then the Gramians exist (and are positive definite).

Proposition 2.4.5 [113] Consider the infinite controllability and observabiltiy Gramians P andQ corresponding to system ΣB, as defined in (2.34) and (2.35). If these matrices exist, then itfollows that

‖ΣB‖2H2 = C

( ∞∑k=1Pk)CT = CPCT , ‖ΣB‖2

H2 = BT( ∞∑k=1Qk)B = BTQB. (2.113)

A generalization of the formula stated in (2.110), was recently introduced in [56]. It requiresthe definition of generalized residues (for multivariate rational functions). Denote the poles ofA with λ1, . . . , λn (assume only simple poles).

Definition 2.4.6 For a kth order transfer function Hk(s1, s2, . . . , sk) associated to the systemΣB, define the residues of it as,

rl1,...,lk = limsk→λlk

(sk − λlk) limsk−1→λlk−1

(sk−1 − λlk−1) · · · lims1→λl1

(s1 − λl1)Hk(s1, . . . , sk).

Then the multivariate rational function Hk can be written as

Hk(s1, . . . , sk) =n∑

l1=1· · ·

n∑lk=1

rl1,...,lk∏ki=1(si − λli)

.

We state the following adaptation to bilinear systems of the result from (2.110).

Proposition 2.4.6 The H2 norm of the system ΣB can be computed as

‖ΣB‖2H2 =

∞∑k=1

n∑l1=1· · ·

n∑lk=1

rl1,...,lkHk(−λl1 , . . . ,−λlk). (2.114)

The literature is quite scarce when it comes to defining appropriate generalizations of the alreadymentioned system norms to the class of quadratic-bilinear systems. The recent contribution in[26] proposes such a definition. We mention an adaptation of the result presented there for thecase of SISO systems and using the notation introduced in Section 2.2.1.

43


Definition 2.4.7 Consider a QB system ΣQB as defined in (1.5). By using the first three kernelsassociated to this system, as defined in (2.83), i.e., h, hN

1 and hQ1 , which correspond to the first

two levels, the following truncated H2 norm is defined ([26])

‖ΣQB‖2H(T )

2=∫ ∞

0h(t1)h(t1)Tdt1 +

∫ ∞0

∫ ∞0

hN(t1, t2)hN(t1, t2)Tdt1dt2

+∫ ∞

0

∫ ∞0

∫ ∞0

hQ(t1, t2, t3)hQ(t1, t2, t3)Tdt1dt2dt3. (2.115)

We propose the following generalization by taking in the account infinite many kernels, i.e., theones which were previously defined in (2.83)).

Definition 2.4.8 Consider a QB system ΣQB as defined in (1.5). The following definition ofan a H2 norm, is introduced as follows:

‖ΣQB‖2H2 =

∞∑k=1

∫ ∞0· · ·

∫ ∞0

hw(t1, t2, . . . , th)hw(t1, t2, . . . , th)Tdt1 · · · dth, |w| = k. (2.116)

As for the previously covered types of dynamical systems ( linear and bilinear to be morespecific), it turns out that the H2 norm can be written in terms of the infinite controllability andobservability Gramians corresponding to ΣQB.

Proposition 2.4.7 For the quadratic system ΣQB, consider the truncated norm defined in(2.115), which can be written in terms of the Gramians introduced in Definitions 2.2.5 and2.2.26, i.e

‖ΣQB‖2H(T )

2= C

(PL + P1

)CT = BT

(QL +Q1

)B. (2.117)

Moreover, the new proposed norm introduced in (2.116), satisfies a similar property to theone stated in Proposition 2.4.5.

Proposition 2.4.8 Consider the infinite controllability and observabiltiy Gramians P and Qcorresponding to system ΣB, as defined in (2.58) and (2.64). If these matrices are finite, then itfollows that

‖ΣQB‖2H2 = C

( ∞∑k=0Pk)CT = CPCT , ‖ΣQB‖2

H2 = BT( ∞∑k=0Qk)B = BTQB. (2.118)

Example 2.4.1 Consider the two dimensional quadratic-bilinear system ΣQB characterized bythe following matrices,

A =[−1 00 −2

], B =

[12

], C =

[1 1

], N =

[0 1

20 0

], Q =

[0 0 0 00 1

212 0

].

The following transfer functions are computed

H(s1) = 1s1 + 1 + 2

s1 + 2 , HN1 (s1, s2) = 1

2 (s1 + 1) (s2 + 1) ,

44


HQ1 (s1, s2, s3) = 1

(s1 + 2) (s2 + 1) (s3 + 2) + 1(s1 + 2) (s2 + 2) (s3 + 1) .

First analyze the controllability infinite Gramians. Hence compute the linear Gramian PL, thetruncated Gramian from [26],i.e., PL + P1, the quadratic-bilinear Gramian we propose, i.e.,P = PL + P1 + P2 + . . ., and the true quadratic-bilinear Gramian from [25], i.e., Ptrue,

PL =[ 1

223

23 1

], PL + P1 =

[ 916

23

23

161144

], P =

[ 47

23

23

22001953

], Ptrue =

[ 47

23

23

133117

]

In addition, compute the H2 norms based on the previously computed controllability Gramiansas,

‖ΣL‖2H2 = CPLCt = 17

6 ≈ 2.83, ‖ΣQB‖2H(T )

2= C

(PL + P1

)CT = 217

72 ≈ 3.01

‖ΣQB‖2H2 = CPCT = 5920

1953 ≈ 3.03, ‖ΣQB‖2Htrue

2= CPtrueCT = 2491

819 ≈ 3.04

Next redirect our attention towards the observability infinite Gramians. Hence computethe linear Gramian QL, the truncated Gramian from [26], i.e., QL +Q1, the quadratic-bilinearGramian we propose, i.e. Q = QL + Q1 + Q2 + . . ., and the true quadratic-bilinear Gramianfrom [26], i.e., Qtrue,

QL =[ 1

213

13

14

], QL +Q1 =

[ 1932

2572

2572

33128

], Q =

[ 132217

97279

97279

831

], Qtrue =

[ 1356722113

169486

169486

727

]

Again compute different H2 norms based on the previously computed observability Gramians as,

‖ΣL‖2H2 = BTQLB = 17

6 ≈ 2.83, ‖ΣQB‖2H(T )

2= BT

(QL +Q1

)B = 217

72 ≈ 3.01

‖ΣQB‖2H2 = BTQB = 5920

1953 ≈ 3.03, ‖ΣQB‖2Htrue

2= BTQtrueB = 2491

819 ≈ 3.04

45

Chapter 3

The Loewner Framework for LinearSystems

3.1 IntroductionIn some applications, input-output measurements are useful since they replace an explicit modelof a to-be-simulated dynamical system. In such cases it is of great interest to be able to efficientlyconstruct state-space models and reduced state-space models from the available data.

The approach we are about to discuss is data driven. It consists of collecting input-output (e.g.frequency response) measurements for some appropriate range of frequencies. Then using thenew methodology known as Loewner framework, we construct models which fit (or approximatelyfit) the data and have reduced dimension. The key is that in contrast to existing interpolatoryapproaches, larger amounts of data than necessary are collected and the essential underlyingsystem structure is extracted appropriately. Thus a basic advantage of this approach is that itis capable of providing the user with a trade-off between accuracy of fit and complexity of themodel. The Loewner framework, was developed in a series of papers. For details we refer thereader to the book [4], as well as [91, 85, 84, 11, 77, 78]. For an overview see [12].

3.2 Set-up of the Loewner interpolation frameworkUsing rational functions, the goal is to show that it is possible to compute models that match(interpolate) given data sets of measurements.

In the context of linear dynamical systems, the starting point is given by data sets composedout of frequency response measurements. We seek the rational function that interpolates thesemeasurements, i.e., the linear dynamical system that models these measurements.

For now, our attention is drawn towards the so-called Lagrange rational interpolation proce-dure, that uses Lagrange basis for the numerator and denominator polynomials and constructsrational interpolants (functions) that match original interpolation conditions. In ([76]), it wasshown that this basis choice leads to algorithms that are numerically robust.

The transfer functions of linear systems are rational functions. The location of their polesdetermines important system properties such as asymptotic stability, transient behavior or damp-ing.

46

3.2 Set-up of the Loewner interpolation framework

3.2.1 Polynomial interpolationWe start with a classic problem from elementary mathematics; the so-called polynomial interpo-lation problem. Given pairs S = (xi, fi)|xi, fi ∈ R, i = 1, 2, . . . , n + 1 of nodes, i.e. xi’s,and points, i.e., fi’s, we seek a polynomial of degree n with real coefficients that interpolates thisgiven data set. Hence, one has to determine the polynomial coefficients ck, k ∈ 0, 1, . . . , n sothat the following n+ 1 interpolation conditions are satisfied

p(xi) = fi, i ∈ 1, 2, . . . , n+ 1 where p(x) =n∑k=0

ckxk. (3.1)

This task turns up to be more or less straightforward; by writing the n+1 interpolation conditionsin matrix format, notice that one has to solve the following linear system to recover the polynomialcoefficients ck, k ∈ 0, 1, . . . , n

1 x1 . . . xn11 x2 . . . xn2...

.... . .

...1 xn+1 . . . xnn+1

︸︷︷︸

V

c0c1...cn

︸︷︷︸

c

=

f1f2...

fn+1

︸︷︷︸

f

.

or in short Vc = f where V ∈ R(n+1)×(n+1) is a Vandermonde matrix (each of its rows containsthe monomial basis 1, x, . . . , xn evaluated at one of the nodes xi corresponding to the index ofthe row, i.e., V(`, h) = xh−1

` , ∀ `, h ∈ 1, 2, . . . , n + 1. Hence, recover the vector of polynomialcoefficients as c = V−1f .

Note that, in general, one would avoid inverting a Vandermonde matrix since the conditionnumber of such matrix increases exponentially, and hence causing to severe ill-conditioning issues.Solving this linear system requires specialized algorithms that try to around this problem.

An alternative way for solving the problem in (3.1) is to avoid using the monomial basismi(x) = xi and hence inverting the Vandermonde matrix V . Instead one can use so-calledLagrange polynomials Li(x) to compute the same polynomial interpolant p as before. Thesepolynomials are of degree n and they form a Lagrange basis

L1(x) = (x− x2)(x− x3) · · · (x− xn+1) =n+1∏

k=1, k 6=1(x− xk),

L2(x) = (x− x1)(x− x3) · · · (x− xn+1) =n+1∏

k=1, k 6=2(x− xk),

...

Ln+1(x) = (x− x1)(x− x2) · · · (x− xn) =n+1∏

k=1, k 6=n+1(x− xk).

47


Note that Li(xk) = 0, ∀ k 6= i, k, i ∈ 1, 2, . . . , n+ 1. Given this basis, directly construct apolynomial p that satisfies the conditions in (3.1)

p(x) =n+1∑i=1

fiLi(xi)︸︷︷︸

bi

Li(x) =n+1∑i=1

biLi(x). (3.2)

Notice that the same polynomial p(x) can be written differently either as ∑n+1i=1 cimi(x) or∑n+1

i=1 biLi(x), depending on what basis we are working with. The argument to advocate forusing the Lagrange basis instead of the monomial basis is that it is avoided to solve a large(in general) possibly ill-conditioned linear system. Moreover, the coefficients ai = fi

Li(xi) arecomputed directly from the given data set D.

Using the Lagrange polynomials, define the degree n polynomial

g(x) =n+1∑i=1

1Li(xi)

Li(x). (3.3)

Note that g(xi) = 1 ∀ i ∈ 1, 2, . . . , n + 1 and let g(x) = g(x) − 1. Hence the g(x) = 0 ∀ x ∈x1, x2, . . . , xn+1. Since g(x) is a degree at most n polynomial with at least n+1 roots, it followsthat g(x) = 0,∀ x ∈ R. Then, conclude that g(x) = 1,∀ x ∈ R and hence p(x) = p(x)

g(x) , ∀ x ∈ R.By putting together (3.2) and (3.3) we come up with the so-called barycentric formula:

p(x) =

n+1∑i=1

fiLi(xi)

Li(x)

n+1∑i=1

1Li(xi)

Li(x). (3.4)

3.2.2 Rational interpolationIn this section we deal with rational Lagrange interpolation which is proven to be a strongalternative to the classical polynomial interpolation.

A rational function r(x) = N (x)D(x) is defined as the ratio between two polynomials, namely, as

the ratio between the numerator polynomial N (x) and the denominator polynomial D(x). Itis said that r is of order n, since deg(N ) = deg(D) = n (and it is also assumed that the twopolynomial do not have any roots in common). In general write

r(x) = βnxn + βn−1x

n−1 + . . .+ β1x+ β0

αnxn + αn−1xn−1 + . . .+ α1x+ α0= N (x)D(x) ,

where βk, αk ∈ R, k ∈ 0, 1, . . . , n and also βn, αn 6= 0.Note that, in general, a rational function is more versatile than a polynomial since it has

both poles (roots of the denominator) and zeros (roots of the numerator). Hence a rationalfunction can model functions with singularities and highly oscillatory behavior more easily thanpolynomials can. Furthermore, rational functions have strong system-theoretic significance asthey represent a natural way of modeling linear dynamical systems in the frequency domain.

48


The Laplace transform of a sum of complex exponentials (of the form e−γt, γ ∈ C) is a rationalfunction (since Le−γt = 1

s+γ ).For polynomial interpolation, it was observed that in order to recover a nth degree polynomial

p(x), one needs n + 1 pairs of sampling nodes/points (since a polynomial p(x) of degree n isdescribed by n + 1 coefficients ck, k ∈ 0, 1, . . . , n). Based on this consideration, one wouldassume that in order to recover a nth order rational function r(x), one needs 2(n + 1) pairs ofsampling nodes/points (since r(x) is described by 2(n+ 1) coefficients αk, βk k ∈ 0, 1, . . . , n).

However, one degree of freedom is redundant since one of these coefficients can be set toequal 1. By dividing both the numerator and denominator by αn, obtain a new set of normalizedcoefficients αk = αk/αn, βk = βk/αn (hence αn = αn/αn = 1).

In order to be able to compute a rational interpolant of order n, one needs to determine 2n+1coefficients so that the following 2n+ 1 interpolation conditions are satisfied

r(xh) = fh, h ∈ 1, 2, . . . , 2n+ 1 where r(x) =∑nk=0 βkx

k∑nk=0 αkx

k. (3.5)

Thus, the data set must contain (at least) 2n+ 1 pairs. Firstly, partition the set of interpolationnodes xh|1 6 h 6 2n+ 1 in two disjoint sets

x1, x2, . . . , x2n+1 = µ1, µ2, . . . , µn ∪ λ1, λ2, . . . , λn+1.

Note that the n nodes λi are used for constructing the Lagrange basis. Secondly, partition theset of interpolation points fh|1 6 h 6 2n + 1 in two disjoint sets (similar to the partition ofthe nodes)

f1, f2, . . . , f2n+1 = v1, v2, . . . , vn ∪ w1, w2, . . . , wn+1.

Alternatively, instead of the monomial basis used in (3.5), we use the Lagrange basis and rewriterational interpolation problem by using the barycentric formula in (3.4)

r(xh) = fh, h ∈ 1, 2, . . . , 2n+ 1 where r(x) =∑n+1i=1 biLi(x)∑n+1i=1 aiLi(x)

, (3.6)

where the Lagrange polynomials are given Li(x) =n+1∏

k=1, k 6=i(x−λk), i ∈ 1, 2, . . . , n+ 1. Hence,

to recover the function r(x), we need to find the coefficinets ai and bi so that r(µj) = vj, j ∈1, 2, . . . , n and r(λi) = wi, i ∈ 1, 2, . . . , n+ 1.

Since `j(λi) = 0, ∀ j 6= i, by evaluating r(x) as defined in (3.6) at the nodes λi, it followsthat r(λi) = bi/ai ⇒ wi = bi/ai ⇒ bi = wiai,∀ i ∈ 1, 2, . . . , n+ 1.

By dividing both of the numerator and denominator of r(x) from (3.6) with the polynomialL (x) = ∏n+1

k=1(x− λk), it follows that the barycentric formula is rewritten as

r(x) =∑n+1i=1

bix−λi∑n+1

i=1ai

x−λi

. (3.7)

Hence, by choosing the numerator coefficients bi to satisfy the condition bi = wiai, it turnsout that the function r(x) automatically matches the values at the Lagrange nodes λi, i.e., by

49


construction. By eliminating the bi’s, rewrite the formula in (3.7) as follows

r(x) =

n+1∑i=1

wiaix− λi

n+1∑i=1

aix− λi

. (3.8)

3.2.3 The Loewner matrixAs pointed in the previous section, the rational interpolation problem has been simplified todetermining the remaining n+1 denominator coefficients a1, a2, . . . , an+1 so that r(µj) = vj,for all j ∈ 1, 2, . . . , n. By evaluating the function r(x) (as defined in (3.7)) at the remainingnodes µj, we write

r(µj) = vj ⇔

n+1∑i=1

wiaiµj − λi

n+1∑i=1

aiµj − λi

= vj, ∀ 1 6 j 6 n ⇔n∑j=1

n+1∑i=1

vj − wiµj − λi

ai = 0 ⇔ La = 0,

where the Loewner matrix is defined as (see [90])

L =

v1−w1µ1−λ1

v1−w2µ1−λ2

. . . v1−wn+1µ1−λn+1

v2−w1µ2−λ1

v2−w2µ2−λ2

. . . v2−wn+1µ2−λn+1

......

. . ....

vn−w1µn−λ1

vn−w2µn−λ2

. . . vn−wn+1µn−λn+1

∈ Rn×(n+1), (3.9)

and the coefficients ai are gathered in the vector a =[a1 a2 . . . an+1

]T. Hence, by comput-

ing the nullspace of the Loewner matrix, we automatically recover the ai coefficients. Then, thenext step is to compute the bi coefficients (by using the relation bi = wiai, ∀i ∈ 1, 2, . . . , n+1).In this way, we uniquely determine the rational function r(x).

Example 3.2.1 We would like to find a rational function r(x) = N (x)D(x) =

∑3i=1

bix−λi∑3

i=1ai

x−λi

that satisfiesthe following interpolation conditions

r(0) = −1, r(1) = 1, r(2) = −1, r(3) = 1, and r(4) = 3.

Hence the given data set consists of the pairs as follows S = (0,−1), (1, 1), (2,−1), (3, 1), (4, 3).First, proceed by partitioning this set as S = SL ∪ SR where SL = (0,−1), (1, 1) and SR =(2,−1), (3, 1), (4, 3). Then identify

µ1 = 0, µ2 = 1, λ1 = 2, λ2 = 3, λ3 = 4v1 = −1, v2 = 1, w1 = −1, w2 = 1, w3 = 3

and also v =[−11

]T, w =

−113

T

.

50


By putting together the data, we form the following Loewner matrix

L =[

0 23 1

−2 0 23

]⇒ ker(L) =

[2 −9 6

]T.

Hence recover a =[

2 −9 6]T

which implies that b =[−2 −9 18

]T. Now, write the

rational interpolant as

r(x) =−2x−2 + −9

x−3 + 18x−4

2x−2 + −9

x−3 + 6x−4

= −7x2 + 22x− 12x2 − 10x+ 12 .

Consider a linear system ΣL characterized by the system matrices (E,A,B,C) (for now considerSISO - one input and one output only). As pointed out in the previous chapter, the transferfunction H(s) is defined as the Laplace transform of the output divided by the Laplace transformof the input, i.e. H(s) = Y(s)/U(s). Also, it follows that H(s) is a rational function that canbe written in terms of the system matrices as follows

H(s) = C(sE−A)−1B. (3.10)

Proposition 3.2.1 Conversely, every rational function r(s) has a realization (E,A,B,C) whichsatisfies the condition r(s) = C(sE−A)−1B; furthermore the realization is minimal if and onlyif the system is both reachable and observable (as defined in chapter 2).

If it is assumed that the sampling points fi = µj ∪ λi are coming from sampling thetransfer function at the sampling nodes xi, we can write that

vj = H(µj) = C(µjE−A)−1B, wi = H(λi) = C(λiE−A)−1B.

Hence write the (j, i) entry of the Loewner matrix defined in (3.9) as follows

L(j, i) = vj − wiµj − λi

= C(µjE−A)−1B−C(λiE−A)−1Bµj − λi

= C(µjE−A)−1 − (λiE−A)−1

µj − λiB

= C(µjE−A)−1 (λiE−A)− (µjE−A)µj − λi

(λiE−A)−1B = −C(µjE−A)−1E(λiE−A)−1B

The matrices O and R are constructed; the jth row of O and the ith column of R are given by

OTj = C(µjE−A)−1, Ri = (λiE−A)−1B.

It follows that the Loewner matrix can be factorized in terms of the matrix E (that correspondsto the underlying system in state-space format)

L = −OER. (3.11)

This opens the road for further discussions and questions such as, can the other factorized systemmatrices, i.e., A,B and C), be written in terms of data matrices as the Loewner matrix (matricesconstructed solely from interpolation points and nodes).

51

3.3 Interpolatory reduction methods and the Loewner framework

As a conclusion, the recovery of a rational functions r that satisfies certain interpolationconditions is strongly related to the Loewner matrix. The number of different solutions r of therational interpolation problem depends on the dimension of the kernel of the Loewner matrix.

Extensive details on the Lagrange interpolation framework in state-space format, connectionbetween the barycentric and state-space form of the system, how to partition the data sets orhow to avoid complex arithmetic, can be found in [76].

3.3 Interpolatory reduction methods and the Loewnerframework

For linear systems, interpolatory reduction can be defined as follows. Consider a full-order systemΣL defined by E ∈ Rn×n, A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, and its transfer function H(s) =C(sE − A)−1B. Given left interpolation points µjqj=1 ⊂ C, with left tangential directions`jqj=1 ⊂ Cp, and right interpolation points λiki=1 ⊂ C, with right tangential directions riki=1⊂ Cm, find a reduced-order systemhatSiL : (E, A, B, C), such that the resulting transfer function H(s) is an approximate tangentialinterpolant to H(s) as follows

`Tj H(µj) = `Tj H(µj), j = 1, . . . , q, and H(λi)ri = H(λi)ri, i = 1, . . . , k . (3.12)

Interpolation points and tangent directions are selected to realize appropriate model reductiongoals. If instead of state space data, we are given input/output data which is either measured orgenerated by DNS − Direct Numerical Simulation). The resulting problem is hence modified.

Given a set of input-output response measurements specified by left driving frequenciesµjqj=1 ⊂ C using left input directions `jqj=1 ⊂ Cp and producing left responses vjqj=1 ⊂Cm, as well as by right driving frequencies λiki=1 ⊂ C using right input directions riki=1 ⊂ Cm

and producing right responses: wiki=1 ⊂ Cp, find (low order) system matrices E, A, B, C suchthat the resulting transfer function, H(s), is an (approximate) tangential interpolant to the data

`Tj H(µj) = vTj , j = 1, . . . , q, and H(λi)ri = wi, i = 1, . . . , k. (3.13)

Interpolation points and tangent directions are determined by the problem under consideration.The Loewner framework will be briefly covered in this chapter. For further details see [12].

Definition 3.3.1 The generalized controllability and observability matrices are written in termsof the original system matrices C,E,A,B and also in terms of the left and right interpolationnodes µjqj=1, λiki=1. The exact definition is presented as follows

R =[

Φ(λ1)Br1 Φ(λ2)Br2 · · · Φ(λk)Brk]∈ Cn×k, (3.14)

O =

`T1 CΦ(µ1)`T2 CΦ(µ2)

...`Tq CΦ(µq)

∈ Cq×n. (3.15)

52


Lemma 3.3.1 Interpolation of linear systems. Let ΣL = (C,E,A,B) be a linear system oforder n. Assume that it is projected to a kth order system by means of X = R and YT = O (asdefined in (3.14) and (3.15) with the condition k = q). The reduced system ΣL = (C, E, A, B),of order k, where

E = YTEX, A = YTAX, B = YTB, C = CX,satisfies the following interpolation conditions

`Tj H(µj) = `Tj H(µj), j ∈ 1, 2, . . . , k, (3.16)H(λi)ri = H(λi)ri, i ∈ 1, 2, . . . , k. (3.17)

Thus, in total 2k moments (interpolation conditions) are matched.Proof of Lemma 3.3.1 We project the linear system ΣL, with X = R ∈ Cn×k, and an arbitrarymatrix Y ∈ Cn×k (so that YTX is nonsingular). Let 1 6 i 6 n; it readily follows that

(a) Φ(λi) Bri = ei.

We make use of the following:

Φ(s)−1 = sE− A = YT (sE−A)R = YTΦ(s)−1R. (3.18)

To prove (a), we first notice that by multiplying R to the right with ei we can write

Rei = Φ(λi)Bri ⇒ Φ(λi)−1Rei = Bri ⇒ YTΦ(λi)−1Rei = YTBri.

Using the notation B = YTB and the result in (3.18), we writeΦ(λi)−1ei = Bri ⇒ Φ(λi)Bri = ei.

By multiplying the equality in (a) with C to the left, it follows that

CΦ(λi) Bri = Cei ⇒ H(λi)ri = C(Rei) = CΦ(λi)Bri ⇒ H(λi)ri = H(λi)ri.

Hence, we have shown that the right-hand conditions (3.17) are satisfied. Similarly, if YT = Oand X is arbitrary, we obtain (for all 1 6 j 6 k)

(b) `Tj C Φ(µj) = eTj .

We also make use of the following

Φ(s)−1 = sE− A = O(sE−A)X = OΦ(s)−1X. (3.19)

To prove (b), start by multiplying O to the left with eTj we can write

eTj O = `Tj CΦ(µj)⇒ eTj OΦ(µj)−1 = `Tj C⇒ eTj OΦ(µj)−1X = `Tj CX.

Using the notation C = CX and the result in (3.19), we write

eTj Φ(µj)−1 = `Tj C⇒ eTj = `Tj CΦ(µj).

53


By multiplying the equality in (b) with B to the right, it follows that

`Tj CΦ(µj) B = eTj B⇒ `Tj H(µj) = (eTj O)B = `Tj CΦ(µj)B⇒ `Tj H(µj) = `Tj H(µj).

Hence, we have shown that the left-hand conditions (3.16) are satisfied. Finally, when usingboth X = R and YT = O, it follows that the interpolation conditions (3.16) and (3.17) aresimultaneously satisfied.

3.3.1 Sylvester equations for O and RWe will formulate the results for the more general tangential interpolation problem. We aregiven the right data (λi; ri,wi), i = 1, · · · , k, and the left data (µj; `Tj ,vTj ), j = 1, · · · , q; it isassumed for simplicity that all points are distinct. The dimensions are as in (3.12), (3.13). Theright data can be written in matrix format as,

Λ = diag [λ1, · · · , λk] ∈ Ck×k, R = [r1, · · · , rk] ∈ Cm×k, W = [w1, · · · ,wk] ∈ Cp×k,

and the same for the left data, since

M = diag [µ1, · · · , µq] ∈ Cq×q, LT = [`1, · · · , `q] ∈ Cq×p, VT = [v1, · · · ,vq] ∈ Cq×m.

Proposition 3.3.1 The generalized controllability and observability matrices R and O definedby (3.14), (3.15), respectively, satisfy the following generalized Sylvester equations

AR+ B R = ERΛ, (3.20)OA + L C = MOE. (3.21)

Proof of Proposition 3.3.1 Multiplying equation (3.20) on the right with the unit vector eiwe obtain

ARi + Bri = λiERi ⇒ Ri = (λiE−A)−1Bri = Φ(λi)Bri. (3.22)

Thus the ith column of the matrix solution of equation (3.20) is indeed equal to the ith columnof the generalized controllability matrix R (as defined (3.14)).

Multiplying equation (3.21) to the left with the unit vector eTj we obtain

OjA + `Tj C = µjOjE⇔ Oj = `Tj C(µjE−A)−1 = `Tj CΦ(µj). (3.23)

Hence it follows that the jth row of the matrix solution of equation (3.21) is indeed equal to thejth row of the generalized observability matrix R (as defined (3.15)).

Corollary 3.3.1 The Sylvester equations (3.20) and (3.21) have unique solutions if the inter-polation points are chosen so that the Sylvester operators

TR = I⊗A−ΛT⊗ E, TO = AT⊗ I− ET⊗M, (3.24)

are invertible, i.e., have no zero eigenvalues. This is equivalent to the condition that the pen-cil (A,E) has no eigenvalues in common with either M or Λ. This in turn means that theinterpolation points µjqj=1 and λiki=1 have to be different than eig(A,E).

54


3.3.2 The Loewner pencilThe associated Loewner pencil consists of the Loewner and shifted Loewner matrices. TheLoewner matrix L ∈ Cq×k is defined as

L =

vT1 r1−`T1 w1µ1−λ1

· · · vT1 rk−`T1 wk

µ1−λk...

. . ....

vTq r1−`Tq w1µq−λ1

· · · vTq rk−`Tq wk

µq−λk

. (3.25)

It readily follows that the Loewner matrix can be factored as

L = −OER. (3.26)

The shifted Loewner matrix Ls ∈ Cq×k is defined as

Ls =

µ1vT1 r1−`T1 w1λ1

µ1−λ1· · · µ1vT1 rk−`T1 wkλk

µ1−λk...

. . ....

µqvTq r1−`Tq w1λ1µq−λ1

· · · µqvTq rk−`Tq wkλkµq−λk

, (3.27)

and can be factored in terms of the generalized controllability/observability matrices as

Ls = −OAR. (3.28)

Note also that the following relations hold:

V = CR, W = OB. (3.29)

It is straightforward to check that the Loewner matrix L and the shifted Loewner matrix Lssatisfy the Sylvester equations (mentioned also in [12])

LΛ−ML = VR − LW, (3.30)LsΛ−MLs = MVR − LWΛ. (3.31)

3.3.3 Construction of interpolants (models)We will distinguish two cases namely, the right amount of data and the more realistic redundantamount of data cases.

Lemma 3.3.2 Right amount of data; assume that k = q, and let (Ls, L), be a regular pencil,such that none of the interpolation points λi, µj are its eigenvalues. Then E = −L, A =−Ls, B = V, C = W, is a minimal realization of an interpolant of the data, i.e., the rationalfunction H(s) = W(Ls − sL)−1V, interpolates the data.

55


If the pencil (Ls, L) is singular we are dealing with the case of redundant data. In this case, thefollowing assumption is made

rank (xL− Ls) = rank[LLs

]= rank [L Ls] = r 6 k, (3.32)

for all x ∈ λi ∪ µj, we consider the following SVD factorizations:

[L Ls] = Y1Σ1XT1 ,

[LLs

]= Y2Σ2XT

2 , (3.33)

where Σ1 ∈ Rk×2k, Σ2 ∈ Rk×k, Y1, X2 ∈ Ck×k for j ∈ 1, 2. By selecting the first r columnsof the matrices Y1 and X2, we come up with projection matrices Y,X ∈ Ck×r. The followingresult holds in general.

Lemma 3.3.3 A rth order realization (E, A, B, C) of an approximate interpolant is given by

E = −YTLX, A = −YTLsX, B = YTV, C = WX. (3.34)

Thus, if we have more data than necessary, we can consider (−Ls, −L, V, W), as a sin-gular exact model of the data, or (E, A, B, C) (previously defined in (3.34)) as a nonsingularapproximated model of the data. An appropriate projection hence yields a reduced system oforder r (as presented in more details in [91, 6]).

As a consequence, the Loewner framework provides a trade-off between accuracy and com-plexity of the reduced-order system, by means of the singular values of the Loewner matrices Land Ls. A recently obtained error bound [12], shows that for the linear case, the 2-norm of theinterpolation error is proportional to the first neglected singular value of L.

3.3.4 Numerical examplesA small RLC network

Consider the RLC circuit in Fig. 3.1. The following equations are derived from basic circuitproperties

x1 = Lx2, u = yR + x1 ⇒ u = RCx1 +Rx2, y = Cx1 + x2.

Write the differential state equations and the input-output equation. Consequently, write a

Figure 3.1: RLC circuit.

56


state-space representation of the underlying linear system that describes its dynamics.

x1 = − 1

RCx1 − 1

Cx2 + 1

RCu

x2 = 1Lx2

y = − 1Rx1 + 1

Ru

⇒

1 00 1

︸︷︷︸

E

x = − 1

RC− 1C

1L

0

︸︷︷︸

A

x + 1

RC

0

︸︷︷︸

B

u

y =[− 1R

0]

︸︷︷︸C

x + 1R︸︷︷︸D

u

Note that the realization (C, E, A, B, D) is equivalent to the new realization (C,E,A,B, 0) (witha zero D term), that is constructed below

Ex = Ax + Buy = CX + Du

⇔

E 00 0

︸︷︷︸

E

x = A 0

0 −D

︸︷︷︸

A

x + B

1

︸︷︷︸

B

u

y =[

C 1]

︸︷︷︸C

x

By choosing R = L = C = 1 (for simplicity), the new system matrices can be written as

C =[−1 0 1

], E =

1 0 00 1 00 0 0

, A =

−1 −1 01 0 00 0 −1

, B =

101

.Note that the two transfer functions (H(s) = C(sE−A)−1B and H(s) = C(sE− A)−1B + D )are the same for the realizations above, i.e., H(s) = H(s) = s2+1

s2+s+1 , ∀ s ∈ C..

The task for this exercise is to recover this transfer function using solely its samples (collectionof pairs of sampling nodes and points). Hence the given data set consists of the pairs as follows

S =(

1, 23

),(j, 0

),(− j, 0

),(− 1, 2

),(

2j, 9 + 6j13

),(− 2j, 9− 6j

13

).

After partitioning, identifyµ1 = 1, µ2 = j, µ3 = −j, λ1 = −1, λ2 = 2j, λ3 = −2jv1 = 2

3 , v2 = 0, v3 = 0, w1 = 2, w2 = 9+6j13 , w3 = 9−6j

13.

By putting together the data, we form the following Loewner matrix as,

L =

−2

3739 −

4 j39

739 + 4 j

39

−1 + j 613 −

9 i13

213 + 3 j

13

−1− j 213 −

3 j13

613 + 9 j

13

.

57


Next, transform the complex Loewner matrix above and the other system matrices, as suggestedin [12]

E = −ZLZT , A = −ZLsZT , B = ZV, C = WZT ,

where Z = blkdiag(1,Z1) ∈ C3×3, and Z1 =√

22

[1 1j −j

]. Thus, we explicitly compute the

real-valued matrices

C =[

2 9√

213

6√

213

], E =

23 −7

√2

394√

239√

2 − 813

1213√

2 − 613 − 4

13

, A =

−4

3 −34√

239 −14

√2

39

−√

2 −2413 −16

13

−√

2 813 −12

13

, B =

23

00

.

Note that the transfer function of the recovered system characterized by (C, E, A, B, D) is thesame as the original transfer function: H(s) = s2+1

s2+s+1 = H(s).

The MNA model

For the next experiment, we are going to analyze a multi-port circuit with voltage sources from[40]. The Modified Nodal Analysis (MNA) equations are written asExn(t) = Axn + Bup,

ip = Cxn.

The port currents and voltages are denoted with ip and up, respectively, and

A =[−R −GGT 0

], E =

[L 00 H

], x =

[vi

],

where v and i are the MNA variables corresponding to the node voltages, inductor and voltagesource currents, respectively. The conductance and susceptance matrices are denoted with Aand E, respectively, while −R , L and H are the resistors, capacitors and inductors matrices,respectively. The entries of G are only 1, -1 and 0 and since the original multi-port is composedof passive linear elements only, L,H and −R are symmetric non-negative definite matrices. Alsonote that B = CT in this context and the E matrix is also symmetric and non-negative definite.

The original system is of order n = 980 and has m = 4 inputs and p = 4 outputs.First choose 200 logarithmically spaced interpolation nodes si’s in the range (3 · 109, 2 · 1010)j

rad/sec. Then sample the transfer function H(s) = C(sE −A)−1B ∈ C4×4 at these nodes (inthis way we come up with the interpolation points).

By assuming that the data set is given, i.e., S = (si, hi)|i ∈ 1, 2, . . . , k, we would like torecover a reduced order model that approximates the response of the original large scale-model.The plot in Fig. 3.2 depicts the decay of the singular values of the Loewner matrix. Note that,by truncating at order r = 56, we attain accuracy of approximately 10−11.

Next compare the frequency response of the original 980th order system with the one of thereduced 56th order system. In Fig. 3.3, the (1, 1) entries of the original transfer function H(s)

58


20 40 60 80 100 120 140 160 180 200

10−10

10−5

Singular values of the Loewner matrix

Figure 3.2: Singular value decay of the Loewner matrix.

109

1010

1011

10−4

10−3

10−2

10−1

100

Magnitude

Frequency (rad/sec)

Frequency response: H(1, 1) and H(1, 1)

Original modelLoewner modelOriginal samplesApproximated samples

Figure 3.3: Frequency response comparison − first frequency band choice.

and the reduced one H(s) are depicted over the range (109, 1011)j. Note that the response is wellapproximated in the sampling range but the matching is poor for higher frequencies.

Now, by changing the sampling interval, and choosing sampling points in a higher frequencyband (i.e (2 · 1010, 6 · 1010)j rad/sec), notice that the overall approximation is improved (on theentire frequency band of interest) as it can be seen in Fig. 3.4.

109

1010

1011

10−4

10−3

10−2

10−1

100

Magnitude

Frequency (rad/sec)


Original modelLoewner modelOriginal samplesApproximated samples

Figure 3.4: Frequency response comparison − second frequency band choice.

The BEAM emodel

We analyze a clamped beam model obtained by spatial discretization of an appropriate partialdifferential equation (see [40]). The input of the system is the force applied to the structure at

59


the free end, and the output is the resulting displacement. The original SISO system is of ordern = 348. As before, proceed with choosing logarithmically spaced interpolation nodes (the si’s).This time select 50 nodes inside (10−2, 2)j rad/sec. The plot in Fig. 3.5 depicts the decay ofthe singular values of the Loewner matrix. Note that, by truncating at order r = 11, we attainaccuracy of approximately 10−6.

5 10 15 20 25 30 35 40 45 50

10−15

10−10

10−5

100



Next compare the frequency response of the original 348th order system with the one of thereduced order system. The response is well approximated inside the sampling range (in Fig. 3.6).

10−2

10−1

100

101

100

101

102

103

Magnitude

Frequency (ω)

Frequency response

Original modelLoewner reduced modelOriginal samplesApproximated samples

Figure 3.6: Frequency response.

In Fig. 3.7, the frequency response of the error system is depicted. Note that, initially, therelative error stagnates at around 10−8 for low frequencies, and then gradually increases outsidewhen turning outside the sampling range (for higher frequencies).

10−2

10−1

100

10110

−10

10−8

10−6

10−4

10−2

100

102

Magnitude

Frequency (ω)

Frequency response of the error system

Figure 3.7: Error analysis in frequency domain.

60

3.4 Enforcing stability in the Loewner framework

The CD-player model

Finally, we look at the CD-player model from [40]. The mechanism treated here consists of aswing arm on which a lens is mounted by means of two horizontal leaf springs. The rotation of thearm in the horizontal plane enables reading of the spiral-shaped disc-tracks, and the suspendedlens is used to focus the spot on the disc. The original system is of order n = 120 and has m = 2inputs and p = 2 outputs. Start by selecting 100 logarithmically spaced interpolation nodesinside (101, 105)j rad/sec.

The plot in Fig. 3.8 depicts the decay of the singular values of the Loewner matrix. Notethat, by truncating at order r = 24, we attain accuracy of approximately 10−9.

0 10 20 30 40 50 60 70 80 90 100

10−15

10−10

10−5

100



Next compare the frequency response of the original 120th order system with the one of thereduced 24th order system. In Fig. 3.9, the (1, 2) entries of the original transfer function H(s)and the reduced one H(s) are depicted over the range (100, 106)j.

100

101

102

103

104

105

106

10−6

10−4

10−2

100

Magnitude

Frequency (rad/sec)


Original modelLoewner reduced modelOriginal samplesApproximated samples

Figure 3.9: Frequency response comparison.

For each input/output pairing (four in total), note that the response is well approximated inthe sampling range as can be observed in Fig. 3.10.

3.4 Enforcing stability in the Loewner frameworkIn a nutshell, the Loewner methodology is an interpolatory model order reduction techniquewhich uses measured or computed data (e.g. measurements of the frequency response of a to-be

61


100

102

104

106

100

Magnitude

Frequency (rad/sec)


100

102

104

106

100

Magnitude

Frequency (rad/sec)


100

102

104

106

100

Magnitude

Frequency (rad/sec)


100

102

104

106

100

Magnitude

Frequency (rad/sec)


Figure 3.10: Frequency response for each input/output combination.

approximated system) instead of the system matrices, and constructs reduced models based ona rank revealing factorization of appropriately constructed matrices.

In some cases the resulting reduced systems may not be stable. This issue is addressed bydeveloping reliable and robust post processing methods to yield a stable reduced model.

Hence the following problem: given an unstable descriptor system with transfer function H,find a stable descriptor system whose transfer function is a optimal (or sub-optimal) approxima-tion of H in the spaces RH2 and RH∞ of real rational functions with bounded H2 and H∞ norms(the so-called (AP2) and (AP∞) problems). For standard systems, this approximation problemwas already approached; for more details: [63],[67]. For an extensive treatment of the problemin hand we refer the reader to [80].

The methods that are going to be used throughout this section were recently developed in [81].There, the so called ε-all pass dilation method that is related to the more general optimal Hankelnorm approximation (see [4],[57]), is generalized to descriptor systems. An explicit algorithmbased on such a dilation is given. The input is a stable system and the output is an antistablesystem so that the H∞ norm of the error system is not higher than the dilation constant.Moreover, the procedure does not require computing a balancing transformation (as in [63]).Note also that all methods and algorithms are susceptible to interchange of the stable andantistable properties of the original system.

3.4.1 Set-upIn the following we will deal with linear systems whose dynamics are described in generalizedstate by the following equations,

Σ :

Ex(t) = Ax(t) + Bu(t),y(t) = Cx(t) + Du(t),

(3.35)

where E,A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, D ∈ Rp×m, u(t) ∈ Rm, y(t) ∈ Rp, x(t) ∈ Rn.Denote the class of all these linear systems with Σn,p,m. We seek reduced systems of the form

Σ :

E ˙x(t) = Ax(t) + Bu(t),y(t) = Cx(t) + Du(t),

(3.36)

62


where E, A ∈ Rk×k, B ∈ Rk×m, C ∈ Rp×k, x(t) ∈ Rk. The number of inputs and outputs mand p, respectively, remain the same, while k n. Hence Σ ∈ Σk,p,m.

Definition 3.4.1 Consider the following classes of descriptor linear systems for m,n, p ∈ N

Σn,p,m = (E,A,B,C,D)|E,A ∈ Rn×n, B ∈ Rn×m,C ∈ Rp×n, D ∈ Rp×m,Σ0n,p,m = (E,A,B,C,D) ∈ Σn,p,m|ρ(E,A) ∩ iR = ∅,

Σ+n,p,m = (E,A,B,C,D) ∈ Σn,p,m|ρ(E,A) ⊂ C+,

Σ−n,p,m = (E,A,B,C,D) ∈ Σn,p,m|ρ(E,A) ⊂ C−,

where Σ+n,p,m and Σ−n,p,m are the sets of the stable and antistable systems, respectively. Moreover,

the E matrix must be regular for the class of stable systems.

Definition 3.4.2 Let Σ = (E,A,B,C,D) ∈ Σn,p,m. Then HΣ : C → Cp×m with HΣ(s) =C(sE−A)−1B + D is the transfer function of Σ. Denote the set of all p×m transfer functionswith R(s)p×m = HΣ|Σ ∈ ∪n∈NΣn,p,m.

Definition 3.4.3 For i ∈ 1, 2 and ni ∈ N, let Σi = (Ei,Ai,Bi,Ci,Di) ∈ Σni,p,m and

Σ1 ⊕Σ2 =( [ E1 0

0 E2

],

[A1 00 A2

],

[B1B2

],[

C1 C2],D1 + D2

).

Then introduce the following notation HΣ1⊕Σ2 = HΣ1 + HΣ2.

Lemma 3.4.1 Let Σ ∈ Σ0n,p,m. Then we can always find two systems Σ+ = (E+,A+,B+,C+,D)

∈ Σ+n+,p,m , Σ− = (E−,A−,B−,C−, 0) ∈ Σ−n−,p,m so that Σ ∼ (Σ+ ⊕Σ−).

Proposition 3.4.1 Consider a linear stable system Σ = (E,A,B,C,D) ∈ Σ−n,p,m. Then thereexist unique and symmetric matrices P ,Q ∈ Rn×n such that

APET + EPAT + BBT = 0, ATQE + ETQA + CTC = 0.

3.4.2 Post processing methodThe following approximation problem (APq) is of interest:

Definition 3.4.4 Let q ∈ 2,∞ and Σ ∈ Σ0n,p,m. We are interested in finding Σ ∈ ∪n∈NΣ+

n,p,m

so that‖HΣ −HΣ‖q = inf

Σ∈∪n∈NΣ+n,p,m

‖HΣ −HΣ‖q. (3.37)

First, we show that solving (APq) for a descriptor system we can equivalently solve (APq) for itsantistable part. The following result was proposed in [81] (Theorem 3.1).Lemma 3.4.2 Let Σ ∈ Σ0

n,p,m. Let q ∈ 2,∞ and γ > 0. Then the following two are equivalent

i).∃ Σ ∈ ∪n∈NΣ+n,p,m : ‖HΣ −HΣ‖ > γ,

ii).∃ Σ ∈ ∪n∈NΣ+n,p,m : ‖HΣ− −HΣ‖ > γ.

If Σ satisfies ii) then Σ⊕Σ+ satisfies i).

63


The optimal approximation in the RH2 space is given by the following result from [81].

Lemma 3.4.3 Let Σ ∈ Σ0n,p,m - then Σ+ solves (AP2) i.e.,

infΣ∈∪n∈NΣ+

n,p,m

‖HΣ −HΣ‖2 = ‖HΣ −HΣ+‖2 = ‖HΣ−‖2. (3.38)

When switching to RH∞, the problem becomes more complex. For a minimal antistable standardsystem S the problem (AP∞), also known as a Nehari problem, was already solved in [63]. Thefollowing more general result from [81] is going to be used in the experiments section

Theorem 3.4.1 Let Σ = (E,A,B,C,D) ∈ Σ−n,p,m and Σ1 be the largest singular value of thissystem. Take γ > 0 so that γ > Σ1 and also consider the P and Q Gramians as defined inProposition 3.4.1. Compute the following transformed matrices

RΣ,γ = QEPET − γ2I, EΣ,γ = ETRΣ,γ, BΣ,γ = ETQB, (3.39)

CΣ,γ = CPET , AΣ,γ = −ATRΣ,γ −CTCΣ,γ. (3.40)

If (EΣ,γ,AΣ,γ) is regular (for γ > Σ1), then Σγ := (EΣ,γ, AΣ,γ,BΣ,γ,CΣ,γ,DΣ,γ) ∈ Σ+n,p,m and

the inequality holds: Σ1 6 ‖HΣ −HΣγ‖∞ 6 γ.

The main result is presented in the following theorem which was first stated in [81], i.e.,

Theorem 3.4.2 Consider Σ ∈ Σ0n,p,m which can be decomposed as stated in Lemma 3.4.1 into

Σ− and Σ+. Let Σ1 the largest singular value of Σ−; then infΣ∈∪n∈NΣ+n,p,m‖HΣ −HΣ‖∞ = Σ1

follows. Let r = rank(AΣ−,Σ1) and V,W ∈ Rn−×r such that WTAΣ−,Σ1V is regular. Then(AP∞) is solved by Σ+ ⊕ (WTΣ−Σ1

V).

This theorem states that, when the dilation coefficient γ is chosen as the value of the largestsingular value Σ1, we are thus able to construct an optimal approximate system in RH∞. Whenincreasing γ beyond the threshold Σ1, sub-optimal systems are instead computed.

As it is shown in the next section, in some cases, these kind of systems are preferred to theoptimal one for practical reasons. For instance, the transfer function of reduced sub-optimalsystems match the initial transfer function at ∞ - this property is not usually satisfied whenγ = Σ1 (as stated in [81]). Also, note that the solution of (AP2) is necessary to compute thesolution of (AP∞) so no further computation is needed.

3.4.3 ExamplesWe will illustrate the performance of four post-processing methods by means of three numericalexperiments. In order to reduce the original systems, we apply the Loewner method. Thecomputed reduced systems are unstable. We use the following notations:

1. ΣLo = Σ− ⊕Σ+ is the Loewner model.

2. Σ− = optimal stable approx. of ΣLo in RH2.

3. Σopt = optimal stable approx. of ΣLo in RH∞.

64


4. Σγsopt = sub-optimal stable approx. of ΣLo in RH∞.

5. Σflp = flipping the antistable poles approx. of ΣLo.

Small artificial example

Assume that the Loewner model transfer function is given

HLo(s) = 1s+ 3︸︷︷︸H−(s)

+ 2s− 2 −

1s− 1︸︷︷︸

H+(s)

.

Note that, in this case, both singular values of the antistable part are equal (i.e σ1 = σ2 = 16).

1. H−(s) = 1s+3 , ‖ΣLo −Σ−‖H∞ = 1

3 ,

2. Hopt(s) = 1s+3 −

16 , ‖ΣLo −Σopt‖H∞ = 1

6 ,

3. Hflp(s) = 1s+3 + 2

s+2 −1s+1 , ‖ΣLo −Σflp‖H∞ = 2

3 ,

4. Hsopt(s) = 1s+3 −

s6(8s2+25s+16) , ‖ΣLo −Σsopt‖H∞ = 49

150 , γ = 76 ,

5. Hcsopt(s) = 1

s+3 −s

(36c2+12c)s2+(108c2+36c+6)s+(72c2+24c) .

The sub-optimal parametrized approximation Hcsopt is written in terms of the free parameter

c = γ − σ1. By substituting c = 1 into Hcsopt(s) we obtain exactly the formula for Hsopt(s)) in 4.

PEEC example

For the second experiment, consider a model that arises from a partial element equivalent circuit(PEEC) of a patch antenna structure (see [40] for further details). It contains 2100 capacitances,172 inductances and 6990 mutual inductances, the circuit can be realized as stable descriptorsystem of dimension n = 480 with one input and one output,i.e., m = p = 1.

The original pencil (A,E) has an infinite eigenvalue. We choose 200 sampling points whichare used as interpolation points to model the system via the Loewner method are logarithmicallyspaced in the range (2,10) rad/sec.The following plot depicts the decay of the singular values of the Loewner matrix. We decideto use truncation order k = 64. In this case, the Loewner model has two unstable poles at1.6265 · 10−3 ± 7.6035i. Table 3.1 contains the computed H∞ norm of all error systems.

‖ΣLo −Σopt‖H∞ ‖ΣLo −Σγsopt‖H∞ ‖ΣLo −Σ−‖H∞ ‖ΣLo −Σflp‖H∞

1.1727 · 10−3 1.8136 · 10−3 2.3392 · 10−3 3.9029 · 10−2

Table 3.1: H∞ norm of the error systems

In Table 3.2 we compute the H2 norm of all error systems (by excluding the optimal approximantwhich has infinite norm)

65


0 20 40 60 80 100 120 140 160 180 20010

−15

10−10

10−5

100


Figure 3.11: Singular values of the Loewner matrix.

‖Σ−Σ−‖H2 ‖Σ−Σγsopt‖H2 ‖Σ−Σflp‖H2

4.4997 · 10−3 4.5017 · 10−3 4.5007 · 10−3

Table 3.2: H2 norm of the error systems

For the suboptimal H∞ stable system Σγsopt we used the offset γ = Σ1 + 10−3 where Σ1 =

‖ΣLo − Σopt‖H∞ = 1.1727 · 10−3. Fig. 3.12 depicts the frequency response of all the reducedsystems that were computed and of the original system. Next vary the upset γ − Σ1 within(10−6, 100) and compare the H∞ norm of the error systems,From Fig. 3.13 one can easily verify that:

limγ→Σ1

‖ΣLo −Σγsopt‖H∞ = ‖ΣLo −Σopt‖H∞ = Σ1,

limγ→∞‖ΣLo −Σγ

sopt‖H∞ = ‖ΣLo −Σ−‖H∞ .

Filter example

For the third experiment we consider real life measurements from an industry application (an all-pass filter). We are given a vector of frequencies as interpolation points (in the range 6.2204 ·1010

to 1.1310 · 1011 rad/sec) and a collection of 1001 2 × 2 S-parameters. The goal is to model andreduce the underlying system (which has two inputs and two outputs) via the Loewner method.The following plot depicts the decay of the singular values of the Loewner matrix. It offers agood estimate of where one should truncate to obtain certain accuracy. In Fig. 3.15 we varied thereduction order from 38 to 110 and we recorded the number of antistable poles for each individualmodel. Table 3.3 contains the H∞ norm of the error systems for the various truncation levels,i.e., k ∈ 68, 86, 107.

Table 3.3: H∞ norm of the error systems

k ‖ΣLo −Σopt‖H∞ ‖ΣLo −Σγsopt‖H∞ ‖ΣLo −Σ−‖H∞ ‖ΣLo −Σflp‖H∞

68 4.5854 · 10−4 8.3481 · 10−4 9.1740 · 10−4 1.7237 · 10−3

86 2.0236 · 10−3 2.7951 · 10−3 4.0467 · 10−3 6.6754 · 10−3

107 2.6945 · 10−3 3.5174 · 10−3 5.3854 · 10−3 6.9719 · 10−3

66


100

101

102

108

106

104

102

Magnitude

Frequency( )

H(i )

Original:Loewner: Lo

Optimal H : opt

Subptimal H : sopt

Optimal H2:Flipped: f lp

Figure 3.12: Frequency response for different methods.

106

105

104

103

102

101

102.9

102.8

102.7

102.6

102.5

102.4

γ −σ

∞ norm of the error systems

Σ −Σopt H∞

Σ −Σ− H∞

Σ −Σγsopt H∞

Σ −Σ f lp H∞

Figure 3.13: Variation of the H∞ norm for different parameters γ.

In Fig. 3.16 the two singular values of the frequency response corresponding to the Loewner modeland the optimal H∞ system of it are compared to the singular values of the original samples.Notice that the Loewner model is well approximating the given samples but the Σopt deviatesfrom the original path. Similarly, in Fig. 3.17 we again compare the singular values but this timefor a sub-optimal H∞ system (γ = Σ1 + 10−3 where Σ1 = ‖ΣLo −Σopt‖H∞ = 2.0236 · 10−3). Inthis case notice that the post-processing approximation faithfully follows the original path.

3.4.4 SummaryOne way to ensure the stability of the reduced model (that could be used for further tasks suchas simulation or control) is to apply a reliable post processing method. In this section, we testedvarious such methods for enforcing stability of the reduced order systems that were build via theLoewner framework.

We have compared several methods that yield stable systems which are close to the initialunstable Loewner model. The performance of the methods was measured in terms of the H∞and H2 norm of the error systems. The poorest approximation is obtained when flipping theunstable poles over the jω axis (an approach which is widely used in the literature). Althoughthe optimal RH∞ method gives the lowest H∞ error norm, we observed that, in some cases,a better approximation is obtained for sub-optimal models (which might be more suitable forpractical purposes). The examples covered in section 3.4.3 verify the theoretical concepts andoffer valuable insights for further development of the Loewner framework.

67


100 200 300 400 500 600 700 800 900 1000

10−8

10−6

10−4

10−2

100


Figure 3.14: Singular values of the Loewner matrix.

40 50 60 70 80 90 100 110

5

10

15

20

25

Number of antistable poles for different reduction orders

Reduction order(k)

Figure 3.15: Variation of the number of antistable poles for different k.

1011

100

100.001

Magnitude

Frequency( )

1(H(i ))

Original samplesLoewner: Lo

Optimal H : opt

1011

10−0.001

100

Magnitude

Frequency( )

2(H(i ))

Figure 3.16: Frequency response comparison when using the optimal method.

1011

100.0001

100.0005

Magnitude

Frequency( )

1(H(i ))

Original samplesLoewner: Lo

Suboptimal H : sopt

1011

100.0005

100

Magnitude

Frequency( )

2(H(i ))

Figure 3.17: Frequency response comparison when using the sub-optimal method.

68

Chapter 4

The Loewner Framework for BilinearSystems

4.1 IntroductionIn this chapter, we focus on extending the Loewner framework to the reduction of bilinear controlsystems. The structure of the chapter is presented below.

A brief introduction as well as a historical account of the contributions to the reduction ofbilinear systems are included in Section 4.1. A short background on bilinear systems follows inSection 4.2. In Section 4.3 we introduce the generalization of the Loewner framework to bilinearsystems. We show how to come up with system matrices solely from samples taken from associ-ated transfer functions. The construction of reduced-order systems, interpolation of complex dataand other issues are addressed here. Also, we present a new procedure for constructing reducedparametrized bilinear models from one-sided interpolation conditions. The multipoint Volterraseries interpolation framework is discussed in Section 4.4. Then the theoretical discussions areillustrated in Section 4.5 via three numerical examples.

4.1.1 Literature overviewModel order reduction (MOR) of linear systems has been treated extensively in recent years. Werefer the reader to the books [4, 9] and the survey [18]. Here is a brief historical account of somecontributions for reducing bilinear dynamical systems by means of Krylov methods.• Phillips gave one of the first ways to apply projection based methods for reducing weakly

nonlinear (including bilinear) systems [101].• Bai, Condon, and collaborators introduced MOR of bilinear systems by expanding the

transfer functions at zero (interpolation problem with multiplicities) [14, 47, 45, 46].• Breiten and Damm made a big step forward by extending model order reduction for bilinear

systems to the case of expansion points different from zero (this also resulted in interpolationwith multiplicities) [35].• Breiten and Benner extended the results above to include interpolation at points different

from zero [23, 33, 22]. In [20], the BIRKA (bilinear iterative rational Krylov algorithm) approachis introduced, which provides reduced models satisfying an H2 optimality criterion.

69

4.2 Bilinear systems

• Flagg and Gugercin [56] provide a novel approach to BIRKA by interpolating infinite sumsof transfer functions. This method is known as Volterra series interpolation (see also [1]).• In addition, balanced truncation related methods for the reduction of bilinear systems have

been developed as well. For details we refer the reader to [66, 24, 37].• Also, several tries were made towards adapting moment matching Krylov-based methods

to reducing MIMO bilinear systems (see [88, 111]).The Krylov-related contributions described above have the shortcoming that no trade-off

between accuracy of fit and complexity for the reduced model is available; in other words, it isnot clear how the reduced model complexity should be chosen to achieve a given accuracy.

The above shortcoming was first addressed in the dissertation [76] using as a main tool theLoewner framework. These considerations on the reduction of bilinear systems will be presentedbelow, together with new results and insights.

4.2 Bilinear systemsWe analyze bilinear control systems ΣB = (C,E,A,N,B) characterized by the following equa-tions:

ΣB : Ex(t) = Ax(t) + Nx(t)u(t) + Bu(t), y(t) = Cx(t), (4.1)

where E, A, N ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and x ∈ Rn, u, y ∈ R. In this chapter wemainly concentrate on reducing systems with nonsingular E. Also, for simplicity of exposition,we will treat the single-input, single-output (SISO) case. The multi-input case is technically moreinvolved, but is based on the same ideas. In this regard, we refer the reader to [35, 20, 33, 22, 56].Bilinear systems are equivalent to the infinite set of systems of the form:

Ex1(t) = Ax1(t) + Bu(t),Ex2(t) = Ax2(t) + Nx1(t)u(t),

...

Exi(t) = Axi(t) + Nxi−1(t)u(t),...

(4.2)

The external representation of the bilinear system ΣB can be expressed in terms of Volterraseries, which describe the nonlinear mapping of admissible inputs u to outputs y ([103, 56]).The solution of (4.1) is x(t) = ∑∞

i=1 xi(t). Moreover, considering xi−1(t) in the ith equation asa pseudo-input, the frequency-domain behavior is described by a series of generalized transferfunctions obtained by taking the multivariate Laplace transform of the degree ` regular kernel(as defined in [103, 56]):

H`(s1, s2, . . . , s`) = C Φ(s1) N Φ(s2) N · · · N Φ(s`) B, l = 1, 2, . . . . (4.3)

where the resolvent of the pencil (A,E) is denoted by

Φ(x) = (xE−A)−1 . (4.4)

70

4.3 The main procedure for extending the Loewner framework from linear to bilinear systems

For the explicit derivation of these types of transfer functions (which is based on the so-calledVolterra series representation) we refer the readers to [103].The characterization of bilinear systems by means of rational functions suggests that reduction ofsuch systems can be performed by means of interpolatory methods, with the Loewner frameworkbeing a prime candidate for achieving this reduction.

4.3 The main procedure for extending the Loewnerframework from linear to bilinear systems

As already stated our purpose is the extension of the Loewner framework to bilinear systems.This section presents the theoretical foundations of this approach while Section 4.5 providesnumerical simulations illustrating the theory.

4.3.1 The generalized controllability and observability matricesConsider a bilinear system ΣB = (C,E,A,N,B) as introduced in (4.1).

Definition 4.3.1 We define the nested right multi-tuples and the nested left multi-tuples

λ =λ(1),λ(2), . . . ,λ(k†)

, µ =

µ(1),µ(2), . . . ,µ(q†)

, (4.5)

composed of the right ith tuples and the left jth tuples:

λ(i) =

λ(i)1 ,

λ(i)2 , λ

(i)1 ,

...

λ(i)mi−1, . . . , λ

(i)2 , λ

(i)1 ,

λ(i)mi, λ

(i)mi−1, . . . , λ

(i)2 , λ

(i)1 ,

, µ(j) =

µ(j)1 ,

µ(j)1 , µ

(j)2 ,

...

µ(j)1 , µ

(j)2 , . . . , µ

(j)pj−1,

µ(j)1 , µ

(j)2 , . . . , µ

(j)pj−1, µ

(j)pj,

(4.6)

where λ(i)mi

, µ(j)pj∈ C and m1 + · · · +mk† = k, p1 + · · · + pq† = q. Here we denote with k† and

q† the number of ”ladder” structures corresponding to the right and left multi-tuples respectively.

Note that these indices satisfy a nestedness property, namely, each row in λ(i) (µ(j)) is containedin the subsequent one. To these tuples the following matrices are associated:

R(i) =[Φ(λ(i)

1 ) B, Φ(λ(i)2 ) N Φ(λ(i)

1 ) B, . . . , Φ(λ(i)mi

) N · · ·N Φ(λ(i)2 ) N Φ(λ(i)

1 ) B],

for i = 1, . . . , k† where R(i) ∈ Cn×mi is attached to λ(i). The matrix

R =[R(1), R(2), . . . , R(k†)

]∈ Cn×k, (4.7)

71


is defined as the generalized controllability matrix of the bilinear system ΣB, associated with theright multi-tuple λ. Similarly, to the left tuple we associate the matrices

O(j) =

C Φ(µ(j)

1 )C Φ(µ(j)

1 ) N Φ(µ(j)2 )

...

C Φ(µ(j)1 ) N Φ(µ(j)

2 ) N · · · N Φ(µ(j)pj

)

∈ Cpj×n, j = 1, . . . , q†,

and the generalized observability matrix:

O =

O(1)

...

O(q†)

∈ Cq×n. (4.8)

Example 4.3.1 For instance, by taking q = 5, q† = 2 and p1 = 2, p2 = 3, the corresponding Omatrix turns out to be:

O =

C Φ(µ(1)1 )

C Φ(µ(1)1 ) N Φ(µ(1)

2 )C Φ(µ(2)

1 )C Φ(µ(2)

1 ) N Φ(µ(2)2 )

C Φ(µ(2)1 ) N Φ(µ(2)

2 ) N Φ(µ(2)3 )

.

The following lemma extends the rational interpolation idea for linear systems approximation(see e.g. [4] Chapter 11.3) to the bilinear case.

Lemma 4.3.1 Interpolation of bilinear systems. Let ΣB = (C,E,A,N,B) be a bilinear systemof order n. Assume that it is projected to a kth order system by means of X = R and YT = O.The reduced system ΣB = (C, E, A, N, B), of order k, where

E = YTEX, A = YTAX, N = YTNV, B = YTB, C = CX,

satisfies the following interpolation conditions:

Hj(µ1, · · · , µj) = Hj(µ1, . . . , µj), Hi(λi, . . . , λ1) = Hi(λi, · · · , λ1), (4.9)

Hj+i(µ1, . . . , µj, λi, . . . , λ1) = Hj+i(µ1, . . . , µj, λi, . . . , λ1), i, j = 1, dots, k. (4.10)

Thus, in total 2k + k2 moments (interpolation conditions) are matched.

Proof of Lemma 4.3.1 For simplicity, assume that we have one set of right multi-tuples, andone set of left multi-tuples with the same number of interpolation points k, namely

λ = λ1, λ2, λ1, . . . , λk, . . . , λ2, λ1 ,µ = µ1, µ1, µ2, . . . , µ1, µ2, . . . , µk .

72


In this case we take k† = q† = 1 and m1 = p1 = k. It follows that the associated generalizedcontrollability and observability matrices are:

R = [ Φ(λ1)B, Φ(λ2)NΦ(λ1)B, · · · , Φ(λk)NΦ(λk−1)N · · · NΦ(λ1)B] ∈ Cn×k,

O =

CΦ(µ1)CΦ(µ1)NΦ(µ2)

...CΦ(µ1)NΦ(µ2)N · · · NΦ(µk)

∈ Ck×n.

We project the bilinear system ΣB with X = R ∈ Cn×k, and an arbitrary matrix Y ∈ Cn×k (sothat YTX is nonsingular). It readily follows that

(a) Φ(λ1) B = e1 and (b) Φ(λi) N ei−1 = ei, i = 2, . . . , k.

where ei = [0, . . . , 1, . . . , 0]T ∈ Rk is the ith unit vector. These equalities imply the left-handconditions (4.9). We make use of the following:

Φ(s)−1 = sE− A = YT (sE−A)R = YTΦ(s)−1R (4.11)

To prove (a), we first notice that by multiplying R to the right with e1 we can write

Re1 = Φ(λ1)B⇒ Φ(λ1)−1Re1 = B⇒ YTΦ(λ1)−1Re1 = YTB

Using the notation B = YTB and the result in (4.11), we write

Φ(λ1)−1Re1 = B⇒ Φ(λ1)B = e1

To prove (b), note that by multiplying R to the right with ei (i > 2) we can write

Rei = Φ(λi)N Φ(λi−1)N · · ·NΦ(λ1)B︸︷︷︸=Rei−1

⇒ Φ(λi)−1Rei = NRei−1 (4.12)

By multiplying (4.12) with YT to the left and then using the notation N = YTNR and theresult in (4.11), we have that

YTΦ(λi)−1Rei = YTNRei−1 ⇒ Φ(λi)−1ei = Nei−1 ⇒ Φ(λi)Nei−1 = ei

Similarly, if YT = O and X is arbitrary, we obtain

(c) C Φ(λ1) = eT1 and (d) eTj−1NΦ(µj) = eTj , j = 2, . . . , k,

which imply the right-hand conditions (4.9). Finally, with X = R, YT = O, and combining(a), (b), (c), (d), interpolation conditions (4.10) follow.

73


Sylvester equations for O and R

The generalized controllability and observability matrices satisfy Sylvester equations. To statethe corresponding result we need to define the following quantities. First we define the matrices

R =[eT1,m1 · · · eT1,m

k†

]∈ R1×k, LT =

[eT1,p1 · · · eT1,p

q†

]∈ R1×q, (4.13)

and the block-shift matrices

SR = blkdiag[Jm1 , . . . , Jm

k†

],

SL = blkdiag[JTp1 , . . . , JTp

q†

].

where J` =

0 1 · · · 0...

.... . .

...0 0 · · · 10 0 · · · 0

∈ R`×`. (4.14)

Here, we use the alternative notation ei,j for the unit vector ei to stress that the vector hasdimension j. Finally we arrange the interpolation points in the diagonal matrices:

M = blkdiag [M1, M2, . . . , Mq† ], Λ = blkdiag [Λ1, Λ2, . . . , Λk† ]. (4.15)

where Mj = diag [µ(j)1 , µ

(j)2 , . . . , µ(j)

pj] and Λi = diag [λ(i)

1 , λ(i)2 , . . . , λ(i)

mi] ; we used the

MATLAB notation ”blkdiag” which outputs a block diagonal matrix with each input entry as ablock. We are now ready to state the following result.

Lemma 4.3.2 The generalized controllability and observability matrices R and O defined by(6.15), (6.17), respectively, satisfy the following generalized Sylvester equations:

AR+ NRSR + B R = ERΛ, (4.16)OA + SLON + L C = MOE. (4.17)

Proof of Lemma 4.3.2 Assume again, for simplicity, that we have one set of right multi-tuples,and one set of left multi-tuples with the same number of interpolation points k - hence takeλ = λ1, λ2, λ1, . . . , λk, . . . , λ2, λ1. Multiplying equation (6.33) on the right with thefirst unit vector e1 we obtain:

AR1 + B = λ1ER1 ⇒ R1 = (λ1E−A)−1B. (4.18)

Thus the first column of the matrix which is the solution of (6.33) is indeed equal to the firstcolumn of the generalized controllability matrix R. Multiplying the same equation on the rightwith the jth unit vector ej, j = 1, . . . , k, we obtain:

ARj + NRj−1 = λjERj ⇒ Rj = (λjE−A)−1NRj−1. (4.19)

Thus the jth column of R is:

Rj = (λjE−A)−1N (λj−1E−A)−1N · · ·N(λ1E−A)−1B,

which implies theR is the generalized controllability matrix. Similarly, it follows that the solutionof (6.39) is the generalized observability matrix.

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As an observation, note that the equations introduced by 4.3.2 reduce to the ones for thelinear case stated in Proposition 3.3.1, if it is considered that N = 0.

Corollary 4.3.1 With ⊗ denoting the Kronecker product, the Sylvester equations (6.33) and(6.39) have unique solutions if the interpolation points in (6.31) are chosen so that the Sylvesteroperators

TR = I⊗A−ΛT⊗ E + STR⊗N, TO = AT⊗ I− ET⊗M + NT⊗ SL,

are invertible, i.e. have no zero eigenvalues.

4.3.2 The generalized Loewner pencilGiven the notations previously introduced, we define the following matrices.

Definition 4.3.2 Consider a bilinear system ΣB, and let R and O be the controllability andobservability matrices associated with the multi-tuples (4.3.1) and defined by (6.15), (6.17) re-spectively. The Loewner matrix L, and the shifted Loewner matrix Ls are defined as

L = −OER, Ls = −OAR . (4.20)

In addition we define the quantities

Ψ = ONR, V = OB and W = CR . (4.21)

Note that L and Ls as defined above are indeed Loewner matrices, that is, they can be expressedas divided differences of appropriate transfer function values of the underlying bilinear system.

Proposition 4.3.1 The following equalities hold:

L(j, i) =Hj+i−1(µ1, . . . , µj , λi−1, . . . , λ1)−Hj+i−1(µ1, . . . , µj−1, λi, . . . , λ1)

µj − λiLs(j, i) =µjHj+i−1(µ1, . . . , µj , λi−1, . . . , λ1)− λiHj+i−1(µ1, . . . , µj−1, λi, . . . , λ1)

µj − λi

, (4.22)

while V(j, 1) = Hj(µ1, . . . , µj−1, µj), W(1, i) = Hi(λi, λi−1, . . . , λ1), andΨ(j, i) = Hj+i(µ1, . . . , µj−1, µj, λi, λi−1, . . . , λ1).

Remark 4.3.1 (a) Selecting the data; Given the jth ordered row and the ith ordered columntuples (µ1, . . ., µj−1, µj), and, (λi, λi−1, . . ., λ1), the table in Tab. 4.1 contains all the transferfunction samples that are matched via the bilinear Loewner procedure: Some properties of thenewly introduced data table (denoted with T ) are presented in the following:

• Its (i, j) entry is Hi+j−2(µ1, . . . , µi−1, λj−1, . . . , λ1), i, j = 1, . . . , k + 1.• It has dimension (k + 1)× (k + 1) for general reduction order k.

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H1(λ1) H2(λ2, λ1) . . . Hk(λk, . . . , λ1)H1(µ1) H2(µ1, λ1) H3(µ1, λ2, λ1) . . . Hk+1(µ1, λk, . . . , λ1)

H2(µ1, µ2) H3(µ1, µ2, λ1) H4(µ1, µ2, λ2, λ1) . . . Hk+2(µ1, µ2, λk, . . . , λ1)...

......

. . ....

Hk(µ1, . . . µk) Hk+1(µ1, . . . , µk, λ1) Hk+2(µ1, . . . , µk, λ2, λ1) . . . H2k(µ1, . . . , µk, λk, . . . , λ1)

Table 4.1: Samples of transfer functions that are matched

• It contains (k + 1)2 − 1 = k2 + 2k samples of transfer functions from the underlying bilinearsystem (up to H2k).(b) Let both left and right interpolation points be singletons, i.e. µj, λi, i, j = 1, . . . , k. In thiscase L, Ls, V, W, are as in the linear case, while Ψ is composed of the k2 values H2(µj, λi),i, j = 1, . . . , k. This shows that the least amount of information needed to reconstruct theoriginal bilinear system consists of samples of H1 and H2, as expected. However, for modelreduction purposes, to increase the accuracy, it is important to match moments of higher ordertransfer functions.(c) With notation (4.4), the following identities have been used in Proposition 4.3.1:Φ(x) E Φ(y) = − (Φ(x)−Φ(y)) /(x − y), Φ(x)A Φ(y) = − (xΦ(x)− yΦ(y)) /(x − y), forx, y ∈ C.

Example 4.3.2 Given the SISO bilinear system (C,E,A,N,B), where A is n × n, considerthe ordered tuples of left and right interpolation points:

[µ1 µ1, µ2

],[λ1, λ2, λ1

].

The associated generalized observability and controllability matrices are

O =[

C(µ1E−A)−1

C(µ1E−A)−1N(µ2E−A)−1

],

R =[

(λ1E−A)−1B, (λ2E−A)−1N(λ1E−A)−1B].

The projected matrices can be written in terms of the samples in the following way:

L = H1(µ1)−H1(λ1)

µ1−λ1

H2(µ1,λ1)−H2(λ2,λ1)µ1−λ2

H2(µ1,µ2)−H2(µ1,λ1)µ2−λ1

H3(µ1,µ2,λ1)−H3(µ1,λ2,λ1)µ2−λ2

= −OER,

Ls = µ1H1(µ1)−λ1H1(λ1)

µ1−λ1

µ1H2(µ1,λ1)−λ2H2(λ2,λ1)µ1−λ2

µ2H2(µ1,µ2)−λ1H2(µ1,λ1)µ2−λ1

µ2H3(µ1,µ2,λ1)−λ2H3(µ1,λ2,λ1)µ2−λ2

= −OAR,

Ψ =[

H2(µ1, µ2) H3(µ1, λ2, λ1)H3(µ1, µ2, λ1) H4(µ1, µ2, λ2, λ1)

]= ONR,

V =[

H1(µ1)H2(µ1, µ2)

]= OB,

W =[

H1(λ1) H2(λ2, λ1)]

= CR.

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It readily follows that given the bilinear system (C,E,A,N,B) a reduced bilinear system oforder two, can be obtained without computation (matrix factorizations or solves) as:

E = OER, A = OAR, N = ONR, B = OB, C = CR.

This reduced system matches eight moments of the original system, namely:

two of H1 : H1(µ1), H1(λ1),three of H2 : H2(µ1, µ2), H2(µ1, λ1), H2(λ2, λ1),

two of H3 : H3(µ1, µ2, λ1), H3(µ1, λ2, λ1), andone of H4 : H4(µ1, µ2, λ2, λ1),

i.e. in total 2k + k2 = 8, for k = 2, moments are matched using this procedure.Alternatively, the tuples can be chosen as

[µ1 µ2

],

[λ2, λ1

], in which case

the associated generalized observability and controllability matrices are

O =[

C(µ1E−A)−1

C(µ2E−A)−1

], R =

[(λ1E−A)−1B, (λ2E−A)−1B

].

Here the associated (V,L,Ls,W) are as in the linear case (Chapter 3) while

Ψ =[

H2(µ1, λ1) H2(µ1, λ2)H2(µ2, λ1) H2(µ2, λ2)

]= ONR,

and the moments matched according to Table 4.1 are as follows: H1(µi), H1(λj), H2(µi, λj)for i, j = 1, 2.

Properties of the Loewner pencil

We will now show that the quantities defined earlier satisfy various equations which generalizethe ones in the linear case. First notice that the Table 4.1 can be divided in certain interest areasthat account for the recovered matrices V,W and Ψ. More exactly, we can write that:

V = T (:, 1), W = T (1, :), Ψ = T (2 : k + 1, 2 : k + 1),

where the MATLAB colon operator ”:” was used.

Proposition 4.3.2 Let K = T (2 : k + 1, 1 : k), J = T (1 : k, 2 : k + 1). It follows that:

K = Ls −ML = LW + SLΨ ∈ Ck×k, J = Ls − LΛ = VR + ΨSR ∈ Ck×k. (4.23)

These equations imply that the Loewner matrices L, Ls satisfy:

Ls = LΛ + VR + ΨSR, Ls = ML + LW + SLΨ, (4.24)

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as well as the Sylvester equations:

ML− LΛ = (VR + ΨSR)− (LW + SLΨ), (4.25)MLs − LsΛ = (MVR + MΨSR)− (LWΛ + SLΨΛ). (4.26)

Proof of Proposition 4.3.2 From lemma 4.3.2, multiplying (6.33) on the left with O weobtain: OAR + (ONR)SR + (OB)R = (OER)Λ. Now substituting the factorization of theunderlying system matrices and of O and R in terms of the reduced system matrices it followsthat: −Ls + ΨSR + VR = −LΛ which proves the first equation.

Similarly, since the generalized observability matrix O satisfies OA + SLON + LC = MOE,multiplying on the right with R we obtain: OAR + SL(ONR) + L(CR) = M(OER). Uponsubstitution of the underlying system matrices and of O and R in terms of the reduced systemmatrices the second equality holds. By subtracting these two equations, we can directly prove(4.25) and (4.26).

As an observation, note that the equations stated in (4.25) and (4.26) reduce to the ones forthe linear case stated in (3.30) and (3.31), if we consider Ψ = 0.

4.3.3 Construction of interpolantsAs we already noted, the interpolation data for the bilinear case is different than for the linearcase, as higher order transfer function values are matched as shown in Section 4.3.1. However,the rest of the procedure remains similar. As in the case of linear systems, the following resultholds:Lemma 4.3.3 Assume that k = q, and let (Ls, L), be a regular pencil, such that none of theinterpolation points λi, µj are its eigenvalues. Then

E = −L, A = −Ls, N = Ψ, B = V, C = W,

is a minimal realization of an interpolant of the data, i.e., the rational functions:

H`(s1, . . . ., s`) = W(Ls − s1L)−1Ψ . . .Ψ(Ls − s`L)−1V,

for ` = 1, . . . , 2k, interpolate the data in Table 4.1, where i, j = 1, . . . , k.In the case of redundant data, the pencil (Ls, L) is singular, and hence we construct projectormatrices X,Y ∈ Rk×r as in the linear case, i.e in (3.33).

Theorem 4.3.1 The quintuple (C, E, A, N, B) given by:

E = −Y∗LX, A = −Y∗LsX, N = Y∗ΨX, B = Y∗V, C = WX,

is the realization of an approximate interpolant of the data. If the truncated singular values areall 0, then the interpolant is exact.

Remark 4.3.2 Thus, as in the linear case, if we have more data than necessary, we can eitherconsider (W,−L,−Ls, Ψ,V) as an exact but singular model of the data or

(WX,−Y∗LX,−Y∗LsX, Y∗ΨX, Y∗V),

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as an approximate (nonsingular) model of the data. The use of the Drazin or Moore-Penrosepseudo inverses holds as in the linear case (see [6]).

Interpolating complex data

In this section we show how to construct real models (W, −Ls, −L, Ψ, V), from complex data,provided that the data is closed under complex conjugation (in other words, complex data isaccompanied by its complex conjugate). The following procedure is the extension of the linearcase (as presented in [12]). Consider the sets of right and left multi-tuples

λ =λ(1), λ(2), . . . , λ(k)

, µ =

µ(1), µ(2), . . . , µ(k)

,

which are nested, that is for each right tuple λ(j) =λ

(j)1 , λ

(j)2 , . . . , λ(j)

pj

, the tuple obtained by

eliminating the first entry belongs to λ as well, i.e.λ

(j)2 , . . . , λ(j)

pj

∈ λ, and similarly for left

tuples. For simplicity consider:

λ(k) = λk, λk−1, . . . , λ2, λ1 and µ(k) = µ1, µ2, . . . , µk−1, µk,

where λi, µj ∈ C, 1 6 i, j 6 k. Hence it follows that:

λ = λ1, λ2, λ1, . . . , λk, . . . , λ2, λ1 and µ = µ1, µ1, µ2, . . . , µ1, µ2 . . . , µk.

Now instead of using the left and right multi-tuples λ and µ, we use new modified multi-tuplesλ and µ that include also the complex conjugates of the singleton frequencies λi and µj:

λ =λ(1), λ(1), λ(2), λ(2), . . . , λ(k), λ(k)

, (4.27)

µ =µ(1), µ(1), µ(2), µ(2), . . . , µ(k), µ(1)

. (4.28)

where λ(k) = λk, λk−1 . . . , λ2, λ1 and µ(k) = µ1, µ2 . . . , µk−1, µk. Define a block diagonalmatrix Zk ∈ C2k×2k that is composed out of k blocks:

Zk = Ik ⊗Z1 ∈ R2k×2k, where Z1 =√

22

[1 1j −j

]. (4.29)

Given the sets of right and left nested multi-tuples and the corresponding samples of the higherorder transfer functions, a 2k bilinear Loewner model is constructed so that it matches the giveninterpolation conditions. In general this model need not be real. By assuming that the interpo-lation points come in complex conjugate pairs, the complex Loewner model can be transformedinto a real model with:

L = ZkLZ∗k , Ls = ZkLsZk, Ψ = ZkΨZ∗k ,V = ZkV, W = WZ∗k , L = ZkL, R = RZk.

(4.30)

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Proposition 4.3.3 Given the complex Loewner model (W,−L,−Ls,Ψ,V), of order 2k con-structed as in (5.48), it follows that the matrices corresponding to the transformed model:

(W,− L,− Ls, Ψ, V),

have only real entries. Furthermore the equations (4.24) hold with real entries, as:

Ls = LΛ + VR + ΨSR, Ls = M L + LW + SLΨ.

4.3.4 One-sided interpolation and parametrized reduced modelsIn this section we discuss the construction of reduced parametrized bilinear systems ΣB :C, I, A, N, B given initial data ν1, . . . , νk, i.e. computed samples of the multivariate func-tions in (4.3). We build reduced kth order bilinear models which satisfy k right interpolationconditions corresponding to the right tuples defined in (4.6)

Hj(λ(i)j , λ

(i)j−1, . . . , λ

(i)2 , λ

(i)1 ) = Hj(λ(i)

j , λ(i)j−1, . . . , λ

(i)2 , λ

(i)1 ) := ν`, (4.31)

where j ∈ 1, . . . ,mi, i ∈ 1, . . . , k†, ` ∈ 1, . . . , k and m1 +m2 + . . .+mk† = k.

Lemma 4.3.4 Given k interpolation points λ` ∈ C and k interpolation values ν` ∈ C, a kth

order reduced parametrized bilinear system ΣB : C, I, A, N, B, satisfying the k interpolationconditions in (4.31), can be constructed as follows:

A = Λ− BR − NSR, Ni,j = ni,j, Bj = bj, Ci = νi, (4.32)

for all 1 6 i, j 6 k, where the k2 +k free parameters are the entries of the B, N matrices (bj andni,j); Λ,SR, R are as previously defined in (4.14) and (6.31).

Proof of Lemma 4.3.4 Let ΣB : C, I, A, N, B be some kth order reduced bilinear systemwhich satisfies the interpolation conditions in (4.31). Now consider the generalized controllabilitybilinear matrix R ∈ Ck×k corresponding to system ΣB. It follows that R satisfies the reducedbilinear Sylvester equation: AR+ NRSR + BR = RΛ.

Since the k interpolation conditions in (4.31) are satisfied by the reduced bilinear system ΣB,it follows that CR = CR = [η1, η2, . . . , ηk]. Since R matrix is not singular, we multiply theabove equation on the left by R−1:

R−1AR+ R−1NRSR + R−1BR = Λ.

Using the notations A = R−1AR, N = R−1NR, B = R−1B, C = CR we write: A + BR +NSR = Λ ⇒ A = Λ − BR − NSR. Since the two reduced systems have the same transferfunctions, it follows that the interpolation conditions in (4.31) are satisfied by ΣB as well.

Application of the proposed parametrization

In the following examples we show that by appropriately choosing some of the parameters con-tained in the B and N matrices, it is possible to place poles and/or zeros of the first transfer

80


function H1 (we consider two choices of interpolation conditions). Then, another option is tochoose the parameters in order to fit the observability Gramian of the system. In this way, thestability of the reduced bilinear model is ensured (see [20]).

Example 4.3.3 a). Consider the right tuples: λ = (λ1), (λ2), . . . , (λk). We construct aparametrized bilinear system of order k which matches the values

H1(λ1) := ν1, H1(λ2) := ν2, . . . , H1(λk) := νk, λj ∈ C, j = 1, . . . , k.

Note that in this case, the given values exclusively correspond to the first transfer function H1.Also, since in this case we have SR = 0k and R = [1 . . . 1], it follows that H1 is written interms of k parameters (bj, j ∈ 1, . . . , k). We rewrite the first transfer function in the followingbarycentric representation

H(s1) = C(s1Ik − A)−1B =∑kj=1

νjbjs1−λj

1 +∑kj=1

bjs1−λj

.

The poles of the system are solutions to 1 + ∑kj=1

bjs1−λj = 0. Hence, to ensure that the system

has the poles ρ1, . . . , ρk, we choose the free parameters collected in B by solving k linearequations as follows: B = −C−1[1 . . . 1 ]T where the Cauchy matrix C ∈ Rn×n is defined asCj,l = 1

ρj−λl, j, l = 1, . . . , k.

b). Consider the right tuples: λ = (λ1), (λ2), . . . , (λk−1), (λk, λk−1). We construct aparametrized bilinear system of order k which matches the values

H1(λ1) := ν1, H1(λ2) := ν2, . . . ,H1(λk−1) := νk−1 H2(λk, λk−1) := νk,

where λj ∈ C, j = 1, . . . , k. Notice that, in this case, the given measurements correspondto the first two transfer functions H1 and H2. Hence, it follows that exactly 2k parameters(bj, nj,1, j ∈ 1, . . . , k) affect H1. Similarly to the previous case, we write a barycentric-likeformula for the linear transfer function

H(s1) =∑kj=1

νjbjs1−λj +∑k−1

j=1νj det(Xj)

(s1−λj)(s1−λk)

1 +∑k−1j=1

bjs1−λj + nk,1

s1−λk+∑k−1

j=1det(Xj)

(s1−λj)(s1−λk)

:= N (s1)D(s1)

,

whereXj :=

[bj nj,1bk nk,1

]⇒ det(Xj) = bjnk,1 − bknj,1.

Since the number of free parameters that enter in H1 is 2k, it follows that we can simultaneouslyplace both the poles and the zeros of the first transfer function H1. Hence we can completelydescribe the linear part of the bilinear system ΣB. To do that, given poles ρ1, . . . , ρk and zerosz1, . . . , zk, by solving a system of 2k nonlinear equations: ¯N (zi) = 0, D(ρi) = 0, i ∈ 1, . . . , k,we recover the parameters bj, nj,1, j ∈ 1, . . . , k.

The remaining k2 − k parameters (that make up for the other columns of the N matrix) canbe freely chosen to satisfy additional constraints.

81


Also notice that if we randomly fix beforehand the first column of the N matrix, we cansimilarly place the k poles by solving k linear equations (as in the first case).

Example 4.3.4 The goal is to construct a real kth order bilinear system which satisfies kgiven interpolation conditions; in addition, the system has a particular predefined observabilityGramian Q. Consider the right tuples λ = (λ1), (λ2, λ1), . . . , (λk, . . . , λ2, λ1). First, constructa parametrized bilinear system of order k which matches the values

H1(λ1) := ν1, H2(λ2, λ1) := ν2, . . . , Hk(λk, . . . , λ1) := νk, λj ∈ C, j = 1, . . . , k,

It follows that the constant matrices are defined as

Λ = diag [λ1, λ2, . . . , λk], R = eT1 , and SR = Jk (as in (4.14)).

Let X = BR + NSR be the unknown matrix which will be recovered iteratively. Notice thatX is written in terms of k2 parameters (the last column of N does not enter X and hence wefix it from the start). The observability Gramian Q satisfies the generalized Lyapunov equationATQ+QA + NTQN + CT C = 0 (see [20]).

Since A = Λ − X, rewrite the above equation as XTQ + QX = NTQN + F, where: F =ΛQ+QΛ + CT C. A solution is found by means of fixed point iteration

X(k + 1) = 12Q−1[N(k)TQN(k) + F

].

i). Given are the following matrices, where x ∈ R

Q = I4, Λ = diag [−1, x, 0, 0], C =[

1 0 0 −1].

The last column of N is taken to be[

0 0 0 0]T

and starting at X0 = 0, after four iterationswe have converged to

A = 12

−(x+ 1

32)2 − 1 0 0 10 − 1

16 0 00 0 −1 01 0 0 −1

, B = 12

(x+ 1

32)2 − 100−1

,

N =

0 0 −1

2 0x+ 1

32 0 0 00 1

4 0 00 0 1

2 0

, C =[

1 0 0 −1].

ii). Given are the following matrices

Q = 125

29 −8 2 −4−8 41 −4 8

2 −4 26 −2−4 8 −2 29

, Λ =

−1 0 0 0

0 1 0 00 0 0 00 0 0 −2

, C =[

1 0 0 −1].

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With the last column of N being e2, we start the iteration at X0 = 0 and after 58 iterations theerror ‖ Xk+1 − Xk ‖2 drops below machine precision. The proposed iteration yields the systemmatrices

A =

−1.5120 −0.2799 −0.1149 −0.4779

0.0045 −0.2494 −0.0286 −0.1298−0.1646 −0.0714 −0.5377 0.0971−0.6343 0.4674 −0.0307 −1.1613

, B =

0.5120−0.0045

0.16460.6343

,

N =

0.2799 0.1149 0.4779 01.2495 0.0286 0.1298 10.0714 0.5367 −0.0971 0−0.4674 0.0307 −0.8386 0

, and hence compute

P =

0.1575 −0.1172 −0.0006 −0.0745−0.1172 0.8748 −0.0367 0.1762−0.0006 −0.0367 0.2842 0.1143−0.0745 0.1762 0.1143 0.3852

⇒ σ2 =

1.80750.43070.20060.1259

,where the entries of σ2 are the squares of bilinear Hankel singular values of the system, in otherwords, the eigenvalues of the product of the Gramians P and Q. Notice also that the eigenvaluesof A are stable, i.e., −1.9197, −0.6947, and −0.4226± 0.1034i.Example 4.3.5 The goal is to construct a real kth order bilinear system which satisfies k giveninterpolation conditions; in addition, the system has a predefined controllability Gramian P .a). First consider the right tuples λ = (λ1), (λ2, λ1), . . . , (λk, . . . , λ2, λ1). Construct aparametrized bilinear system of order k which matches the values

H1(λ1) := ν1, H2(λ2, λ1) := ν2, . . . , Hk(λk, . . . , λ1) := νk, λj ∈ C, j = 1, . . . , k.

It follows that the constant matrices are defined as

Λ = diag [λ1, λ2, . . . , λk], R = eT1 , and SR = Jk (as in (4.14)).

Again, consider X = BR + NSR to be the unknown matrix which will be recovered iteratively.Exactly k2 free parameters are used to write X (the last column of N does not enter X andhence we fix it from the start). Next rewrite the system matrices in term of X as follows

A = Λ− X, B = Xe1, N = XJTk + L, (4.33)

where L = eTk ⊗(Nek

)is a constant matrix (the last column of N is fixed). The controllability

Gramian P satisfies the following generalized Lyapunov equation

AP + PAT + NPNT + BBT = 0.

By substituting the relation from (4.33) into this equation, it follows that,

(Λ− X)P + P(Λ− X)T + (XJTk + L)P(XJTk + L)T + XKXT = 0 ⇒X(JTkPLT − P)+(LPJk − P)XT +X(JTkPJk + K)XT +(ΛP + PΛ + LPLT ) = 0,

83


where K = e1eT1 . Denote with D = ΛP +PΛ + LPLT , F = JTkPJk + K, G = JTkPLT −P . Byrewriting the last equation in terms of the above defined matrices, it turns out that the unknownX matrix satisfies the following T-Ricatti equation

XG + GT XT + XFXT + D = 0.

One option to solve such an equation is Newton’s method. Let F : Rn×n → Rn×n be the mappingF(X) = XG + GT XT + XFXT + D. The Frechet derivative of this mapping in the Z directioncan be written as

F ′X(Z) = ZG + GTZT + XFZT + ZFXT .

Newton’s method defines Z as the solution of F(X) + F ′X(Z) = 0, and replaces X by X + Z asdescribed below. Starting at X0 = O4, we implement the following iteration

Xk+1 = Xk − (F ′Xk)−1F(Xk)⇒ F(Xk) + F ′Xk

(Xk+1 − Xk) = 0.

The ensuing iterative process is characterized by

(GT + XkF)XTk+1 + Xk+1(G + FXT

k ) = XkFXTk −D,

which is yields T-Sylvester equation in Xk+1 (for details on this kind of equations see e.g. [51]).At each step we compute the solution as

Xk+1 = 12(XkFXT

k −D)(G + FXTk )−1.

For k = 4, construct the parametrized 4th order system ΣB : (C, A, N, B), where

A = Λ− BR − NS =

λ1 − b1 −n11 −n12 −n13−b2 λ2 − n21 −n22 −n23−b3 −n31 λ3 − n32 −n33−b4 −n41 −n42 λ4 − n43

, B =

b1b2b3b4

,

C =[ν1 ν2 ν3 ν4

], N =

n11 n12 n13 n14n21 n22 n23 n24n31 n32 n33 n34n41 n42 n43 n44

.

We are now given the following interpolation conditions H(−0.5) = 1, H2(0.5,−0.5) =0, H3(0, 0.5,−0.5) = 0, H4(−1, 0, 0.5,−0.5) = −1; hence the matrices Λ and C are constructed

Λ = diag[− 0.5, 0.5, 0, − 1

], C =

[1 0 0 −1

].

84


Moreover, the following controllability Gramian is given

P = 125

29 −8 2 −4−8 41 −4 8

2 −4 26 −2−4 8 −2 29

.

We obtain the solution (after 7 iterations the error ‖ Xk+1−Xk ‖2 drops below machine precision):

X =

−0.4392 −0.1520 0.0039 0.0542

0.1066 0.7618 0.0358 −0.1583−0.0185 −0.0486 −0.0018 0.0313

0.0155 0.1775 −0.0104 −0.7735

.

Hence we recover the following system matrices

A =

−0.0608 0.1520 −0.0039 −0.0542−0.1066 −0.2618 −0.0358 0.1583

0.0185 0.0486 0.0018 −0.0313−0.0155 −0.1775 0.0104 −0.2265

, B =

−0.4392

0.1066−0.0185

0.0155

,

N =

−0.1520 0.0039 0.0542 0

0.7618 0.0358 −0.1583 0−0.0486 −0.0018 0.0313 0

0.1775 −0.0104 −0.7735 0

, C =[

1 0 0 −1].

Note that the matrix A is Hurwitz (all poles are in the complex left half plane −0.2051±0.1862i,−0.1323 and −0.0047).

b). Now consider instead the right tuples λ = (λ1), (λ2, λ1), . . . , (λ2k†−1), (λ2k† , λ2k†−1) wherek = 2k†. Construct a parametrized bilinear system of order k which matches the values

H1(λ1) := ν1, H2(λ2, λ1) := ν2, . . . , H1(λ2k†−1) := ν2k†−1, H2(λ2k† , λ2k†−1) := ν2k† ,

and λj ∈ C, j = 1, . . . , k. It follows that the constant matrices are defined as

Λ = diag [λ1, λ2, . . . , λk], R = [1Tk† 0Tk† ], and SR =[

0k† Ik†0k† 0k†

].

As before, consider X = BR+NSR to be the unknown matrix which will be recovered iteratively.Exactly (k2 + 2k)/4 free parameters are used to write X. Next rewrite the system matrices interm of X as follows:

A = Λ− X, B = B(X) = Xe1, N = N(X) = XSTR + L, (4.34)

where L = STRN is a constant matrix (we consider the last k† columns of N to be fixed). Since thecontrollability Gramian satisfies the Lyapunov equation in P : AP +PAT + NPNT + BBT = 0,

85


by substituting (4.34) onto this equation, we write

(Λ− X)P + P(Λ− X)T + N(X)PN(X)T + B(X)B(X)T = 0,

By denoting G(X) = N(X)PN(X)T + B(X)B(X)T +(ΛP+PΛ), we rewrite the above equationby means of the Kronecker product as,

(P ⊗ Ik + Ik ⊗ PΠ)Vec(X) = Vec(G(X)),

where Π is a permutation matrix. Also, one has to account for the fact that the first two columnsof X are the same. So augment the matrix M1 = P⊗Ik+Ik⊗PΠ to fit the additional conditions,

M = [P ⊗ Ik + Ik ⊗ PΠ ; Ik†2 − Ik†2 Ok†2 Ok†2 ].

Then iteratively find a solution by starting with an initial guess X0 (which is usually taken thezero matrix 0), where M+ = MT (MMT )−1 is the pseudo-inverse of M, i.e.,

Vec(Xk+1) = M+Vec(G(Xk)).

Now, for k = 4 (and k† = 2), construct the parametrized 4th order system ΣB : A, N, B, C:

A = Λ− BR − NS =

λ1 − b1 −b2 −n11 −n12−b2 λ2 − b2 −n21 −n22−b3 −b3 λ3 − n31 −n32−b4 −b4 −n41 λ4 − n42

, B =

b1b2b3b4

,

C =[η1 η2 η3 η4

], N =

n11 n12 n13 n14n21 n22 n23 n24n31 n32 n33 n34n41 n42 n43 n44

.

Let the following interpolation conditions be given:

H1(−0.5) = 1, H1(0.5) = 0, H2(0.5,−0.5) = 0, H2(−1, 0.5) = −2.

which determine the matrices

Λ = diag[− 1

2 ,12 ,

12 , − 1

], R = [1 1 0 0], and SR =

0 0 1 00 0 0 10 0 0 00 0 0 0

.Choose the following Gramian, P = diag [1, 2, 3, 4]. We obtain the following solution (after38 iterations the error ‖ Xk+1 − Xk ‖2 drops below machine precision):

X =

1−√

2 1−√

2 0 02−√

2 2−√

2 0 00 0 3−

√6 0

0 0 0 2− 2√

2

.

86

4.4 Volterra series interpolation

The following system matrices ensue (the last two columns of N are taken to be 0):

A =

√

2− 32

√2− 1 0 0√

2− 2√

2− 32 0 0

0 0√

6− 52 0

0 0 0 2√

2− 3

, B =

1−√

22−√

200

,

N =

0 0 0 00 0 0 0

3−√

6 0 0 00 2− 2

√2 0 0

, C =[

1 0 0 −1].

In this case the eigenvalues of the matrix A are −0.0858± 0.4925i, −0.0505 and −0.1716.

4.4 Volterra series interpolationThe goal of the usual interpolatory approach for bilinear systems is to find a reduced systemwhose leading kth order transfer functions interpolates those of the original one (i.e. 4.31) on agrid of sample points. However, instead of interpolating some of the leading subsystem transferfunctions, the purpose of ”multipoint Volterra series interpolation” (as introduced in [56]) is tocapture the response of the whole infinite Volterra series (more details in the book [103]) withrespect to a collection of frequencies.

In [5], a simplification of the Volterra series method was introduced (in the sense that theweight matrices U and S in [56], Lemma 3.1 were taken to be diagonal). This idea has been alsoused in [1], where the authors treat reduction of bilinear systems as reduction of parametrizedlinear systems with low-rank variation in the state matrix - this method is called implicit Volterraseries interpolation.

As pointed out in [56], important system properties are measured by Volterra series (i.e.weighted sums of the kth order transfer functions evaluated at all possible combinations of theinterpolation points). For instance, one example is given by the closed form expression of theH2 norm of a SISO bilinear system (see [56], Theorem 2.2). Another example is representedby the multipoint Volterra series interpolation conditions that are satisfied by the H2 optimalkthorder approximation of the original bilinear system (see [56], Theorem 4.2). This particularreduced system is computed via BIRKA which was first introduced in [20]. Here, the authorsstate equivalent optimality conditions which do not involve any Volterra series. In [1], the authorsgeneralize the multipoint Volterra series interpolation conditions to the MIMO case.

The interpolation framework in [56] is fundamentally different from the usual one. The bridgethat connects the two approaches is provided by the generalized Sylvester equations (6.33) and(6.39). Define the generalized resolvent of the triple (E,A,N) for x, y ∈ C as follows:

Ξ(x, y) = (xE−A− yN)−1 . (4.35)

87


Introduce the following multivariate transfer functions:

G1(s1, z) = CΞ(s1, z)BG2(s1, s2, z) = CΞ(s1, z)NΞ(s2, z)B

...

Gj(s1, s2, . . . , sj, z) = CΞ(s1, z)NΞ(s2, z)N · · ·NΞ(sj, z)B... (4.36)

where s1, s2, . . . , sj ∈ C are interpolation points and z ∈ C is a complex scalar.

Lemma 4.4.1 Provided that the condition ‖zNΦ(si)‖ < 1 is satisfied for all i 6 j, one canwrite the newly introduced transfer functions Gj in terms of the original transfer functions cor-responding to the bilinear system ΣB (as defined in (4.3)), as:

Gj(s1, . . . , sj, z) =∞∑`1=0· · ·

∞∑`j=0

z`1+...+`jH`1+...+`j+j(s1, . . . , s1︸︷︷︸`1+1

, . . . , sj, . . . , sj︸︷︷︸`j+1

). (4.37)

Proof of Lemma 4.4.1

G1(s1, z) = CΞ(s1, z)B = C(s1E−A− zN)−1B = C(Φ(s1)−1 − zN)−1B

= CΦ(s1)(I− zNΦ(s1))−1B = CΦ(s1)[ ∞∑`=0

(zNΦ(s1))`]B

= CΦ(s1)B + zCΦ(s1)NΦ(s1)B + z2CΦ(s1)NΦ(s1)NΦ(s1)B + . . .

=∞∑`1=0

z`H`1+1(s1, . . . , s1︸︷︷︸`1+1

).

The above result can be easily extended for Gj, where j > 2, as:

Gj(s1, s2, . . . , sj, z) = CΞ(s1, z)NΞ(s2, z)N · · ·NΞ(sj, z)B= CΦ(s1)(I− zNΦ(s1))−1N · · ·NΦ(sj)(I− zNΦ(sj))−1B

= CΦ(s1)[ ∞∑`1=0

(zNΦ(s1))`1]N · · ·NΦ(sj)

[ ∞∑`j=0

(zNΦ(sj))`j]B

=∞∑`1=0

∞∑`2=0· · ·

∞∑`j=0

z`1+...+`jH`1+...+`j+j(s1, . . . , s1︸︷︷︸`1+1

, . . . , sj, . . . , sj︸︷︷︸`j+1

).

To the right tuples in (4.6) the following matrices are associated:

R(i)v =

[Ξ(λ(i)

1 , ηi) B, Ξ(λ(i)2 , ηi) N Ξ(λ(i)

1 , ηi) B, . . . ,

88


. . . Ξ(λ(i)mi, ηi) N · · ·N Ξ(λ(i)

2 , ηi) N Ξ(λ(i)1 , ηi) B

],

for i = 1, . . . , k† where R(i) ∈ Cn×mi is attached to λ(i). The matrix

Rv =[R(1)

v , R(2)v , . . . , R(k†)

v

]∈ Cn×k, (4.38)

is defined as a Volterra controllability matrix of the bilinear system ΣB, associated with the rightmulti-tuple λ. Similarly, to the left tuples in (4.6) we associate the matrices

O(j)v =

C Ξ(µ(j)

1 , τj)C Ξ(µ(j)

1 , τj) N Ξ(µ(j)2 , τj)

...

C Ξ(µ(j)1 , τj) N Ξ(µ(j)

2 , τj) N · · · N Ξ(µ(j)pj, τj)

∈ Cpj×n, j = 1, . . . , q†,

and the Volterra observability matrix:

Ov =

O(1)

v...

O(q†)v

∈ Cq×n. (4.39)

As before, assume for simplicity that we have only one set of right multi-tuples, and one set ofleft multi-tuples with the same number of interpolation points k, namely

λ = λ1, λ2, λ1, . . . , λk, . . . , λ2, λ1 ,µ = µ1, µ1, µ2, . . . , µ1, µ2, . . . , µk .

In this case we need only two weights η, τ ∈ C. The following lemma extends the rationalinterpolation idea for the special rational functions defined in (4.37).

Lemma 4.4.2 Let ΣB = (C,E,A,N,B) be a bilinear system of order n. Assume that it isprojected to a kth order system by means of X = Rv and YT = Ov. The reduced systemΣB : (C, E, A, N, B), of order k, where

E = YTEX, A = YTAX, N = YTNV, B = YTB, C = CX,

satisfies the following 2k interpolation conditions ( i, j = 1, . . . , k ):

Gj(µ1, . . . , µj, τ) = Gj(µ1, . . . , µj, τ), Gi(λi, . . . , λ1, η) = Gi(λi, . . . , λ1, η)and another k2 mixed interpolation conditions. If one chooses equal weight values (i.e. η = τ),then the latter conditions can be written ( i, j = 1, . . . , k )

Gj+i(µ1, . . . , µj, λi, . . . , λ1, τ) = Gj+i(µ1, . . . , µj, λi, . . . , λ1, τ),

Remark 4.4.1 Consider a bilinear system ΣB : (C,E,A,N,B), and let Rv and Ov bethe controllability and observability matrices associated with the multi-tuples (4.3.1) anddefined by (4.38), (4.39) respectively. The generalized Loewner matrix Lv, and the gen-eralized shifted Loewner matrix Lsv corresponding to the Volterra series approach, are

89


defined as Lv = −Ov ERv, Lsv = −Ov ARv. In addition we define the quantitiesΨv = Ov NRv, Vv = Ov B and Wv = CRv.

4.4.1 One-sided interpolation in the Volterra frameworkIn this section we propose a procedure that extends the parametrization from Section 4.3.4 todifferent types of one sided interpolation conditions (based on the new introduced functions in(4.37)). We will hence construct reduced order models which satisfy one-sided interpolationconditions only (corresponding to the right tuples).

Again, the new generalized controllability matrix satisfies a Sylvester equation. Given thescalars η1, η2, . . . , ηk†, the vector

Rv =[eT1,m1 . . . eT1,m

k†

]∈ R1×k (4.40)

the block-shift matrix

SvR = blkdiag

[Sm1 + η1Im1 ,Sm2 + η2Im2 , . . . , Sm

k†+ ηk†Imk†

]∈ Rk×k (4.41)

and the block-diagonal matrix

Λv = blkdiag [Λ1, Λ2, . . . , Λk† ]. (4.42)

where Λi = diag [λ(i)1 , λ

(i)2 , . . . , λ

(i)mi

] (we arrange the interpolation points as in (6.31)), I` ∈ R`×`

is the identity matrix and S` is previously defined in (4.14).

Remark 4.4.2 The matrix R defined in (4.38) is the solution to the following generalizedSylvester equation:

AR+ NRSvR + B Rv = ERΛv (4.43)

We will discuss the construction of reduced parametrized bilinear systems of the type ΣB :C, I, A, N, B given initial data ν1, . . . , νk, i.e. computed samples of the newly introducedmultivariate functions in (4.36). We build reduced kth order bilinear models which satisfy k rightinterpolation conditions corresponding to the right tuples defined in (4.6) and to the scalarsη1, η2, . . . , ηk†

Gj(λ(i)j , λ

(i)j−1, . . . , λ

(i)2 , λ

(i)1 , ηi) = Gj(λ(i)

j , λ(i)j−1, . . . , λ

(i)2 , λ

(i)1 , ηi) := ν`, (4.44)

where j ∈ 1, . . . ,mi, i ∈ 1, . . . , k†, ` ∈ 1, . . . , k and m1 +m2 + . . .+mk† = k.

Lemma 4.4.3 Given k interpolation points λ` ∈ C and k interpolation values ν` ∈ C, a kth

order reduced parametrized bilinear system ΣB : C, I, A, N, B, satisfying the k interpolationconditions in (4.44), is constructed:

A = Λv − BRv − NSvR, Ni,j = ni,j, Bj = bj, Ci = νi, (4.45)

for all 1 6 i, j 6 k and Rv, SvR,Λ

v as previously defined in (4.40), (4.41) and (4.42). Noticethat the parametrization contains k2 +k+k† parameters which are included in the B, N matrices(bj and ni,j) and in η1, η2, . . . , ηk†.

90


Proof of Lemma 4.4.3 Similar to the proof of Lemma 4.3.4.The next result automatically follows:

Theorem 4.4.1 The interpolation conditions (4.44) satisfied by the parametrized reduced bilin-ear system ΣB = (C, I, A, N, B) can be written equivalently in the following infinite Volterraseries representation (provided that the conditions ‖ηiNΦ(λ(i)

j )‖ < 1 and ‖ηiNΦ(λ(i)j )‖ < 1 are

satisfied for all i ∈ 1, 2, . . . , k† and j ∈ 1, 2, . . . ,mi)∞∑`1=0

∞∑`2=0· · ·

∞∑`j=0

η`1+...+`ji H`1+...+`j+j(λ

(i)1 , . . . , λ

(i)1︸︷︷︸

`1+1

, · · · , λ(i)j , . . . , λ

(i)j︸︷︷︸

`j+1

)

=∞∑`1=0

∞∑`2=0· · ·

∞∑`j=0

η`1+...+`ji H`1+...+`j+j(λ

(i)1 , . . . , λ

(i)1︸︷︷︸

`1+1

, · · · , λ(i)j , . . . , λ

(i)j︸︷︷︸

`j+1

), (4.46)

for all i ∈ 1, 2, . . . , k† and j ∈ 1, 2, . . . ,mi.

Proof of Theorem 4.4.1 This result is a consequence of Lemmas 4.4.1 and 4.4.2.

Remark 4.4.3 Notice that when ηi = 0, ∀i ∈ 1, . . . , k† the above infinite series is simplifiedto only one term (i.e for l1 = l2 = . . . = lj = 0):

Gj(λ(i)j , λ

(i)j−1, . . . , λ

(i)2 , λ

(i)1 , 0) = Hj(λ(i)

j , λ(i)j−1, . . . , λ

(i)2 , λ

(i)1 ) (4.47)

which means that this procedure represents a generalization of the parametrization given inSection 4.4.

4.4.2 Two-sided mixed interpolation conditionsFrom the above considerations, it follows that it is possible to combine left interpolation con-ditions of the regular type (4.31) with right interpolation conditions of the Volterra series type(4.44). In a simple case, given the SISO bilinear system ΣB : (C,E,A,N,B), by choosing theordered tuples of left and right interpolation points:

[µ1 µ1, µ2

],[λ1, λ2, λ1

]and

the weight η1, we construct the associated generalized observability and controllability matrices:

O =[

CΦ(µ1)CΦ(µ1)NΦ(µ2)

], R =

[Ξ(λ1, η1), Ξ(λ2, η1)NΞ(λ1, η1)B

].

The reduced system ΣB = (C, E, A, N, B), of order 2, where

E = OER, A = OAR, N = ONR, B = OB, C = CR,

is guaranteed to satisfy the following 4 interpolation conditions:

H1(µ1) = H1(µ1), H2(µ1, µ2) = H2(µ1, µ2),G1(λ1, η1) = G1(µ1, η1), G2(λ2, λ1, η1) = G2(λ2, λ1, η1).

91

4.5 Numerical experiments

4.5 Numerical experimentsIn this section we illustrate the new method by means of three numerical examples. We comparethe new Loewner approach to the so called BIRKA approach proposed in [20, 33]. This procedureessentially constructs iteratively the best (in terms of the H2 norm) kth order reduced bilinearsystem and it is the extension of IRKA (see [74]) to the bilinear case.

4.5.1 Bilinear controlled heat transfer systemOn the unit square we consider the heat equation ∂x

∂t= ∆x with mixed Dirichlet and Robin

boundary conditions:

n · ∇x = u1(x− 1), on Γ1 = 0×c0, 1b,n · ∇x = u2(x− 1), on Γ2 =c0, 1b×0,

x = 0, on Γ3 = 1 × [0, 1] and Γ4 = [0, 1]× 1.

The heat transfer coefficients u1 and u2 at the lower boundaries Γ1 and Γ2 are the input variables.They can be interpreted as spraying-intensities of a cooling fluid acting on these boundaries.The temperature of the fluid is normalized to the value 1 and the heat flow over the boundaryis proportional to the difference of temperatures x− 1 on the boundary. Note that the inputs ujenter these conditions bilinearly. By a finite difference discretization of the Poisson equation onan equidistant n×n mesh (with meshsize h = 1

n+1 and nodes xij), we obtain the Poisson matrix:

P = I⊗Tk + Tk ⊗ I, Tk = tridiag [1, − 2, 1].The dynamics of the heat flow are thus given by a bilinear system described by:

x = Ax + u1N1x + u2N2x + Bu,

where x = vec(xij), u = [u1 u2]T and the system matrices are determined as follows:

A = 1h2 (I⊗Tk + Tk ⊗ I + E1 ⊗ I + I⊗ Ek), Ej = ejeTj ,

N1 = 1h

E1 ⊗ I, N2 = 1h

I⊗ Ek, b1 = 1h

E1 ⊗ e, b2 = 1h

e⊗ Ek, e = [1, 1, . . . , 1]T ,

B = [b1 b2], C = 1k2 (e⊗ e)T .

In the following experiment we assume that u1 = u2. We denote by ΣB the 2500th order systemobtained by discretizing the heat equation. This bilinear system will be reduced by means of thefollowing three methods:

1. The two-sided Loewner framework to obtain Σ1 of order 28.

2. The two-sided mixed approach (i.e. the Loewner approach on the left and the Volterraseries approach on the right) to obtain Σ2 of order 28.

3. The BIRKA approach to obtain Σ3 of order 28.

92


The first step is to collect samples from generalized bilinear transfer functions up to order two(this corresponds to mi = pj = 2 ∀i ∈ 1, . . . , k† and j ∈ 1, . . . , q† in 4.6). The 120 inter-polation points µj and λi are logarithmically chosen inside [10−3, 104]j. Since we are consideringcomplex conjugate pairs it follows that k† = q† = 60. The weights νi are chosen randomly in(0, 1). The following plot depicts the decay of the singular values corresponding to the ensuingLoewner pencil. It offers a good estimate of where one should truncate to obtain certain accuracy.

As illustrated in Fig. 4.1, we notice that σ1 = 1, σ28 ≈ 10−15, i.e., the 28th singular valueattains machine precision. We choose the reduced order k = 28 for all reduced systems.

10 20 30 40 50 60 70 80 90 100 110 120

10−25

10−20

10−15

10−10

10−5

100

Singular values of bilinear Loewner matrices

Loewners-Loewner

Figure 4.1: Singular values of the Loewner pencil.

Next, we compare the time-domain output of ΣB, and the outputs of the reduced systemswhen the input signal is chosen to be a decaying oscillation u(t) = 0.2 sin (4t)e−t/2 +e−t/2. Noticethat the response of the original system is accurately matched by the responses of all reducedsystems (Fig. 4.2).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1Time domain simulation: output signals

Time(t)

full bilinear, N = 2500Loewner bilinear, k1 =28Mixed conditions, k2 =28BIRKA, k3 =28

Figure 4.2: Time domain simulation - output signals.

In Fig. 4.3, the error between the response of ΣB and the responses of each reduced systemis depicted. Notice that our proposed methods produce slightly lower error than the BIRKAmethod.

4.5.2 Burgers’ equationThe second example for illustrating the bilinear modeling and reduction concepts proposed inthis chapter is the viscous Burgers equation:

93


0.5 1 1.5 2 2.5 3 3.5 4 4.5

10−16

10−14

10−12

10−10

10−8

Output error between the initial bilinear system and the reduced ones

Time(t)

Error

LoewnerMixedBIRKA

Figure 4.3: Time domain simulation - approximation errors.

∂v(x, t)∂t

+ v(x, t)∂v(x, t)∂x

= ∂

∂x

(νv(x, t)∂x

), (x, t) ∈ (0, 1)× (0, T ) , (4.48)

subject to the initial and boundary conditions given by:v(x, 0) = f(x), x ∈ [0, 1], v(0, t) = u(t), v(1, t) = 0, t > 0 .

The above system occurs in the area of fluid dynamics where it can be used for modeling gasdynamics and traffic flow. The solution v(x, t) can be interpreted as a function describing thevelocity at (x, t). In general, the viscosity coefficient ν(x, t) might depend on space and time aswell.

Some simplifications are performed: the viscosity coefficient ν(x, t) = ν is assumed to beconstant. Furthermore, a zero initial condition on the system, i.e. f(x) = 0, is considered.Finally, we assume that the left boundary is subject to a control.

Start with a spatial discretization of equation (4.48), using an equidistant step size h = 1n+1

where n denotes the number of interior points of the interval (0, 1). Use the following spacialderivative approximations:

∂v

∂x= v(x+ h)− v(x)

h,∂2v

∂x2 = v(x+ h)− 2v(x) + v(x− h)h2 ,

to obtain the nonlinear model:

vk =

−v1v2

2h + νh2 (v2 − 2v1) + ( v1

2h + νh2 )u, k = 1,

− vk2h(vk+1 − vk−1) + ν

h2 (vk+1 − 2vk + vk−1), 2 6 k 6 n− 1,−vnvn−1

2h + νh2 (−2vn + 2vn−1), k = n.

Next we use the Carleman linearization technique to approximate the above nth nonlinear systemwith a bilinear system of order N = n2 + n.

Denote with ΣB the 2550th order initial bilinear system obtained by means of the Carlemanlinearization. As for the first example, ΣB is reduced using Loewner and BIRKA to obtain Σ1and Σ3 respectively (both of order 30).

The first step is to collect samples from generalized bilinear transfer functions up to ordertwo - the 120 interpolations points are chosen logarithmically spaced in the interval [10−1, 101]j.

94


10 20 30 40 50 60 70 80 90 100 110 12010

−20

10−15

10−10

10−5

100


Loewners-Loewner


As illustrated in Fig. 4.4, we notice that σ1 = 1, σ30 ≈ 10−15, i.e., the 30th singular valueattains machine precision. We choose the reduced order k = 30 for all reduced systems.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Time(t)

full bilinear, N = 2550Loewner bilinear, k1 =30BIRKA, k2 =30


Next, we compare the time-domain output of ΣB, and the outputs of the reduced systemswhen the input signal is chosen to be a decaying oscillation u(t) = 0.2 sin (4t)e−t/2 +e−t/2. Noticethat the response of the original system is accurately matched by the responses of both reducedsystems (Fig. 4.5).

In Fig. 4.6, the error between the response of ΣB and the responses of each reduced system isdepicted. Our method produces, as in the first experiment, lower error than the BIRKA method.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

12

1010

108

106

104


Err

or

Figure 4.6: Time domain simulation- approximation errors.

95


4.5.3 Chafee-Infante equationNext, we consider the one-dimensional Chafee-Infante equation. The equation exhibits a cubicnonlinearity:

∂v(x, t)∂t

+ v3(x, t) = ∂2v(x, t)∂x2 + v(x, t), (x, t) ∈ (0, 1)× (0, T ) , (4.49)

and is subject to similar initial and boundary conditions as the Burgers equationv(x, 0) = v0(x), x ∈ (0, 1), αv(0, t) + βv(0, t) = u(t), v(1, t) = 0, t ∈ (0, T ) .

We again use a finite difference scheme for the spatial discretization. The resulting system ofnonlinear ODEs then has to be transformed to bilinear structure. This is done by means ofCarleman linearization. Starting with the boundary controlled equation with T = 10 and a zeroinitial condition v0(x) = 0 - use the value at the right boundary as the output. The discretizationwas done with n = 50 points. Hence, after transformation to bilinear form, the system consists of2550 states. The 60 interpolations points are logarithmically spaced in [10−3, 103]j. As illustrated

10 20 30 40 50 60

10−30

10−20

10−10

100


Loewners-Loewner


in Fig. 4.7, the 12th singular value attains machine precision - the decay is more pronounced thanfor the other two examples in this section. We choose the reduced order k = 10 for both reducedsystems. Next, we compare the time-domain output of ΣB, and the outputs of the reducedsystems when the input signal is chosen to be a decaying oscillation u(t) = 0.5(1 + cos(πt)).Notice that the response of the original system is accurately matched by the responses of both

0 1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2


Time(t)

full bilinear, N = 2550Loewner bilinear, k1 =10BIRKA, k2 =10


96


reduced systems (Fig. 4.8). In Fig. 4.9, the error between the response of ΣB and the responsesof each reduced system is depicted and as shown the new approach produced better results thanBIRKA.

0 1 2 3 4 5 6 7 8 9 1010

−12

10−10

10−8

10−6

10−4


Time(t)

Error

LoewnerBIRKA

Figure 4.9: Time domain simulation- approximation errors.

97

Chapter 5

The Loewner Framework forQuadratic-Bilinear Systems

5.1 IntroductionWe introduce a data-driven model order reduction (MOR) approach that represents an extensionof the Loewner framework for linear and bilinear systems to the case of quadratic-bilinear (QB)systems. For certain types of nonlinear systems, one can always find an equivalent QB modelwithout performing any approximation whatsoever.

An advantage of the Loewner framework is that information about the redundancy of thegiven data is explicitly available, by means of the singular values of the Loewner matrices. Thisfeature is also valid for the proposed generalization. As for the linear and bilinear cases, thesematrices can be directly computed by solving Sylvester equations.

We begin by defining generalized higher order transfer functions for QB systems. These mul-tivariate rational functions play an important role in the MOR process. We construct reducedorder systems for which the associated transfer functions match those corresponding to the orig-inal system at selected tuples of interpolations points. Another benefit of the proposed approachis that it is data-driven oriented, in the sense that one would only need computed or measuredsamples to construct a reduced order QB system. We illustrate the practical applicability of theproposed method by means of several numerical experiments.

After the introduction in Section 5.1, which includes an overview of some contributions onthe reduction of QB systems, we present a short background and some general properties ofsuch systems in Section 5.2. Then, in Section 5.3 we introduce the proposed methodology, ageneralization of the Loewner framework for QB systems. The reduced system is described bymatrices constructed using computed data (samples of specifically chosen generalized transferfunctions of the underlying system). The theoretical discussions are illustrated in Section 5.4 viathree numerical examples.

5.1.1 Literature overviewWe begin with a short historical account of various contributions to the reduction of quadratic-bilinear systems. Most of the methods mentioned below are projection based.

98

5.2 Quadratic-bilinear systems

• One of the first attempts in reducing this class of dynamical systems was made by Chen.He adapted the Arnoldi algorithm for building one sided projectors which can be applied tothe dimension reduction of QB systems by using a Krylov subspace generated from linearizedanalysis ([43, 44]).• Li and Pileggi introduced a compact nonlinear MOR method (known as NORM) suitable

mostly to weakly nonlinear systems that can be well characterized by low-order Volterra func-tional series. It is based on moment matching of nonlinear transfer functions by projection ofthe original system onto a set of minimum Krylov subspaces ( [86]).• Gu introduced the QLMOR framework which is a projection-based moment matching MOR

approach that uses the quadratic-linear representation of nonlinear systems. The method wasproven to preserve local passivity and also to provide an upper bound on the number of quadraticDAEs derived from a polynomial system ( [69, 70, 71, 72]).• Van Beeumen and Meerbergen adapted the widely used balanced truncation method to the

class of linear systems with quadratic output ([19]).• Benner and Breiten extended the results from Gu by introducing rational Krylov subspace-

based methods for quadratic-bilinear approximations of nonlinear systems. In particular, appli-cation to QB differential algebraic equations ( [21, 22, 34]).• In recent years, increased attention has been allocated to MOR by means of (symmetric)

tensor decomposition. Such methods were applied for reducing QB systems ([89, 52]).• A number of contributions were proposed by Benner, Goyal and collaborators who managed

to adapt two very well established MOR techniques, i.e., balanced truncation as well as the IRKAmethod, to the class of QB systems (see [25, 26]). We also mention some results on reducingStokes-type QB systems in descriptor format ([2]).

5.2 Quadratic-bilinear systemsWe analyze quadratic-bilinear control systems ΣQB = (E,A,Q,N,B,C) characterized by thefollowing equations:

ΣQB :Ex(t) = Ax(t) + Q

(x(t)⊗ x(t)

)+ Nx(t)u(t) + Bu(t), y(t) = Cx(t), (5.1)

where E, A, N ∈ Rn×n, Q ∈ Rn×n2 , B,CT ∈ Rn and x ∈ Rn, u, y ∈ R. We discuss the approx-imation of systems in (5.1), by constructing reduced order models ΣQB = (E, A, Q, N, B, C),described by:

ΣQB :E ˙x(t) = Ax(t) + Q

(x(t)⊗ x(t)

)+ Nx(t)u(t) + Bu(t), y(t) = Cx(t), (5.2)

where E, A, N ∈ Rk×k, Q ∈ Rk×k2 , B, CT ∈ Rk and x ∈ Rk, y ∈ R.For simplicity of exposition, we treat the single-input, single-output (SISO) case. The multi-

input case is technically more involved but it is based on the same ideas.In Section 3.4 from [103], a procedure is proposed that allows a state-equation description to

be obtained for each degree-k homogeneous subsystem in the input/output representation. Theadvantage is that although this equation is coupled nonlinearly to the equations for the lower-

99

5.2 Quadratic-bilinear systems

degree subsystems, each of the equations has identical linear terms. This procedure is widelyknown as the variational analysis approach.

In the following it is assumed that the original system ΣQB consists of a series of homogeneoussubsystems. Hence, proceed with explicitly identifying them. The first step is to consider as aninput, the scaled signal au(t), where a > 0 is an arbitrary scalar. Then, it is assumed that thesolution x(t) can be written as a power series, as follows:

x(t) =∞∑`=0

a`x`(t), (5.3)

where x` ∈ Rn. By substituting the equality in (5.3) into the original differential equation in(5.1) and using the appropriate input signal au(t), we obtain that:

E( ∞∑`=0

a`x`(t))

= A( ∞∑`=0

a`x`(t))

+ Q( ∞∑`=0

a`x`(t))⊗( ∞∑`=0

a`x`(t))

+ N( ∞∑`=0

a`+1x`(t)u(t))

+ aBu(t). (5.4)

By equating the coefficients of a` for all exponents ` > 0, from both sides of the equality in (5.4),it follows that the differential equation in (5.1) can be split into an infinite series of equations,similar to the splitting for the case of bilinear systems in (4.2), Chapter 4, Section 4.2:

Ex0(t) = Ax0(t) + Bu(t), ` = 0Ex1(t) = Ax1(t) + Q

(x0 ⊗ x0

)+ Nx0(t)u(t), ` = 1,

...

Ex`(t) = Ax`(t) + Q( `−1∑k=0

xk ⊗ x`−k−1)

+ Nx`−1(t)u(t), ` > 1,

(5.5)

where the solution of the equation at level `− 1 is used as an additional input for the equationat level ` (for ` > 1).

Considering x`−1(t) in the `th equation above as a pseudo-input, it follows that one canrecursively compute a closed form solution of the system variable x(t), as x(t) = ∑∞

`=0 x`(t).This allows the identification of multivariate input-output mappings, first in time domain, andafterwards also in frequency domain (by means of applying the multivariate Laplace transform).In particular, one can come up with appropriate rational transfer functions, as described inChapter 4, Section 2.3.3 and Definition 2.3.10, in particular.

These rational functions are divided in sub-categories or levels. Each level contains 2` func-tions, for ` > 0. By starting with the linear transfer function corresponding level 0, we writeHε

0(s) = H(s) = C Φ(s) B. Moving on to level 1, it follows that we can write

H(N)1 (s1, s2) = C Φ(s1) N Φ(s2)B, H(Q)

1 (s1, s2, s3) = C Φ(s1) Q(Φ(s2)B⊗Φ(s3)B

),

where again, the resolvent of the pencil (A,E) is denoted by Φ(x) = (xE−A)−1.

100

5.3 The main procedure for extending the Loewner framework to QB systems

Thus, in general a `th level transfer function the proposed framework is a multivariate rationalfunction depending on h variables si, i = 1, · · · , h (where `+ 1 6 h 6 2`+ 1). For more detailson the exact structure of these functions, see Definition 2.3.10 in Chapter 2.

The direct consequence is that the characterization of QB systems by means of rationalfunctions suggests that reduction of such systems can be performed by means of interpolatorymethods, with the Loewner framework being a prime candidate for achieving this goal.

5.3 The main procedure for extending the Loewnerframework to QB systems

As stated in the introduction, the goal of this chapter is to extend the Loewner framework to theclass of nonlinear system, and in particular to quadratic-bilinear systems. This section presentsthe theoretical foundations of this approach while Section 5.3 provides numerical simulationsillustrating the theory.

5.3.1 Arranging the data in the required formatConsider a quadratic-bilinear system ΣQB = (E,A,Q,N,B,C) as in (5.1). Let S =ω1, ω2, . . . , ωm be the set of interpolation points, with ωi ∈ C, 1 6 i 6 m.

Proceed by partitioning this set into two disjoint sets Sµ and Sλ, as S = Sµ ∪ Sλ. Theleft points are collected into the set Sµ = µ1, µ2, . . . , µq, while the right points are given bySλ = λ1, λ2, . . . , λk. We denote with q, the number of left interpolation points, while k denotesthe number of right interpolation points, and hence q+ k = m. Moreover, for simplicity, assumethat both q and k are multiples of 3. Consequently, let k†, q† ∈ N so that k = 3k†, q = 3q†.Rename the left and right interpolation points, as follows:

Sµ =⋃

16j6q†µ(j)

1 , µ(j)2 , µ

(j)3 , Sλ =

⋃16i6k†

λ(i)1 , λ

(i)2 , λ

(i)3 . (5.6)

where λ(i)1 , λ

(i)2 , λ

(i)3 , µ(j)

1 , µ(j)2 , µ

(j)3 ∈ C for 1 6 i 6 k† and 1 6 j 6 q†.

Definition 5.3.1 We introduce the set of right multi-tuples λ, composed of right interpolationpoints from Sλ. Additionally, let µ be the set of left multi-tuples composed of left interpolationpoints from Sµ . The sets λ and µ are composed of right ith multi-tuples λ(i) and, respectively,of left jth multi-tuples µ(j), as follows:

λ =λ(1),λ(2), . . . ,λ(k†)

, µ =

µ(1),µ(2), . . . ,µ(q†)

. (5.7)

where the explicit definitions of λ(i) and µ(j) are presented below for 1 6 i 6 k† and 1 6 j 6 q†,

λ(i) =

(λ(i)

1 ),

(λ(i)2 , λ

(i)1 ),

(λ(i)3 , λ

(i)1 , λ

(i)1 ).

, µ(j) =

(µ(j)

1 ),

(µ(j)1 , µ

(j)2 ),

(µ(j)1 , λ

(j)1 , µ

(j)3 ).

. (5.8)

101


For i = 1, . . . , k†, associate the matrix R(i) ∈ Cn×3 to the ith multi-tuple λ(i) in (5.8), as follows

R(i) =[Φ(λ(i)

1 ) B, Φ(λ(i)2 ) N Φ(λ(i)

1 ) B, Φ(λ(i)3 )Q

(Φ(λ(i)

1 )B⊗Φ(λ(i)1 )B

)]. (5.9)

The matrix R ∈ Cn×k is composed of the blocks R(i) from (5.9), in the following way

R =[R(1), R(2), · · · , R(k†)

]. (5.10)

We will refer to R, as the generalized controllability matrix of the quadratic-bilinear systemΣQB in (5.1), associated with the right set of multi-tuples λ in (5.7). Similarly, for j = 1, . . . , q†,associate the matrix O(j) ∈ C3×n to the jth multi-tuple µ(j) in (5.8), as

O(j) =

C Φ(µ(j)

1 )C Φ(µ(j)

1 ) N Φ(µ(j)2 )

C Φ(µ(j)1 ) Q

(Φ(λ(j)

1 )B⊗Φ(µ(j)3 )) . (5.11)

The matrix O ∈ Cq×n is composed of the blocks O(j) from (5.11):

O =[ (O(1)

)T (O(2)

)T. . .(O(q†)

)T ]T∈ Cq×n, (5.12)

and will be referred to as the generalized observability matrix of the quadratic-bilinear systemΣQB in (5.1), associated with the left set of multi-tuples µ in (5.7).

Example 5.3.1 For instance, assume q = 6, q† = 2. The corresponding O matrix introduced in(5.12) can be written

O =

C Φ(µ(1)1 )

C Φ(µ(1)1 ) N Φ(µ(1)

2 )C Φ(µ(1)

1 ) Q(Φ(λ(1)

1 )B⊗Φ(µ(1)3 )

)C Φ(µ(2)

1 )C Φ(µ(2)

1 ) N Φ(µ(2)2 )

C Φ(µ(2)1 ) Q

(Φ(λ(2)

1 )B⊗Φ(µ(2)3 )

)

.

5.3.2 The interpolation propertyIn this section we discuss the extension of rational interpolation from linear systems approxi-mation (see e.g. [4] Chapter 11.3) to the quadratic-bilinear systems case. First, introduce thenotations:

Definition 5.3.2 For 1 6 i 6 k†, 1 6 j 6 q† and h, ` ∈ 1, 2, 3, denote with λ(i)(h) the hth

tuple of multi-tuple λ(i) and with µ(j)(`) the `th tuple of multi-tuple µ(j). For example, if h = 1and ` = 2, write λ(i)(1) = (λ(i)

1 ), while µ(j)(2) = (µ(j)1 , µ

(j)2 ).

Lemma 5.3.1 Let ΣQB = (E,A,Q,N,B,C) be a quadratic-bilinear system of order n andassume that Q is written in the symmetric format that allows it to satisfy (2.39). By means of

102


projection, the system ΣQB is approximated by a kth order QB system. The projection matricesX = R and YT = O, are chosen as in (5.10) and, respectively, in (5.12) for q = k. The reducedsystem Σ = (C, E, A, Q, N, B), of order k, where

E = YTEX, A = YTAX, Q = YTQ(X⊗X

), N = YTNX, B = YTB, C = CX,

satisfies the following interpolation conditions (where w = (ε,N,Q), w = (ε, N, Q), i, j ∈1, 2, . . . , k†, `, h, h1, h2 ∈ 1, 2, 3 with h1 ∨ h2 = 1):

k conditions:

Hε

0(µ(j)(1)) = Hε0(µ(j)(1))

HN1 (µ(j)(2)) = HN

1 (µ(j)(2))HQ

1 (µ(j)(3)) = HQ1 (µ(j)(3))

, or Hw(`)|w(`)|(µ(j)(`)) = Hw(`)

|w(`)|(µ(j)(`)),(5.13)

k conditions:

Hε

0(λ(i)(1)) = Hε0(λ(i)(1))

HN1 (λ(i)(2)) = HN

1 (λ(i)(2))HQ

1 (λ(i)(3)) = HQ1 (λ(i)(3))

, or Hw(h)|w(h)|(λ

(i)(h)) = Hw(h)|w(h)|(λ

(i)(h)),(5.14)

k2 conditions: Hw(`)Nw(h)|w(`)|+|w(h)|+1(µ(j)(`) λ(i)(h)) = Hw(`)Nw(h)

|w(`)|+|w(h)|+1(µ(j)(`) λ(i)(h)),(5.15)

δ(k) conditions: Hw(`)Qw(h1)w(h2)|w(`)|+|w(h1)|+|w(h2)|+1(µ(j)(`) λ(i)(h1) λ(i)(h2))

= Hw(`)Qw(h1)w(h2)|w(`)|+|w(h1)|+|w(h2)|+1(µ(j)(`) λ(i)(h1) λ(i)(h2)), (5.16)

where δ(k) = (k+1)k2

2 − (2k+3)k2

9 . Thus, by means of the above procedure, we match a total numberof 2k + k2 + δ(k) interpolation conditions that can directly be written in terms of the transferfunctions in (2.97).

Remark 5.3.1 The predicted total number of moments matched is 2k+k2+k3 (2k correspondingto the linear part, k2 to the purely bilinear part and k3 to the purely quadratic part).

The Q matrix is considered to be transformed according to Proposition 2.2.14. Hence, itmeans that some of the moments are counted twice (because of the properties stated in (2.39)).By excluding those particular ones, it follows that the total number of moments matched usingthis procedure should be instead, 2k + k2 + (k+1)k2

2 .Now, not all these moments can be directly written as samples of the generalized transfer

functions in (2.97). By excluding those which do not posses this property, we reach to theconclusion that 2k + k2 + δ(k) is the correct number of interpolation conditions.

5.3.3 Sylvester-type equations satisfied by the matrices O and RThe generalized controllability matrix R and observability matrix O defined in (5.10) and, re-spectively in (5.12), satisfy certain matrix equations. More explicitly, we will show that Rsatisfies a Sylvester-type equation with quadratic term, while O can be written as the solutionof a generalized (linear) Sylvester equation.

Before stating the corresponding results, define some additional constant matrices that con-tain only 0 and 1 entries. Start with the column vectors

103


R =[eT1,3 · · · eT1,3

]∈ R1×k, LT =

[eT1,3 · · · eT1,3

]∈ R1×q, (5.17)

and the block-shift matricesZR = Ik† ⊗ e1,3 ⊗ eT2,3 ∈ Rk×k, ZL = Iq† ⊗ eT1,3 ⊗ e2,3 ∈ Rq×q, (5.18)

YR =k†∑j=1

e3j−2,k ⊗ e3j−2,k ⊗ eT3j,k ∈ Rk2×k, YL =q†∑j=1

eT3j−2,q ⊗ eT3j−2,q ⊗ e3j,q ∈ Rq×q2. (5.19)

as well as the matrices,

Y(j)R = e3j−2,k ⊗ e3j−2,k ⊗ eT3j,k ∈ Rk2×k, Y(j)

L = eT3j−2,q ⊗ eT3j−2,q ⊗ e3j,q ∈ Rq×q2, (5.20)

and hence write YR = ∑k†

j=1 Y(j)R and also YL = ∑q†

j=1 Y(j)L . Next, arrange the interpolation

points in diagonal matrices format (as for the linear and bilinear cases), i.e.,

M = blkdiag [M1, M2, · · · , Mq† ], Λ = blkdiag [Λ1, Λ2, · · · , Λk† ]. (5.21)

where Mj = diag [µ(j)1 , µ

(j)2 , µ

(j)3 ] and Λi = diag [λ(i)

1 , λ(i)2 , λ

(i)3 ].

Additionally, introduce the following matrices (for j ∈ 1, 2, . . . , q†)

X(j) = ej,q† ⊗ eTj,q† ⊗ e1,3 ⊗ eT3,3 ∈ Rq×q, (5.22)T(j) = In ⊗ e3j−2,k ∈ Rnk×n, (5.23)U(j) = e3j−2,k ⊗ Ik ∈ Rk2×k. (5.24)

Lemma 5.3.2 The generalized controllability matrix R defined in (5.10) satisfies the followingSylvester-type matrix equation with quadratic term:

AR+ Q(R⊗R

)YR + NRZR + B R = ERΛ. (5.25)

Proof of Lemma 5.3.2 Multiply equation (5.25) to the right with the unit vector e3j−2,k(1 6 j 6 k†)

AR3j−2 + B = λ3j−2ER3j−2 ⇔ R3j−2 = (λ3j−2E−A)−1B = Φ(λ3j−2)B. (5.26)

Thus the (3j − 2)th column of the matrix which is the solution of (5.25) is indeed equal to the(3j− 2)th column of the generalized controllability matrix R. By multiplying the same equationon the right with the unit vector e3j−1,k obtain

AR3j−1 + NR3j−2 = λ3j−1ER3j−1 ⇔ R3j−1 = (λ3j−1E−A)−1NR3j−2. (5.27)

By substituting (5.26) into (5.27), we get that

R3j−1 = (λ3j−1E−A)−1N(λ3j−2E−A)−1B = Φ(λ3j−1)NΦ(λ3j−2)B.

104


By again multiplying equation (5.25) to the right, this time with the unit vector e3j,k, write

AR3j + Q(R⊗R

)(e3j−2,k ⊗ e3j−2,k

)= λ3jER3j ⇔ (λ3jE−A)R3j = Q

(R3j−2 ⊗R3j−2

)⇔ R3j = (λ3jE−A)−1Q

(R3j−2 ⊗R3j−2

)= Φλ3jQ

(R3j−2 ⊗R3j−2

). (5.28)


R3j = Φ(λ3j)Q(Φ(λ3j−2)B⊗Φ(λ3j−2)B

).

By putting together all the results above, it follows that indeed, the generalized controllabilitymatrix R constructed in (5.10) satisfies equation (5.25).

Lemma 5.3.3 The generalized observability matrix O defined in (5.12) satisfies the followinggeneralized Sylvester equation:

OA +q†∑j=1

X(j)OQ(R3j−2 ⊗ I

)+ ZLON + L C = MOE, (5.29)

or equivalently

OA +q†∑j=1

X(j)OQ(I⊗R

)T(j) + ZLON + L C = MOE. (5.30)

Proof of Lemma 5.3.3 Multiply equation (5.29) to the right with the row vector eT3j−2,k (1 6j 6 k†)

OT3j−2A + C = µ3j−2OT3j−2E ⇔ OT3j−2 = C(µ3j−2E−A)−1 = CΦ(µ3j−2). (5.31)

Multiplying the same equation on the right with the row vector eT3j−1,k obtain:

OT3j−1A +OT3j−2N = µ3j−1OT3j−1E ⇔ OT3j−1 = OT3j−2N(µ3j−1E−A)−1. (5.32)


OT3j−1 = C(µ3j−2E−A)−1N(µ3j−1E−A)−1 = CΦ(µ3j−2)NΦ(µ3j−1).

Finally, multiplying equation (5.29) to the right, this time with the row vector eT3j,k, write

OT3jA +OT3j−2Q(R⊗ I

)= µ3jOT3jE ⇔ OT3j(µ3jE−A) = OT3j−2Q

(R3j−2 ⊗ I

)⇔ OT3j = OT3j−2Q

(R3j−2 ⊗ I

)(µ3jE−A)−1 = OT3j−2Q

(R3j−2 ⊗Φ(µ3j)

). (5.33)


O3j = CΦ(µ3j−2)Q(Φ(λ3j−2)B⊗Φ(µ3j)

).

105


By putting together all the results above, it follows that indeed, the generalized observabilitymatrix O construcetd in (5.12) satisfies equation (5.29). We can write for j ∈ 1, 2, . . . , k†

Q(R3j−2 ⊗ I

)= Q

(I⊗R3j−2

)= Q

(I⊗Re3j−2,k

)= Q

(I⊗R

)(I⊗ e3j−2,k

)= Q

(I⊗R

)T(j).

Hence, it is shown that the equation (5.29) can be rewritten as (5.30).Proposition 5.3.1 Moreover, equation (5.29) can be further simplified by replacing the Q matrixwith Q(−1) (which was introduced in Definition 2.2.19) as follows,

OA + YL(O ⊗RT

)(Q(−1)

)T+ ZLON + L C = MOE. (5.34)

Proof of Proposition 5.3.1 First show that, for all j ∈ 1, 2, . . . , q†, we have that X(j) ⊗eT3j−2,k = Y(j)

L . Note that, using the original definition of X(j) and Y(j)L from (5.22) and (5.20)

as well as the result in (2.44), one can write the following

X(j) ⊗ eT3j−2,k = ej,k† ⊗ eTj,k† ⊗ eT1,3 ⊗ e3,3 ⊗ eT3j−2,k = ej,k† ⊗ eT3j−2,k ⊗ e3,3 ⊗ eT3j−2,k

=(ej,k†eT3j−2,k

)⊗(e3,3 ⊗ eT3j−2,k

)=(ej,k† ⊗ e3,3

)(eT3j−2,k ⊗ eT3j−2,k

)= e3j,k

(eT3j−2,k ⊗ eT3j−2,k

),

Y(j)L = eT3j−2,k ⊗ eT3j−2,k ⊗ e3j,k = e3j,k

(eT3j−2,k ⊗ eT3j−2,k

).

We apply the result in (2.47) for v = R3j−2, j ∈ 1, 2, . . . , k†, i.e.,

Q(R3j−2 ⊗ I

)=(I⊗RT

3j−2

)(Q(−1)

)T.

By multiplying to the left with X(j)O (for j ∈ 1, 2), it follows that


)= X(j)O

(I⊗RT

3j−2

)(Q(−1)

)T= X(j)

(O ⊗RT

3j−2

)(Q(−1)

)T= X(j)

[(IkO

)⊗(eT3j−2,kRT

)](Q(−1)

)T= X(j)

(Ik ⊗ eT3j−2,k

)(O ⊗RT

)(Q(−1)

)T=(X(j) ⊗ eT3j−2,k︸︷︷︸

Y(j)L

)(O ⊗RT

)(Q(−1)

)T= Y(j)

L

(O ⊗RT

)(Q(−1)

)T. (5.35)

By substituting (5.35) into (5.29), it follows that the equation which characterizes the observ-ability matrix O, can be rewritten as in (5.34).

Corollary 5.3.1 The Sylvester equation in (5.34) has unique solution if the interpolation pointsin (5.21)are chosen so that the Sylvester operator

TO = AT ⊗ In − ET ⊗M +(Q(−1) ⊗YL

)Π(vec(RT )⊗ In2

)+ NT ⊗ ZL

is invertible, i.e., has no zero eigenvalues. Here, Π ∈ Rn4×n4 is a permutation matrix that allowsto write the vectorization of the Kronecker product between any two matrices U,V ∈ Rn as

106


followsvec(U⊗V) = Π

(vec(V)⊗ In2

)vec(U).

In this context, it is assumed that the controllability matrix R has been already computed.

The results mentioned in this section, in particular Lemmas 5.3.2 and 5.3.3 represent possibleextensions of the equations corresponding to the linear and bilinear cases. More exactly, byconsidering Q = 0, the equation (5.25) simplifies to AR + NRZR + B R = ERΛ which wasmentioned in Chapter 4, Lemma 4.3.2. Additionally, assuming that N = 0, the equation (5.25)further simplifies to the Sylvester equation AR+B R = ERΛ which was mentioned in Chapter3, Proposition 3.3.1.

5.3.4 The generalized Loewner pencilGiven the notations introduced in Section 5.3.1, we introduce appropriate generalizations of theLoewner matrices to the quadratic-bilinear case.

Definition 5.3.3 Consider a quadratic-bilinear system ΣQB as in (5.1), and let R and O bethe controllability and observability matrices associated with the multi-tuples (5.7) and defined by(5.10) and, respectively, by (5.12). The Loewner matrix L, and the shifted Loewner matrix Lsare defined as

L = −OER, Ls = −OAR . (5.36)

In addition we define the quantities

Ω = OQ (R⊗R), Ψ = ONR, V = OB and W = CR . (5.37)

The fact that the Loewner matrices are factorized in terms of the pairs of matrices (E,A) and(O,R) is an inherent property of the Loewner framework. Consequently, the formulas in (5.36)are exactly the same as the ones for the linear (see Chapter 3, Section 3.3.2, identities 3.26 to3.29) and bilinear (see Chapter 4, Definition 4.3.2) cases.The next example illustrates the procedure for the case of reduction order k = 3. We approximatea given QB system of dimension n, by means of interpolation.

Example 5.3.2 Given the SISO quadratic-bilinear system ΣQB characterized by the collectionof matrices (E,A,Q,N,B,C), where E,A,N ∈ Rn×n,Q ∈ Rn×n2 and B,CT ∈ Rn, consider 6interpolation points ω1, ω2, . . . , ω6. First partition this set into two disjoint sets correspondingto right and left points, as λ1, λ2, λ3 ∪ µ1, µ2, µ3. Next, consider the multi-tuples λ and µ,composed of tuples arranged in the required format as,

λ =

(λ1), (λ2, λ1), (λ3, λ1, λ1), µ =

(µ1), (µ1, µ2), (µ1, λ1, µ3)

.

The choice of multi-tuple sets above corresponds to k† = 1, k = 3 and q† = 1, q = 3. Theassociated generalized observability and controllability matrices can be written as:

R =[

Φ(λ1)B Φ(λ2)NΦ(λ1)B Φ(λ3)Q(Φ(λ1)B⊗Φ(λ1)B

) ],

107


O =

CΦ(µ1)

CΦ(µ1)NΦ(µ2)CΦ(µ1)Q

(Φ(λ1)B⊗Φ(µ3)

) .

Next, the Loewner matrix L = −OER ∈ R3×3 is a divided difference matrix, which is written

L =

H(µ1)−H(λ1)

µ1−λ1. . .

HQ1 (µ1,λ1,λ1)−HQ

1 (λ3,λ1,λ1)µ1−λ3

HN1 (µ1,µ2)−HN

1 (µ1,λ1)µ2−λ1

. . . HN,Q2 (µ1,µ2,λ1,λ1)−HN,Q

2 (µ1,λ3,λ1,λ1)µ2−λ3

HQ1 (µ1,λ1,µ3)−HQ

1 (µ1,λ1,λ1)µ3−λ1

. . .HQ,Q

2 (µ1,λ1,µ3,λ1,λ1)−HQ,Q2 (µ1,λ1,λ3,λ1,λ1)

µ3−λ3

.

Similarly, the shifted Loewner matrix Ls ∈ R3×3 is also a divided difference matrix,

Ls =

µ1H(µ1)−λ1H(λ1)

µ1−λ1· · · µ1HQ

1 (µ1,λ1,λ1)−λ3HQ1 (λ3,λ1,λ1)

µ1−λ3

µ2HN1 (µ1,µ2)−λ1HN

1 (µ1,λ1)µ2−λ1

. . . µ2HN,Q2 (µ1,µ2,λ1,λ1)−λ3HN,Q

2 (µ1,λ3,λ1,λ1)µ2−λ3

µ3HQ1 (µ1,λ1,µ3)−λ1HQ

1 (µ1,λ1,λ1)µ3−λ1

. . .µ3HQ,Q

2 (µ1,λ1,µ3,λ1,λ1)−λ3HQ,Q2 (µ1,λ1,λ3,λ1,λ1)

µ3−λ3

.

Note that the V and W vectors can be written in terms of samples corresponding to linear andlevel 1 transfer functions only, as

V =

H(µ1)HN

1 (µ1, µ2)HQ

1 (µ1, λ1, µ3)

, W =[

H(λ1) HN1 (λ2, λ1) HQ

1 (λ3, λ1, λ1)].

Next, construct the matrices Ξ and Ω, corresponding to the bilinear dynamics, and, respectivelyto the quadratic dynamics, as follows:

Ξ =

HN

1 (µ1, λ1) HN,N2 (µ1, λ2, λ1) HN,Q

2 (µ1, λ3, λ2, λ1)HN,N

2 (µ1, µ2, λ1) HN,N,N3 (µ1, µ2, λ2, λ1) HN,N,Q

3 (µ1, µ2, λ3, λ2, λ1)HQ,N

2 (µ1, λ1, µ3, λ1) HQ,N,N3 (µ1, λ1, µ3, λ2, λ1) HQ,N,Q

3 (µ1, λ1, µ2, λ3, λ2, λ1)

,

Ω =

HQ

1 (µ1, λ1, λ1) HQ,N2 (µ1, λ1, λ2, λ1) HQ,Q

2 (µ1, λ1, λ3, λ1, λ1) · · ·

HN,Q2 (µ1, µ2, λ1, λ1) HN,Q,N

3 (µ1, µ2, λ1, λ2, λ1) HN,Q,Q3 (µ1, µ2, λ1, λ3, λ1, λ1)

. . .

HQ,Q2 (µ1, λ1, µ3, λ1, λ1) HQ,Q,N

3 (µ1, λ1, µ3, λ1, λ2, λ1) HQ,Q,Q3 (µ1, λ1, µ3, . . . , λ3, λ1, λ1) · · ·

.Some entries of the Ω matrix can not be directly written in terms of samples of transfer functionsdefined in (2.97). Nevertheless, we can overcome this issue by altering the Q matrix. In this waywe successfully keep the simplified format of transfer functions from (2.97). Next, we show howto rewrite the entries of Ω; for brevity choose only one such example:

Ω(1, 5) = O1Q(R2 ⊗R2

)= CΦ(µ1)Q

(Φ(λ2)NΦ(λ1)B⊗Φ(λ2)NΦ(λ1)B

)= CΦ(µ1)Q

(Φ(λ2)NΦ(λ1)B⊗Φ(λ2)

)NΦ(λ1)B

108


= CΦ(µ1) Q(Φ(λ2)N⊗ I

)︸︷︷︸

Q

(Φ(λ1)B⊗Φ(λ2)

)NΦ(λ1)B = HQ,N

2 (µ1, λ1, λ2, λ1).

It readily follows that given the system ΣQB, a reduced system ΣQB := (−L,−Ls,Ω,Ψ,V,W)of order k = 3 can be constructed. This reduced system matches 2k+k2 + (k+1)k2

3 − (2k+3)k2

9 = 24moments of the original system that can be directly written in the format defined in (2.97). Next,we enumerate these values: two corresponding to the linear transfer function, six correspondingto level 1 transfer functions, eight corresponding to level 2 transfer functions and finally, eightcorresponding to level 3 transfer functions, as

two of Hε0 : H(µ1), H(λ1),

three of HN1 : HN

1 (µ1, µ2), HN1 (µ1, λ1), HN

1 (λ2, λ1),three of HQ

1 : HQ1 (µ1, λ1, µ3), HQ

1 (µ1, µ2, λ1), HQ1 (µ1, λ2, λ1),

two of HN,N2 : HN,N

2 (µ1, µ2, λ1), HN,N2 (µ1, λ2, λ1),

two of HN,Q2 : HN,Q

2 (µ1, λ3, λ2, λ1), HN,Q2 (µ1, µ2, λ1, λ1),

two of HQ,N2 : HQ,N

2 (µ1, λ1, µ3, λ1), HQ,N2 (µ1, λ1, λ2, λ1),

two of HQ,Q2 : HQ,Q

2 (µ1, λ1, λ3, λ1, λ1), HQ,Q2 (µ1, λ1, µ3, λ1, λ1),

eight of Hw3 : HN,N,N

3 (µ1, µ2, λ2, λ1), HN,N,Q3 (µ1, µ2, λ3, λ2, λ1),

. . . , HQ,Q,Q3 (µ1, λ1, µ3, λ3, λ1, λ3, λ1, λ1), w ∈ Υ3.

5.3.5 Sylvester equations satisfied by the Loewner matrices

We will show that the matrices defined in (5.36) satisfy various relations and Sylvester equa-tions which generalize the ones that were already introduced for the linear case, i.e, in [91] andalso for the bilinear case in [10].Proposition 5.3.2 The Loewner and shifted Loewner matrices satisfy the following relations

Ls = LΛ + ΩYR + ΞZR + VR, (5.38)

Ls = ML +k†∑j=1

X(j)ΩU(j) + ZTLΞ + LW. (5.39)

Proof of Proposition 5.3.2 By multiplying equation (5.25) with O to the left, it follows thatwe can write:

OAR︸︷︷︸−Ls

+OQ(R⊗R︸︷︷︸Ω

)YR +ONR︸︷︷︸

Ψ

ZR +OB︸︷︷︸V

R = OER︸︷︷︸−L

Λ.

Now by substituting the projected matrices defined in (5.36) and (5.37) onto the above equation,it directly follows that the relation (5.38) is verified. Moreover, by multiplying equation (5.29)with R to the right, it follows that we can write

OAR+k†∑j=1


)R+ ZT

LONR+ LCR = MOER. (5.40)

109


Using the Kronecker product properties in Proposition 2.2.15, the following holds for j ∈ 1, 2(R3j−2 ⊗ I

)R = R3j−2 ⊗R = (Re3j−2,k)⊗ (RIk) = (R⊗R)(e3j−2,k ⊗ Ik) = (R⊗R)U(j).

Now replacing this equality into (5.40), we write

OAR︸︷︷︸−Ls

+k†∑j=1

X(j)OQ(R⊗R

)︸︷︷︸

Ω

U(j) + ZTLONR︸︷︷︸

Φ

+L CR︸︷︷︸W

= MOER︸︷︷︸−L

.

Again, by substituting the projected matrices defined in (5.36) and (5.37) onto the above equa-tion, it directly follows that the relation (5.39) is verified.

Proposition 5.3.3 The Loewner matrix satisfies the following Sylvester equation

ML− LΛ = (VR + ΞZR + ΩY)− (LW + ZTLΞ +

k†∑j=1

X(j)ΩU(j)). (5.41)

Proof of Proposition 5.3.3This result directly follows by subtracting equation (5.38) from equation (5.39).

Proposition 5.3.4 The shifted Loewner matrix satisfies the following Sylvester equation:

MLs − LsΛ = (MVR + MΞZR + MΩY)− (LWΛ + ZTLΞΛ +

k†∑j=1

X(j)ΩU(j)Λ). (5.42)

Proof of Proposition 5.3.4 This result directly follows by subtracting equation (5.39) multi-plied with Λ to the right from equation (5.38) multiplied with M to the left.

5.3.6 Construction of reduced order modelsAs it was already specified, the structure of the interpolation data for the quadratic-bilinear caseis more complex than that of the data corresponding to the bilinear case. This is because higherorder transfer function of purely quadratic as well as mixed quadratic-bilinear transfer functionsfunctions need to be taken into account. However, the procedure is similar to that from thebilinear case in Chapter 4, Section 4.3.3 and to that from the linear case in Chapter 3, Section3.3.3.

As in the case of linear and bilinear systems the following result states that by putting thedata together, one can automatically construct a model for a reduced order QB system thatmatches the interpolation conditions mentioned before.

Proposition 5.3.5 Assume that k = q, and let (Ls, L), be a regular pencil, such that none ofthe interpolation points λi, µj are its eigenvalues. Then it follows that the matrices

E = −L, A = −Ls, Q = Ω, N = Ψ, B = V, C = W, (5.43)

110


form a realization of a QB system that matches the interpolation conditions in (5.13) - (5.16).

Proof of Proposition 5.3.5 This result follows directly from Lemma 5.3.1 by using the nota-tions introduced in Definition 5.3.3.

If the pencil (Ls, L) is singular, then we are dealing with the case of redundant data. Inpractical applications such as simulation, control and optimization, it is better to eliminate theredundancies of the model before proceeding to these tasks. One way to do so is by means ofprojection. The projector matrices are selected from the singular vector matrices of the Loewnermatrices, as it is described in the following procedure.

Procedure 5.3.1 Let τ > 0 be a positive real scalar which will be referred to as the tolerancevalue. Consider the following singular value factorizations of two matrices composed of theLoewner matrix L and of the shifted Loewner matrix Ls, as follows

[L Ls

]= Y1Σ1XT

1 ,

[LLs

]= Y2Σ2XT

2 , (5.44)

where Σ1,Σ2 ∈ Rk×k are diagonal matrices so that Σi = diag[σ(1)i , σ

(2)i , . . . , σ

(k)i ] for i ∈ 1, 2.

Also, the eigenvector matrices are Y1, X2 ∈ Ck×k and Y2, X1 ∈ C2k×k.Additionally let r ∈ N be the largest positive integer so that σ(j)

1 > τ, ∀j = 1, r. Next,additively split the singular value decompositions in (5.44) in two parts, as:

[L Ls

]= Y1Σ1XT

1 = Y(1)1 Σ(1)

1

(X(1)

1

)T+ Y(2)

1 Σ(2)1

(X(2)

1

)T, (5.45)[

LLs

]= Y2Σ2XT

2 = Y(1)2 Σ(1)

2

(X(1)

2

)T+ Y(2)

2 Σ(2)2

(X(2)

2

)T, (5.46)

where Σ(1)i = diag[σ(1)

i , σ(2)i , . . . , σ

(r)i ], Σ(2)

i = diag[σ(r+1)i , σ

(r+2)i , . . . , σ

(k)i ] for i ∈ 1, 2. Also,

the eigenvector matrices are Y(1)1 ,X(1)

2 ∈ Ck×r, Y(2)1 ,X(2)

2 ∈ Ck×(k−r), Y(1)2 ,X(1)

1 ∈ C2k×r andY(2)

2 ,X(2)1 ∈ C2k×(k−r). By selecting the first r columns of the matrices Y1 and X2, introduce the

projection matrices Y(1)1 , X(1)

2 ∈ Rk×r.The projected system Σ = (E, A, Q, N, B, C), composed of the following matrices

E = −(Y(1)

1

)∗LX(1)

2 , A = −(Y(1)

1

)∗LsX(1)

2 , Q =(Y(1)

1

)∗Ω(X(1)

2 ⊗X(1)2

),

N =(Y(1)

1

)∗ΨX(1)

2 , B =(Y(1)

1

)∗V, C = WX(1)

2 ,(5.47)

is the realization of a reduced QB system that approximately matches the interpolation conditionsin (5.13) - (5.16).

If the truncated singular values included on the diagonal of the matrices Σ(2)1 and Σ(2)

2 , i.e.σ(r+1)

1 , . . . , σ(k)1 ∪ σ

(r+1)2 , . . . , σ

(k)2 , are all zero, then the interpolation is exact.

Remark 5.3.2 As in the linear case treated in [91], if we have more data than necessary, wecan consider (E, A, Q, N, B, C) in (5.43) as a singular model that matches the interpolationconditions. Alternatively, consider (E, A, Q, N, B, C) in (5.47) as an nonsingular model thatapproximately matches the interpolation conditions.

111


Next, we show how to construct real models (Wreal, −Lsreal, −Lreal, Ωreal,Ψreal, Vreal) fromcomplex data, provided that the data is closed under complex conjugation. In other words,complex sample points and values are accompanied by their complex conjugate counterparts.The procedure proposed is a natural extension of the one for bilinear systems, in 4.

In applications, assuming that the underlying system is real, we seek reduced order systemsdescribed by real valued matrices only. To attain this, it must be assumed that, given a certainset of data, the complex conjugate data is also provided. The procedure is similar to the onealready proposed for linear systems in [12], Section ”Real Loewner matrices from complex data”and alss to the one described in Chapter 4, Section ”Interpolating complex data”.

Next, a 2k quadratic-bilinear Loewner model is constructed so that it matches the giveninterpolation conditions. By assuming that the interpolation points come in complex conjugatepairs, the complex Loewner model can be transformed to a real model, as follows

Lreal = ZkLZ∗k , Lsreal = ZkLsZk, Ωreal = ZkΩ(Z∗k ⊗Z∗k

),

Ψreal = ZkΨZ∗k , Vreal = ZkV, Wreal = WZ∗k ., (5.48)

where the block diagonal matrix Zk ∈ C2k×2k is composed of k blocks (k = 3k†), as in (4.29)

Zk = Ik ⊗Z1 ∈ R2k×2k, where Z1 =√

22

[1 1j −j

]. (5.49)

To ensure that our model is composed of real-valued matrices, the dimension of the Loewnermatrices increases to 2k, instead of k as for the original procedure described in Section 5.3.


5.4.1 Burgers’ equationThe first example we present in this section is the viscous Burgers’ equation which was alreadystudied in the context of model reduction for Carleman linearized large-scale bilinear systems inChapter 4, Section 4.5.2.

We denote by Σ0 the original nonlinear system for which the state variable has dimensionn = 50; furthermore, ΣB denotes 2550th order approximation of Σ0, obtained by means of theCarleman linearization, and ΣQB denotes the quadratic bilinear form of Σ0 (no approximationinvolved) of order 50. The system will be reduced by means of the following four methods:

1. ΣB is reduced using Loewner (as in Chapter 4) to obtain Σ1 of order 30.

2. ΣQB is reduced using Loewner (as in Chapter 5) to obtain Σ2 of order 16.

3. Σ0 is reduced using standard POD method to obtain Σ3 of order 16.

4. Σ0 is reduced using discrete empirical interpolation method (or DEIM, see [41, 42]) toobtain Σ4 of order 16.

The first step is to collect samples from appropriately defined generalized transfer functionsand plot the singular values of the ensuing Loewner pencil. As illustrated in Fig. 5.1, we notice

112


10 20 30 40 50 60

10−15

10−10

10−5

100Singular values of bilinear Loewner matrices

Loewners-Loewner

10 20 30 40 50 60

10−15

10−10

10−5

100

Singular values of quadratic bilinear Loewner

Loewners-Loewner

Figure 5.1: Singular values of the Loewner pencil; (a) bilinear; (b) quadratic-bilinear.

that σ1 = 1, σ30 ≈ 10−15, i.e., the 30th singular value attains machine precision. We choose thereduced order rb = 30 for the bilinear case. For the quadratic=bilinear case, a steeper drop insingular values is noticed. The order rq = 16 was chosen instead.

In Fig. 5.2, the distribution of the poles corresponding to both ΣB and ΣQB is depicted. Notethat the A matrix corresponding to both reduced systems is Hurwitz (since all the poles are inthe left half of the complex plane).

−35 −30 −25 −20 −15 −10 −5 0−1.5

−1

−0.5

0

0.5

1

Re(eig(A,E))

Im(eig(A

,E))

poles of the reduced bilinear systempoles of the reduced quadratic bilinear system

Figure 5.2: The poles of the reduced Loewner models.

Next, we compare the time-domain response of the original nonlinear system against theresponses of the reduced models , when the input is u(t) = 0.2(cos(2πt) + cos(4πt)). For thePOD based approximation, collect snapshots of the true solution for the training input u1(t) =10 sin(4t)e−t/2, and then compute the projection by taking the 16 most dominant basis vectors.

In Fig. 5.3 we depict the respective outputs. By examining the plot, observe that all but oneoutputs seem to follow the same path; the one corresponding to the bilinear Loewner methoddeviates from the original output, i.e., the dotted red curve does not follow the black curve.

Finally, in Fig. 5.4, we depict the error between the response of Σ0 and the responses of allthe reduced systems. We notice that the error when applying the quadratic-bilinear Loewnerprocedure is slightly lower than the error for the POD type methods.

5.4.2 Nonlinear RLC networkThe nonlinear transmission line circuit (for which the schematic is depicted in Fig. 5.5) is a verycommonly used circuit for testing nonlinear model reduction techniques (see [14, 86, 69, 33]).

113


0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.005

0

0.005

0.01

0.015

0.02

0.025

Time(t)

non-linearLoewner bilinearLoewner quadraticPODPOD-DEIM

Figure 5.3: Time domain simulations − original vs. reduced systems.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

−14

10−12

10−10

10−8

10−6

10−4

10−2

Time(t)

Error

Output error between the initial nonlinear system and the reduced systems

Loewner bilinear, rb =30Loewner quadratic, rq =16POD, q1 = 16POD-DEIM, q1 = 16, q2 = 16

Figure 5.4: Time-domain approximation error between original and reduced systems.

Consider all resistors and capacitors to be set to 1 and the diode to be characterized by thefollowing nonlinear current/voltage dependency iD = g(vD) = e40vD − 1. The input is set to thecurrent source and the output is the voltage at node 1. By writing the corresponding equations,we construct a nonlinear system in state space representation that characterize the dynamics ofthe circuit, as

ΣN :

v(t) = f(v(t)) + Bu(t),y(t) = Cv(t),

where in this context, identify

f(vk) =

−g(v1)− g(v1 − v2), for k = 1g(xk−1 − xk)− g(vk − vk+1), for 1 < k < N

g(vN−1)− g(vN), for k = N

, B = CT = e1.

As was pointed out in [71], a transformation to quadratic-bilinear form is easily obtained byintroducing additional state variables xi = e40vi and zi = e−40vi . It follows that the state variableof the new transformed system will have dimension 3N , i.e., v = [v ; x ; z] ∈ R3N .

An alternative to this is presented in [33], where the new state variables are introducedx1 = v1, xk = vk − vk+1, z1 = e40v1 − 1 and zk = e40xk for k ∈ 2, . . . , N. Hence it turns outthat it is possible to construct an equivalent quadratic-bilinear system of a lower dimension (2Nto be precise) where, the new variable v = [x ; z] ∈ R2N is defined. This alternative final system

114


Figure 5.5: Circuit schematic [69].

is quadratic-bilinear, and its dynamic is determined by the equations

z1 = 40(z1 + 1)(−x1 − x2 − z1 − z2 + u︸︷︷︸x1

),

...

zk = 40(zk + 1)(zk−1 − 2zk + wk+1 + zk−1 − 2zk + zk+1︸︷︷︸xk

), 2 < k < N,

...

zN = 40(zN + 1)(xN−1 − 2xN + zN−1 − 2zN︸︷︷︸xN

).

Hence a quadratic-bilinear representation of the original non-linear system is computed (oforder n = 2N). This will be considered as the original system in the following computations; itwill be reduced to a much smaller dimension by means of the QB Loewner framework and of theTQB-IRKA method (as introduced in [26]).

First, proceed by collecting samples from generalized linear, quadratic, bilinear and quadratic-bilinear transfer functions up to level 3 (the procedure which was described in Section 5.3).

In total consider 60 interpolations points that are logarithmically spaced inside [10−3, 103]j.As illustrated in Fig. 5.6, the 18th singular value (of the Loewner matrix) attains machine pre-cision. We choose the reduced-order k = 12 for the Loewner reduced system as well as the oneobtained via the TQB-IRKA procedure.

5 10 15 20 25 30 35 40 45 50 55 60

10−30

10−20

10−10

100

Singular values of QB Loewner matrices

Loewners-Loewner

Figure 5.6: Singular value decay of the QB Loewner pencil.

115


As it can be observed Fig. 5.7, the responses corresponding to both reduction methods seem tofaithfully duplicate the original response. The control input was chosen as u(t) = (1+cos(πt))/2.

0 1 2 3 4 5 6 7 8 9 10

0.005

0.01

0.015

0.02

0.025

Time domain simulation: output signals

Time(t)

Original system, n = 500Loewner QB, k1 =12QIRKA, k2 =12

Figure 5.7: Time-domain simulation.

When analyzing the magnitude of the relative error between the original response and thetwo responses of the reduced systems, notice that the method that we propose produces slightlybetter approximation than TQB-IRKA does (in Fig. 5.8).

0 1 2 3 4 5 6 7 8 9 10

10−8

10−7

10−6

10−5

10−4

Relative error of the time-domain response

Time(t)

Error

Loewner QBTQB-IRKA

Figure 5.8: Relative error between the response of the original system and of the reduced ones.

5.4.3 Chafee-Infante equationNext, consider the same example as in Section 4.5.3, i.e., the one-dimensional Chafee-Infanteequation with cubic nonlinearity. By means of a finite difference scheme (with n equidistantpoints over the length), construct a semi-discretized quadratic-bilinear system of order 2n. Theoutput y(t) is chosen to be the response at the right boundary. Take n = 500 which will resultin a 1000th initial QB system.

Previously in Section 4.5.3, the discretized quadratic-bilinear system was approximated by apurely bilinear system by means of the Carleman linearization technique. There, two reductionprocedures were applied for decreasing the dimension of the aforementioned bilinear system.

Now, the linearization step can skipped, and hence we directly apply the MOR tools for theQB finite element model. As for the bilinear case, proceed by collecting samples from variousgeneralized transfer functions. This time, the transfer functions are more diverse and are writtenin terms of both the N and the Q matrices.

116


5 10 15 20 25 30 35 40

10−20

10−15

10−10

10−5

100

Singular values of QB Loewner matrices

Loewners-Loewner

Figure 5.9: Singular value decay of the QB Loewner pencil.

Consider in total 40 interpolations points that are logarithmically spaced inside [10−2, 102]j.As illustrated in Fig. 5.9, the 12th singular value attains machine precision. We choose thereduced-order k = 10 for both reduced systems.

Again, both reduction methods seem to produce good approximations of the original transientresponse (as it can be observed in Fig. 5.10). In this case, the control input was chosen as in[26], i.e., as a decaying oscillatory exponential u(t) = (1 + sin(πt))e−t/5.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1


Time(t)

Original system, n = 500Loewner QB, k1 =10TQB-IRKA, k2 =10

Figure 5.10: Time-domain simulation.

By analyzing the absolute value of the offset between the original response and the tworesponses of the reduced systems, notice that the proposed method that again produces betterapproximation than the truncated quadratic-bilinear generalization of IRKA (in Fig. 5.11).

0 1 2 3 4 5 6 7 8 9 10

10−8

10−6

10−4

Relative error of the time-domain response

Time(t)

Error

Loewner QBTQB-IRKA

Figure 5.11: Relative error between the response of the original system and of the reduced ones.

117

Chapter 6

The Loewner Framework for LinearSwitched Systems

The Loewner framework for model reduction is extended to the class of linear switched systems.One advantage of this framework is that it introduces a trade-off between accuracy and com-plexity. Moreover, through this procedure, one can derive state-space models directly from datawhich is related to the input-output behavior of the original system. Hence, another advantage ofthe framework is that it does not require the initial system matrices. More exactly, the data usedin this framework consists in frequency domain samples of input-output mappings of the originalsystem. The definition of generalized transfer functions for linear switched systems resemblesthe one for bilinear systems. A key role is played by the coupling matrices, which ensure thetransition from one active mode to another.

6.1 IntroductionHybrid systems are a class of nonlinear systems which result from the interaction of continuoustime dynamical sub-systems with discrete events. More precisely, a hybrid system is a collection ofcontinuous time dynamical systems. The internal variable of each dynamical system is governedby a set of differential equations. Each of the separate continuous time systems are labeled as adiscrete mode. The transitions between the discrete states may result in a jump in the continuousinternal variable. Linear switched systems (in short LSS) constitute a subclass of hybrid systems;the main property is that these systems switch among a finite number of linear subsystems. Also,the discrete events interacting with the sub-systems are governed by a piecewise continuousfunction called the switching signal.

Hybrid and switched systems are powerful models for distributed embedded systems designwhere discrete controls are routinely applied to continuous processes. However, the complexityof verifying and assessing general properties of these systems is very high so that the use of thesemodels is limited in applications where the size of the state-space is not too large. To cope withcomplexity, abstraction and reduction are useful techniques. In this chapter we mainly analyzethe reduction part.

In the past years, hybrid and switched systems have received increasing attention in thescientific community. For a detailed characterization of this relatively new class of dynamical

118

6.1 Introduction

systems, we refer the readers to the books [87], [108], [109] and [64]. Such systems are used inmodeling, analysis and design of supervisory control systems, mechanical systems with impact,circuits with relays or ideal diodes.

The study of the properties of hybrid systems in general and switched systems in particular isstill the subject of intense research, including the problems of stability (see [49, 108]), realizationincluding observability/controllability (see [97, 98]), analysis of switched differential-algebraicequations, or in short DAEs (see [93, 110]) and of its numerical solutions (see [75]). Recently,considerable research has been dedicated to the problem of MOR for linear switched systems.The most prolific method that has been applied is balanced truncation (or a Gramian-basedderivation of it). Techniques that are based on balancing have been considered in the following:[61, 39, 30, 106, 94, 100, 96]. Also, another class of methods involve matching of generalizedMarkov parameters (known also as time domain Krylov methods) such as the ones in [16, 15];H∞ type of reduction methods were developed in [115, 31, 116]. Finally, we mention somepublications that are focused on the reduction of discrete LSS, such as [112, 32].

A linear switched system involves switching between a number of linear systems (the modesof the LSS). Hence, to apply balanced truncation techniques to a switched linear system, onemay seek a basis of the state space such that the corresponding modes are all in balanced form.

It may happen that some state components of the LSS are difficult to reach and observe insome of the modes yet easy to reach and observe in others. In that case, deciding how to truncatethe state variables and obtain a reduced order model is not trivial. A solution to this problemis proposed in [94] where it is shown that the average Gramian can be used to obtain a reducedorder model. This method will be used as a comparison tool for our new MOR method.

In this chapter, we focus on extending the Loewner framework (see [12]) for reducing linearswitched systems. This method can be viewed as a special subclass of rational Krylov methods,also referred to as moment matching or interpolatory methods. Roughly speaking, in the linearcase, interpolatory methods seek reduced models whose transfer function (and possibly someof its derivatives) matches the transfer function (and possibly some of its derivatives) of theoriginal system at selected frequencies. For the nonlinear case, these methods require appropriatedefinitions of transfer functions.

We construct LSS reduced models by means of matching samples of input-output mappingsin frequency domain, corresponding to the original LSS and evaluated at particular samplingpoints (as opposed to other approaches, i.e. [15, 16], where the behavior at infinity is studiedinstead, i.e. by matching Markov parameters). For the explicit derivation of higher order input-output mappings in time domain, based on the so-called Volterra series representation, we referthe reader to [103].

The chapter is organized as follows. In Section 6.2, the formal definition of continuous-timelinear switched systems is provided. Furthermore, we introduce the generalized transfer functionsfor LSS as input-output mappings in frequency domain. In Section 6.3, we present the extensionof the Loewner framework for LSS with 2 modes. This is done in order to familiarize thereader with basic ideas without having to use heavy notation. Then, in Section 6.4, we proposeextensions of the results introduced in the previous section to the general case of LSS that switchamongst D > 2 modes. Afterwards, in Section 6.5, we discuss the practical applicability ofthe new introduced method by means of three numerical examples (one of which is large scale).In those examples, we compare our algorithms with the balanced truncation algorithm of [94].Finally, in Section 6.6, we present a summary of the findings.

119

6.2 Linear switched systems

6.2 Linear switched systemsDefinition 6.2.1 A continuous time linear switched system (LSS) is a dynamical systemdesribed by the following equations

Σ :

Eσ(t)x(t) = Aσ(t)x(t) + Bσ(t)u(t), x(0) = x0,

y(t) = Cσ(t)x(t),(6.1)

where σ(t) is the switching signal (σ : R → M), u is the input, x is the state, and y is theoutput. Additionally, M = 1, 2, . . . , D, D > 1, is the set of discrete modes.

The system matrices Eq,Aq ∈ Rnq×nq , Bq ∈ Rnq×m, Cq ∈ Rp×nq , where q ∈M, correspond tothe linear system active in mode q ∈M, and x0 ∈ Rnqs is the initial state. Here, n1, n2, . . . , nD,mand p are positive integers and qs ∈M is the mode in which the system is initialized. We considerthe Eq matrices to be invertible. Furthermore, the transition from one mode to another is madevia the so called switching or coupling matrices Kq,q ∈ Rnq×nq where q, q ∈M .

Remark 6.2.1 The case for which the coupling is made between identical modes is excluded,Hence, when q = q, consider that the coupling matrices are identity matrices, i.e. Kq,q = Inq .

The notation Σ = (n1, n2, . . . , nD, (Eq,Aq,Bq,Cq)|q ∈ M, Kq,q|q, q ∈ M,x0) is used asa short-hand representation for LSSs described by the equations in (6.1). The vector n ∈ ND,where n =

[n1 n2 · · · nD

]is the dimension (order) of Σ. The linear system which is active

in the qth mode of Σ is denoted with Σq and it is described by (where 1 6 q 6 D)

Σk :

Eqxq(t) = Aqxq(t) + Bqu(t), x(tk) = xk,y(t) = Cqxq(t).

(6.2)

Remark 6.2.2 Certain issues might arise when the matrices Eq are allowed to be singular suchas the existence of a solution of the LSS (Definition 6.2.2) or the formal definition of time-domain kernels in (6.4). The case of descriptor LSS (with singular Eq matrices) was treated indetail in [110]. In this chapter we assume that the matrices Eq are invertible to avoid furthercomplications.

The restriction of the switching signal σ(t) to a finite interval of time [0, T ] can be interpretedas a finite sequence of elements of M× R+ of the form:

ν(σ) = (q1, t1)(q2, t2) . . . (qk, tk),

120


where q1, . . . , qk ∈ M and 0 < t1 < t2 < · · · < tk ∈ R+, t1 + · · · + tk = T , such that for allt ∈ [0, T ] we have:

σ(t) =

q1 if t ∈ [0, t1),q2 if t ∈ [t1, t1 + t2),...qi if t ∈ [t1 + . . .+ ti−1, t1 + . . .+ ti−1 + ti),...qk if t ∈ [t1 + . . .+ tk−1, t1 + . . .+ tk−1 + tk).

In short, by denoting Ti := t1 + . . .+ ti−1 + ti, T0 := 0, Tk := T , write:

σ(t) =

q1 if t ∈ [0, T1),qi if t ∈ [Ti−1, Ti), i > 2.

Denote by PC(R+,Rn), Pc(R+,Rn), the set of all piecewise-continuous, and piecewise-constant functions, respectively.

Definition 6.2.2 A tuple (x,u, σ,y), where x : R+ →⋃Di=1 Rni, u ∈ PC(R+,Rm), σ ∈

Pc(R+,M), y ∈ PC(R+,Rp) is called a solution, if the following conditions simultaneously hold:

1. The restriction of x(t) to [Ti−1, Ti) is differentiable, and satisfies Eqix(t) = Aqix(t)+Bu(t).

2. Furthermore, when switching from mode qi to mode qi+1 at time Ti, the following holdsEqi+1x(Ti) = Kqi,qi+1 lim

tTix(t).

3. Moreover, for all t ∈ R, y(t) = Cσ(t)x(t) holds.

The switching matrices Kqi,qi+1 allow having different dimensions for the subsystems active indifferent modes. For instance, the pencil (Aqi ,Eqi) ∈ Rnqi×nqi , while the pencil (Aqi+1 ,Eqi+1) ∈Rnqi+1×nqi+1 where the values nqi and nqi+1 need not be the same. If the Kqi,qi+1 matrices are notexplicitly given, it is considered that they are identity matrices.

The input-output behavior of an LSS can be formalized in time domain as a map f(u, σ)(t).This particular map can be written in generalized kernel representation (as suggested in [99])using the unique family of analytic functions: gq1,...,qk : Rk

+ → Rp and hq1,...,qk : Rk+ → Rp×m with

q1, . . . , qk ∈M, k > 1 such that for all pairs (u, σ) and for T = t1 + t2 + · · ·+ tk we can write:

f(u, σ)(t) = gq1,q2,...,qk(t1, t2, . . . , tk) +k∑i=1

∫ ti

0hqi,qi+1,...,qk(ti − τ, ti+1, . . . , tk)u(τ + Ti−1)dτ,

where the functions g,h are defined for k > 1, as follows,

gq1,q2,...,qk(t1, t2, . . . , tk) = CqkeAqk

tkKqk−1,qkeAqk−1 tk−1Kqk−2,qk−1 · · · Kq1,q2e

Aq1 t1x0, (6.3)

hq1,q2,...,qk(t1, t2, . . . , tk) = CqkeAqk

tkKqk−1,qkeAqk−1 tk−1Kqk−2,qk−1 · · · Kq1,q2e

Aq1 t1B1. (6.4)

121


Note that, for the functions defined in (6.3) and (6.4) we consider the Eqi matrices to beincorporated into the Aqi and Bqi matrices (i.e. Aqi = E−1

qiAqi , Bqi = E−1

qiBqi). Moreover, the

transformed coupling matrices are written accordingly Kqi,qi+1 = E−1qi+1

Kqi,qi+1 .

In the rest of the chapter, the LSS we are studying are assumed to have zero initial conditions,i.e. x0 = 0. Hence, only the h functions in (6.4) are relevant for characterizing the input-outputmapping f .

The behavior of the input-output mappings in frequency domain is in turn characterized bya series of multivariate rational functions obtained by taking the multivariate Laplace transformof the regular kernels in (6.4), as for

Hq1(s1) = Cq1Φq1(s1)Bq1 , Hq1,q2(s1, s2) = Cq1Φq1(s1)Kq2,q1Φq2(s2)Bq2 ,

Hq1,q2,q3(s1, s2, s3) = Cq1Φq1(s1)Kq2,q1Φq2(s2)Kq3,q2Φq3(s3)Bq3 , · · ·

In general, for k > 3, write the level k generalized transfer function associated to the switchingsequence (q1, q2, . . . , qk), and evaluated at the points (s1, s2, . . . sk) as,

Hq1,q2,...,qk(s1, s2, . . . , sk) = Cq1Φq1(s1)Kq2,q1Φq2(s2) · · ·Kqk,qk−1Φqk(sk)Bqk , (6.5)

where Φq(s) = (sEq − Aq)−1, qj ∈ 1, 2, . . . , D, 1 6 j 6 k. The functions defined in (6.5)will be referred as the generalized transfer functions of the linear switched system Σ. By usingtheir samples, we are able to directly construct (reduced) switched models that interpolate theoriginal model, i.e. the extension of the Loewner framework to LSS.

Remark 6.2.3 For the classical Loewner framework applied to linear systems (see [91]), it isnot mandatory that the samples used in the modeling step, come from systems with an invertibleE matrix. We believe that similarly to the LTI case, the Loewner framework can be extended toLSSs for which the matrices Ek are not invertible. Since one needs to evaluate the multi-variabletransfer functions on certain frequency grids, the only condition that is mandatory is that thepencils (Ai,Ei) are regular for all i ∈ M. This is because the resolvent of such pencils entersthe transfer functions for various interpolation points s, as described in (6.5).

Remark 6.2.4 Conceptually, the initial state of an LSS is part of its definition, since it isimpossible to define the input-output map of an LSS without fixing such a starting state. In thischapter we consider only the case when the initial state is zero. Note that the classical Loewnerframework (as introduced in [91]) uses only the input-output behavior from the zero initial state.

The proposed extension of the Loewner framework aims at finding a reduced system whosegeneralized transfer functions match those of the original LSS. That is, the proposed methodtakes into account only the input-output behavior from the zero initial state.

For the case of LSS, one could include non-zero initial states by including them as an addi-tional column into one of the matrices Bi, but the system theoretic interpretation of the thusobtained generalized transfer functions and their moments remains unclear.

Remark 6.2.5 The structure of the transfer functions in (6.5) is similar to the one of the func-tions introduced in Chapter 4, Section 4.2, for the class of bilinear systems. This is because onecan formulate an LSS as a bilinear system, by introducing additional input signals. Nevertheless,

122

6.3 The Loewner framework for LSS - the case D=2

there are some restrictions, i.e. this works only for LSSs with no coupling matrices and that havethe same state dimension in each mode. For the case D = 2, introduce the signal u(t) = q − 1,if the system operates in mode q ∈ 1, 2. Then, write the dynamics of the LSS by merging theindividual dynamics of the two modes, as

x(t) = A1x(t) + (A2 −A1)x(t)u(t) + B1u(t)(1− u(t)) + B2u(t)u(t),

or equivalently, to emphasize the bilinear multiple input format, as

x(t) = Abilx(t) + Nbil1 x(t)u(t) + Nbil

2 x(t)u(t) + Nbil3 x(t)u(t)u(t) + Bbil

1 u(t) + Bbil2 u(t) + Bbil

3 u(t)u(t),

where Abil = A1, Nbil2 = A2−A1, Nbil

1 = Nbil3 = 0, Bbil

1 = B1, B2 = 0, Bbil3 = B2−B1. Hence,

rewrite the above differential equation equivalently as,

x(t) = Abilx(t) +3∑i=1

Nbili x(t)ui(t) +

3∑i=1

Bbili ui(t),

where the three control inputs are u1 = u, u2 = u and u3 = uu. That is, solutions of LSSs aresolutions of a bilinear system with a very specific structure and with specially chosen inputs. Inparticular, the continuous input u and the switching signal have to be merged into a new artificialinput u3. Note that not all solutions of the bilinear system correspond to solutions of an LSS:for the correspondence to hold, u2 should take values 0, 1 and u3 should satisfy u3 = u2u1.

6.3 The Loewner framework for LSS - the case D=2The characterization of linear switched systems by means of rational functions suggests thatreduction of such systems can be performed by means of interpolatory methods. In the followingwe propose a way to generalize the Loewner framework to LSS by interpolating appropriatelydefined transfer functions on a chosen grid of frequencies (interpolation points).

As for the linear case, the given set of sampling (interpolation) points is first partitioned intothe two following categories: left interpolation points: µj|1 6 j 6 ` ⊂ C and right interpolationpoints: λi|1 6 i 6 k ⊂ C.

In this chapter we consider the case of SISO linear switched systems and thus, the left andright tangential directions can be considered to be scalar (i.e. taking the value 1). The transferfunctions which will be matched are not single variable functions anymore (they depend onmultiple variables as described in (6.5)). Hence, the structure of the interpolation points usedin the new framework will be modified. Instead of having singleton values (as for the linear casetreated in Chapter 3), we will use instead n-tuples that include multiple singleton values (similarto the bilinear case in Chapter 4).

For simplicity of the exposition, we first consider the simplified case D = 2 (the systemswitches between two modes only). This situation is encountered in most of the numericalexamples in the literature that we encountered. Nevertheless, all the results presented in thissection can be generalized for higher number of modes in a direct way (as presented in Section

123


6.4). Depending on the switching signal σ(t), we either have,

Σ1 :

E1x1(t) = A1x1(t) + B1u(t),y(t) = C1x1(t),

or Σ2 :

E2x2(t) = A2x2(t) + B2u(t),y(t) = C2x2(t),

where dim(Σ1) = n1 (i.e. x1 ∈ Rn1 and E1,A1 ∈ Rn1×n1 ,B1,CT1 ∈ Rn1) and also dim(Σ2) = n2

(i.e. x2 ∈ Rn2 and E2,A2 ∈ Rn2×n2 ,B2,CT2 ∈ Rn2).

Denote, for simplicity, with K1 the coupling matrix when switching from mode 1 to mode2 (instead of K1,2) and, with K2, the coupling matrix when switching from mode 2 to mode 1(instead of K2,1) with K1 ∈ Rn2×n1 and K2 ∈ Rn1×n2 .

Let Φq(s) = (sEq −Aq)−1, for q ∈ 1, 2, s ∈ C so that det(sEq −Aq) 6= 0. The generalizedtransfer functions corresponding to the first three levels, are written as:

Level 1

H1(s1) = C1Φ1(s1)B1 H2(s2) = C2Φ2(s2)B2,

Level 2

H1,2(s1, s2) = C1Φ1(s1)K2Φ2(s2)B2 H2,1(s2, s1) = C2Φ2(s2)K1Φ1(s1)B1,

Level 3 H1,2,1(s1, s2, s3) = C1Φ1(s1)K2Φ2(s2)K1Φ1(s3)B1,

H2,1,2(s1, s2, s3) = C2Φ2(s1)K1Φ1(s2)K2Φ2(s3)B2.

Definition 6.3.1 Consider two LSS Σ = (n1, n2, (Ei, Ai, Bi, Ci)|i ∈ M, Ki,j|i, j ∈ M,0)and Σ = (n1, n2, (Ei, Ai, Bi, Ci)|i ∈ M, Ki,j|i, j ∈ M,0) with M = 1, 2. These systemsare said to be equivalent if there exist non-singular matrices ZL

j ,ZRj ∈ Rnj×nj so that

Ej = ZLj EjZR

j , Aj = ZLj AjZR

j , Bj = ZLj Bj, Cj = CjZR

j , j ∈ 1, 2,

and also K1 = ZL2 K1ZR

1 , K2 = ZL1 K2ZR

2 . In this configuration, one can easily show that thetransfer functions defined above are the same for each LSS and for all sampling points sk.

6.3.1 The generalized controllability and observability matricesLet Σ be an LSS as described in (6.1) with dim(Σk) = nk for k = 1, 2 and let K1 ∈ Rn2×n1 andK2 ∈ Rn1×n2 be the coupling matrices. Before stating the general definitions, we first clarify howthe newly introduced matrices are constructed through a simple self-explanatory example.

Example 6.3.1 Let U be a set with ` = 12 elements, interpreted as left interpola-tion points, where U = µ1, µ2, . . . , µ12. Partition U as U = U1 ∪ U2, where U1 =µ(1)

1 , µ(1)3 , µ

(2)1 , µ

(2)3 , µ

(2)5 , µ

(3)1 and U2 = µ(1)

2 , µ(1)4 , µ

(2)2 , µ

(2)4 , µ

(2)6 , µ

(3)2 . Here, Ui contains points

124


associated to mode i. Introduce the nested multi-tuples corresponding to each mode of the LSS,as:

Mode 1 : µ(1)1 =

(µ

(1)1

)(µ

(1)2 , µ

(1)3

) , µ(2)1 =

(µ

(2)1

)(µ

(2)2 , µ

(2)3

)(µ

(2)1 , µ

(2)4 , µ

(2)5

) , µ(3)1 =

(µ

(3)1

),

Mode 2 : µ(1)2 =

(µ

(1)2

)(µ

(1)1 , µ

(1)4

) , µ(2)2 =

(µ

(2)2

)(µ

(2)1 , µ

(2)4

)(µ

(2)2 , µ

(2)3 , µ

(2)6

) , µ(3)2 =

(µ

(3)2

).

We explicitly write the generalized observability matrices O1 and O2 as follows:

O1 =

C1 Φ1(µ(1)1 )

C2 Φ2(µ(1)2 ) K1 Φ1(µ(1)

3 )C1 Φ1(µ(2)

1 )C2 Φ2(µ(2)

2 ) K1 Φ1(µ(2)3 )

C1 Φ1(µ(2)1 ) K2 Φ2(µ(2)

4 ) K1 Φ1(µ(2)5 )

C1 Φ1(µ(3)1 )

, O2 =

C2 Φ2(µ(1)2 )

C1 Φ1(µ(1)1 ) K2 Φ2(µ(1)

4 )C2 Φ2(µ(2)

2 )C1 Φ1(µ(2)

1 ) K2 Φ2(µ(2)4 )

C2 Φ2(µ(2)2 ) K1 Φ1(µ(2)

3 ) K2 Φ2(µ(2)6 )

C2 Φ2(µ(3)2 )

.

For an interpolation point µ(i)j , the subscript j is related to the mode with which the point is

associated to. This mode is given by the residue η(j), where η(j) = 1, if j is odd, and η(j) =2, if j is even. The superscript i stands for the block index. In this particular example, weconsidered three such blocks for each of the two modes with the following dimensions: p1 =2, p2 = 3 and p3 = 1 (here pi represents the dimension of the block index i for i ∈ 1, 2, 3).

Definition 6.3.2 Given a non-empty set X , denote with X i the set of all ith tuples with elementsfrom X . Introduce the concatenation of two tuples composed of elements (symbols) α1, . . . , αi,and β1, . . . , βj from X as the mapping : X i ×X j → X i+j with the following property:(

α1, α2, . . . , αi)(β1, β2, . . . , βj

)=(α1, α2, . . . αi, β1, β2, . . . βj

).

In the following we denote the `th element of the ordered set µ(i)j with µ(i)

j (`) (where j ∈ Mand i > 1). For instance, µ(2)

1 (3) :=(µ

(2)1 , µ

(2)4 , µ

(2)5

). For convenience, use the notation

H1,2,1(µ(2)1 , µ

(2)4 , µ

(2)5 ) instead of H(1,2,1)

((µ(2)

1 , µ(2)4 , µ

(2)5 )

), when referring to the function evalu-

ation:H1,2,1(µ(2)

1 , µ(2)4 , µ

(2)5 ) = C1Φ1(µ(2)

1 )K2Φ2(µ(2)4 )K1Φ1(µ(2)

5 )B1.

Definition 6.3.3 Let V = λ1, λ2, . . . , λk ⊂ C be a set composed of k right interpolation points.Partition V in two sets V1 and V2, as V = V1 ∪ V2, where:

V1 = λ(i)2g−1|1 6 g 6 mi, 1 6 i 6 k†, V2 = λ(i)

2g |1 6 g 6 mi, 1 6 i 6 k†. (6.6)

125


Here, V1 and V2 correspond to interpolation points associated to the Ist mode and, respectively,associated to the IInd mode. Let k† > 1 be a positive integer. For each i = 1, . . . , k†, introducethe blocks of right ith tuples in terms of the points from (6.6), as:

λ(i)1 =

(λ

(i)1

),(

λ(i)3 , λ

(i)2

),(

λ(i)5 , λ

(i)4 , λ

(i)1

),

...(λ

(i)2mi−3, . . . , λ

(i)4 , λ

(i)1

),(

λ(i)2mi−1, λ

(i)2mi−2, . . . , λ

(i)3 , λ

(i)2

).

, λ(i)2 =

(λ

(i)2

),(

λ(i)4 , λ

(i)1

),(

λ(i)6 , λ

(i)3 , λ

(i)2

),

...(λ

(i)2mi−2, . . . , λ

(i)3 , λ

(i)2

),(

λ(i)2mi , λ

(i)2mi−3, . . . , λ

(i)4 , λ

(i)1

).

, (6.7)

where k† is the number of blocks, and mi is the dimension of the index i block so that the equalitym1 + · · · +mk† = k holds. Finally define the nested right multi-tuples, as:

λ1 =λ

(1)1 ,λ

(2)1 , . . . ,λ

(k†)1

, λ2 =

λ

(1)2 ,λ

(2)2 , . . . ,λ

(k†)2

. (6.8)

Note that the right tuples in (6.7) are constructed based on the following recurrence relations (where λ(i)

1 (1) =(λ

(i)1

)and λ(i)

2 (1) =(λ

(i)2

))

λ(i)1 (g) =

(λ

(i)2g−1

) λ(i)

2 (g − 1), λ(i)2 (g) =

(λ

(i)2g

) λ(i)

1 (g − 1). (6.9)

Definition 6.3.4 Let U = µ1, µ2, . . . , µk ⊂ C be a set composed of k left interpolation points.Partition U in two sets U1 and U2, as U = U1 ∪ U2, where:

U1 = µ(i)2h−1|1 6 h 6 pj, 1 6 j 6 `†, U2 = µ(j)

2h |1 6 h 6 pj, 1 6 j 6 `†. (6.10)

Here, U1 and U2 correspond to interpolation points associated to the Ist mode and, respectively,associated to the IInd mode. Let `† > 1 be a positive integer. For each j = 1, . . . , `†, introducethe blocks of left jth tuples in terms of the points from (6.10), as:

µ(j)1 =

(µ

(j)1

),(

µ(j)2 , µ

(j)3

),(

µ(j)1 , µ

(j)4 , µ

(j)5

),

...(µ

(j)1 , µ

(j)4 , . . . , µ

(j)2pj−3

),(

µ(j)2 , µ

(j)3 , . . . , µ

(j)2pj−2, µ

(j)2pj−1

).

µ(j)2 =

(µ

(j)2

),(

µ(j)1 , µ

(j)4

),(

µ(j)2 , µ

(j)3 , µ

(j)6

),

...(µ

(j)2 , µ

(j)3 , . . . , µ

(j)2pj−2

),(

µ(j)1 , µ

(j)4 , . . . , µ

(j)2pj−3, µ

(j)2pj

).

, (6.11)

126


where `† > 1 is the number of blocks, and pj is the dimension of the index j block. Additionally,note that the equality p1 + · · · + p`† = `, holds. Finally define the nested left multi-tuples, as:

µ1 =µ

(1)1 ,µ

(2)1 , . . . ,µ

(`†)1

, µ2 =

µ

(1)2 ,µ

(2)2 , . . . ,µ

(`†)2

. (6.12)

Note that the left tuples are constructed based on the following recurrence relations (whereµ

(j)1 (1) =

(µ

(j)1

)and µ(j)

2 (1) =(µ

(j)2

))

µ(j)1 (h) = µ

(j)2 (h− 1)

(µ

(j)2h−1

), µ

(j)2 (h) = µ

(j)1 (h− 1)

(µ

(j)2h

). (6.13)

Condition 6.3.1 The right interpolation points λu, u ∈ 1, 2, . . . , k are chosen in such a wayso that they do not coincide with the poles of any of the subsystems Σ1 or Σ2. More exactly,the following conditions are imposed for all i = 1, . . . , k† and g = 1, . . . ,mi,

det(λ(i)2g−1E1 −A1) 6= 0, det(λ(i)

2gE2 −A2) 6= 0. (6.14)

We associate the following matrices to the set of right tuples in (6.7), as

R(i)1 =

[Φ1(λ(i)

1 ) B1, Φ1(λ(i)3 ) K2 Φ2(λ(i)

2 ) B2, . . . , Φ1(λ(i)2mi−1) K2 · · ·K1 Φ1(λ(i)

3 ) K2 Φ2(λ(i)2 ) B2

],

R(i)2 =

[Φ2(λ(i)

2 ) B2, Φ2(λ(i)4 ) K1 Φ1(λ(i)

1 ) B1, . . . , Φ2(λ(i)2mi) K1 · · ·K2 Φ2(λ(i)

2 ) K1 Φ1(λ(i)1 ) B1

],

where i = 1, . . . , k† and R(i)q ∈ Cnq×mi is attached to Λ(i)

q for q ∈ 1, 2.

Definition 6.3.5 For the LSS Σ in (6.1), introduce the generalized controllability matrices R1and R2 associated to the right multi-tuples λ1, and, respectively λ2, as follows:

R1 =[R(1)

1 , R(2)1 , . . . , R(k†)

1

]∈ Cn1×k, R2 =

[R(1)

2 , R(2)2 , . . . , R(k†)

2

]∈ Cn2×k. (6.15)

Condition 6.3.2 The left interpolation points µv, v ∈ 1, 2, . . . , ` are chosen in such a way sothat they do not coincide with the poles of any of the subsystems Σ1 or Σ2. More exactly, thefollowing conditions are imposed for all j = 1, . . . , `† and h = 1, . . . , pj,

det(µ(j)2h−1E1 −A1) 6= 0, det(µ(j)

2hE2 −A2) 6= 0. (6.16)

Next, associate the following matrices to the set of right tuples in (6.11), as

O(j)1 =

C1 Φ1(µ(j)

1 )C2 Φ2(µ(j)

2 ) K1 Φ1(µ(j)3 )

...

C2 Φ2(µ(j)2 ) K1 Φ1(µ(j)

3 ) K2 · · · K1 Φ1(µ(j)2pj−1)

∈ Cpj×n1 , j = 1, . . . , `†,

127


O(j)2 =

C2 Φ2(µ(j)

1 )C1 Φ1(µ(j)

1 ) K2 Φ2(µ(j)4 )

...

C1 Φ1(µ(j)1 ) K2 Φ2(µ(j)

2 ) K1 · · · K2 Φ2(µ(j)2pj)

∈ Cpj×n2 , j = 1, . . . , `†.

Definition 6.3.6 For the LSS Σ in (6.1), introduce the generalized observability matrices O1and O2 associated to the right multi-tuples µ1, and, respectively µ2, as follows:

O1 =

O(1)

1...

O(`†)1

∈ C`×n1 , O2 =

O(1)

2...

O(`†)2

∈ C`×n2 . (6.17)

Definition 6.3.7 For ν ∈ 1, 2, let Mν,+ and M+,ν be the ordered sets containing all tuplesthat can be constructed with symbols from the M = 1, 2 and that start (and respectively end)with the symbol ν. Also, no two consecutive characters are allowed to be the same. Hence,explicitly write the new introduced sets as follows:

M1,+ = (1), (1, 2), (1, 2, 1), . . ., M2,+ = (2), (2, 1), (2, 1, 2), . . ., (6.18)M+,1 = (1), (2, 1), (1, 2, 1), . . ., M+,2 = (2), (1, 2), (2, 1, 2), . . .. (6.19)

Remark 6.3.1 In the following we denote the `th element of the ordered setMν,+ withMν,+(`).For example, one writes M1,+(4) := (1, 2, 1, 2). Moreover, we have M+,2(3) M1,+(2) =(2, 1, 2, 1, 2).

The compact notation HM+,1(µ1(2)) is used instead of H2,1(µ2, µ3), where µ1(2) :=(µ2, µ3

).

Definition 6.3.8 Let the ith unit vector of length k be denoted with ei,k = [0 . . . , 1, . . . , 0]T ∈ Rk.Additionally, let 0k,` ∈ Rk×` be an all zero matrix.

In the following, use the notation H to emphasize that we are referring to the generalizedtransfer functions corresponding to the LSS Σ.

Definition 6.3.9 We say that an LSS Σ = (n1, n2, (Ei, Ai, Bi, Ci)|i ∈ M, Ki,j|i, j ∈M,0) matches the data associated with the right tuples λ(1)

a , . . . ,λ(k†)a as well as with

the left tuples µ(1)b , . . . ,µ

(`†)b , a, b ∈ M and corresponding to the original LSS Σ =

(n1, n2, (Ei,Ai,Bi,Ci)|i ∈M, Ki,j|i, j ∈M,0), if the following 2(k2 + 2k) relations

HM+,1(h)(µ(j)1 (h)) = HM+,1(h)(µ(j)

1 (h)), HM+,2(h)(µ(j)2 (h)) = HM+,2(h)(µ(j)

2 (h)),HM1,+(g)(λ(i)

1 (g)) = HM1,+(g)(λ(i)1 (g)), HM2,+(g)(λ(i)

2 (g)) = HM2,+(g)(λ(i)2 (g)),

HM+,1(h)M2,+(g)(µ(j)1 (h) λ(i)

2 (g)) = HM+,1(h)M2,+(g)(µ(j)1 (h) λ(i)

2 (g)),HM+,2(h)M1,+(g)(µ(j)

2 (h) λ(i)1 (g)) = HM+,2(h)M1,+(g)(µ(j)

2 (h) λ(i)1 (g)),

(6.20)

128


hold for j = 1, . . . , k†, h = 1, . . . , pj and i = 1, . . . , k†, g = 1, . . . ,mi, where

p1 + p2 + . . .+ pk† = m1 +m2 + . . .+mk† = k.

The following lemma extends the rational interpolation idea for linear systems approximation tothe linear switched system case.

Lemma 6.3.1 Interpolation of LSS. Let Σ = (n1, n2, (Ei,Ai,Bi,Ci)|i ∈ M, Ki,j|i, j ∈M,0) be an LSS of order (n1, n2). Consider that the number of left and right interpolationpoints is the same for each mode, i.e ` = k. Additionally, assume the matrices in (6.15) and(6.17) have full rank, i.e. rank(Ri) = rank(Oi) = k, i ∈ 1, 2. An order k reduced LSSΣ = (n1, n2, (Ei, Ai, Bi, Ci)|i ∈ M, Ki,j|i, j ∈ M,0) is constructed by projection, i.e. byusing right and left projectors chosen as

X1 = R1, X2 = R2 and YT1 = O1, YT

2 = O2.

The projected matrices corresponding to the first subsystem Σ1 are computed as,

E1 = YT1 E1X1, A1 = YT

1 A1X1, B1 = YT1 B1, C1 = C1X1, K1 = YT

2 K1X1, (6.21)

while the projected matrices corresponding to the second subsystem Σ2 can be computed as,

E2 = YT2 E2X2, A2 = YT

2 A2X2, B2 = YT2 B2, C2 = C2X2, K2 = YT

1 K2X2. (6.22)

It follows that the reduced-order system Σ matches the data of the system Σ (as it was previouslyintroduced in Definition 6.3.9).

Proof of Lemma 6.3.1 For simplicity, assume that we have one set of right multi-tuples,and one set of left multi-tuples with k interpolation points for each mode. This corresponds tothe case `† = k† = 1 and m1 = p1 = k (using the notations introduced in Definitions 6.3.3 and6.3.4). For the first mode, write down the interpolation points as follows: λ1 =

(λ1),(λ3, λ2

), . . . ,

(λ2k−1, · · · , λ3, λ2

),

µ1 =(µ1),(µ2, µ3

), . . . ,

(µ2, µ3, · · · , µ2k−1

).

(6.23)

For the second mode, write down the interpolation points as follows: λ2 =(λ2),(λ4, λ1

), . . . ,

(λ2k, · · · , λ2, λ1

),

µ2 =(µ2),(µ1, µ4

), . . . ,

(µ1, µ4, · · · , µ2k

).

(6.24)

129


It follows that the interpolation conditions stated in Definition 6.3.9, can be rewritten by takinginto account the aforementioned simplification as,

2k conditions:

HM+,1(j)(µ1(j)) = HM+,1(j)(µ1(j))HM+,2(j)(µ2(j)) = HM+,2(j)(µ2(j))

, j ∈ 1, . . . , k, (6.25)

2k conditions:

HM1,+(i)(λ1(i)) = HM1,+(i)(λ1(i))HM2,+(i)(λ2(i)) = HM2,+(i)(λ2(i))

, i ∈ 1, . . . , k, (6.26)

k2 conditions:HM+,1(j)M2,+(i)(µ1(j) λ2(i)) = HM+,1(j)M2,+(i)(µ1(j) λ2(i)), (6.27)

k2 conditions:HM+,2(j)M1,+(i)(µ2(j) λ1(i)) = HM+,2(j)M1,+(i)(µ2(j) λ1(i)). (6.28)

With the assumptions in (6.23) and (6.24), it follows that the associated generalized control-lability and observability matrices defined previously in (6.15) and (6.17), are rewritten as:

R1 = [ Φ1(λ1)B1, Φ1(λ3)K2Φ2(λ2)B2, . . . , Φ1(λ2k−1)K2 · · · K2Φ2(λ2)B2] ∈ Cn×k,

R2 = [ Φ2(λ2)B2, Φ2(λ4)K1Φ1(λ1)B1, . . . , Φ2(λ2k)K1 · · · K1Φ1(λ1)B1] ∈ Cn×k,

O1 =

C1Φ1(µ1)C2Φ2(µ2)K1Φ1(µ3)

...C2Φ2(µ2)K1Φ1(µ3) · · ·K1Φ1(µ2k−1)

, O2 =

C2Φ2(µ2)C1Φ1(µ1)K2Φ2(µ4)

...C1Φ1(µ1)K2Φ2(µ4) · · ·K2Φ2(µ2k)

,

with O1, O2 ∈ Ck×n. Additionally, introduce the notation Φi(s) = (sE− A)−1.From (6.21) and (6.22), using that Xi = Ri for i = 1, 2, it readily follows that:

(a) Φ1(λ1) B1 = e1,k and (b) Φ1(λ2i−1) K2 ei−1,k = ei,k, i = 2, . . . , k,

(c) Φ2(λ2) B2 = e1 and (d) Φ2(λ2i) K1 ei−1,k = ei,k, i = 2, . . . , k.

These equalities imply the right-hand conditions in (6.26). Similarly, from (6.21) and (6.22),using that YT

j = Oj for j = 1, 2, it follows that:

(e) C1 Φ1(µ1) = eT1,k and (f) eTj−1,kK2Φ2(µ2j) = eTj,k, j = 2, . . . , k,

(g) C2 Φ2(µ2) = eT1,k and (h) eTj−1K1Φ1(µ2j−1) = eTj,k, j = 2, . . . , k,

which imply left-hand conditions in (6.25). Finally, with X = R, YT = O, and combining(a)-(h), all interpolation conditions in (6.27) and (6.28) are hence satisfied.

Remark 6.3.2 For instance, in Example 6.3.2, the conditions stated in (6.43) are satisfied.

130


Sylvester equations for O and R

The motivation behind this section is closely related to building parametrized reduced ordermodels. The idea is that, one can use only one sided interpolation conditions, either left as in(6.25) or right as in (6.26), to reduce the original LSS. Then, one can choose the free parametersto impose additional conditions (not necessarily interpolatory).

Further development of this strategy was studied in Chapter 4, Section 4.3, for the case ofgeneralized Sylvester equations for bilinear systems.

The generalized controllability and observability matrices satisfy Sylvester equations. To statethe corresponding result we need to define particular quantities. First introduce the vectors

R =[eT1,m1 · · · eT1,m

k†

]∈ R1×k, LT =

[eT1,p1 · · · eT1,p

`†

]∈ R1×`, (6.29)

where m1 + . . .+mk† = k and p1 + . . .+ p`† = `. Next, introduce the block-shift matrices

SR = blkdiag[Jm1 , . . . , Jm

k†

],

SL = blkdiag[JTp1 , . . . , JTp

`†

].

where Jn =

0 1 · · · 0...

.... . .

...0 0 · · · 10 0 · · · 0

∈ Rn×n. (6.30)

Finally we arrange the left interpolation points in the diagonal matrices M1,M2 ∈ R`×` as,

M1 = blkdiag [M(1)1 , M(2)

1 , . . . , M(`†)1 ], M2 = blkdiag [M(1)

2 , M(2)2 , . . . , M(`†)

2 ], (6.31)

where M(j)1 = diag [µ(j)

1 , µ(j)3 , . . . , µ

(j)2pj−1] and M(j)

2 = diag [µ(j)2 , µ

(j)4 , . . . , µ

(j)2pj ]. Also arrange

the right interpolation points in the diagonal matrices Λ1, Λ1 ∈ Rk×k as,

Λ1 = blkdiag [Λ(1)1 , Λ(2)

1 , . . . , Λ(k†)1 ], Λ2 = blkdiag [Λ(1)

2 , Λ(2)2 , . . . , Λ(k†)

2 ], (6.32)

where Λ(i)1 = diag [λ(i)

1 , λ(i)3 , . . . , λ

(i)2mi−1] and Λ(i)

2 = diag [λ(i)2 , λ

(i)4 , . . . , λ

(i)2mi ]. The next

results represent extensions of the linear case and hence follow naturally.

Lemma 6.3.2 Consider that the assumption in Condition 4.1 holds, i.e. (6.14) is valid. Then,the generalized controllability matrices R1,R2 defined in (6.15) are the unique solutions of thefollowing coupled Sylvester equations:A1R1 + K2R2SR + B1R = E1R1Λ1,

A2R2 + K1R1SR + B2R = E2R2Λ2.(6.33)

Proof of Lemma 6.3.2 Assume again, for simplicity of the proof, that the assumptions madein (6.23)-(6.24) are valid. Hence, we have one set of right multi-tuples for each of the two modeswith same number of interpolation points k (with k even). Multiplying the first equation in(6.33) on the right with the unit vector e1,k we obtain:

A1R(1)1 + B1 = λ1E1R(1)

1 ⇔ R(1)1 = (λ1E1 −A1)−1B1 = Φ1(λ1)B1. (6.34)

131


where R(j)i is the jth column of Ri (with j 6 k and i ∈ 1, 2). Thus the first column of the

matrix which is the solution of the first equation in (6.33) is indeed equal to the first column ofthe generalized controllability matrix R1. By multiplying the second equation in (6.33) on theright with the unit vector e1,k, we obtain:

A2R(1)2 + B2 = λ2E2R(1)

2 ⇔ R(1)2 = (λ2E2 −A2)−1B2 = Φ2(λ2)B2. (6.35)

Thus the first column of the matrix which is the solution of the second equation in (6.33) isindeed equal to the first column of the generalized controllability matrix R2. By multiplyingfirst equation in (6.33) on the right with the jth unit vector ej,k, we obtain:

A1R(j)1 + K2R(j−1)

2 = λ2j−1E1R(j)1 ⇔ R(j)

1 = (λ2j−1E1 −A1)−1K2R(j−1)2 . (6.36)

By multiplying the second equation in (6.33) on the right with the jth unit vector ej,k, write:

A2R(j)2 + K1R(j−1)

1 = λ2jE2R(j)2 ⇔ R(j)

2 = (λ2jE2 −A2)−1K1R(j−1)1 . (6.37)

From (6.36) and (6.37) derive the following linear recursive system of equations:R(j)1 = Φ1(λ2j−1)K2R(j−1)

2 ,

R(j)2 = Φ2(λ2j)K1R(j−1)

1 .(6.38)

with initial conditions (6.34) and (6.35). Hence, by solving the recursive system of equations,we conclude that any solution of (6.33) is given by pairs of generalized controllability matricesdefined as in (6.15). Conversely, it automatically follows that the matrices defined in (6.15)satisfy the relations in (6.33). In this case, the assumption made in Condition 4.1 insures thatthe equations are solvable.

This proof can be straightforward adapted from the simplified case in (6.23)-(6.24) to themore general case of interpolation tuples considered in (6.7)-(6.11).

Lemma 6.3.3 Consider that the assumption in Condition 4.2 holds, i.e. (6.16) is valid. Then,the generalized observability matrices O1 and O2 defined by (6.17) satisfy the following coupledgeneralized Sylvester equations:O1A1 + SLO2K1 + LC1 = M1O1E1,

O2A2 + SLO1K2 + LC2 = M2O2E2.(6.39)

Proof of Lemma 6.3.3 Similar to the proof of Lemma 6.3.3.

6.3.2 The generalized Loewner pencilDefinition 6.3.10 Given a linear switched system Σ as defined in (6.1), let R1,R2 andO1,O2 be the controllability and observability matrices defined in (6.15), (6.17) respectively,and associated with the multi-tuples in (6.8), (6.12) respectively. The Loewner matrices L1 andL2 are defined as

L1 = −O1 E1R1, L2 = −O2 E2R2 . (6.40)

132


Additionally, the shifted Loewner matrices Ls1 and Ls2 are defined as

Ls1 = −O1 A1R1, Ls2 = −O2 A2R2. (6.41)

Also define the quantitiesW1 = C1R1

W2 = C2R2,

V1 = O1 B1

V2 = O2 B2and

Ξ1 = O2 K1R1

Ξ2 = O1 K2R2. (6.42)

Remark 6.3.3 In general, the Loewner matrices defined above need not have only real entries.For instance, it may happen that the samples points are purely imaginary values (on the jω axis).In this case, we refer the readers to Section 4.3.3 in Chapter 4. We propose a similar methodto enforce all system matrices have only real entries. In short, the sampling points have to bechosen as complex conjugate pairs; after the data is arranged into matrix format, use projectionmatrices as in equation (4.26) in [10] to multiply the matrices in (6.40), (6.41) and (6.42) to theleft and to the right. In this way, the LSS does not change as pointed out in Definition 6.3.1.

Remark 6.3.4 Note that Lk and Lsk (where k ∈ 1, 2), as defined above, are indeed Loewnermatrices, that is, they can be expressed as divided differences of appropriate transfer functionvalues of the underlying LSS (see the following example).

Example 6.3.2 Given the LSS described by (Cj,Ej,Aj,Bj) (D = 2 and j ∈ 1, 2), considerthe ordered tuples of left interpolation points:

(µ1), (µ2, µ3)

,

(µ2), (µ1, µ4)

and right

interpolation points

(λ1), (λ3, λ2),

(λ2), (λ4, λ1)

. The associated generalized observabilityand controllability matrices are computed as follows

O1 =[

C1Φ1(µ1)C2Φ2(µ2)K1Φ1(µ3)

], O2 =

[C2Φ2(µ2)

C1Φ3(µ1)K2Φ2(µ4)

],

R1 =[

Φ1(λ1)B1 Φ1(λ3)K2Φ2(λ2)B2], R2 =

[Φ2(λ2)B2 Φ2(λ4)K1Φ1(λ1)B1

].

The projected Loewner matrices can be written in terms of the samples in the following way:

L1 =

H1(µ1)−H1(λ1)µ1−λ1

H1,2(µ1,λ2)−H1,2(λ3,λ2)µ1−λ3

H2,1(µ2,µ3)−H2,1(µ2,λ1)µ3−λ1

H2,1,2(µ2,µ3,λ2)−H2,1,2(µ2,λ3,λ2)µ3−λ3

= −O1E1R1,

L2 =

H2(µ2)−H2(λ2)µ2−λ2

H2,1(µ2,λ1)−H2,1(λ4,λ1)µ2−λ4

H1,2(µ1,µ4)−H1,2(µ1,λ2)µ4−λ2

H1,2,1(µ1,µ4,λ4)−H1,2,1(µ1,λ4,λ1)µ4−λ4

= −O2E2R2.

133


The projected shifted Loewner matrices can also be written in terms of the samples as:

Ls1 =

µ1H1(µ1)−λ1H1(λ1)µ1−λ1

µ1H1,2(µ1,λ2)−λ3H1,2(λ3,λ2)µ1−λ3

µ3H2,1(µ2,µ3)−λ1H2,1(µ2,λ1)µ3−λ1

µ3H2,1,2(µ2,µ3,λ2)−λ3H2,1,2(µ2,λ3,λ2)µ3−λ3

= −O1A1R1,

Ls2 =

µ2H2(µ2)−λ2H2(λ2)µ2−λ2

µ2H2,1(µ2,λ1)−λ4H2,1(λ4,λ1)µ2−λ4

µ4H1,2(µ1,µ4)−λ2H1,2(µ1,λ2)µ4−λ2

µ4H1,2,1(µ1,µ4,λ4)−λ4H1,2,1(µ1,λ4,λ1)µ4−λ4

= −O2A2R2.

The same property applies for the Vi and Wj vectors and Ξj matrices:

V1 =[

H1(µ1)H2,1(µ2, µ3)

]= O1B1, V2 =

[H2(µ2)

H1,2(µ1, µ4)

]= O2B2,

W1 =[

H1(λ1) H1,2(λ3, λ2)]

= C1R1, W2 =[

H2(λ2) H2,1(λ4, λ1)]

= C2R2,

Ξ1 =[

H2,1(µ2, λ1) H2,1,2(µ2, λ3, λ2)H1,2,1(µ1, µ4, λ1) H1,2,1,2(µ1, µ4, λ3, λ2)

]= O2K1R1,

Ξ2 =[

H1,2(µ1, λ2) H1,2,1(µ1, λ4, λ1)H2,1,2(µ2, µ3, λ2) H2,1,2,1(µ2, µ3, λ4, λ1)

]= O1K2R2.

It readily follows that, given the original system Σ, a reduced LSS of order two can beobtained without computation (matrix factorizations or solves) as:

Ek = OER, A = OAR, N = ONR, B = OB, C = CR.

This reduced system matches sixteen moments of the original system, namely:

four of H1/H2 : H1(µ1), H2(µ2), H1(λ1), H2(λ2),three of H1,2 : H1,2(µ1, µ4), H1,2(µ1, λ2), H1,2(λ3, λ2),three of H2,1 : H2,1(µ2, µ3), H2,1(µ2, λ1), H2,1(λ4, λ1),

...one of H1,2,1,2 : H1,2,1,2(µ1, µ4, λ3, λ2),one of H2,1,2,1 : H2,1,2,1(µ2, µ3, λ4, λ1).

(6.43)

i.e. in total 2(2k + k2) = 16 moments are matched using this procedure.

Properties of the Loewner pencil

We will now show that the quantities defined earlier satisfy various equations which generalizethe ones in the linear or bilinear case.

The equations that are be presented in this section are used to automatically find the Loewnerand shifted Loewner matrices by means of solving Sylvester equations (instead of building thedivided difference matrices from the computed samples at the sampling points).

134


Proposition 6.3.1 The Loewner matrix L1 and the shifted Loewner matrix Ls1 (correspondingto mode 1) satisfy the following relations (where L,R,Λk,Mk,SL,SR are given in (6.29),(6.30)and(6.31)):

Ls1 = L1Λ1 + V1R + Ξ2SR, (6.44)Ls1 = M1L1 + LW1 + SLΞ1. (6.45)

The Loewner matrix L2 and the shifted Loewner matrix Ls2 (corresponding to mode 2) satisfythe following relations:

Ls2 = L2Λ2 + V2R + Ξ1SR, (6.46)Ls2 = M2L2 + LW2 + SLΞ2. (6.47)

Proof of Proposition 6.3.1 By multiplying the first equation in (6.33) with O1 to the left weobtain:

O1A1R1 +O1K2R2SR +O1B1R = O1E1R1Λ1 ⇒ −Ls1 + Ξ2SR + V1R = −L1Λ1,

and hence relation (6.44) is proven. Similarly we prove (6.46). By multiplying the first equationin (6.39) with R1 to the right we obtain:

O1A1R1 + SLO2K1R1 + LC1R1 = M1O1E1R1 ⇒ −Ls1 + SLΞ1 + LW1 = −M1L1,

and hence relation (6.45) is proven. Similarly we prove (6.47).

Proposition 6.3.2 The Loewner matrices L1 and L2 satisfy the following Sylvester equations:

M1L1 − L1Λ1 = (V1R − LW1) + (Ξ2SR − SLΞ1), (6.48)M2L2 − L2Λ2 = (V2R − LW2) + (Ξ1SR − SLΞ2). (6.49)

Proof of Proposition 6.3.2 By subtracting equation (6.44) from (6.45) we directly obtain(6.48) and also, by subtracting equation (6.46) from (6.47) we directly obtain (6.49).

Proposition 6.3.3 The shifted Loewner matrices Ls1 and Ls2 satisfy the following Sylvesterequations:

M1Ls1 − Ls1Λ1 = (M1V1R − LW1Λ1) + (M1Ξ2SR − SLΞ1Λ1), (6.50)M2Ls2 − Ls2Λ2 = (M2V2R − LW2Λ2) + (M2Ξ1SR − SLΞ2Λ2). (6.51)

Proof of Proposition 6.3.3 By subtracting equation (6.44) after being multiplied with M1to the left from equation (6.45) after being multiplied with Λ1 to the right, we directly obtain(6.50). Similar procedure is applied to prove (6.51).

Remark 6.3.5 The right hand side of the equations (6.48) - (6.51) contains constant 0, 1matrices (i.e. R,L,SR,SL) as well as matrices (i.e. Vj,Wj,Ξj, j ∈ 1, 2) which are directlyconstructed by putting together the given samples values as presented in Example 6.3.2.

135


6.3.3 Construction of reduced order modelsAs we already noted, the interpolation data for the LSS has different format than the one usedfor the linear case without switching, as higher order transfer function values are matched asshown in the previous sections. However, the procedure is similar to the one presented in Lemma3.3.2 from Chapter 3, Section 3.3.3.

Lemma 6.3.4 Assume that k = ` and that the interpolation points are chosen to satisfy theconditions in (6.14) and (6.16). Moreover, assume that the Loewner matrices L1 and L2 to beinvertible. Then, a realization of a reduced order LSS Σ that matches the data of the originalLSS Σ (as introduced in Definition 6.3.9) is given by the following matrices,E1 = −L1, A1 = −Ls1, B1 = V1, C1 = W1,

E2 = −L2, A2 = −Ls2, B2 = V2, C2 = W2and K1 = Ξ1, K2 = Ξ2.

If k = n, then the proposed realization is equivalent to the original one (as in Definition 6.3.1).

Proof of Lemma 6.3.4 This result directly follows from Lemma 6.3.1 by taking into consider-ation the notations introduced in (6.40-6.42).

In the case of redundant data, at least one of the pencils (Lsj, Lj) is singular (for j ∈ 1, 2),and hence construct pairs of projectors (Xj,Yj) (corresponding to mode j) similar to the linearand bilinear cases. The MOR procedure for approximate data matching is presented as follows.Procedure 6.1 Consider the rank revealing singular value factorization of the following matricescomposed of the Loewner and shifted Loewner matrices corresponding to mode j ∈ 1, 2, as

[Lj Lsj

]=[

Y(1)j Y(1)

j

] [ Σ(1)j OO Σ(1)

j

] [X(1)j X(1)

j

]T= Y(1)

j Σ(1)j (X(1)

j )T + Y(1)j Σ(1)

j (X(1)j )T

[LjLsj

]=[

Y(2)j Y(2)

j

] [ Σ(2)j OO Σ(2)

j

] [X(2)j X(2)

j

]T= Y(2)

j Σ(2)j (X(2)

j )T + Y(2)j Σ(2)

j (X(2)j )T ,

(6.52)

where Y(i)j ,X

(i)j ∈ Rk×rj and Σ(i)

j ∈ Rrj×rj for i ∈ 1, 2 . The projected system matricescorresponding to subsystem Σj are computed as,

Ej = −(Y(1)j )TLjX(2)

j , Aj = −(Y(1)j )TLsjX

(2)j , Bj = (Y(1)

j )TVj, Cj = WjX(2)j , for j ∈ 1, 2

Moreover, the projected coupling matrices are computed in the following way

K1 = (Y(1)2 )TΞ1X(2)

1 , K2 = (Y(1)1 )TΞ2X(2)

2 .

By choosing rj as the numerical rank of the Loewner matrix Lj (i.e. the largest neglected singularvalue corresponding to index rj + 1 is less than machine precision ε), ensure that the Ej matricesare not singular. Hence, construct a reduced order LSS denoted with Σ, that approximatelymatches the data of the original LSS Σ. If the truncated singular values are all 0 (the ones onthe main diagonal of the matrices Σ(i)

j ), then the matching is exact.

136

6.4 The Loewner framework for linear switched systems - the general case

We provide a qualitative rather than quantitative result for the projected Loewner case. Thequality of approximation is directly linked to the singular values of the Loewner pencils whichrepresent an indicator of the desired accuracy. For linear systems with no switching, an errorbound is provided in [12] as a quantitative measure.

The dimensions of the subsystems Σ1 and Σ2, corresponding to the reduced order LSS, neednot be the same (i.e. r1 6= r2). In this case the coupling matrices are not square anymore.

The projectors are computed via singular value factorization of the Loewner matrices.

Remark 6.3.6 For an error bound that links the quality of approximation to the singular valuesof the Loewner pencil (which is valid only at the interpolation points µj and λi), we refer thereaders to [12].

6.4 The Loewner framework for linear switched systems- the general case

In this section we are mainly concerned with generalizing some of the results presented in Section6.3. Most of the findings can be smoothly extended to the cases with more complex switchingpatterns (more modes). By enforcing a prefix/suffix closure structure in the proposed framework,we can show that all interpolation conditions can be written in matrix equation format.

Definition 6.4.1 Let Γ and Θ be finite sets of tuples so that Γ,Θ ⊆∞⋃k=1Mk ×Ck so that Γ has

the prefix closure property, i.e.

(q1, q2, . . . , qi, λ1, . . . , λi) ∈ Γ⇒ (q2, . . . , qi, λ2, . . . , λi) ∈ Γ ∀i > 2,

and Θ has the suffix closure property, i.e.

(q1, q2, . . . , qj, µ1, . . . , µj) ∈ Θ⇒ (q1, . . . , qj−1, µ1, . . . , µj−1) ∈ Θ ∀j > 2.

Now consider the specific subset Γq (for any q ∈M) of the set Γ, defined in the following way:

Γq = (q1, q2, . . . , qi, λ1, . . . , λi) ∈ Γ | q1 = q, i 6 δΓ, δΓ = max(|w|)w∈Γ

/2.

Denote the cardinality of Γq with kq = card(Γq) and explicitly enumerate the elements of this setas follows: Γq = w(1)

q , w(2)q , . . . , w(kq)

q . Consider the following function (mapping) r : Γq → Cnq×1

that maps a word form Γq into a column vector of size nq:

r((q, q2, . . . , qi, λ1, . . . , λi)) = Φq(λ1)Kq2,qΦq2(λ2) · · ·Kqi,qi−1Φqi(λi)Bqi .

Next, construct the controllability matrix Rq corresponding to the mode q of the system Σ asfollows:

Rq =[

r(w(1)q ) r(w(2)

q ) · · · r(w(kq)q )

]∈ Cnq×kq . (6.53)

Similarly, define the specific subset Θq (for any q ∈M) of the set Θ in the following way:

137


Θq = (q1, q2, . . . , qj, µ1, . . . , µj) ∈ Γ | qj = q, j 6 δΘ, δΘ = max(|w|)w∈Θ

/2.

Consider the cardinality of Θq to be the same as the one of Γq, i.e. kq = card(Θq). Althoughthis additional constraint is not necessarily needed, we would like to enforce the construction ofreduced systems with square matrices Ak and Ek. Next we explicitly enumerate the elements ofthis set as follows: Θq = v(1)

q , v(2)q , . . . , v(kq)

q . Consider the following mapping o : Θq → C1×nq

that maps a word form Θq into a row vector of size nq:

o((q1, q2, . . . , qj−1, q, µ1, . . . , µj)) = Cq1Φq1(µ1)Kq2,q1Φq2(µ2) · · ·Kq,qj−1Φq(µj).

Next, construct the observability matrix Oq ∈ Ckq×nq corresponding to the mode q of the systemΣ as follows

Oq =[

o(v(1)q )T o(v(2)

q )T · · · o(v(kq)q )T

]T∈ Ckq×nq . (6.54)

Consider the following example to show how the general procedure is extended from the linearcase (no switching) to the case when switching occurs.

Example 6.4.1 Take D = 3 (3 active modes) and hence M = 1, 2, 3. The following inter-polation points are given: s1, s2, . . . , s18 ⊂ C. The first step is to partition this set into twodisjoint subsets (each having 9 points):

left interpolation points : µ1, µ2, . . . , µ9, right interpolation points : λ1, λ2, . . . , λ9.

The set Γ is composed of three subsets Γ = Γ1⋃Γ2

⋃Γ3 which are defined by imposing thepreviously defined suffix closure property, as

Γ1 = (1, λ1), (1, 3, λ4, λ3), (1, 3, 2, λ7, λ6, λ2),Γ2 = (2, λ2), (2, 1, λ5, λ1), (2, 1, 3, λ8, λ4, λ3),Γ3 = (3, λ3), (3, 2, λ6, λ2), (3, 2, 1, λ9, λ5, λ1).

To the sets Γj, we attach the following controllability matrices Rj

R1 =[

Φ1(λ1) Φ1(λ4)K3,1Φ3(λ3)B3 Φ1(λ7)K3,1Φ3(λ6)K2,3Φ2(λ2)B2],

R2 =[

Φ2(λ2) Φ2(λ5)K1,2Φ1(λ1)B1 Φ2(λ8)K1,2Φ1(λ4)K3,1Φ3(λ3)B3],

R3 =[

Φ3(λ3) Φ3(λ6)K2,3Φ2(λ2)B2 Φ3(λ9)K2,3Φ2(λ5)K1,2Φ1(λ1)B1].

In the same manner, the set Θ is composed of three subsets Θ = Θ1⋃Θ2

⋃Θ3. These are definedby means of imposing the previously defined prefix closure property, as follows

Θ1 = (1, µ1), (3, 1, µ3, µ4), (1, 2, 1, µ1, µ5, µ7),Θ2 = (2, µ2), (1, 2, µ1, µ5), (2, 3, 2, µ2, µ6, µ8),Θ3 = (3, µ3), (2, 3, µ2, µ6), (3, 1, 3, µ3, µ4, µ9).

138


To the sets Θi, we attach the observability matrices Oi defined as follows:

O1 =

C1Φ1(µ1)C3Φ3(µ3)K1,3Φ1(µ4)

C1Φ1(µ1)K2,1Φ2(µ5)K1,2Φ1(µ7)

, O2 =

C2Φ2(µ2)C1Φ1(µ1)K2,1Φ2(µ5)

C2Φ2(µ2)K3,2Φ3(µ6)K2,3Φ2(µ8)

,

O3 =

C3Φ3(µ3)C2Φ2(µ2)K3,2Φ3(µ6)

C3Φ3(µ3)K1,3Φ1(µ4)K3,1Φ3(µ9)

.

6.4.1 Sylvester equations for Rq and OqIn this section we would like to generalize the results presented in Lemmas 6.3.2 and 6.3.4, andhence extend the framework to a general number of operational modes denoted with D.

Definition 6.4.2 Introduce the special concatenation of tuples composed of mixed elements (sym-bols) from the sets M and C, as the mapping with the following property:

(α1 β1

)(α2 β2

)=((α1 α2

) (β1 β2

)),

where αk ∈Mik and βk ∈ Cjk for ik, jk > 1 and k = 1, 2.

Definition 6.4.3 For g, i = 1, . . . , D, let S(g)i =

[S(g)i (1) . . . S(g)

i (kg)]∈ Rki×kg be constant

matrices that contain only 0/1 entries constructed so that S(g)i (1) = 0ki,1 and for u = 2, . . . , kg,

we write:

S(g)i (u) =

eu−1,ki , if ∃ λ ∈ C, s.t. w(u)g = (g, λ)w(u−1)

i ,

0ki,1, else. (6.55)

Also, introduce the matrices R(i) and Λi that are defined similarly as in (6.29) and (6.32),

R(i) =[eT1,m1 · · · eT1,m

k†

]∈ R1×ki , Λi = blkdiag [Λ(1)

i , Λ(2)i , . . . , Λ(`†)

i ] ∈ Rki×ki , (6.56)

where the diagonal matrices Λ(a)i , a = 1, . . . , k† contain the right interpolation points associated

to mode i. For a set Γ with general structure (as in Defintion 5.1), it follows that the control-lability matrices Ri ∈ Rni×ki , 1 6 i 6 D satisfy the following system of generalized Sylvesterequations:

A1R1 +D∑i=1

Ki,1RiS(1)i + B1R(1) = E1R1Λ1,

A2R2 +D∑i=1

Ki,2RiS(2)i + B2R(2) = E2R2Λ2,

...

ADRD +D∑i=1

Ki,DRiS(D)i + BDR(D) = EDRDΛD.

(6.57)

139


Note that S(i)i = 0ki,ki , and if k1 = k2 = · · · = kD = k, the above defined matrices S(g)

i satisfythe following equality ∀g ∈M :

D∑i=1

S(g)i = blkdiag

[Jm1 , . . . ,Jmk†

]. (6.58)

where Jl is the Jordan block of size l defined in (6.30).To directly find Rg, g = 1, 2, 3 for the case presented in Example 6.4.1, we have to solve the

following system of coupled generalized Sylvester equationsA1R1 + K3,1R3S(1)

3 + B1R = E1R1Λ1,

A2R2 + K1,2R1S(2)1 + B2R = E2R2Λ2,

A3R3 + K2,3R2S(3)2 + B3R = E3R3Λ3.

where:

Λ1 =

λ1 0 00 λ4 00 0 λ7

, Λ2 =

λ2 0 00 λ5 00 0 λ8

, Λ3 =

λ3 0 00 λ6 00 0 λ9

,

R =[

1 0 0], S(1)

3 = S(2)1 = S(3)

2 =

0 1 00 0 10 0 0

.This corresponds to the case k1 = k2 = k3 = 3, k† = 1 and m1 = 3.

Definition 6.4.4 For h, j = 1, . . . , D, let T(h)j =

[ (T(h)j

)T(1) . . .

(T(h)j

)T(kh)

]T∈ R`h×`j be

constant matrices that contain only 0/1 entries constructed so that(T(h)j

)T(1) = 0`j ,1 and for

v = 2, . . . , kg, we write:

(T(h)j

)T(v) =

ev−1,kj , if ∃ µ ∈ C, s.t. w(v)h = w(v−1)

j (h, µ),0`j ,1, else

. (6.59)

Also, introduce the following matrices(L(j)

)T=[eT1,p1 · · · eT1,p

`†

]∈ R1×`j , Mj = blkdiag [M(1)

j , M(2)j , . . . , M(`†)

j ] ∈ R`j×`j , (6.60)

where the diagonal matrices M(v)j for v = 1, . . . , `j contain the left interpolation points associated

to mode j. For a set Θ with general structure (as in Defintion 5.1), one can conclude that theobservability matrices Oj ∈ R`j×nj , 1 6 j 6 D satisfy the following system of generalized

140


Sylvester equations:

O1A1 +D∑j=1

T(1)j OjK1,j + L(1)C1 = M1O1E1,

O2A2 +D∑j=1

T(2)j OjK2,j + L(2)C2 = M2O2E2,

...

ODAD +D∑j=1

T(D)j OjKD,j + L(D)CD = MDODED.

(6.61)

Note that T(j)j = 0`j ,`j , and if `1 = `2 = · · · = `D = `, the square matrices T(h)

j ∈ R`×` satisfythe following equality, ∀h ∈M:

D∑j=1

T(h)j = blkdiag

[Jp1 , . . . ,Jp`†

]T. (6.62)

Again to find the matricesOh, h = 1, 2, 3 in Example 6.4.1, it is required to solve the followingsystem of coupled generalized Sylvester equations

O1A1 + T(1)3 O3K1,3 + T(1)

2 O2K1,2 + LC1 = M1O1E1,

O2A2 + T(2)1 O1K2,1 + T(2)

3 O3K2,3 + LC2 = M2O2E2,

O3A3 + T(3)2 O2K3,2 + T(3)

1 O1K3,1 + LC3 = M3O3E3.

where:

M1 =

µ1 0 00 µ4 00 0 µ7

, M2 =

µ2 0 00 µ5 00 0 µ8

, M3 =

µ3 0 00 µ6 00 0 µ9

,

T(1)3 = T(2)

1 = T(3)2 =

0 0 01 0 00 0 0

, T(1)2 = T(2)

3 = T(3)1 =

0 0 00 0 00 1 0

, L = e1,3.

This corresponds to the case `1 = `2 = `3 = 3, `† = 1 and p1 = 3. Note that the relation in(6.62) hold, i.e. T(1)

2 + T(1)3 = T(2)

1 + T(2)3 = T(3)

1 + T(3)2 = JT3 .

Throughout this section, we chose cycling switching to make the exposition more comprehen-sible and to directly relate it to the case D = 2. The cyclic property of the switching scenarioscomes naturally in many applications (for example in the boost power converter model in [17],Section 5.2), but does not necessarily need to be enforced in our framework (Definition 6.4.1 isnot constrained to only such type of switching). Consequently, in Example 5.1, the constructionof the sets Γk is indeed cyclic, while the sets Θk do not posses this property.

141


6.4.2 The Loewner matricesFor the case of linear switched systems with D active modes, the generalization of the Loewnerframework includes one important feature. Instead of only one pair of Loewner matrices (as inthe linear case without switching which is covered in Chapter 3, Section 3.3), we define a pair ofLoewner matrices for each individual active mode; hence in total D pairs of Loewner matrices.

Definition 6.4.5 Given a linear switched system Σ, let Ri|i ∈ M and Oj|j ∈ M bethe controllability and observability matrices associated with the multi-tuples Γi and Θj. TheLoewner matrices Li| i ∈M are defined as

L1 = −O1 E1R1, L2 = −O2 E2R2, . . . , LD = −OD EDRD, (6.63)

Additionally, the shifted Loewner matrices Lsi| i ∈M are defined as

Ls1 = −O1 A1R1, Ls2 = −O2 A2R2, . . . , LsD = −OD ADRD. (6.64)

Also introduce the matrices ∀i, j ∈M

Wi = CiRi, Vj = Oj Bj, and Ξi,j = Oj Ki,jRi.

Remark 6.4.1 The number of Loewner matrices, shifted Loewner matrices, Wi row vectors andVj column vectors is the same as the number of active modes (i.e D). On the other hand, thenumber of matrices Ξi,j increases quadratically with D (i.e in total D2 matrices).

Remark 6.4.2 Note that the matrices Li and Lsi as defined in (6.63) and (6.64) (for i ∈1, 2, . . . , D) are indeed Loewner matrices, that is, they can be expressed as divided differencesof generalized transfer function values of the underlying LSS.

Proposition 6.4.1 The Loewner matrices Lh satisfy the following Sylvester equations:

MhLh − LhΛh = (VhR − LWh) +D∑j=1

(Ξj,hS(h)

j −T(h)j Ξh,j

), h ∈M. (6.65)

Proposition 6.4.2 The shifted Loewner matrices Lsh satisfy the following Sylvester equations:

MhLsh − LshΛh = (MhVhR − LWhΛh) +D∑j=1

(MhΞj,hS(h)

j −T(h)j Ξh,jΛh

), h ∈M. (6.66)

Remark 6.4.3 The proof of the results stated in (6.65)-(6.66) is performed in a similar manneras for the results obtained for the special case D = 2 in Section 6.3 (i.e. for (6.48)-(6.51)).

6.4.3 Construction of reduced order modelsThe general procedure for the case with D switching modes is more or less similar to the onecovered in Section 6.3.3 (where D = 2).

142


Lemma 6.4.1 Let Lj be invertible matrices for 1 6 j 6 D, such that none of the interpolationpoints λi, µk are eigenvalues of any of the Loewner pencils (Lsj, Lj). Then, the matrices

Ej = −Lj, A1 = −Lsj, Bj = Vj, Cj = Wj, Ki,j = Ξi,j, i, j ∈ 1, . . . , D,

form a realization of a reduced order LSS Σ that matches the data of the original LSS Σ. Ifkj = nj for 1 6 j 6 D, the proposed realization is equivalent to the original one.

The concept of an LSS matching the data of another LSS in the case D > 2 is formulated in asimilar manner as to the case D = 2, which is covered in Definition 6.3.9. Also, the definition ofequivalent LSS for the case D > 2 is formulated similarly as to Definition 6.3.1.

In the case of redundant data, at least one of the pencils (Lsj, Lj) is singular (for j ∈1, . . . , D). The main procedure is presented as follows.Procedure 6.2 Consider the rank revealing singular value factorization of the matrices com-posed of the Loewner matrices Lj and of the shifted Loewner matrices Lsj as in (6.52) , this timefor j ∈ 1, . . . , D. Again, X(`)

j ,Y(`)j ∈ Rkj×rj , Σ(`)

j ∈ Rrj×rj , j = 1, . . . , D and ` = 1, 2.Here, choose rj as the numerical rank of the Loewner matrix Lj (i.e. the largest neglected singu-lar value corresponding to index rj + 1 is less than machine precision ε). The projected systemmatrices computed as

Ej = −(Y(1)j )TLjX(2)

j , Aj = −(Y(1)j )TLsjX

(2)j , Bj = (Y(1)

j )TVj, Cj = WjX(2)j , for j ∈ 1, . . . , D,

and the projected coupling matrices computed in the following way

Ki,j = (Y(1)j )TΞi,jX(2)

i , ∀i, j ∈ 1, . . . , D,

form a realization of a reduced order LSS denoted with Σ that approximately matches the dataof the original LSS Σ. Each reduced subsystem Σj has dimension rj, j ∈ 1, . . . , D.

Remark 6.4.4 If the truncated singular values are all 0 (the ones on the main diagonal of thematrices Σj), then the interpolation is exact.

6.5 Numerical experimentsIn this section we illustrate the new method by means of three numerical examples. We usea certain generalization of the balanced truncation (BT) method for LSS (as presented in [94]to compare the performance of our new introduced method. The main ingredient of the BTmethod is to compute the the controllability and observability Gramians Pi and Qi (wherei ∈ 1, 2, . . . , D) as the solutions of the following Lyapunov equations:

AiPiETi + EiPiAT

i + BiBTi = 0, (6.67)

ATi QiEi + ET

i PiAi + CTi Ci = 0. (6.68)

143


6.5.1 The balanced truncation method proposed in [94]In [94] it was shown that, if certain conditions are satisfied, the technique of simultaneous bal-anced truncation can be applied to switched linear systems. In some special cases, the existenceof a global transformation matrix Vbal is guaranteed, provided that (Corollary IV.3 in [94]):

1. The matrices PiQi and PjQj commute for all i, j ∈ 1, 2, . . . , D;

2. The conditions PiQj = PjQi are satisfied for all i, j ∈ 1, 2, . . . , D.

Hence, it follows that:VbalPiVT

bal = V−TbalQiV−1bal = Ui, (6.69)

where Ui are diagonal matrices. Although conceptually attractive as a MOR method, in generalthe conditions are rather restrictive in practice. This motivates the search for a more generalMOR approach for the case where simultaneous balancing cannot be achieved.

The problem of finding a balancing transformation for a single linear system can be formulatedas finding a nonsingular matrix such that the following cost function is minimized (see [4]):

f(V) = trace[VPVT + V−TQV−1]. (6.70)

For the class of LSS with distinct operational modes, we hence have to minimize not one but anumber of D cost functions:

fi(V) = trace[VPiVT + V−TQiV−1], i ∈ 1, 2, . . . , D. (6.71)

If the conditions of Corollary IV.3 from [94] hold, simultaneous balancing is possible, and thereexists a transformation V which simultaneously minimizes fi for all i = 1, 2, . . . , D. Instead ofhaving D functions as in (6.71), one can introduce a single overall cost function (i.e the averageof the cost functions of the individual modes). Define the function fav as in [94]:

fav(V) = 1D

D∑i=1

trace[VPiVT + V−TQiV−1] = trace[VPavVT + V−TQavV−1], (6.72)

wherePav = 1

D

D∑i=1Pi, Qav = 1

D

D∑i=1Qi. (6.73)

In the case of LSS, the BT method computes a basis where the sum of the eigenvalues of Pi andQi over all modes is minimal. Hence, minimizing the proposed overall cost function provides anatural extension of classical BT to the case of LSS.

It follows that the transformation V that minimizes the cost function in (6.72) is preciselythe one which balances the pair (Pav,Qav) of average Gramians.

By applying V to the individual modes and truncating, a reduced order model is obtained.After applying the transformation V, the new state space representations of the individual modesneed not be balanced. Nevertheless, as stated in [94], it is expected to be relatively close to beingbalanced. Moreover, a downside of this method is given by the fact that it does not allow differentstate-space dimensionality for different modes.

144


6.5.2 First exampleAs first example we consider the simple model of an evaporator vessel from [95]. There is aconstant inflow of liquid into a tank as well as an outflow that depends on the pressure in thetank and the Bernoulli resistance Rb. To keep the level of fluid in the evaporator vessel at orbelow a pre-specified maximum, an overflow mechanism is activated when the level of fluid Lin the evaporator exceeds the threshold value Lth. This causes a flow through a narrow pipewith resistance Rp and inertia I that builds up flow momentum p. The system is modeled intwo distinct operation modes: mode 1, where there is no overflow (the fluid level is below theoverflow level), and mode 2, where the overflow mechanism is active. The ordinary differentialequations describing the system in the two operation modes are given by[

I 00 C

] [pL

]=[−Rp 0

0 −1/Rb

] [pL

]+[

0fin

](mode 1),[

I 00 C

] [pL

]=[−Rp 1−1 −1/Rb

] [pL

]+[

0fin

](mode 2).

Additionally, note that the observed output y is chosen to be the average of the two systemvariables p and L, i.e. y = (p+L)/2 (for both modes). Assuming the system is initially in mode1, the inflow then causes the tank to start filling, which causes an outflow through resistance Rb.In this mode the outflow through the narrow pipe is zero. If L exceeds the level Lth, a switchfrom mode 1 to mode 2 occurs at the point in time when L = Lth.

Figure 6.1: Schematic of the evaporator vessel [95].

In the following, use the parameters Rb = 1, Rp = 0.5, I = 1, C = 1, fin = 1, Lth = 0.08 andcompute the following system matrices:

Mode 1 : E1 =[

1 00 1

], A1 =

[−1

2 00 −1

], B1 =

[01

], C1 =

[12

12

],

Mode 2 : E2 =[

1 00 1

], A2 =

[−1

2 1−1 −1

], B2 =

[01

], C2 =

[12

12

].

145


The coupling matrices are chosen to be identity matrices, i.e. K1 = K2 = I2. Next, consider thefollowing tuples of left and right interpolation points corresponding to each mode, asλ1 = (−1.5), (−2, 1),

µ1 = (2), (0, 0.5).,

λ2 = (1), (1.5,−1.5),µ2 = (0), (2,−0.5).

.

Hence, following the procedure described in Section 6.3, we recover the following system matrices:

Mode 1 : E1 =[−1

3 −23240

−23 −1

8

], A1 =

[ 13

19240

23

18

], B1 =

[ 1613

], C1 =

[−1 −13

48

],

Mode 2 : E2 =[ 3

16 −13

7120 −

19

], A2 =

[− 5

1612

− 17120

730

], B2 =

[ 1215

], C2 =

[516 −

12

].

Note that the recovered realization is equivalent to the original one (no reduction has beenenforced since the task was to recover the initial system only). The coupling matrices are alsocomputed:

K1 =[−1 − 3

16−2

5 −23360

], K2 =

[ 980 −

845

18 −2

9

].

6.5.3 Second exampleFor the next experiment, consider the CD player system from the SLICOT benchmark examplesfor MOR (see [40]). This linear system of order 120 has two inputs and two outputs. We considerthat, at any given instance of time, only one input and one output are active (the others are notfunctional due to mechanical failure). More precisely, consider mode j to be activated wheneverthe jth input and the jth output are simultaneously failing (where j ∈ 1, 2).

In this way, we construct an LSS with two operational modes (hence D = 2). Both subsystemsare stable SISO linear systems of order 120. This initial linear switched system Σ will be reducedby means of the Loewner framework to obtain ΣL and by means of the balanced truncationmethod proposed in [94] to obtain ΣB. Denote with ΣLj and ΣBj the jth linear subsystemcorresponding to ΣL, and respectively, to ΣB.

The frequency response of each original subsystem is depicted in Fig. 6.2. Note that, anamplitude scaling of the original system was performed, in order to enforce a similar gain foreach of the two subsystems.

100 101 102 103 104 105 106

Frequency(ω)

10-10

10-5

100

Frequency response of the original LSS

Ist subsystem

IInd subsystem

Figure 6.2: Frequency response of the original subsystems.

146


For the Loewner method, we choose 120 logarithmically distributed interpolation points inthe interval [101, 105]j. In Fig. 6.3 (a), we depict the singular value decay of the appendedLoewner matrices, i.e. [Lj Lsj], j ∈ 1, 2. These matrices are used in Procedure 6.1, i.e. in(6.52). Note that the Loewner matrices Lj and the shifted Loewner matrices Lsj are definedas in Section 6.3.2. Additionally, Fig. 6.3 (a) depicts the decay of the Hankel singular values ofthe averaged Gramians corresponding to Σ, as defined in (6.73). We observe that the 80th valueattains machine precision (ε ≈ 10−16) in the case of the Loewner matrices, while in the case ofthe balancing procedure, the same truncation order provides a 10−9 decay. The decay presentedin Fig. 6.3 is a good indicator for choosing the desired truncation order.

10 20 30 40 50 60 70 80 90 100 110 120

10-15

10-10

10-5

100Singular value decay

Mode 1 (LSS)Mode 2 (LSS)Averaged Gramians

(a) Appended Loewner matrices + averaged Gramians.

0 10 20 30 40 50 60 70 80

10-15

10-10

10-5

100

Singular value decay

Mode 1 (LSS)Mode 2 (LSS)Mode 1 (no switch)Mode 2 (no switch)

(b) Appended Loewner matrices.

Figure 6.3: Decay of the singular values for different matrices.

For both reduced order systems, ΣL and ΣB, we decide to truncate at order k1 = k2 = 27 forthe two subsystems. This choice was made so that the neglected singular values correspondingto all three curves in Fig. 6.3 (a) are less than the chosen tolerance value, i.e. ε1 = 10−6. Morespecifically, this corresponds to the following values for the 27th singular value: σ(1)

27 = 1.5498·10−7

for the first Loewner subsystem and σ(2)27 = 8.5741 · 10−8 for the second Loewner subsystem. The

last kept singular value corresponding to the averaged balanced model is higher, i.e. σ(3)27 =

1.5516 · 10−6. Hence, we would like to emphasize that the singular value decay is faster forLoewner compared to BT and thus certain errors can be achieved with lower order models.

Additionally, in Fig. 6.3 (b) we present the singular value decay of appended Loewner matrices[Lj Lsj], j ∈ 1, 2, defined as in Section 6.3.2 (the LSS case for which samples of higherorder transfer functions are used in the process). The first and second curves correspond tothese quantities. In the same figure, we present the singular value decay of appended Loewnermatrices defined as in Section 3.3.2 (the classical linear case with no switching and in whichonly samples of first order linear transfer functions are used in the process). The third and forthcurves correspond to these quantities. Note that, the singular value decay of the latter matricesis slightly slower than that of the first matrices.

We assess the performance the two MOR methods mentioned above in the following ways:depicting the frequency domain simulation error, computing the H2 and H∞ norms of the errorsubsystems and by depicting the time domain simulation error.

First, start by comparing the quality of approximation of the frequency response. In Fig. 6.4the frequency domain error is depicted for both MOR methods (Loewner and BT).

Observe that the error curve corresponding to the Loewner method is lower than that of theBT method for most of the frequency points considered in this experiment. This behavior is

147


100 101 102 103 104 105 106

Frequency(ω)

10-8

10-6

10-4

Error in frequency domain - mode 1

LoewnerBT

100 102 104 106

Frequency(ω)

10-7

10-5

Error in frequency domain - mode 2LoewnerBT

Figure 6.4: Frequency domain approximation error.

especially noticeable in the low frequency range. If one compares the maximum of the error, i.e.the H∞ norm, then one can notice that for mode 1 the Loewner method produces a higher peakerror, while for mode 2, the Loewner method produces a lower peak error.

Next, we explicitly compute the relative approximation errors for each mode individually, forboth reduction methods, with respect to the H2 norm. More specifically, ‖ΣLj −Σj‖H2/‖Σj‖H2

for the Loewner method, and ‖ΣBj − Σj‖H2/‖Σj‖H2 where j ∈ 1, 2 for the BT method.Additionally, we compute the H∞ norm relative errors, i.e. ‖ΣLj − Σj‖H∞/‖Σj‖H∞ for theLoewner method, and ‖ΣBj −Σj‖H∞/‖Σj‖H∞ where j ∈ 1, 2 for the BT method. The resultsare presented in Tab. 6.1. Note that, for mode 1, the balanced truncation method producesslightly lower errors, while for mode 2, the errors corresponding to the Loewner method arelower.

H2 Loewner Bal TruncMode 1 3.9210 · 10−5 2.0303 · 10−5

Mode 2 9.9040 · 10−6 2.2922 · 10−5

H∞ Loewner Bal TruncMode 1 3.3888 · 10−5 5.5266 · 10−6

Mode 2 2.6036 · 10−7 6.3798 · 10−7

Table 6.1: Relative approximation error for the two modes in the H2 and H∞ norms

Finally, we compare the time domain response of the original linear switched system againstthe ones corresponding to the two reduced models. We first use a simple sinusoidal signal, i.e.u(t) = cos(t)/10 as the control input. The piecewise-constant signal in the upper part of Fig. 6.5(a) represents the switching signal. The switching times are chosen within the interval [0,10]seconds so that fast switching is also exhibited. More precisely, note that in the range [5.5,7]s,this signal switches between modes 1 and 2 much more frequently then in the rest of the timeaxis.

As it can be seen in the lower part of Fig. 6.5 (a), the output of the LSS is well approximatedfor both MOR methods (all three curves are indistinguishable from one another). The bluecircles located on the observed output curve in the lower part of Fig. 6.5 (a) are used to markthe switching times.

Finally, by inspecting the time domain error between the original response and the responsescoming from the two reduced models (depicted in Fig. 6.5 (b)), we notice that the error curvecorresponding to the Loewner method is two orders of magnitude below the error curve corre-sponding to the BT method for most of the points on the time axis.

148


0 1 2 3 4 5 6 7 8 9 10

1

1.5

2

Switching signal σ(t)

0 1 2 3 4 5 6 7 8 9 10Time(t)

-2

0

2Time domain simulation - Output

Original LSSReduced LSS - LoewnerReduced LSS - BT

(a) The switching signal and the output.

0 1 2 3 4 5 6 7 8 9 10Time(t)

10-12

10-10

10-8

10-6

10-4

Error

Output error in time domain

LoewnerBT

(b) Approximation error of the output signal.

Figure 6.5: Time domain simulation - first choice of input.

Additionally, we repeat the time domain simulation experiment depicted in Fig. 6.5, by con-sidering another control input signal. Let u(t) = (∑10

k=1 sin(ωkt))/10 be a richer frequency spec-trum signal, where the frequency points ωk are logarithmically spaced in the interval [1, 50]. Wealso choose another switching signal σ(t), with random switching times (and with no particularimposed conditions to the frequency of switching). In Fig. 6.6 we depict the switching signal, theobserved output and the approximation error. Similar conclusions to the ones mentioned abovefor the results in Fig. 6.5 can be drawn by inspecting Fig. 6/6.

0 1 2 3 4 5 6 7 8 9 10

1

1.5

2

Switching signal σ(t)

0 1 2 3 4 5 6 7 8 9 10Times(t)

-5

0

5

Time domain simulation - OutputOriginal LSSReduced LSS - LoewnerReduced LSS - BT

(a) The switching signal and the output.

0 1 2 3 4 5 6 7 8 9 10Time(t)

10-10

10-8

10-6

10-4

Magnitude

Output error in time domain LoewnerBT

(b) Approximation error of the output signal.

Figure 6.6: Time domain simulation - second choice of input.

6.5.4 Third exampleFor the last experiment, consider a large scale LSS constructed as in [83] from the original machinestand example given in [60]. In this example, the system variability is induced by a moving toolslide on the guide rails of the stand (see Fig. 6.7). The aim is to determine the thermally drivendisplacement of the machine stand structure. Following the model setting in [60], consider theheat equation with Robin boundary conditions. Using a finite element (FE) discretization anddenoting the external influences as the system input z, we obtain the dynamical heat model

Ethx(t) = Ath(t)x(t) + Bth(t)z(t), (6.74)

149


Figure 6.7: Schematic of the tool slide on the guide rails of the stand [83].

describing the deformation independent evolution of the temperature field x with the systemmatrices Eth,Ath(t) and Bth(t). The variability of the model is described by time dependentmatrices Ath(t) and Bth(t). This leads directly to the linear time varying system describedby (6.74). Since model reduction for linear time-varying (LTV) systems is a highly storageconsuming procedure, the authors in [83] exploit properties of the spatially semi-discretizedmodel to set up an LSS consisting of LTI subsystems only. As described in [60], the guide rails ofthe machine stand are modeled as 15 equally distributed horizontal segments (see Fig. 6.7). Anyof these segments is said to be completely covered by the tool slide if its midpoint lies within theheight of the slide. On the other hand, each segment whose midpoint is not covered is treatedas not in contact and therefore the slide always covers exactly 5 segments at each time. This infact allows the stand to reach 11 distinct, discrete positions given by the model restrictions. Inthis way, one can define the subsystems of the LSS as follows:

Σ` :

Ethx = A`thx + B`

thz`,y = Cx,

(6.75)

where ` ∈ 1, ..., 11. Note that the change of the input operator Bth(t) is hidden in the inputitself, since it is sufficient to activate the correct boundary parts by choosing the correspondingcolumns in Bth via the input z`. Therefore, the input operator Bth(t) := Bth becomes constantand the input variability is represented by the input z`

z`i :=

zi, segment i is in contact,0, otherwise,

, for i ∈ 1, . . . , 15. (6.76)

Here, zi ∈ R is the thermal input as described in [83]. The only varying part influencing themodel reduction process left in the dynamical system is the system matrix Ath(t) := A`

th.Since the application of this example is to study the thermally driven deformation at particular

points, the output equation y = Cz is used to explicitly select these points (such as the oneslocated around the tool center point or around the connections to neighboring assembly groups).The rows of the matrix C are unit vectors with only 0 and 1 values and hence, the output ycontains selected entries of the internal variable x.

150


After the finite element discretization was performed, obtain an LSS with 11 active modes(denote it with Σ). Each subsystem has dimension n = 16626. The E and C matrices arethe same for all modes of the LSS. The B matrices have 6 columns (corresponding to differentinputs) and the C matrix has 9 rows (corresponding to different outputs). All the aforementionedmatrices are saved in sparse format.

The proposed extension of the Loewner method to LSSs (described in Section 6.3 and 6.4)can be generalized in a straightforward manner to the MIMO case similarly to the linear case(see [12]), by introducing left and right tangential direction vectors.

In the following experiments, we take into consideration three active modes (the first, thethird and the fifth). This corresponds to the particular case of D = 3 (treated in Section 6.4).Furthermore, consider only the pairs of the first and third inputs as well as the first and thirdoutputs to be activated at any time, for each of the three modes. Hence, the measurements usedin the Loewner framework are 2× 2 matrices.

We analyze a simplified model composed of three modes and certain input/output pairs inorder to ensure that the numerical results can be depicted in a clear and distinguishable manner(without overcrowding the figures).

All subsystems are stable linear systems of order 16626 in sparse format. The large scale LSSΣ will be again reduced, as in the second example, by means of the Loewner method and of thebalanced truncation method proposed in [94].

We perform a time domain simulation to investigate the approximation quality of the observedoutput, which in this case has two components. More exactly, the choice of outputs is as follows:the first output represents the 9163th entry of the deformation vector x, while the second outputis the 9814th entry of x.

The singular values of the frequency response of each original subsystem are depicted inFig. 6.8, More exactly, for all three modes and for each frequency point jω, compute the twosingular values corresponding to Hk(jω) ∈ C2×2.

10-6 10-4 10-2 100

Frequency(ω)

10-6

10-4

10-2

Frequency response of the original LSS

Ist subsystem

IInd subsystem

IIIrd subsystem

Figure 6.8: Frequency response of the original subsystems.

For the Loewner method, we choose 200 logarithmically spaced interpolation points in theinterval [10−6, 5 · 100]j. The decay of the singular values of the appended Loewner matrices[Lj Lsj] corresponding to mode j, for j ∈ 1, 2, 3, is presented in Fig. 6.9. We notice that allthree curves are close to each other and already the 100th singular values attain machine precision.Additionally, in the same figure one can observe the decay of the approximate averaged Hankelsingular values corresponding to Σ. Of course, one can not compute exact Gramians for the

151


large order system, but only approximate low rank factors by using, for instance, such softwaretools as in [104].

50 100 150 200 250

10-15

10-10

10-5

100Singular value decay

Mode 1 (LSS)Mode 2 (LSS)Mode 3 (LSS)Averaged Gramians

Figure 6.9: Decay of the singular values of the different matrices.

For the Loewner reduced order LSS (i.e Σ1), we decide to truncate at order k = 66 for allthree subsystems. This corresponds to eliminating the singular values that are smaller than10−13 (for each of the three appended Loewner matrices). The same truncation order is chosenfor the reduced order model computed via BT for which the last kept singular value (before thebalancing truncation procedure is applied) is σavg

66 = 8.0693 · 10−11.In the upper part of figure Fig. 6.10, the control input signals u1 and u2 are depicted.

u1(t) = 12 sin(t/20)e−t/500 + 1

20e−t/500, u2(t) = 1

10 . (6.77)

As it can be observed in the x axis of this figure, the running time of the performed experimentis 1 hour (the control is active form time ts = 0s to te = 3600s). In the lower part of Fig. 6.10,the switching signal σ : R→ 1, 2, 3 is presented. Note that the time axis is restricted from 380to 440 seconds. The switching signal follows a simple periodical rule, by repeating the sequenceof modes (1, 2, 3, 2, 1).

0 500 1000 1500 2000 2500 3000 3500-0.4-0.2

00.20.40.6

The control input signalsInput 1Input 2

380 390 400 410 420 430 440Time(t)

1

2

3Switching signal σ(t)

Figure 6.10: The control and switched input signals.

Next, compare the time domain response of the original LSS against the ones correspondingto the two reduced models. Notice that the two outputs of the LSS are well approximated forboth MOR methods, as it can be seen in Fig. 6.11 (the upper part depicts the first observedoutput, while in the lower part, the second output is shown). As for Fig. 6.5 in Section 6.5.3,the blue circles included on the output curves are used to mark the exact time instances whenswitching occurs. Again, the time axis is restricted from 380 to 440 seconds.

152

6.6 Summary

380 390 400 410 420 430 440

0

0.2

0.4

0.6

Deformationin

[µm]

Time domain simulation - Output 1

380 390 400 410 420 430 440Time(t)

0

2

4

6

Time domain simulation - Output 2 Original LSSReduced LSS - LoewnerReduced LSS - BT

Figure 6.11: Time domain simulation.

Finally, we inspect the time domain error between the original response and the responsesof the two reduced models (depicted in Fig. 6.12). Note that the time axis is again restrictedto one minute, in between [380, 440] seconds. We observe that, the error curve corresponding toour proposed method is always below the error curve corresponding to the BT method, in theconsidered time range.

380 390 400 410 420 430 440

10-610-410-2

Time domain simulation error - Output 1

LoewnerBT

380 390 400 410 420 430 440Time(t)

10-610-410-2

Time domain simulation error - Output 2

Figure 6.12: Time domain approximation error.

6.6 SummaryIn this chapter we address the problem of model reduction of linear switched systems fromdata consisting of values of high order transfer functions. The underlying philosophy of theLoewner framework is to collect data and then extract the desired information. Having therequired data, the next step would be to arrange it into matrix format. We have shown thatthe Loewner matrices (which represent the recovered E and A matrices of the underlying LSS)can be automatically calculated as solutions of Sylvester equations. In the proposed framework,the transition/coupling matrices can be recovered from the given computed data as well. Sincethese matrices need not be square, they allow having different dimensions of the reduced statespace in different modes.

Three numerical examples demonstrate the effectiveness of the proposed approach. Thequality of approximation for the reduced models was determined by performing both frequencyand time domain tests. We have chosen a generalization of the classical balanced truncationmethod to LSS for comparison purposes. As opposed to most of the balancing methods weencountered in the literature ([39], [30], [106] and [100]), the method we choose (i.e [94]) does

153

6.6 Summary

not require solving systems of LMI (linear matrix inequalities) which might be difficult for verylarge systems such as the one in Section 6.5.4.

The results of the new proposed method turned out to be overall better than the ones obtainedwhen using the BT method. More precisely, we would like to emphasize that, the Loewner methodseems to be able to achieve similar approximation accuracy to the BT method, but with lowerdimension of the ROMs constructed. This observation is confirmed by the faster decay of thesingular values computed for our method as compared to the decay of the (averaged) Hankelsingular values corresponding to the BT method (presented in Fig. 6.3 and Fig. 6.9).

In Section 6.5.3, we also investigated switching signals σ(t) that exhibit a fast switchingbehavior (hence smaller dwell times) in a particular time interval (Fig. 6.5 (b)). We concludethat there are no reasons to suspect that our method will fail for such cases.

In general, stability preservation of the reduced order model is still an open issue for momentmatching MOR methods, even for linear systems. In fact, even for linear systems, the Loewnerframework does not guarantee preservation of stability. Since the model reduction method pre-sented in this chapter is a direct extension of the Loewner framework for linear systems, ingeneral, it will not preserve stability. Although it is likely that under suitable assumptions theproposed method could be modified to preserve stability, this remains a topic of future research.

The Loewner framework is conceptually a data driven MOR method that builds reducedorder models that interpolate the frequency response of the large scale original system. Inprinciple, this method requires only data and not solving any type of equations. In the linearcase, the data can be measured using VNAs (vector network analyzers). In the nonlinear case(as for bilinear, quadratic-bilinear or linear switched), the data is obtained by direct numericalsimulation (DNS). In the LSS case, no equation needs to be solved (the data is gathered andput together in the fashion described in the chapter). The computational complexity of ourproposed method is related to the DNS process. Hence, one needs to compute samples of thegeneralized transfer functions (in the case for which these values are not provided via real timemeasurements). This can be performed in a fast way by avoiding the explicit inversion of thematrices ωjEk −Ak and using, for example, Gaussian elimination instead.

154

Chapter 7

Conclusion

In this thesis we address the problem of model reduction of some classes of dynamical systemsfrom data consisting of values of high-order transfer functions. The underlying philosophy is asfollows: collect data and extract the desired information. This is accomplished by extending theLoewner framework to the reduction of some classes of nonlinear systems. Its main features are• Given input-output data, one can construct with basically no computation, a singular model

in generalized state-space form. In applications the singular pencil (Ls,L) must be reduced atsome stage.• In this framework the singular values of the pencil (Ls, L) offer a trade-off between accuracy

of fit and complexity of the reduced system.• This approach to model reduction, first developed for linear time-invariant systems (see

[12] for a survey), was later extended to linear parametrized systems (in [11, 77, 78, 76]), and tobilinear systems (in [10]).

In this thesis, the following research developments were accomplished:• We tried to offer a solution to the fact that the computed reduced order models via the

Loewner framework for linear systems, need not be in general stable (even if the underlying linearsystem is stable).• We proposed an extension of the Loewner framework to the class of bilinear systems. One

of the results obtained for this class of systems includes a parametrization of all bilinear systemswhich satisfies one-sided interpolation conditions (of usual or of Volterra-series type).• We proposed an extension of the Loewner framework to the class of quadratic-bilinear

systems.•We proposed an extension of the Loewner framework to the class of linear switched systems.• We compared the performance of the Loewner method against the performance of well

established state of the art MOR techniques such as balanced truncation (BT), proper orthogonaldecomposition (POD) or the more recent versions of IRKA, i.e. BIRKA and TQB-IRKA.• In all of the numerical examples performed, the Loewner method produced comparable

results in terms of quality of approximation and, in some of them, even outperformed the othermethods.• We applied the Loewner framework to a class of distributed parameter systems with irra-

tional transfer functions.

155

Appendix A

The Loewner Framework Applied to aVibrating Beam Model

The Loewner framework for model order reduction is applied to the class of infinite-dimensionsystems that have an irrational transfer function. Indeed, this transfer function can be expressedas an infinite series of rational functions.

The main advantage of our method is the fact that reduced orders models are constructeddirectly from input-output measurements. The method can be directly applied to the origi-nal transfer function or to the one coming from the finite element discretization of the PDE.Significantly better results are obtained when using it directly.

The classes of systems which can be treated in our approach range from linear to bilinearand also quadratic systems. For simplicity, consider SISO systems (m, p = 1). We will deal onlywith linear systems whose dynamics are described in generalized state by the equations in (1.3).

Notice that the transfer function of a linear system is a rational function, or a rational matrixfunction in the case of multiple inputs and outputs, m and p (as defined in (2.86)).

Model order reduction (MOR) plays a vital role in numerical simulation of large-scale complexdynamical systems. These dynamical systems are governed by ordinary differential equations(ODEs), or partial differential equations (PDEs), or both. To capture the essential informationabout the dynamics of the systems, a fine semi-discretization of these governing equations in thespatial domain is often required.

Since the solution of the PDE reflects the distribution of a physical quantity such as thetemperature of a rod or the deflection of a beam, these systems are often called distributed-parameter systems (DPS).

The transfer functions of DPS systems are irrational functions as opposed to the transferfunctions of systems modeled by ordinary differential equations which are rational functions.Another difference is that the state space is infinite dimensional, usually a Hilbert space. Conse-quently, DPS are also called infinite-dimensional systems. The analysis of rational and irrationaltransfer functions differ in a number of important aspects. A difference between rational andirrational transfer functions is given by the fact that the later type often have infinitely manypoles and/or zeros.

A simple example of transverse vibrations in a structure is a beam, where the vibrations canbe considered to occur only in one dimension.

156

Consider a homogeneous beam of length L experiencing small transverse vibrations. Forsmall deflections the plane cross-sections of the beam remains planar during bending. Under thisassumption, we obtain the classic Euler-Bernoulli beam model for the deflection w(x, t)

∂2w(x, t)∂t2

+ EI∂4w(x, t)∂x4 = 0. (A.1)

Consider the Kelvin-Voigt damping model, sometimes referred to as the Rayleigh dampingwhich leads to the PDE

∂2w(x, t)∂t2

+ EI∂4w(x, t)∂x4 + cdI

∂5w(x, t)∂x4∂t

= 0, (A.2)

with the boundary conditions for the clamped end, w(0, t) = 0, ∂w(0,t)∂x

= 0, and at the free end

EI∂2w(L, t)∂x2 + cdI

∂3w(L, t)∂x2∂t

= 0, −EI ∂3w(x, t)∂x3 − cdI

∂4w(L, t)∂x3∂t

= u(t).

Here u(t) represents an applied force at the tip. Also take the observation (output) to bey(t) = ∂w

∂t(L, t)and let z(t) = w(L, t).

This model corresponds to the case of a clamped-free beam with shear force control coveredin [48]. By taking different boundary conditions, one might analyze different models such asclamped-free beam with torque control or pinned-free beam with shear force control. For thisreport, we restrict our attention only to the first model. As derived in [48], the transfer functioncan be written in the following way

Horig(s) = sN(s)(EI + scdI)m3(s)D(s) , (A.3)

in terms of the following nonlinear functions

m(s) =( −s2

EI + scdI

) 14,

N(s) = cosh(Lm(s)) sin(Lm(s))− sinh(Lm(s)) cos(Lm(s)),D(s) = 1 + cosh(Lm(s)) cos(Lm(s)).

The exact derivation of these types of irrational transfer functions is described in [73]. The polesof the transfer function (A.3) are the solution of the equation s2 + cdIα

4ks + EIα4

k = 0 and canbe written as

µ±k =−cdIα4

k ±√

(cdI)2α8k − 4EIα4

k

2 , (A.4)

where αk’s are the real positive roots of the hyperbolic equation in α 1 + cosh(Lα) cos(Lα) = 0.When k →∞, these values converge to (2k+1)π

2L . The complex poles µ±k approach the imaginaryaxis as cd → 0 (the case for 0 damping). Notice that there is also a real pole at − E

cd.

157

The zeros of the transfer function are the solution of the equation s2 + cdIγ4ks + EIγ4

k = 0where γk’s are the roots of the equation in γ cosh(Lγ) sin(Lγ) − sinh(Lγ) cos(Lγ) = 0. Whenk →∞, these values converge to (4k + 1)π/4L.

The original irrational transfer function is written as an infinite partial fraction expansion

Horig(s) =∞∑k=1

rks− µk

+ r−ks− µ−k

=∞∑k=1

4ss2 + cdIα4

ks+ EIα4k

, (A.5)

where rk = 4µkµ−k−µk

, r−k = 4µ−kµ−k−µk

are the residues of the poles µ±k.

We consider the beam to have length L = 0.7 m and that it is build of aluminum. Hencethe Young modulus elasticity constant is taken to be E = 69GPa = 6.9 × 1010N/m2. Then,the height and base of the rectangular cross section of the beam are taken to be h = 8.5mmand b = 70mm. Then, since the moment of inertia can be calculated precisely in terms of thesetwo quantities, i.e., I = bh3/12, it follows that I = 3.58 × 10−9m4. Finally, take the dampingcoefficient to be cd = 5× 10−4Ns/m2.

One way to proceed in modeling the dynamics of the beam is by computing a finite elementdiscretization (get rid of the variable x by approximating the spatial derivatives). Assume thatthe beam has been divided in N +2 intervals of length h = L/(N +2); the resulting variables arew(kh, t), k ∈ 0, 1, . . . , N + 2. Consider wk = 0 for k > N + 3. For simplicity, use wk insteadof w(kh, t). Approximate the time derivatives of w(x, t) at x = kh as,

w(1)k ≈

wk+1 − wkh

, w(2)k ≈

wk+1 − 2wk + wk−1

h2 , (A.6)

w(3)k ≈

wk+2 − 3wk+1 + 3wk − wk−1

h3 , w(4)k ≈

wk+2 − 4wk+1 + 6wk − 4wk−1 + wk−2

h4 . (A.7)

The first two boundary conditions give us the following constraints w0 = 0 and w1−w0h

= 0 whichmeans that we can eliminate the first two variables from the state vector (w0 = w1 = 0). Theother two boundary conditions give us the following

EI(wN+2 − 2wN+1 + wN

h2

)+ cdI

(wN+2 − 2wN+1 + wN

h2

)= 0, (A.8)

−EI(wN+2 − 3wN+1 + 3wN − wN−1

h3

)− cdI

(wN+2 − 3wN+1 + 3wN − wN−1

h3

)= u. (A.9)

It follows that

EIwN+1 + cdIwN+1 = EI(2wN − wN−1) + cdI(2wN − wN−1) + h3u, (A.10)EIwN+2 + cdIwN+2 = EI(3wN − 2wN−1) + cdI(3wN − 2wN−1) + 2h3u. (A.11)

Hence, we are left with N − 1 variables in the state vector v = [w2, w3, . . . , wN ]T ∈ RN−1. Foreach variable, write down the corresponding ODE w

(2)k + EIw

(4)k + cdIw

(4)k = 0, 2 6 k 6 N

where w(j)k is defined above. Then, the next step is to rewrite these ODEs into matrix format

Mv + Uv + Kv = fu. Since y(t) = w(L, t), the output is chosen as, y = wN+2.

158

A.1 Experiments

A.1 ExperimentsUsing the Loewner framework, we construct reduced order linear models approximating theoriginal transfer function of the clamped-free beam with shear force control model in section3. We will do so directly by using measurements of the original transfer function or by firstperforming a finite element semi-discretization of the PDE and then taking measurements. Thedimension of all reduced models is taken to be constant throughout all experiments, i.e., r = 32.

For the choice of the parameters in section 3, we start with sampling the original transferfunction in (A.3) on a logarithmic frequency range between [100, 1012]j. The original frequencyis depicted in Fig. A.1. Only when reaching 1010 Hz the peaks seem to stagnate at a value below10−5.

100

102

104

106

108

1010

1012

10−6

10−4

10−2

100

Frequency response of the original beam model

Frequency(Hz)

Figure A.1: Original frequency response.

The first 32 poles and the first 31 zeros of the original transfer function are intertwined aswe can observe in Fig. A.2. Also, note that the absolute value of the imaginary part is less than2.5 · 10−5 for all poles. The imaginary part of the dominant pair of poles is −4.6113 · 10−11.Notice that as the damping constant cd becomes smaller and smaller, the poles approach moreand more the imaginary axis.

−2.5 −2 −1.5 −1 −0.5 0x 10

−5

−8

−6

−4

−2

0

2

4

6

8x 104

Poles of the original modelZeros of the original model

Figure A.2: Poles and zeros of the original transfer function.

Next use the proposed data driven MOR technique to find a linear model that interpolatesthe original transfer function at some chosen frequencies in some particular interest range, i.e.,

159

A.1 Experiments

[101, 104.5]j. The number of interpolation points is taken to be 400. The linear model that isconstructed based on these input-output measurements is numerically singular.

The decay of the singular values of the Loewner matrix (Fig. A.3) is an indicator that themodel can be compressed to a smaller dimension; the 40th singular value is below machineprecision 10−16).

50 100 150 200 250 300 350 400

10−20

10−15

10−10

10−5

100

Singular values of the Loewner matrix - Original beam model

Figure A.3: Decay of the Loewner singular values - original samples.

Choose reduction order r = 32. Note that the initial samples are accurately matched. Ofcourse, outside the interest range, our linear model is not able to faithfully reproduce the peaks(see Fig. A.4).

100

101

102

103

104

105

106

10−4

10−2

100

Frequency(Hz)

Initial modelLoewner reduced model - r = 32

Figure A.4: Frequency response comparison - original vs. Loewner.

Observe that the poles and the zeros of the reduced linear model are matching the onesof the original transfer function. Only the least three dominant poles/zeros are not accuratelyreproduced (see Fig. A.5).

Next, we would like to compare our reduction method with other available methods for re-ducing systems with irrational input output mappings. A candidate method is modal truncation(see [4]). When applying this method, certain poles are cut from the infinite partial fractionexpansion of Horig. Hence, only the first dominant 2N poles are kept, as presented below

Hmod(s) =N∑k=1

4ss2 + cdIα4

ks+ EIα4k

.

160

A.1 Experiments

−2.5 −2 −1.5 −1 −0.5 0

x 10−5

−8

−6

−4

−2

0

2

4

6

8

x 104

Poles of the original model

Poles of the Loewner reduced model

−2 −1.5 −1 −0.5 0

x 10−5

−6

−4

−2

0

2

4

6

8x 10

4

Zeros of the original model

Zeros of the Loewner reduced model

Figure A.5: Poles and zeros.

By choosing N = 16, notice that the first peaks are accurately approximated. The accuracy islost outside the interest range as we can see in Fig. A.6.

100

101

102

103

104

105

106

10−4

10−2

100

Frequency(Hz)

Initial modelReduced modal model - r = 32

Figure A.6: Frequency response comparison - original vs. modal.

In the next experiment, a comparison between the Loewner method (applied to data comingfrom the original transfer function) and the modal truncation method is made by analyzing theapproximation error of the two methods inside the target range of frequencies. We notice thatthe error coming from the Loewner method is lower than the one coming from modal truncation,as it is depicted in Fig. A.7, i.e., by seven orders of magnitude.

100

101

102

103

104

105

106

10−15

10−10

10−5

Frequency(Hz)

Error plots

Loewner reduced modelModal reduced model

Figure A.7: Error in frequency domain - Loewner vs. modal.

161

A.2 Summary

In the following experiments, we sample the transfer function corresponding to the finiteelement model constructed as described in section 3. Clearly, the discretization introduces anapproximation on its own that will negatively influence the accurate matching of the originalresponse.

Again, the first step is to take samples of HFEM(s) in the same interest range as before. Fora discretization with N = 1000 elements, take as before 400 samples of the transfer function.The first major difference when using this approach is the decay of singular values which is notas steep as when using samples of the original transfer function. For reduction order 32, thesmallest singular value is around 10−6 (see Fig. A.8).

50 100 150 200 250 300 350 40010

−15

10−10

10−5

100

Singular values drop of the Loewner matrix - FE beam model

Figure A.8: Decay of the Loewner singular values - FE samples.

The frequency response corresponding to both the direct Loewenr method, as well as to theLoewner method for finite element approximation, are compared against the original samples(see Fig. A.9). The matching error is also compared. Note that the approximation is much

100

101

102

103

104

105

106

10−4

10−2

100

Frequency(Hz)

Original modelLoewner reduced model - originalLoewner reduced model - FE

Figure A.9: Frequency response comparison for various methods.

better when applying Loewner to the original data (see Fig. A.10).

A.2 SummaryThe aim of this work was to study the applicability of the Loewner framework to approximating aparticular class of systems, i.e., infinite-dimensional systems. It has been shown from the variousexperiments performed that, indeed, the Loewner method is able to yield better approximation

162

A.2 Summary

100

101

102

103

104

105

106

10−15

10−10

10−5

100

Frequency(Hz)

Error plots

Loewner reduced model - originalLoewner reduced model - FE

Figure A.10: Frequency response comparison - Loewner vs. Loewner FE.

than other methods such as modal truncation. An advantage of the Loewner method is that itrequires only input-output measurements to construct reduced order models of the data.

The experiments performed showed that, in terms of the accuracy of the approximation,it is advisable to use measurements coming directly from the original transfer function. Theintroduction of the semi-discretization step that transforms the PDE into a series of ODEs addswith it additional errors that drastically decrease the quality of approximation. Of course, inpractice, the above mentioned step can not always be avoided.

Further challenges and research interests include topics such as optimal choice of samplingpoints or recovering from noise corrupted measurements.

163

Appendix B

Various Proofs

B.0.1 Proof of Proposition 2.2.17W will specifically concentrate on the equality stated in (2.47) from Proposition 2.2.17.

We will prove the equality of the two n×n matrices by showing that each column is the samefor both; this is done by multiplying the equality in (2.47) to the right with the unit vector ek,for k ∈ 1, 2, . . . , n. By using the third identity from Proposition 2.2.15, we have that

Q(−1)(In ⊗w

)ek =

(wT ⊗ In

)QTek ⇔ Q(−1)

(ek ⊗w

)=(wT ⊗ In

)QTek. (B.1)

We further writeQ(−1)

(ek ⊗w

)= Qtens

k w, (B.2)

where Qtensk is the kth frontal slice of tensor Q; the matrix Q(−1) is the mode-1 matricization

of the same tensor Q (see Definition 2.2.19). Moreover, from the same definition, we have thatQTek = vec(Qtens

k ). Combining this result with the first identity from Proposition 2.2.15, itfollows (

wT ⊗ In)QTek =

(wT ⊗ In

)vec(Qtens

k ) = vec(InQtens

k w)

= Qtensk w, (B.3)

since Qtensk w ∈ Rn. From (B.1), (B.2) and (B.3) it follows that

Q(−1)(In ⊗w

)ek =

(wT ⊗ In

)QTek, ∀ k ∈ 1, 2, . . . , n,

and hence (2.47) is proven. The proof for (2.46) follows in a similar manner.

B.1 Proof of Proposition 2.2.21For the linear controllability Gramian PL = P0, one can write

APL + PLAT =∫ ∞

0(AeAtBBT eAT t + eAtBBT eAT tAT )dt

=∫ ∞

0

(deAt

dtBBT eAT t + eAtBBT de

AT t

dt

)dt

164

B.1 Proof of Proposition 2.2.21

=∫ ∞

0

d(eAtBBT eAT t)dt

dt = eAtBBT eAT t

∣∣∣∣∞0

= −BBT .

For the level 1 Gramians, i.e., PN1 and PQ

1 , one can also write

APN1 + PN

1 AT =∫ ∞

0

∫ ∞0

(AeAt1NeAt2BBT eAT t2NeAT t1 + eAt1NeAt2BBT eAT t2NeAT t1AT )

=∫ ∞

0

(deAt1

dtNPLNT eAT t1 + eAt1NPLNT de

AT t1

dt1

)dt1

=∫ ∞

0

d(eAt1NPLNT eAT t1)dt1

dt1 = eAt1NPLNT eAT t1

∣∣∣∣∞0

= −NPLNT .

APQ1 + PQ

1 AT =∫ ∞

0

∫ ∞0

∫ ∞0

(AeAt1F(eAt2B⊗ eAt3B

)(BT eAT t2 ⊗BT eAT t3

)QT eAT t1 + . . .

=∫ ∞

0

∫ ∞0

∫ ∞0

(AeAt1Q(eAt2B⊗ eAt3B

)(BT eAT t2 ⊗BT eAT t3

)QT eAT t1 + . . .

=∫ ∞

0

(deAt1

dtQ(PL ⊗ PL

)QT eAT t1 + eAt1Q

(PL ⊗ PL

)QT de

AT t1

dt1

)dt1

=∫ ∞

0

d(eAt1Q(PL ⊗ PL

)QT eAT t1)

dt1dt1 = eAt1Q

(PL ⊗ PL

)QT eAT t1

∣∣∣∣∞0

= −Q(PL ⊗ PL

)QT .

Hence write two equations for the first level controllability GramiansAPN1 + PN

1 AT + NPLNT = 0,APQ

1 + PQ1 AT + Q

(PL ⊗ PL

)QT = 0.

By adding these equations it follows that

AP1 + P1AT + Q(PL ⊗ PL

)QT + NPLNT = 0. (B.4)

Similarly, we write four equations for the second level Gramians

APN,N2 + PN,N

2 AT + NPN1 NT = 0,

APN,Q2 + PN,Q

2 AT + NPQ1 NT = 0,

APQ,N2 + PQ,N

2 AT + Q(PN

1 ⊗ PL)QT = 0,

APQ,Q2 + PQ,Q

2 AT + Q(PQ

1 ⊗ PL)QT = 0.

.

By adding these equations it follows that

AP2 + P2AT + Q(P1 ⊗ PL

)QT + NP1NT = 0. (B.5)

To prove the general formula, one proceeds by means of mathematical induction (and usingProposition 2.2.20).

165


B.2 Proof of Proposition 2.2.23For the linear observability Gramian QL = Q0, one can write

ATQL +QLA =∫ ∞

0(AT eAT tCTCeAt + eAT tCTCeAtA)dt

=∫ ∞

0

(deAT t

dtCTCeAt + eAT tCTC

deAt

dt

)dt

=∫ ∞

0

d(eAT tCTCeAt)dt

dt = eAT tCTCeAt∣∣∣∣∞0

= −CTC.

For the level 1 observability Gramians, i.e., QN1 and QQ

1 write

ATQN1 +QN

1 A =∫ ∞

0

∫ ∞0

(AT eAT t2NT eAT t1CTCeAt1NeAt2 + eAT t2NT eAT t1CTCeAt1NeAt2A)

=∫ ∞

0

(deAT t2

dt2NTQLNeAt2 + eAT t2NTPLN

deAt2

dt2

)dt2

=∫ ∞

0

d(eAT t2NTQLNeAt2)dt2

dt2 = eAT t2NTQLNeAt2∣∣∣∣∞0

= −NTQLN.

ATQQ1 +QQ

1 A =∫ ∞

0

∫ ∞0

∫ ∞0

AT(BT eAT t2 ⊗ eAT t3

)QT eAT t1CTCeAt1Q

(eAt2B⊗ eAt3

)+ . . .

=∫ ∞

0

∫ ∞0

∫ ∞0

AT eAT t3(BT eAT t2 ⊗ In

)QT eAT t1CTCeAt1Q

(eAt2B⊗ eAt3

)+ . . .

=∫ ∞

0

∫ ∞0

deAT t3

dt3

(BT eAT t2 ⊗ In

)QTQLQ

(eAt2B⊗ eAt3

)dt2dt3

+∫ ∞

0

∫ ∞0

(BT eAT t2 ⊗ In

)QTQLQ

(eAt2B⊗ In

)deAt3

dt3dt2dt3

=∫ ∞

0

∫ ∞0

d(eAT t3(BT eAT t2 ⊗ In

)QTQLQ

(eAt2B⊗ In

)dt3

dt3dt2

= −∫ ∞

0

(BT eAT t2 ⊗ In

)QTZT

L︸︷︷︸F(t2)T

ZLQ(eAt2B⊗ In

)︸︷︷︸

F(t2)

dt2

= −∫ ∞

0F(t2)TF(t2)dt2.

where ZL is the Cholesky factor of QL, i.e., QL = ZTLZL. Note that,∫ ∞

0F(t2)F(t2)Tdt2 =

∫ ∞0

ZLQ(eAt2B⊗ In

)(BT eAT t2 ⊗ In

)QTZT

Ldt2

= ZLQ( ∫ ∞

0

(eAt2BBT eAT t2 ⊗ In

)dt2

)QTZT

L

= ZLQ(PL ⊗ In

)QTZT

L.

166


The following holds in general:∫ ∞0

F(t2)TF(t2)dt2 6=∫ ∞

0F(t2)F(t2)Tdt2.

Using the result in (2.46) (for v = eAt2B), one can write:

ATQQ1 +QQ

1 A = −∫ ∞

0

(BT eAT t2 ⊗ In

)QTZT

LZLQ(eAt2B⊗ In

)dt2

= −∫ ∞

0Q(2)

(eAt2B⊗ In

)ZTLZL

(BT eAT t2 ⊗ In

)(Q(2)

)Tdt2

= −∫ ∞

0Q(2)

(eAt2B⊗ ZT

L

)(BT eAT t2 ⊗ ZL

)(Q(2)

)Tdt2

= −∫ ∞

0Q(2)

(eAt2BBT eAT t2 ⊗ ZT

LZL

)(Q(2)

)Tdt2 = −Q(2)

(PL ⊗QL

)(Q(2)

)T.

Using the result in (2.47) (for w = eAt2B), one can write:

ATQQ1 +QQ

1 A = −∫ ∞

0

(BT eAT t2 ⊗ In

)QTZT

LZLQ(eAt2B⊗ In

)dt2

= −∫ ∞

0Q(−1)

(In ⊗ eAt2B

)ZTLZL

(In ⊗BT eAT t2

)(Q(−1)

)Tdt2

= −∫ ∞

0Q(−1)

(ZTL ⊗ eAt2B

)(ZL ⊗BT eAT t2

)(Q(−1)

)Tdt2

= −∫ ∞

0Q(−1)

(ZTLZL ⊗ eAt2BBT eAT t2

)(Q(−1)

)Tdt2

= −Q(−1)(QL ⊗ PL

)(Q(−1)

)T.

Hence write two equations for the first level observability GramiansATQN1 +QN

1 A + NTQLN = 0,ATQQ

1 +QQ1 A + Q(−1)

(QL ⊗ PL

)(Q(−1)

)T= 0.

By adding these equations it follows that:

ATQ1 +Q1AT + Q(−1)(QL ⊗ PL

)(Q(−1)

)T+ NTQLN = 0. (B.6)

Similarly, we write four equations for the second level Gramians:

ATQN,N2 +QN,N

2 +¯

NTQN1 N = 0,

ATQN,Q2 +QN,Q

2 A + Q(−1)(QN

1 ⊗ PL)(

Q(−1))T

= 0,ATQQ,N

2 +QQ,N2 A + NTQQ

1 N = 0,ATQQ,Q

2 +QQ,Q2 A + Q(−1)

(QQ

1 ⊗ PL)(

Q(−1))T

= 0.

By adding these equations it follows that:

ATQ2 +Q2A + Q(−1)(Q1 ⊗ PL

)(Q(−1)

)T+ NTQ1N = 0. (B.7)

167

B.3 Proof of Lemma 5.3.1

To prove the general formula stated in (2.65), one proceeds by means of mathematical inductionusing Proposition 2.2.22.

B.3 Proof of Lemma 5.3.1We project the quadratic-bilinear system ΣQB with X = R ∈ Cn×k, and with an arbitrary matrixY ∈ Cn×k (so that YTX is nonsingular). It readily follows that, for i ∈ 1, 2, . . . , k

(a) Φ(λ(i)1 ) B = e3i−2 (b) Φ(λ(i)

2 ) N e3i−2 = e3i−1 and (c) Φ(λ(i)3 ) Q

(e3i−2 ⊗ e3i−2

)= e3i.

We make use of the following result:

Φ(s)−1 = sE− A = YT (sE−A)R = YTΦ(s)−1R. (B.8)

To prove (a), we first notice that by multiplying R to the right with e3i−2 we can write

Re3i−2 = Φ(λ(i)1 )B⇒ Φ(λ(i)

1 )−1Re3i−2 = B⇒ YTΦ(λ(i)1 )−1Re3i−2 = YTB.

Using the notation B = YTB and the result in (B.8), we write

Φ(λ(i)1 )−1Re3i−2 = B⇒ Φ(λ(i)

1 )B = e3i−2.

To prove (b), note that by multiplying R to the right with e3i−1, we can write

Re3i−1 = Φ(λ(i)2 )NΦ(λ(i)

1 )B⇒ Φ(λ(i)2 )−1Re3i−1 = NRe3i−2. (B.9)

By multiplying (B.9) with YT to the left and then using the notation N = YTNR and the resultin (B.8), we have that

YTΦ(λ(i)2 )−1Re3i−1 = YTNRe3i−2 ⇒ Φ(λ(i)

2 )−1e3i−1 = Ne3i−2 ⇒ Φ(λ(i)2 )Ne3i−2 = e3i−1.

To prove (c), note that by multiplying R to the right with e3i, we can write

Re3i = Φ(λ(i)3 )Q

(Φ(λ(i)

1 )B⊗Φ(λ(i)1 ))⇒ Φ(λ(i)

3 )−1Re3i = Q(Re3i−2 ⊗Re3i−2

). (B.10)

By multiplying (B.10) with YT to the left and then using the notation Q = YTQ(R⊗R

)as

well as the result in (B.8), we have that

YTΦ(λ(i)3 )−1Re3i = YTQ

(R⊗R

)(e3i−2 ⊗ e3i−2

)⇒ Φ(λ(i)

3 )−1e3i = Q(e3i−2 ⊗ e3i−2

)⇒ Φ(λ(i)

3 )Q(e3i−2 ⊗ e3i−2

)= e3i.

The equalities in (a),(b) and (c) imply the right-hand conditions in (5.14). For instance, bymultiplying the relation stated in (a) with C to the left we obtain:

CΦ(λ(i)1 )B = Ce1 = CRe3i−2 = CΦ(λ(i)

1 )B⇒ Hε0(λ(i)

1 ) = Hε0(λ(i)

1 ).

168

B.3 Proof of Lemma 5.3.1

Also, by multiplying the relation stated in (b) with C to the left we obtain

CΦ(λ(i)2 )Ne3i−2 = Ce3i−1 = CRe3i−1 = CΦ(λ(i)

2 )NΦ(λ(i)1 )B⇒ HN

1 (λ(i)2 , λ

(i)1 ) = HN

1 (λ(i)2 , λ

(i)1 ).

Finally, by multiplying the relation stated in (c) with C to the left, we write

CΦ(λ(i)3 )Q

(e3i−2 ⊗ e3i−2

)= Ce3i = CRe3i = CΦ(λ(i)

2 )Q(Φ(λ(i)

1 )B⊗ λ(i)1 )B)

⇒ HQ1 (λ(i)

3 , λ(i)1 , λ

(i)1 ) = HQ

1 (λ(i)3 , λ

(i)1 , λ

(i)1 ).

Similarly, if YT = O and X arbitrary, we have that the following holds for j ∈ 1, 2, . . . , k, i.e.,

(d) C Φ(µ(j)1 ) = eT3j−2 (e) eT3j−2NΦ(µ(j)

2 ) = eT3j−1 and (f) eT3j−2Q(Φ(λ(j)

1 )B⊗Φ(µ(j)3 ))

= eT3j,

which imply the left-hand conditions (5.13). Finally, with X = R, YT = O, and combining (a)- (f), interpolation conditions (5.15) and (5.16) follow.

For instance, by fixing i, j ∈ 1, 2, . . . , k and `, h ∈ 1, 2, 3, we would like to show that theequalities in (5.15) hold. By choosing ` = 2 and h = 3, it follows that we would need to showthe following equality

HN,N,Q3 (µ(j)

1 , µ(j)2 , λ

(i)3 , λ

(i)1 , λ

(i)1 ) = HN,N,Q

3 (µ(j)1 , µ

(j)2 , λ

(i)3 , λ

(i)1 , λ

(i)1 ). (B.11)

From (c) and (e), we write

CΦ(µ(j)1 )NΦ(µ(j)

2 ) = eT3j−1 and Φ(λ(i)3 ) Q

(Φ(λ(i)

1 )B⊗ Φ(λ(i)1 )B

)= e3i.

By multiplying these two relations, we hence write that(CΦ(µ(j)

1 )NΦ(µ(j)2 ))N(Φ(λ(i)

3 ) Q(Φ(λ(i)

1 )B⊗ Φ(λ(i)1 )B

)= eT3j−1Ne3i = eT3j−1

(OTNR

)e3i

=(eT3j−1OT

)N(Re3i

)= CΦ(µ(j)

1 NΦ(µ(j)2 N

(Φ(λ(i)

3 ) Q(Φ(λ(i)

1 )B⊗Φ(λ(i)1 )B

),

which proves the equality in (B.11). In general, the interpolation conditions in (5.15) hold

Hw(`)N ˆw(h)|w(`)|+|w(h)|+1(µ(j)(`) λ(i)(h)) = eT3j−3+`Ne3i−3+h = eT3j−3+`

(OTNR

)e3i−3+h

=(eT3j−3+ÒT

)N(Re3i−3+h

)= Hw(`)Nv(h)

|w(`)|+|v(h)|+1(µ(j)(`) λ(i)(h)).

Also, it can be shown that the interpolation conditions in (5.16) hold in general (by choosingh1, h2 ∈ 1, 2, 3 so that h1 ∨ h2 = 1)

Hw(`)Qw(h1)w(h2)|w(`)|+|w(h1)|+|w(h2)|+1(µ(j)(`) λ(i)(h1) λ(i)(h2)) = eT3j−3+`Q

(e3i−3+h1 ⊗ e3i−3+h2

)= eT3j−3+ÒTQ

(R⊗R

)(e3i−3+h1 ⊗ e3i−3+h2

)= eT3j−3+ÒTQ

(Re3i−3+h1

)⊗(Re3i−3+h2

)= Hw(`)Qw(h1)w(h2)

|w(`)|+|w(h1)|+|w(h2)|+1(µ(j)(`) λ(i)(h1) λ(i)(h2)).

169

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