From classical absolute stability tests
towards
a comprehensive robustness analysis
Von der Fakultät Mathematik und Physik der Universität
Stuttgart
zur Erlangung der Würde eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
Vorgelegt von
Matthias Fetzer
aus Rottweil
Hauptberichter: Prof. Dr. Carsten W. Scherer
Mitberichter: Prof. Dr. Arjan van der Schaft
Tag der mündlichen Prüfung: 23.11.2017
Institut für Mathematische Methoden in den Ingenieurwissenschaften,
Numerik und geometrische Modellierung der Universität Stuttgart
2017
ii
Acknowledgement
First and foremost, I want to express my deep gratitude to myadvisor Prof. Dr. Carsten Scherer, who introduced me to the
exciting field of mathematical systems theory. It is a privilege to study
control, and mathematics as a whole, under the guidance of such an
extraordinary teacher. I am very grateful for countless enlightening
discussions and in particular his confidence in a complete novice without
any background in control.
Furthermore, I want to acknowledge that this project has partly
been supported by the German Research Foundation (DFG) within the
Cluster of Excellence in Simulation Technology (Grant number: EXC
310/2) at the University of Stuttgart.
A special thanks goes to my colleagues, in particular, Joost Veenman,
Elisabeth Schaettgen, Julien Chaudenson, Tobias Holicki and Christian
Rösinger for fruitful collaborations and support yet more importantly
for making my time here both productive and enjoyable.
But most of all, I am eternally grateful to my wife Michèle Aicher
for her love, unwavering support and the great fortune of being her
husband. Thank you for being my best friend.
Stuttgart, June 2017 Matthias Fetzer
iii

Abstract
In this thesis, we are concerned with the stability and performance anal-ysis of feedback interconnections comprising a linear (time-invariant)
system and an uncertain component subject to external disturbances.
Building on the framework of integral quadratic constraints (IQCs),
we aim at verifying stability of the interconnection using only coarse
information about the input-output behavior of the uncertainty.
In the first part of the thesis, we establish a comprehensive frame-
work for global stability and performance analysis on general function
spaces that significantly widens the range of applications if compared
to standard IQC theory. Furthermore, our novel approach allows to
flexibly combine and also improve on all multiplier based stability cri-
teria available in the literature for the classical problem of absolute
stability analysis, i.e., the case where the uncertain system is defined
via a slope-restricted or sector-bounded nonlinearity.
By forging a strong and very general link to the theory of dissipation
as developed by Willems, we demonstrate for the first time in the
second part of the present thesis that general IQC theory can indeed
be extended towards local analysis of feedback interconnections. This
is achieved by a reformulation of IQC theory in a trajectory based
setting that opens the way for the application of standard Lyapunov
type arguments. Hence, we can now employ input-output descriptions
of uncertainties in order to robustly verify and guarantee hard state and
v
output constraints on the linear part of the interconnection depending
on the set of possible disturbances.
vi
Zusammenfassung
Das zentrale Thema dieser Arbeit ist die Stabilitäts- und Güteana-lyse von Rückkopplungssystemen, die aus einem linearen (zeitin-
varianten) Element und einer unsicheren Komponente bestehen und
darüber hinaus externen Störungen ausgesetzt sind. Auf die Theorie
der sogenannten integral quadratic constraints (IQCs) aufbauend, ver-
folgen wir das Ziel Stabilitätsaussagen lediglich auf Basis relativ grober
Informationen über das Eingangs-Ausgangsverhalten der Unsicherheit
zu treffen.
Der erste Teil dieser Arbeit ist der Entwicklung einer strukturierten
und umfassenden Vorgehensweise zur globalen Stabilitäts- und Güteana-
lyse auf allgemeinen Funktionenräumen gewidmet. Hierdurch kann ein
weitaus größeres Anwendungsgebiet als durch klassische IQC Theorie
erschlossen werden. Für den konkreten Fall, dass die Unsicherheit durch
eine sektoriell- oder steigungsbeschränkte Nichtlinearität definiert ist,
ermöglicht es unsere neuartige Herangehensweise, alle in der Literatur
verfügbaren und auf Multiplikatoren basierenden Stabilitätskriterien
flexibel zu kombinieren und darüber hinaus zu verbessern.
Indem wir einen direkten und sehr allgemeinen Zusammenhang zur
Willems’schen Dissipationstheorie herstellen, zeigen wir im zweiten Teil
unserer Arbeit zum ersten Mal, dass allgemeine IQC Theorie sogar auf
die lokale Analyse von Rückkopplungssystemen angewendet werden kann.
Die Grundlage hierfür bildet eine Trajektorien basierte Formulierung
der IQC Theorie, welche uns eine Kombination mit Standard Lyapunov
vii
Argumenten erlaubt. Infolgedessen ist es uns nun möglich, lediglich auf
Basis von Eingangs-Ausgangsbeschreibungen der Unsicherheiten, harte
Zustands- oder Ausgangsbeschränkungen an den durch eine Zustands-
raumdarstellung gegebenen linearen Teil des Rückkopplungssystems zu
garantieren.
viii
Contents
1 Motivation and contributions 1
1.1 Some words on robustness analysis . . . . . . . . . . . . 1
1.2 Main goal of the thesis . . . . . . . . . . . . . . . . . . . 5
1.3 Outline and contribution . . . . . . . . . . . . . . . . . . 5
I A comprehensive framework for global analysis
of feedback interconnections 9
2 Integral quadratic constraints 13
2.1 A historical perspective . . . . . . . . . . . . . . . . . . 13
2.2 Integral quadratic constraints . . . . . . . . . . . . . . . 23
2.2.1 The setting of Megretski and Rantzer . . . . . . 23
2.2.2 The Popov criterion in the IQC framework . . . 27
2.3 A note on the application of Theorem 2.6 . . . . . . . . 29
2.4 Connection to other approaches . . . . . . . . . . . . . . 31
3 A first generalization of the IQC theorem 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 A generalized IQC theorem . . . . . . . . . . . . . . . . 34
3.2.1 Basic definitions . . . . . . . . . . . . . . . . . . 35
3.2.2 Stability theorem . . . . . . . . . . . . . . . . . . 36
3.3 A class of sampled-data systems . . . . . . . . . . . . . 39
3.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . 40
ix
3.3.2 Lifting and frequency domain . . . . . . . . . . . 42
3.3.3 IQC description and reduction to finite dimensions 44
3.4 Application to PWM feedback systems . . . . . . . . . . 51
3.4.1 Definition of a PWM . . . . . . . . . . . . . . . . 51
3.4.2 Direct approach to PWM analysis . . . . . . . . 52
3.4.3 Averaging approach to PWM . . . . . . . . . . . 55
3.4.4 Computational results . . . . . . . . . . . . . . . 61
3.5 Summary and possible extensions . . . . . . . . . . . . . 64
4 A general analysis framework 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Function spaces, causality and boundedness . . . . . . . 68
4.3 Fundamental stability result . . . . . . . . . . . . . . . . 71
4.4 From stability to performance analysis . . . . . . . . . . 74
4.5 Application to Sobolev spaces . . . . . . . . . . . . . . . 77
4.5.1 Sobolev spaces . . . . . . . . . . . . . . . . . . . 77
4.5.2 Systems and quadratic forms on H r . . . . . . . 78
4.5.3 Verification of stability and performance . . . . . 79
4.5.4 Application to parametric uncertainties . . . . . 82
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Slope-restricted nonlinearities 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Application to slope-restricted nonlinearities . . . . . . . 90
5.2.1 Slope-restricted nonlinearities . . . . . . . . . . . 90
5.2.2 Signal spaces and operators . . . . . . . . . . . . 91
5.2.3 Well-posedness . . . . . . . . . . . . . . . . . . . 92
5.2.4 Quadratic forms and verification of constraints . 94
5.2.5 Sketch of procedure . . . . . . . . . . . . . . . . 95
5.3 Derivation and application of multipliers . . . . . . . . . 95
5.3.1 Full-block multipliers for the circle criterion . . . 95
5.3.2 Classical Popov criterion . . . . . . . . . . . . . . 99
5.3.3 Full-block Zames-Falb criterion . . . . . . . . . . 104
x
5.3.4 Combination of multipliers in the frequency domain113
5.4 General Popov and Yakubovich criteria . . . . . . . . . 117
5.4.1 Full-block Yakubovich criterion . . . . . . . . . . 118
5.4.2 Popov criterion for D 6= 0 . . . . . . . . . . . . . 121
5.4.3 Combination of multipliers in the state space . . 122
5.5 Related stability tests . . . . . . . . . . . . . . . . . . . 126
5.5.1 Multiplier proposed by Park . . . . . . . . . . . . 126
5.5.2 Stability criterion by Hu et al. . . . . . . . . . . 127
5.5.3 Zames-Falb implementation by Turner et al. . . . 127
5.6 Numerical examples . . . . . . . . . . . . . . . . . . . . 128
5.7 Summary and recommendations . . . . . . . . . . . . . 133
6 Absolute stability of discrete-time interconnections 135
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 138
6.3 Principles of stability multipliers . . . . . . . . . . . . . 140
6.3.1 Methods based on polytopic bounding . . . . . . 140
6.3.2 Subgradient based arguments . . . . . . . . . . . 144
6.4 Relation to multipliers in the literature . . . . . . . . . . 150
6.4.1 Zames-Falb multipliers of order one . . . . . . . 150
6.4.2 Redundant multiplier combinations . . . . . . . 152
6.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . 153
6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 157
II From input-output properties to the analysis
of internal behavior 159
7 Invariance with dynamic multipliers 163
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 168
7.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . 168
7.2.2 Performance specifications . . . . . . . . . . . . . 171
xi
7.2.3 Technical motivation of contributions . . . . . . 175
7.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.4 Relation to existing local IQC results . . . . . . . . . . . 182
7.5 Application to regional performance criteria . . . . . . . 184
7.5.1 Invariance with general dynamic IQC multipliers 185
7.5.2 Invariance using regionally valid IQCs . . . . . . 188
7.5.3 Excitation through nonzero initial conditions . . 189
7.6 A selection of concrete applications . . . . . . . . . . . . 191
7.6.1 Real parametric uncertainties . . . . . . . . . . . 192
7.6.2 Locally stable saturated systems . . . . . . . . . 194
7.6.3 Unbounded nonlinearities . . . . . . . . . . . . . 199
7.7 Summary and recommendations . . . . . . . . . . . . . 203
8 Hard Zames-Falb factorizations for invariance 205
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.2 Local analysis with hard and soft IQCs . . . . . . . . . 208
8.2.1 Two local IQC results . . . . . . . . . . . . . . . 209
8.3 Zames-Falb multipliers . . . . . . . . . . . . . . . . . . . 213
8.4 Concrete numerical example . . . . . . . . . . . . . . . . 220
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9 Concluding remarks 223
III Appendices 225
A Explanation of symbols 227
B List of terms 231
C Some additional proofs 233
References 250
Declaration 273
xii
Chapter 1
Motivation and contributions
1.1 Some words on robustness analysis
One of the most important aspects of control theory is the under-standing of the behavior of a dynamical system, which is usually
defined by its processing properties that define the response or output
to some external excitation or input (see Figure 1.1). In this setup, both
the input and the output are functions of time and also the dynamic
properties of the system may vary over time. A frequently taken ap-
proach to understanding the system characteristics relies on stimulating
it with some input and observing the corresponding output, which
motivates the abstract description of a dynamical system as an operator
mapping input signals into outputs. This rather general viewpoint is
one of the major reasons for the successful application of control theory
concepts far beyond engineering applications, and in such diverse areas
as, e.g., the social behavior of humans [30, 31], the manipulation of
quantum states [114, 41, 99], and system theoretical understanding of
biological processes [47, 179, 73].
Very often such abstract maps that transform an input signal into
the output, can be described using differential (or difference) equations
1
2 Chapter 1. Motivation and contributions
SystemOutput Input
Figure 1.1: Dynamical system
with fixed initial conditions. In a first step towards understanding and
also predicting the dynamical behavior of a system, we, hence, stimulate
the system and measure its corresponding response in order to model
the underlying generating principles, i.e., we describe its input-output
behavior using differential equations.
Another approach of obtaining a system model that is often used
in engineering applications aims to deduce the overall behavior of the
system from physical models of the individual components, which are
again formulated as differential equations. Obviously, this approach
relies on the a priori knowledge of the internal structure of the system
that is not necessary for the above described input-output approach.
However, independent of the chosen path we will not be able to arrive
at an exact quantification, which gives rise to discrepancies between
the true dynamical behavior of the system and that of our approximate
model. These intrinsic deviations are the fundamental reason behind
the classical paradigm of robust control, the assumption of a so-called
uncertain system; a nominal part of the system N , our best guess, and
an uncertain component ∆, with which we take care of all effects that
may have led to a wrong nominal system such as, e.g., modelling errors
and measuring inaccuracies.
A further complication arises from the fact that highly accurate
models are typically rather hard or even impossible to handle in terms
of simulation, analysis or control. Thus we strive for models that describe
the true behavior sufficiently well, while keeping the level of complexity
as low as possible. Here, a natural first step is to assume a linear and
also time-invariant, nominal model behavior, described by linear time-
invariant differential equations. This allows us to rely on the large variety
of tools developed in the well-established field of linear control theory.
1.1. Some words on robustness analysis 3
However, there are no linear systems in real life. In addition, the
nonlinear behavior of a system often critically influences the whole
dynamics. As a consequence, the assumption that N is linear is paid
for by subsuming nonlinearities like saturations, delays, parametric
uncertainties or other unmodeled dynamics into the uncertain part of
the system. In conclusion, ∆ now comprises the intrinsic nonlinear
behavior in addition to the mismatch between the real-life system and
the model. Translated into the standard configuration of robust control,
we approximate the setting depicted in Figure 1.1 by considering a
nominal linear time-invariant (LTI) system N interconnected with an
uncertainty ∆ which is assumed to be contained in a set of possible
systems ∆. This leads to the interconnection depicted in Figure 1.2.
Having obtained such a system model, we may now now proceed to
the analysis of its behavior. Here the most important question are those
of stability and performance; does the system operate in a safe way and
does it respond to external disturbances in a desired way. In view of
our uncertain system approach, stability and performance analysis for
the system in Figure 1.1 now translate into robust stability and robust
performance analysis of the interconnection in Figure 1.2, i.e., into the
verification of stability and performance of the interconnection for all
∆ ∈∆.
Uncertain System
N
∆
InputOutput
Figure 1.2: Approximated dynamical system
The input-output perspective of a dynamical system as a black box
that somehow processes input data is now of pivotal importance as it
4 Chapter 1. Motivation and contributions
allows to capture the nonlinear, uncertain, time-varying (or in any other
way troubling) part of the system solely by means of their input-output
relations. As a consequence, we do not need to have any information
about the internal dynamics of the uncertainty ∆. The ability to capture
the true and often complex system behavior by means of easier to handle
yet possibly coarse input-output descriptions, is one of the reason for
the successful application of robust analysis ideas in all areas of control.
Furthermore, the obtained robustness guarantees are indispensable in
many practical applications. A framework specifically designed for the
analysis of systems containing an LTI part and a troubling one, that is
only know to satisfy some constraints on its inputs and outputs, is that
of integral quadratic constraints (IQCs) as briefly reviewed and put into
its historical perspective in Chapter 2. Here, the exclusive focus lies on
the so-called global robust stability and performance analysis, i.e., the
analysis of the input-output behavior of the uncertain system without
any restriction on the input signals.
However, this global perspective also comes at the expense of a
major disadvantage. If we describe N and ∆ only by their input-output
relations it is intrinsically impossible to quantify the consequences for
the internal dynamics in N due to external stimuli d and uncertain
dynamics ∆ ∈∆. In many practical applications, the nominal system
N is given by a (finite dimensional) state-space description, where the
states have physical meaning. Furthermore, the inputs are often know
to satisfy certain properties which allows to focus on a very particular
set of signals. In conclusion, it is desirable to guarantee not only a
certain input-output behavior but also the satisfaction of robust state
constraints, i.e., the guaranty that the states do not exceed some limit
value, under the assumption that the inputs are restricted to a given
set. The extension of the framework of IQCs, which is exceptionally
well suited for the previously described global analysis of feedback
interconnections to this so-called local analysis is, however, an open
problem [29].
1.2. Main goal of the thesis 5
1.2 Main goal of the thesis
Robust or absolute stability analysis of interconnections as depicted
in Figure 1.2 is a widely and still actively researched field that has its
roots in the 1960s. However, despite the early and also later efforts to
achieve a comprehensive and unifying analysis theory, the last decades
have witnessed the decoupling of this research area into various only
loosely connected fractures. This is, at least in part, due to the fact
that there does not seem to exist a framework that is, on the one
hand, general enough to provide answers for a wide range of research
problems while, at the same time, being powerful enough to be able to
outperform specialized approaches. This motivates us to formulate the
key objectives of the present thesis as follows:
The main goal of this thesis is to develop a comprehensive
approach for robust stability and performance analysis. It
is aimed for a framework that permits both global and lo-
cal analysis of feedback interconnections, and furthermore
provides the means for their efficient numerical verification.
1.3 Outline and contribution
The above described goals are also reflected in the two parts of the
present thesis where we focus on global and local analysis separately.
In the first part, which is concerned with the verification of global
properties, we focus on the extension and improvement of the classical
framework of integral quadratic constraints. The presented general-
izations comprise both a widening of the area of applications for IQC
theory as well as an enhancing of the analysis tools themselves. In
the second part, we consider local stability and performance analysis
by establishing a connection between classical global IQC theory and
local Lyapunov techniques by means of dissipation arguments. In the
remainder of this section, we discuss the individual contributions in
the respective chapters in some detail and finally point the reader to
6 Chapter 1. Motivation and contributions
the already published or submitted contributions that, to some extent,
constitute this thesis.
In Chapter 2 we discuss the historical roots and gradual development
of the framework of IQCs as finally presented in [110]. This chapter is
also designed to highlight the struggle for a unifying theory of stability
and to discuss the milestone results achieved in this area.
In Chapter 3 we introduce a first generalization of classical IQC
theory that significantly widens its range of applications. In particular,
the following contributions are made:
 a generalization of the IQC framework that allows for the rigorous
inclusion of uncertainties admitting sampling behavior;
 a lossless reduction of the occurring infinite dimensional FDI into
a finite dimensional LMI that permits the verification of stability
using standard solvers;
 a treatment of pulse-width modulators within the novel framework.
In Chapter 4 we extend the ideas outlined in Chapter 3 and present
a comprehensive framework for global robust stability and performance
analysis that extends the classical one in several respects as it
 allows for very general function spaces;
 permits the seamless incorporation of operators, constraints and
performance measures on Sobolev spaces;
 provides the means for numerical verification of stability and
performance in this more general setting.
In Chapter 5, we illustrate how the many advantages offered by our
novel formulation come to flourish for the particular case of continuous-
time feedback interconnections containing repeated slope-restricted
nonlinearities. The contributions are
 a novel full-block Yakubovich criterion;
1.3. Outline and contribution 7
 an asymptotically exact parametrization of full-block Zames-Falb
multipliers;
 a combined application of all multiplier based stability criteria
available;
 an extension of the Popov and Yakubovich criteria to not strictly
proper LTI systems.
With Chapter 6 we conclude the first part of this thesis by highlight-
ing the simplicity with which the IQC framework, and also our novel
extension thereof, allows to translate results from the continuous-time
setting into the discrete-time one. The contributions of this chapter are
 a novel unstructured polytopic criterion that combines Yakubovich
and circle criterion multipliers;
 a classification of various stability results in the literature;
 a combined application of all multiplier based stability criteria
available similar to the continuous-time case.
In the second part of the thesis, we shift our focus towards local
analysis of feedback interconnections by establishing in Chapter 7 a
strong and very general link to classical dissipation theory which allows
us to connect operator based IQC descriptions, as employed in the
first part, to trajectory based Lyapunov arguments. This relies on the
following contributions
 a local IQC stability theorem;
 a novel link between general dynamic multipliers and Lyapunov
theory;
 a complete framework for local stability and performance analysis
within IQC theory;
 several local performance criteria and their derivation from our
local IQC theorem.
8 Chapter 1. Motivation and contributions
In Chapter 8 we conclude the contributions of this thesis by embed-
ding the Zames-Falb multipliers discussed in Chapter 5 into the local
analysis framework of Chapter 7. This is based on
 a simple and computationally effective hard IQC factorizations of
causal and anti-causal Zames-Falb multipliers;
 a refined local analysis framework that allows for non-conservative
incorporations of arbitrary hard IQCs.
As the topics discussed in the individual chapters are rather diverse,
we aim at keeping the chapters reasonably self-contained such that each
may be read independently from the others. Of course, the price to pay
is some overlap between the chapters which is kept to a minimum.
Finally, we emphasize that the findings presented in the first part
are the result of several papers that have already been published in
conference proceedings and journals and some parts of the text overlap.
In particular, this comprises the references [33, 55] for Chapter 3, [58, 56]
for Chapters 4 and 5, as well as [57] for Chapter 6. Furthermore,
the results presented in Chapters 7 and 8 have been submitted for
publication in [60] and [59] and are currently under review.
Part I
A comprehensive framework for global
analysis of feedback interconnections
9

Introduction to Part I
The first part of this thesis is devoted to the problem of globalstability and performance analysis of feedback interconnections
containing an LTI systemM and an uncertain part ∆ subject to external
stimuli u, v (see Figure 1.3).
As already mentioned in the introduction, we do not require explicit
knowledge about the system ∆ but only rather coarse information
concerning its input-output properties. These typically apply to a whole
set of uncertain operators ∆. Moreover, we consider the external signals
u, v as disturbances acting on the feedback interconnection and aim
at concluding stability for all possible inputs and all ∆ ∈ ∆. Our
main goal will be the development of a general analysis framework
that still permits the application of standard optimization tools for the
efficient numerical verification of stability and also of given performance
specifications.
M
∆ +
+
u
v
z
w
Figure 1.3: Uncertain feedback interconnection
11
12
As this problem has been the subject of a multitude of papers,
their contributions ranging from the presentation of general analysis
frameworks to the consideration of very particular settings, we illustrate
the historical development of (robust) stability tests in some detail in
Chapter 2. In order to concisely highlight the fundamental concepts
common to all multiplier based stability results, we restrict our attention
to the successive evolvement of the framework of integral quadratic
constraints and, thus, provide the foundation for all subsequent chapters
of this thesis.
Our refinement of the standard IQC approach in Chapter 3 that
allows to incorporate uncertainties exhibiting sampling behavior into
the theory, ultimately leads to a general framework for stability and
performance analysis presented in Chapter 4. The central characteristics
of our novel framework is its ability to incorporate disturbances and
operators defined on very general function spaces. As one of the main
contributions of Part I, we subsequently present a unifying approach
to the classical and very fundamental problem of absolute stability
analysis in Chapter 5. Here ∆ is defined via a nonlinearity ϕ : R→ R
that is either sector-bounded or slope-restricted. Our comprehensive
framework allows to subsume all multiplier based stability criteria and
thus presents the least conservative stability estimates available in the
literature. Finally, we illustrate how the results developed in Chapter 5
carry over to discrete-time interconnections in Chapter 6 thus allowing
to classify and outperform the multitude of classical and more recently
proposed stability tests.
Chapter 2
The framework of integral quadratic con-
straints
Before we present the main contributions of this thesis in the subse-quent chapters, let us first take one step back and highlight some
of the many earlier results this work is build upon. As the framework
of integral quadratic constraints (IQCs) is at the very heart of this
thesis, we devote this chapter to its historical roots in absolute stability
analysis, the classical framework itself and also related approaches in
the literature.
2.1 A historical perspective
The goal of robust stability analysis within the input-output framework
is probably best described by Zames [195]:
"It seems possible, from only coarse information about a
system, and perhaps even without knowing details of in-
ternal structure, to make useful assessments of qualitative
behavior."
Historically, robust or absolute stability analysis may be traced back
to the paper of Lurye and Postnikov [106]. Due to the overwhelming
13
14 Chapter 2. Integral quadratic constraints
number of contributions in this rather actively researched field, it is
hopeless to give an exhaustive overview of the developments in the
past 60 years. Consequently, we will only discuss the major milestones
that sparked the development of the framework of IQCs as eventually
formulated in [110]. We also recommend the insightfully written intro-
duction to the problem given in the monograph [10]. Another attempt
to give a comprehensive summary of the historical development was
made by Liberzon [105] that, unfortunately, loses much of its readability
by trying to do justice to all the contributors in the field. Nevertheless,
it remains a valuable source of information.
The problem posed by Lurye and Postnikov [106] can be stated as
follows (for a collection of early papers see [107] and also the excellent
monograph [8]). Let ϕ : R→ R be a Lipschitz continuous nonlinearity
confined to the sector sec[0, β], i. e., there exist β > 0 such that
ϕ(x)(βx− ϕ(x)) ≥ 0 for all x ∈ R. (2.1)
With real matrices A ∈ Rn×n, B ∈ Rn×1 and C ∈ R1×n and ∆ϕ defined
through ∆ϕ(z)(t) := ϕ(z(t)) for all locally square integrable signals
z ∈ L2e and almost all t ∈ [0,∞), consider the feedback system
x˙ = Ax+Bw, x(0) = x0, w = ∆ϕ(z),
z = Cx. (2.2)
The objective is to determine conditions on the linear part of the
interconnection, i.e., A, B and C, that guarantee global asymptotic
stability of the trivial solution x = 0 by exploiting the input-output
properties of ∆ϕ. If M denotes the linear part in (2.2) and we visualize
the initial condition as an input to M , the interconnection (2.2) may
be depicted as in Figure 2.1.
Popov obtained a sufficient condition, by only exploiting (2.1) and
perfectly fitting with the characterization of Zames quoted above, for
the stability of (2.2) that only involves the transfer function of the
2.1. A historical perspective 15
M
∆ϕ
x0
wz
Figure 2.1: Lurye feedback interconnection
linear part, i.e., M(iω) := C(iωI −A)−1B for ω ∈ R. In fact, stability
is guaranteed if there exists a positive ε and a real1 λ such that(
M(iω)
I
)∗(
0 1− λiω
1 + λiω − 2β
)(
M(iω)
I
)
4 −εI for all ω ∈ R.
(2.3)
The solution proposed by Popov is remarkable in several respects. First,
like the other celebrated frequency-domain criterion known at the time,
the Nyquist criterion [119], it has a geometric counterpart that allows
to verify stability based on a graphical test (see, e.g., [8, p. 53]). As
the transfer function of a system can often be extracted from physical
experiments, this provides the means for efficient stability analysis.
From a more global perspective, the Popov criterion marked the first
instance of the use of so-called multipliers for the analysis of feedback
stability, thus paving the way for many central stability principles based
on the so-called passivity theorem.
In order to proceed one step further towards a unifying theory of
stability that comprises but also extends Popov’s result, let us now
discuss the contribution of Yakubovich. Shortly after Popov proposed
his stability theorem, several others emerged among which are the
circle criterion (see, e.g., [132, 18, 141, 142, 196]) and the small-gain
theorem [194, 142]. These results raised the fundamental question
whether there exists a general underlying principle that is common to
1Popov actually required λ > 0 in his seminal paper [126], but this was immedi-
ately recognized to impose an unnecessary restriction (see, e.g., [8]).
16 Chapter 2. Integral quadratic constraints
all stability criteria. One attempt to unify the aforementioned criteria
was made by Yakubovich who developed a general framework [189] that
allows to merge the circle, small-gain and Popov criterion in the case
where M is LTI. The key aspect of this generalization is the collection
of all information about the nonlinearity using quadratic forms. For
example, the so-called sector constraint (2.1) implies the validity of the
pointwise quadratic constraint(
z(t)
∆ϕ(z)(t)
)T (
0 1
1 − 2β
)(
z(t)
∆ϕ(z)(t)
)
≥ 0 for all z ∈ L2 (2.4)
and almost all t ∈ [0,∞). Note that the middle matrix in (2.4) exactly
matches the one in (2.3) for λ = 0, a key fact in the proof of Popov’s
stability result. Extending this idea, Yakubovich assumes that ∆ϕ
satisfies the (integral) quadratic constraint2∫ T
0
(
z(t)
∆ϕ(z)(t)
)T
Π
(
z(t)
∆ϕ(z)(t)
)
≥ 0 for all T ≥ 0, z ∈ L2 (2.5)
defined by a constant symmetric matrix Π. Simply put, Yakubovich
proves that stability of (2.2) is then guaranteed if there exists ε > 0 such
that the stable LTI system M satisfies the frequency-domain inequality
(FDI) (
M(iω)wˆ(iω)
wˆ(iω)
)∗
Π
(
M(iω)wˆ(iω)
wˆ(iω)
)
≤ −εwˆ(iω)∗wˆ(iω) (2.6)
for all ω ∈ R and all w ∈ L2. We emphasize that in the above
formulation the matrix Π does not depend on time (or frequency) as is
the case in (2.3). Hence, Popov’s criterion is not contained so far. By a
somewhat artificial addition of a frequency dependent term, Yakubovich
was able to extend his criterion such that it also incorporates the
2The major stability theorem in [189] is formulated with pointwise quadratic
constraints. Yet, as remarked by Yakubovich, the proof only requires the less
restrictive integral constraint (2.5).
2.1. A historical perspective 17
one proposed by Popov. Although the proof of stability within this
framework relies on a Lyapunov argument, the ideas developed in [189]
are essential for the later established framework of integral quadratic
constraints. In particular, the approach by Yakubovich already allows
to combine different individual stability criteria in order to enhance the
resulting stability test.
Equally fundamental as the contributions by Popov and Yakubovich
are those by Sandberg [141, 142, 143] and Zames [195, 196], who, in
contrast to the predominantly pursued Lyapunov approach, applied
functional analytic methods to the robust stability problem. As these
papers are obviously closely related, and appeared at roughly the same
time, it is now impossible to assign the individual contributions to
the respective author. In the following, we discuss the general theory
outlined in the seminal papers by Zames [195, 196].
The key problem considered by Zames and Sandberg can be sum-
marized as follows. Given two dynamical systems M and ∆, defined as
arbitrary maps from one function space into another3 that are intercon-
nected as
z = M(w) + v, w = ∆(z) + u, (2.7)
with external disturbances u, v, find conditions on M, ∆ such that
the maps from v → z and u→ w are bounded. The pivotal difference
to the by then already well developed Lyapunov method lies in the
mathematical formulation of this problem in the input-output setting
which completely avoids details of the internal structure and considers
the interconnection as an open system, able to interact with its sur-
roundings. This point of view instrumentally relied on the novel concept
of extended spaces, that allows to analyze situations where the signals
in the loop are not a priori bounded (contained in a normed space) but
are shown to have this property a posteriori; this is then interpreted as
input-output stability.
3Zames actually defines systems more generally as relations, i.e., their graphs are
subsets of the product of the input and the output space.
18 Chapter 2. Integral quadratic constraints
Most noteworthy among the many contributions and deep insight
in [195, 196] are the formulation of three fundamental stability theorems:
the small-gain theorem, the conicity (or conic relation) theorem and,
formulated in [195] as a corollary of the latter, the passivity (or positivity)
theorem. Intuitively, the small gain theorem states that if the loop gain
is less than one, then the interconnection is stable. The conicity theorem
generalizes this concept to the case where stability is guaranteed if M
and ∆ satisfy certain conic relations, while the passivity theorem is then
obtained as a limit case of the conicity theorem.
It is exceptional that all three theorems sparked the development
of entire fields within control theory. As already indicated in [196], the
applicability of the passivity theorem is immensely enhanced by the
idea of not only considering the systems M and ∆ but by allowing a
factorization of the loop into two parts that then have to satisfy the
passivity conditions. This idea allowed Zames to merge his theory with
the results of Popov [126] as well as Brockett and Willems [22, 23]
and was further developed within multiplier theory (see, e.g., [197,
44]). In addition, the small gain theorem proved instrumental for the
development of robust and H∞ optimal control [62, 48, 49]. And finally,
his result on conic relations was generalized by Safonov [138, 137] in
his seminal work on topological separation that eventually lead to the
formulation of the framework of integral quadratic constraints (see
also [159, 70, 89, 88]).
It is also remarkable that many of the results following these early
developments can be seen as natural extensions thereof. The celebrated
theory of dissipativity as formulated by Willems in [182, 183], e.g.,
extends the results by Yakubovich (as well as those by Brockett and
Willems [22, 23]) based on Lyapunov theory towards open systems. The
pioneering contributions by Willems were made possible by merging
Lyapunov theory with the concept of extended spaces as proposed by
Zames and Sandberg. However, in contrast to Zames’ work, knowledge
about the interior dynamics, i.e., the assumption of a state-space de-
scription, is essential to the approach of Willems; thus it is a genuine
2.1. A historical perspective 19
extension of the work of Yakubovich [189]. The strength and general
applicability of the ideas presented by Willems were recognized by many
researches (see, e.g., [77, 78]) and they still play a fundamental role in
the understanding of systems behavior [101, 12, 172]. Yet, similarly to
the results by Yakubovich, and to a certain extent also those by Safonov
that are discussed next, dissipation theory suffers from the drawback
that it relies on quadratic constraints of the form (2.5) that are often
limiting.
In contrast to the approach by Willems, the very elegant stability
theorem proposed by Safonov completely avoids the concept of interior
dynamics. In [138, 137] Safonov generalized the conic relation theorem
of Zames by introducing separating functionals. Following the approach
by Zames, the central result is formulated on general extended spaces
and applicable to systems defined through relations. For simplicity,
let us illustrate the key ideas of [138, 137] for the special case where
M and ∆ are operators defined on the extended space L2e. Then,
stability of (2.7) is guaranteed if there exists a separating functional
σ : L2 ×L2 → R such that for all T > 0 the following two conditions
hold4
a) σ(zT , (∆(z))T ) ≥ 0 for all z ∈ L2e;
b) there exists some ε > 0 such that
σ((Mw)T , wT ) ≤ −ε
(‖wT ‖2 + ‖(Mw)T ‖2) for all w ∈ L2e.
As a significant contribution, the general version of this result [137,
Theorem 2.1] is not only applicable in the input-output setting proposed
by Zames but also encompasses the Lyapunov based stability results
derived by Yakubovich. Thus Safonov’s contribution may be seen as the
4uT denotes the truncation of u ∈ L2e, i.e., uT = u on [0, T ] and u = 0 on
(T,∞); see Definition 2.1.
20 Chapter 2. Integral quadratic constraints
unification (and also generalization) of both classical theories. Indeed,
by choosing
σ(z, w) :=
∫ ∞
0
(
z(t)
w(t)
)T
Π
(
z(t)
w(t)
)
dt for z, w ∈ L2,
we obtain the stability criterion by Yakubovich for LTI systems M .
Moreover, the conditions formulated by Safonov reduce the stability
theorem to its essence, a positivity constraint on the graph of one
operator and a strict negativity constraint on the inverse graph of the
other. It is this fundamental principle that all known stability theorems
follow.
However, in terms of the formulation of novel stability criteria, the
major issue concerning applicability lies of course in the choice of the
functional σ. In order to recover most of the classical stability results,
Safonov uses functionals defined via an inner product and globally
Lipschitz functions S1, S2 as
σ(zT , wT ) = 〈S1(zT ), S2(wT )〉 for all z, w ∈ L2e. (2.8)
Yet, as we will see in the subsequent section, this definition (and also the
more general stability criteria in [137, 159, 70]) does not allow to recover
all classical results, where the one proposed by Zames and Falb [197]
poses a well-known exception.
Let us now proceed one step further towards the framework estab-
lished in this thesis by formulating a graph separation result, similar in
spirit to the one by Safonov, that also allows for quadratic forms as in
(2.8). To this end, consider the following scenario: With maps M, ∆
on L2e and an external disturbance d ∈ L2e we study the feedback
interconnection
z = M(w) + d, w = ∆(z) (2.9)
as depicted in Figure 2.2. All results presented in this thesis funda-
2.1. A historical perspective 21
M
∆
+
d
wz
Figure 2.2: Feedback interconnection with external disturbance
mentally rely on the concept of causality that is typically defined using
truncation operators.
Definition 2.1.
Let T > 0. Then the truncation operator (or past projection) PT :
L2e → L2e is defined as
(PTu)(t) := uT (t) :=
{
u(t), t ∈ [0, T ],
0, t > T
for all u ∈ L2e and almost all t ∈ [0,∞). For brevity of notation we
write uT := PTu for u ∈ L2e. An operator S : L2e → L2e is said to be
causal if PTS = PTSPT holds for any T > 0 on L2e. ?
For the separating functionals Σ : L2 → R, we only assume the
following property:
∃c > 0 : Σ(u+v)−Σ(u) ≤ 2c‖u‖‖v‖+c‖v‖2 for all u, v ∈ L2. (2.10)
With these definitions, all prerequisites are assembled in order to state
an intermediate stability result in the spirit of [137, 159], but already
formulated it in such a way that it provides a transition to the integral
quadratic constraints framework. This allows us to highlight the ad-
vantages and also the disadvantages if compared to the later presented
classical IQC theory. We give a direct proof here (due to Scherer [146])
that will serve as a starting point for later more general IQC results.
22 Chapter 2. Integral quadratic constraints
Theorem 2.2 ([146])
Suppose that M : L2e → L2e is causal and bounded, ∆ : L2e → L2e is
causal, Σ : L2 → R satisfies (2.10), D ⊂ L2e, and that
a) there exist  > 0 and m0 such that
Σ
(
M(w)T
wT
)
≤ −‖wT ‖2 +m0 for all T > 0, w ∈ L2; (2.11)
b) there exists δ0 ≥ 0 with
Σ
(
zT
∆(z)T
)
≥ −δ0 for all T > 0, z ∈M(L2) +D . (2.12)
Then there exist γ > 0 and γ0 ∈ R such that for any d ∈ D and any
response z ∈ L2e satisfying (2.9), we have
‖zT ‖2 ≤ γ2‖dT ‖2 + γγ0 for all T > 0. (2.13)
If M is linear one can choose γ0 = m0 + δ0.
Proof. A proof is found in Appendix C.1.1.
Remark 2.3.
In accordance with [195, 137, 159] neither existence nor uniqueness of
a solution to (2.9) is assumed. Instead, the conclusion is formulated
for all those disturbances d ∈ D for which the feedback interconnection
does have a response. ?
Remark 2.4.
In the terminology of [110], Theorem 2.2 can be interpreted as a
hard IQC stability result. For the particular choices Σ(xT ) :=∫ T
0
x(t)Π(t)x(t) dt (with some appropriately chosen Hermitian val-
ued and essentially bounded function Π : [0,∞)→ Rn×n), D = L2 and
δ0 = 0, the constraint (2.12) reads as∫ T
0
(
z(t)
∆(z)(t)
)T
Π(t)
(
z(t)
∆(z)(t)
)
≥ 0 for all T > 0, z ∈ L2 (2.14)
2.2. Integral quadratic constraints 23
which is a generalization of (2.5) to time varying Π and coincides with
the hard IQC constraint defined in [110]. The term hard IQC originates
from the fact that (2.14) is required to hold for all T > 0 and not just
for T =∞ (as is the case for soft (time-domain) IQCs). ?
It is important to note that the general formulation of Theorem 2.2
for arbitrary ∆ and bounded M typically renders the (numerical) verifi-
cation of stability for a given separating functional Σ and two nonlinear
systems M and ∆ impossible. This problem is circumvented in the sub-
sequent section by assuming a particular structure of Σ and restricting
the attention to LTI systems M .
2.2 Integral quadratic constraints
Let us now introduce the framework of IQCs as established in the
seminal papers [129, 110, 130, 93] by Megretski, Rantzer and Jönsson.
The subsequent sections are devoted to the description of the underlying
setting, the highlighting of their major contributions and also the
numerical verification of stability using linear matrix inequalitys (LMIs).
2.2.1 The setting of Megretski and Rantzer
As already mentioned, the results by Megretski and Rantzer were largely
motivated by the general framework developed by Yakubovich in the
1960s. By the 1990s it was very well-known (see, e.g., [182, 74, 137, 78])
that hard IQC constraints could be exploited in order to guarantee
the existence of a Lyapunov function, thus providing an alternative to
the functional analytic approach taken by Zames and Sandberg. Yet,
there remained annoying exceptions, such as the celebrated Zames-Falb
stability criterion [197], for which no hard IQC representation is known.
Moreover, the application of the corresponding, typically non-causal,
multipliers for stability analysis required factorizations [197, 44]. This
inspired the formulation of a more general stability criterion that covers
all hard IQC results as special cases. Aiming at a framework that allows
24 Chapter 2. Integral quadratic constraints
for efficient numerical verification of stability, Megretski and Rantzer
restricted their setting to the case where M is an LTI system. The
underlying setting can then be formulated as follows.
Given an stable LTI system M as realized by
x˙ = Ax+Bw, x(0) = 0,
z = Cx+Dw
with A being Hurwitz, and a causal and bounded operator ∆ : L2e →
L2e, we consider the feedback interconnection
z = Mw +Mu+ v, w = ∆(z) (2.15)
with external disturbances u, v ∈ L2. Due to the linearity of M , (2.15)
is equivalent to the canonical configuration (2.7) and may be depicted
as in Figure 2.3.
M
∆ +
+
u
v
z
w
Figure 2.3: Feedback interconnection in the input-output framework
In accordance with the classical papers [195, 196] we split the ques-
tions of existence and uniqueness of solutions from the one of stability of
the interconnection. Yet, in contrast to these papers which completely
avoid the subject of well-posedness, we will assume it as a prerequisite.
The necessary assumptions were most concisely already formulated by
Zames [195] who stated in view of his work with relations instead of
operators that:
"For the results to be practically significant, it must usually
be known from some other source that solutions exist and
are unique (and have infinite escape times)."
2.2. Integral quadratic constraints 25
All this will be reflected in the following definition of well-posedness of
(2.15).
Definition 2.5 ([110]).
The interconnection (2.15) is well-posed if for each (u, v) ∈ L2e×L2e
and for each τ ∈ [0, 1] there exists a unique z ∈ L2e satisfying z −
Mτ∆(z) = Mu+ v and such that the correspondingly defined response
map (u, v) 7→ z = Rτ (u, v) is causal:
Rτ (u, v)T = Rτ (uT , v)T for all T ≥ 0, τ ∈ [0, 1], (u, v) ∈ L2e ×L2e.
The feedback system (2.15) is stable if, in addition, R1 : L2×L2 → L2
is bounded. ?
Thus, well-posedness not only requires existence and uniqueness of a
solution of (2.15), but for all interconnections (2.15) where ∆ is replaced
by τ∆ and τ ∈ [0, 1]. This stronger assumption is introduced in order to
deal with soft IQCs that are defined as follows: Two signals z, w ∈ L2
with Fourier transforms zˆ, wˆ are said to satisfy the IQC defined by a
multiplier Π = Π∗ ∈ RL∞, if
ΣΠ
(
z
w
)
=
∫ ∞
−∞
(
zˆ(iω)
wˆ(iω)
)∗
Π(iω)
(
zˆ(iω)
wˆ(iω)
)
dω ≥ 0. (2.16)
A causal operator ∆ : L2 → L2 satisfies the IQC imposed by Π, in
short ∆ ∈ IQC(Π), in case that
ΣΠ
(
z
∆(z)
)
≥ 0 for all z ∈ L2. (2.17)
With these preparations, we can formulate the central result in [110].
Theorem 2.6 ([110, Theorem 1])
Assume that the interconnection (2.15) is well-posed. Then it is also
stable if
a) τ∆ satisfies the IQC defined by Π for all τ ∈ [0, 1];
26 Chapter 2. Integral quadratic constraints
b) there exists some ε > 0 such that the following IQC holds:
ΣΠ
(
Mw
w
)
≤ −ε‖w‖2 for all w ∈ L2. (2.18)
Remark 2.7.
In [110] the constraint in Theorem 2.6 b) is rewritten in terms of the
following FDI. There exists ε > 0 such that(
M(iω)
I
)∗
Π(iω)
(
M(iω)
I
)
4 −εI for all ω ∈ R. (2.19)
Both conditions can be shown to be equivalent if the left-hand side
in (2.19) is continuous and bounded as is the case here. In the sequel, we
will typically work with (2.18) since it is more suitable for generalizations
and also nicely displays the symmetry in the treatment ofM and ∆. Yet,
as will be demonstrated in Section 2.3, (2.19) allows for the immediate
translation of standard IQCs into LMIs and, thus, the verification of
stability using standard optimization solvers. ?
Remark 2.8.
The choice of the separating function in (2.16) is of course rather
particular. However, it is explicitly tailored for numerical verification
and thus the practical application of Theorem 2.6. Moreover, as another
contribution in [110], it contains a rather long list of relevant choices
of Π, thus enabling the application of IQC theory for a wide range of
problems. ?
If we compare Theorems 2.6 and 2.2 it emerges that the generalization
toward soft IQCs is paid for by requiring both the boundedness of ∆ and
the well-posedness of (2.15) for ∆ replaced by τ∆ and all τ ∈ [0, 1]. The
latter is an essential ingredient to the proof that relies on a homotopy
argument connecting the stable (trivial) interconnection for τ = 0 with
the interconnection under consideration for τ = 1. There exists several
generalizations that weaken both the assumptions on boundedness of
∆ [152] and also the well-posedness of (2.15) [109].
2.2. Integral quadratic constraints 27
One of the major drawbacks of Theorem 2.6 is that the restriction
to multipliers Π ∈ RL∞ does not allow for Popov multipliers as in (2.3)
since they are obviously not bounded on iR. The remedy proposed by
Jönsson is addressed in the following section.
2.2.2 The Popov criterion in the IQC framework
As in the case of the framework proposed by Yakubovich (but due
to different reasons), also the IQC framework does not allow for the
immediate inclusion of Popov’s stability criterion. A first, rather obvious
reason is the fact that the middle matrix, i.e., the multiplier, in (2.3) is
unbounded on the imaginary axis (for λ 6= 0). In addition, the frequency
dependent part of this multiplier given by
Πλ(iω) =
(
0 −iωλ
iωλ 0
)
for some real λ = λT and all ω ∈ R
(2.20)
was introduced by Popov in order to exploit constraints of the from∫ ∞
0
w(t)Tλz˙(t) dt ≥ 0 where w = ∆(z). (2.21)
However, in order for this integral to make sense, the signal z has
to be differentiable. Thus, if we want to capture the operation of ∆
using constraints as in (2.21), the formulation (2.17), that requires
non-negativity for all z ∈ L2, is too restrictive.
A solution to both problems was proposed by Jönsson. We discuss
the extension of the IQC framework outlined in [93, 92] in some detail
since these represent, in fact, a first step towards our subsequently
derived general framework. In order to render z in (2.15) differentiable,
Jönsson restricted his attention to interconnections of a strictly proper
and stable LTI system M with some causal and bounded uncertainty ∆
x˙ = Ax+Bw, x(0) = x0, w = ∆(z) + u, u ∈ L2
z = Cx. (2.22)
28 Chapter 2. Integral quadratic constraints
Note that, due to the nonzero initial condition, M is not linear and
hence we cannot consider this interconnection within the IQC frame-
work directly. However, by setting V =
{
CeA•x0
∣∣ x0 ∈ Rn}, (2.22) is
equivalent to
x˙ = Ax+Bw, x(0) = 0, w = ∆(z) + u, (u, v) ∈ L2 × V
z = Cx+ v. (2.23)
Thus, if compared to the setting of Megretski and Rantzer, Jönsson, on
the one hand, allowed for non-zero initial conditions, but on the other
hand, confines the originally free input v to the set V containing the
response due to x0. By further exploiting the filtering property of the
strictly proper plant M , the signal z in the loop (2.15) is now indeed
differentiable. This leads to the following definition of well-posedness.
Definition 2.9 ([93, Definition 2]).
The interconnection (2.22) is well-posed if for any τ ∈ [0, 1], any initial
condition x0, and for any input u ∈ L2e there exists a solution (x, z)
such that (x, x˙, z) ∈ L2e × L2e × L2e, where ∆ is replaced by τ∆.
Furthermore, the map from u to (x, z) should be causal. ?
In order to circumvent the second stumbling block in applying
Theorem 2.6, the fact that the multiplier (2.20) is unbounded, Jönsson
added a second (bounded) multiplier Πb = Π∗b ∈ RL∞, which allowed
him to prove the following theorem.
Theorem 2.10 ([93, Theorem 1])
Assume that the interconnection (2.15) is well-posed. Further let
a) there exist δ > 0 such that for all z ∈ L2 with z˙ ∈ L2 the following
constraint is satisfied:
ΣΠλ
(
z
∆(z)
)
+ ΣΠb
(
z
∆(z)
)
≥ −δ‖Cx(0)‖2; (2.24)
b) there exist ε > 0 such that (2.19) holds with Π = Πλ + Πb.
2.3. A note on the application of Theorem 2.6 29
Then the interconnection (2.15) is stable, i.e., there exist positive con-
stants γ, γ0 such that
‖z‖2 ≤ γ‖u‖2 + γ0‖x0‖2 for arbitrary x0 ∈ Rn and u ∈ L2.
In conclusion, Jönsson’s approach allows for constraints that are
not valid on the full space L2 but on a subspace that is somehow
compatible with the set of multipliers and the feedback interconnection
under consideration. We will exploit this idea in much greater generality
in the subsequent chapters.
2.3 A note on the application of Theorem 2.6
Let us now focus on the application of Theorem 2.6 with particular
emphasis on the numerical verification of stability. Readers interested in
a more thorough presentation are referred to the elaborate tutorial [176]
on IQCs.
Assume that we are given an uncertainty ∆ and a class of multipliers
Π ⊂ RL∞ such that (2.17) holds for all Π ∈ Π and all τ∆ with τ ∈ [0, 1].
Moreover, we assume that (2.15) is well-posed. The approach outlined
in this section immediately extend to the Popov criterion formulated
above, even though the requirement Π ⊂ RL∞ is not satisfied, as
discussed in detail in Section 5.3.4.
First note that any Π = Π∗ ∈ RL∞ can be factorized as
Π = Ψ∗PΨ with a real P = PT and Ψ ∈ RH∞. (2.25)
Indeed, we can choose some (large) η such that Π+ηI  0 on C∞0 ; if ψ is
a spectral factor with Π+ηI = ψ∗ψ [62], we get (2.25) for Ψ := col(ψ, I)
and P := diag(I,−ηI). This insight motivates to parameterize the class
Π with fixed Ψ ∈ RH∞ as
Π = {Ψ∗PΨ | Ψ ∈ RH∞ and P ∈ P}
30 Chapter 2. Integral quadratic constraints
for some subset P of the real symmetric matrices. Then, then intercon-
nection (2.15) is stable, if there exits some P ∈ P such that
∃ε > 0 :
(
M
I
)∗
Ψ∗PΨ
(
M
I
)
4 −εI on C0. (2.26)
The celebrated KYP lemma [187, 180, 128, 14] equivalently characterises
(2.26) as an LMI feasibility problem. A very general version is derived
in [14] and stated below.
Lemma 2.11 (Generalized KYP Lemma [14])
Let A ∈ Rn×n, B ∈ Rn×m and suppose that the real symmetric matrix
K is structured as K =
(
K11 K12
KT12 K22
)
∈ Sn+m. Then the following
statements are equivalent:
a) There exists X ∈ Sn such that(
I 0
A B
)T (
0 X
X 0
)(
I 0
A B
)
+K ≺ 0;
b) K22 ≺ 0 and for all ω ∈ R the following implication holds:
(
iωI −A B)(x
w
)
︸ ︷︷ ︸
6=0
= 0 =⇒
(
x
w
)T
K
(
x
w
)
< 0.
Typically, it will suffice to work with the following particular version,
that is an easy corollary of Lemma 2.11 under the additional assumption
that A has no eigenvalues on the imaginary axis.
Lemma 2.12 (KYP Lemma [128])
Let P ∈ Sm and assume that the realization (A,B,C,D) of the LTI
system G satisfies eig(A) ∩ C0 = ∅. Then the following statements are
equivalent:
2.4. Connection to other approaches 31
a) There exists X ∈ Sn such that I 0A B
C D
T  0 X 0X 0 0
0 0 P
 I 0A B
C D
 ≺ 0;
b) There exists ε > 0 such that G∗PG 4 −εI on C0.
In conclusion, if P admits an LMI description, the search for P ∈
P satisfying (2.26) is characterized through Lemma 2.12 with G =
Ψ col(M, I) as an LMI feasibility problem and is thus verifiable with
standard optimization tools. Note that all LMIs in this thesis are solved
using MATLAB’s LMI Lab.
2.4 Connection to other approaches
Let us emphasize that this chapter is solely devoted to a very brief expo-
sition of the evolution of the framework of integral quadratic constraints
and its practical application. For brevity of display, many related topics
were only touched upon or even completely omitted in the presentation.
In particular, we did not elaborate on the connection between classi-
cal multiplier theory (see, e.g., [44, 197, 181, 22, 23]) and the framework
of IQCs as this is already the subject of several papers (see, e.g., [63]
and also [72, 71]) and is also discussed by Megretski and Rantzer [110].
Moreover, Jönsson [92] devoted an insightfully written section in his
thesis on this subject (see also [94]) that highlights all major links
between the two fields.
Furthermore, the links between dissipation theory, the IQC frame-
work and the stated KYP results are completely omitted. Yet, as one
of the major contributions of this thesis is the merging of dissipation
theory with integral quadratic constraints, we postpone the discussion
on their precise connection to Chapter 7.
32 Chapter 2. Integral quadratic constraints
Chapter 3
A first generalization of the IQC theorem
with applications to a class of sampled-data
systems
3.1 Introduction
Inspired by the classical IQC framework as outlined in Section 2.2, weshow in the present chapter that it is still possible to prove stability
under weaker assumptions on the multiplier and the IQC description of
the uncertainty. By taking the interconnection structure into account,
we can relax the assumptions on the uncertainty such that they are no
longer required to hold on the full signal space. Our generalizations are
largely stimulated by system interconnections containing uncertainties
∆ that exhibit sampling behavior, which means in our context that they
satisfy ∆ = ∆Sh with the sample and hold operator Sh. By means
of our extension of Theorem 2.6 (and also its discrete-time counter-
part [97, Theorem 1]), we show how to handle such uncertainties within
the IQC framework. Moreover, we present an exact reduction of the
resulting infinite dimensional frequency-domain inequality (FDI) to a
finite dimensional linear matrix inequality (LMI) feasibility problem,
thus rendering the stability test computational. As an example of an
33
34 Chapter 3. A first generalization of the IQC theorem
uncertainty exhibiting sampling behavior, we consider a pulse-width
modulator (PWM). Applications of such interconnections are manifold
and vary from power converters (see, e.g., [80, 98, 64] and references
therein) and biological models (see, e.g., [38, 43, 113, 34]) to attitude
control of satellites as considered, for example, in [1, 108]. For back-
ground information on the physical modeling of satellite thrusters using
pulse-width modulation and a discussion on the resulting stability issues,
we refer the reader to our preliminary work in [33]. The advantages of
the generalized framework presented in this chapter is demonstrated by
comparing our results with the analysis techniques introduced in [69]
and [82].
The chapter is structured into three parts. In Section 3.2 our IQC
stability result is proven in a general setting. We then illustrate in
Section 3.3 how to incorporate systems into the proposed framework
where the uncertainty shows sampling behavior. Furthermore, we
reduce our stability test to deciding the feasibility of a standard finite
dimensional LMI. In Section 3.4 the derived results are applied to
a feedback interconnection including a PWM, and some numerical
illustration is provided. Finally, we emphasize that the results in this
chapter have already appeared in [55] and large portions of the text
overlap.
3.2 A generalized IQC theorem
Let us first prepare the stage for our generalized discrete-time version
of Theorem 2.6 by discussing the underlying setup.
3.2. A generalized IQC theorem 35
3.2.1 Basic definitions
LetH be a Hilbert space, k ∈ N, andH k the k-fold Cartesian product.
Then `(H k) := (H k)N0 is the set of all H k-valued sequences. With
the standard norm on H k we set
‖u‖2 :=
∞∑
n=0
‖u(n)‖2H k for u ∈ `(H k)
and `2(H k) :=
{
u ∈ `(H k) ∣∣ ‖u‖ <∞} . For T ∈ N0 let PT denote the
(discrete-time) truncation operator on `(H k), i.e.,
uT := PTu := (u(0), u(1), . . . , u(T ), 0, . . .) for u ∈ `(H k).
We infer that PTu ∈ `2(H k) for all u ∈ `(H k) and T ∈ N0. With a
subspace Ee ⊂ `(H k) satisfying
(Ee)T := {zT | z ∈ Ee} ⊂ Ee for all T ∈ N0, (3.1)
a dynamical system S on Ee is a mapping S : Ee ⊂ `(H k) → `(H l)
which takes any input u ∈ Ee into the output y = S(u) ∈ `(H l). The
system S is said to be linear if the map is; it is causal if S(u)T =
S(uT )T for all T ∈ N0 and u ∈ Ee. The `2-gain ‖S‖ of the system S is
the infimal real number γ ≥ 0 for which there exists some γ0 ∈ R with
‖S(u)T ‖ ≤ γ‖uT ‖+ γ0 for all T ∈ N0, u ∈ Ee.
For linear systems one can take γ0 = 0, and if ‖S‖ is finite we say that
S is bounded.
Concerning the separating map Σ we further generalize the prop-
erty (2.10) in order to allow for a broader class of constraints. As visible
from the proof of the following theorem, it is sufficient to only require
the existence of σij ∈ R with
Σ
(
w
Mw +Mu+Nv
)
−Σ
(
w
Mw
)
≤ (?)T
 0 σ12 σ13σ12 σ22 σ23
σ13 σ23 σ33
‖w‖‖u‖
‖v‖

(3.2)
36 Chapter 3. A first generalization of the IQC theorem
for all w, u ∈ `2(H k), v ∈ V . This is indeed true if (2.10) holds.
3.2.2 Stability theorem
The interconnection structure for our problem is depicted in Figure 3.1.
Here we assume that M : `(H k) → `(H l) and ∆ : Ee ⊂ `(H l) →
`(H k) are causal and bounded, while M is linear. The two systems are
interconnected as
z = Mq + d2 and q = ∆(z) + d1
with d1 ∈ `(H k), d2 ∈ `(H l). Since M is linear, this reduces to
z = Mw+ (Md1 +d2) and w = ∆(z) with the single external input d =
Md1 + d2. In order to adapt this setting to sampled-data applications,
M
∆ +
+
d1
d2
qz ⇐⇒
M
∆
+
w
d
z
Figure 3.1: External disturbances in the interconnection
we further introduce a bounded linear filter N : `2(H m)→ `2(H l) and
consider the case where the disturbance d is confined to M`(H k)+NV
with some subset V ⊂ `2(H m). In our application, M,N will be
strictly proper and stable LTI systems acting as prefilters on the external
disturbances and allowing for subsequent sampling. The feedback system
under consideration in this chapter is hence described by
z −M∆(z) = Mu+Nv with (u, v) ∈ `(H k)× V (3.3)
and depicted in Figure 3.2. Since ∆ is only defined on Ee ⊂ `(H l), we
need to ensure that z ∈ Ee for all possible inputs (u, v). Note that we
3.2. A generalized IQC theorem 37
M
∆
+Mu+Nv
z w
Figure 3.2: General interconnection
treat u and v differently, as we consider extended spaces only for u and
z but not for v. Yet, in our definition of well-posedness and the later
proof of our stability result, it suffices to consider finite energy signals
v ∈ V .
Definition 3.1.
The interconnection (3.3) is well-posed if for each (u, v) ∈ `(H k)×V
and for each τ ∈ [0, 1] there exists a unique z ∈ Ee satisfying z −
Mτ∆(z) = Mu+Nv and such that the map (u, v) 7→ z = Rτ (u, v) is
causal in the first argument:
Rτ (u, v)T = Rτ (uT , v)T for all T ∈ N0, τ ∈ [0, 1], (u, v) ∈ `(H k)×V .
The feedback system (3.3) is stable if, in addition, R1 : `2(H k)×V →
`2(H l) is bounded.
Remark 3.2.
Classically (see Definition 2.5), well-posedness is defined through exis-
tence and causality of the inverse (I − τM∆)−1 for all τ ∈ [0, 1] [110].
We recover this condition as a special case, by removing the distinction
between u and v, i.e., we set V = `2(H m), N = I, and require causality
in both arguments of the response map; moreover, we assume that ∆ is
defined on the full signal space `(H l). Jönsson [93] already relaxed the
aforementioned concept of well-posedness (Definition 2.9). The further
generalization in Definition 3.1 is mostly due to our definition of ∆ only
on the subspace Ee. ?
Remark 3.3.
The introduction of V offers flexibility beyond sampled-data appli-
38 Chapter 3. A first generalization of the IQC theorem
cations. Indeed, for H = R and by setting N = C as well as
V := {v ∈ `2(Rn) | v(t) = Atx0} we may treat initial conditions of an
LTI system M as discussed in Section 2.2.2. ?
Let us now state the central stability result of this chapter.
Theorem 3.4
Assume that M : `(H k) → `(H l) and ∆ : Ee ⊂ `(H l) → `(H k)
are causal and bounded, while M is linear, and that Σ satisfies (3.2).
Suppose, in addition, that
a) the feedback system (3.3) is well-posed;
b) there exists ε > 0 such that
Σ
(
Mw
w
)
≤ −‖w‖2 for all w ∈ `2(H k);
c) there exists some function δ0 : V → [0,∞) with
Σ
(
z
τ∆(z)
)
≥ −δ0(v) for all τ ∈ [0, 1]
and all z = Mu+Nv with (u, v) ∈ `2(H k)× V .
Then there exists some γ > 0 (only depending on M, N, and Σ) such
that
‖R1(u, v)‖2 ≤ γ2
(‖u‖2 + ‖v‖2)+γδ0(v) for all (u, v) ∈ `2(H k)×V .
(3.4)
Proof. A proof is found in Appendix C.2.1.
In contrast to Theorem 2.6 (and also its discrete-time version [97,
Theorem 1]), all assumptions on ∆ are only required to hold on a
subspace Ee ⊂ `(H l) with the key property Rτ (`(H k)× V ) ⊂ Ee for
τ ∈ [0, 1] that ensures well-posedness. In our application the uncertainty
3.3. A class of sampled-data systems 39
will be a PWM. As we will see, this operator is not well-defined on L2
and unbounded even on C[0,∞) ∩L2. However, we can still choose a
suitable subspace E such that it becomes bounded and even passive.
Hence, only by considering the PWM on a smaller set of signals, IQC
theory becomes applicable.
Remark 3.5.
The function δ0(.) is introduced in order to cover the Popov criterion as
discussed in Section 2.2.2 and generalizes the right hand side in (2.24).
?
Remark 3.6.
The derived framework is easily translated to the continuous-time setting.
With the continuous-time truncation operator (Definition 2.1), we can
define well-posedness and stability as in Definition 3.1 by exchanging
L k2e and L k2 for `(H k) and `2(H k), respectively. Apart from these
modifications, the statement of Theorem 3.4 stays unchanged for systems
on L2e and the proof proceeds in an analogous fashion. However, we
will present in Chapter 4 a further generalization for continuous-time
interconnections that allows for operators and constraints on much more
general function spaces. ?
3.3 A class of sampled-data systems
Let us now illustrate how the generalizations in Theorem 3.4 are em-
ployed to incorporate uncertainties exhibiting sampling behavior into
our framework. As we will see, this heavily relies on a loop transfor-
mation, which renders the system time varying. By using a lifting
formalism we can then return to an LTI description that allows for a
straightforward transformation to the frequency domain. Verification of
the IQC in the frequency domain finally leads to an infinite dimensional
FDI, which we reduce without any loss to a finite dimensional LMI that
can be checked efficiently using standard techniques.
40 Chapter 3. A first generalization of the IQC theorem
3.3.1 Motivation
Consider the feedback interconnection of a strictly proper and stable LTI
system M realized by (A,B,C, 0) with eig(A) ⊂ C− and an uncertainty
∆ : Ee ⊂ L2e → L2e as described by
x˙ = Ax+Bw, x(0) = 0, w = ∆(z),
z = Cx+ d.
In the context of Theorem 3.4, we confine the external disturbance d to
ML2e+NV , where N is a strictly proper, stable and finite dimensional
LTI system and V ⊂ L2 (see left-hand side of Figure 3.3). Moreover, let
an equidistant time grid on [0,∞) be given by a fixed sampling period
h > 0, i.e., tn := nh for n ∈ N0. We then set In := [tn, tn+1) and define
the sample and hold operator Sh by
Sh(z)(t) = z(tn) for z ∈ PC[0,∞), t ∈ In. (3.5)
Since the sampling period is assumed to be fixed, we drop the subscript h
in the sequel. For the uncertainty, we assume the following key property
to hold on Ee ⊂ PC[0,∞):
∆ = ∆S. (3.6)
Here the filter N is essential since it ensures that d ∈ PC[0,∞) and
hence ∆(z) = ∆(Sz) is well defined.
In IQC theory we are interested in describing the input-output
behavior of uncertainties as accurately as possible. Property (3.6) states
that the output of ∆ over the time interval In is completely defined
by the value of the input at the time instance tn. Hence all inputs
coinciding at tn lead to the same output ∆(z)(t) for all t ∈ In and thus
to the same square integral on In. On the other hand, the square integral
of the input signal can change arbitrarily on this interval. Consequently,
IQC relations and especially gain bounds for uncertainties with (3.6)
are either hard or even impossible to derive, or very conservative.
3.3. A class of sampled-data systems 41
An elegant way to bypass this problem relies on two key ideas. First
we exploit (3.6) and move the sample and hold operator from ∆ = ∆S
to M , i.e., we consider Sz as an input to the uncertainty (Figure 3.3).
Then we take advantage of the freedom offered by Theorem 3.4 and
choose as the domain Ee of ∆ the space of functions that are constant
in each time interval. We will demonstrate in the examples that these
ideas significantly simplify the derivation of suitable IQCs and open the
way for the application of IQC theory.
M
∆
+
Mu+Nv
z w
⇐⇒
SM
∆
+
SMu+ SNv
Sz w
Figure 3.3: Equivalent interconnections
Note that this approach, in principle, amounts to interpreting S
as a classical multiplier that enables the application of some stability
criterion to the transformed loop. In order to conclude stability of the
original interconnection, it suffices to prove stability for the system
interconnection on the right in Figure 3.3. Indeed, assume we used
Theorem 3.4 to show that the interconnection on the right is stable
(with δ0(.) = 0), i.e., there exists γ > 0 such that ‖Sz‖ ≤ γ(‖u‖+ ‖v‖)
for all (u, v) ∈ L2 × V . Since w = ∆(Sz) = ∆(z) is the same signal in
both configurations and ∆, M are bounded, we infer
‖z‖ = ‖M∆(z)+Mu+Nv‖ = ‖M∆(Sz)+Mu+Nv‖ ≤ γ˜u‖u‖+ γ˜v‖v‖
for all (u, v) ∈ L2 × V and some positive constants γ˜u, γ˜v. Hence the
interconnection on the left is stable.
42 Chapter 3. A first generalization of the IQC theorem
3.3.2 Lifting and frequency domain
In order to deal with the time-varying system SM in the IQC framework,
we rely on the lifting formalism. Lifting procedures, have originally been
introduced for sampled-data systems (see, e.g., [191, 192, 161, 15, 16])
as they provide a means of transforming the sampled system SM into
an LTI system on a lifted space. The cited approaches mainly differ in
the way they capture the inter-sampling behavior and define the state
space of the lifted system. For our purpose, it will be essential that the
state space is finite dimensional, which is a distinguishing feature of
the approach presented in [16]. Since lifting as well as transformation
to the frequency domain are standard in the theory of sampled-data
systems [36], we state only the required results. Both topics are discussed
in more detail in [16, 50, 111].
Lifting
Let
L˜2 :=
{
u˜ : N0 → L2[0, h)
∣∣∣∣‖u˜‖2L˜2 := ∞∑
n=0
‖u˜(n)‖2L2[0,h) <∞
}
.
Then the lifting operator L˜ : L2 → L˜2, given by
(L˜u)(n, τ) := u˜(n, τ) := u(τ + nh) for τ ∈ [0, h) and n ∈ N0,
is an isometric isomorphism between the spaces L2 and L˜2. We state
the following result, which is a combination of those derived in [35]
and [15] for a stable LTI operator M represented as
x˙ = Ax+Bu, x(0) = 0,
y = Cx (3.7)
with A being Hurwitz.
3.3. A class of sampled-data systems 43
Lemma 3.7
Let M be given by (3.7). Then SM is bounded and S˜M := L˜SML˜−1
can be described as
xd(n+ 1) = Aˇxd(n) + Bˇu˜(n), xd(0) = 0 ∈ Rn,
z˜(n) = Cˇxd(n) (3.8)
for u˜ ∈ L˜2, z˜ = S˜Mu˜, and
Aˇ ∈ Rn×n, Aˇ := eAh,
Bˇ : L2[0, h)→ Rn, Bˇψ :=
∫ h
0
eA(h−τ)Bψ(τ)dτ,
Cˇ : Rn → L2[0, h), (Cˇξ)(τ) := Cξ, for all τ ∈ [0, h).
Note that the input and output spaces are infinite dimensional, while
the state dimension is invariant under lifting.
Frequency domain
In order to exploit the fact that the lifted system is time invariant, we
now employ the z-transform to obtain a frequency-domain description.
Let H2 denote the class of analytic functions uˆ mapping the open unit
disc D into L2[0, h) such that
‖uˆ‖2H2 := sup
0≤r<1
1
2pi
∫ 2pi
0
‖uˆ(reiω)‖2L2[0,h)dω <∞.
This space is complete and can be associated with L˜2 via the z-transform.
For u˜ ∈ L˜2 the one-sided z-transform Z : L˜2 → H2 is defined by
uˆ(z) := (Zu˜)(z) :=
∞∑
n=0
u˜(n) zn for |z| < 1,
where we used the symbol z to distinguish the frequency-domain variable
from the signal z. For Banach space valued sequences many of the
44 Chapter 3. A first generalization of the IQC theorem
familiar results persist to hold, such as that the spaces L˜2 and H2 are
isomorphic via the z-transform [50, Prop. 2.9] or that the nontangential
limit for r → 1 exists pointwise almost everywhere [154]. With the
pointwise limit, the inner product on H2 as given by
〈uˆ, vˆ〉H2 :=
1
2pi
∫ 2pi
0
〈
uˆ(eiω), vˆ(eiω)
〉
L2[0,h)
dω
is well defined (see, e.g., [135]). Moreover, by Parseval’s theorem (see,
e.g., [111]), we infer 〈uˆ, vˆ〉H2 = 〈u˜, v˜〉L˜2 . In [50] it is shown that the
transfer function Tˆ ∈ H∞ associated with S˜M as represented in (3.8)
is given by
Tˆ (z) = Cˇ z(I − z Aˇ)−1Bˇ for z ∈ D.
Hence the time-domain equation z˜ = S˜Mw˜ corresponds in the frequency
domain to zˆ = Tˆ wˆ for wˆ, zˆ ∈ H2. Moreover, if A is Hurwitz, then Aˇ is
Schur stable and, consequently, the definition of Tˆ (z) can be extended
to D := D ∪ T.
3.3.3 IQC description and reduction to finite dimen-
sions
From IQCs to FDIs
In the following application of Theorem 3.4 we restrict our attention to
uncertainties ∆ : L k2e → L k2e and bounded quadratic forms Σ defined
by a multiplier P ∈ R2k×2k as
ΣP
(
z˜
w˜
)
:=
〈(
z˜
w˜
)
, P
(
z˜
w˜
)〉
L˜2
:=
〈(
z˜
w˜
)
,
(
p11I p12I
p12I p22I
)(
z˜
w˜
)〉
L˜2
.
(3.9)
This particular structure of P covers the standard static multipliers
corresponding to small gain, passivity, or circle criteria. By Parseval’s
theorem, Theorem 3.4.b) is then equivalent to〈(
Tˆ wˆ
wˆ
)
, P
(
Tˆ wˆ
wˆ
)〉
H2
≤ −‖wˆ‖2 for some ε > 0 and all wˆ ∈ H2,
3.3. A class of sampled-data systems 45
which is guaranteed by the infinite dimensional FDI(
Tˆ (z)
I
)∗
P
(
Tˆ (z)
I
)
4 −εI for some ε > 0 and all z ∈ T. (3.10)
To simplify the notation in this chapter, for an arbitrary operator X
we sometimes use X ≺ε 0 if there exists ε > 0 such that X 4 −εI.
Ultimately, we would like to employ Lemma 2.12 in order to represent
(3.10) as an LMI. Since KYP results for general Hilbert spaces [190]
do not allow for the reduction to finite dimensional (computationally
tractable) LMIs, we first reduce (3.10) to a finite dimensional FDI. In
order to keep this reduction lossless, it is crucial that the state space
dimension is finite after lifting, as this translates into Bˇ, Cˇ having finite
rank. This property was first exploited in [16] and used to calculate
the H∞-norm of a sampled-data system by decomposing Bˇ, Cˇ. We will
now extend this approach to the IQC setting with static multipliers.
Decomposition of the input and the output space
Following [16] we decompose L2[0, h) into an infinite and a finite di-
mensional part as
L2[0, h) = Ker(Bˇ)⊕Ker(Bˇ)⊥ =: UB ⊕ VB with dim(VB) <∞.
This naturally induces the isometry TB : L2[0, h)→ UB × VB ,
TBw :=
(
T 1Bw
T 2Bw
)
=
(
u
v
)
with w = u+ v, u ∈ UB , v ∈ VB ,
and the embedding
JB : UB × VB → L2[0, h), JB
(
u
v
)
:= u+ v = J1Bu+ J
2
Bv,
where UB × VB is equipped with the usual inner product. Throughout
this chapter we write elements of such product spaces as column vectors.
By standard computations we infer that T ∗B = JB and
BˇBˇ∗ = BB
∗
=
∫ h
0
eAtBBT eA
T tdt with B := Bˇ|VB .
46 Chapter 3. A first generalization of the IQC theorem
Consequently, BˇBˇ∗ has a representation as an n× n matrix which is
invertible if the pair (A,B) is controllable. Figure 3.4 illustrates the
relation between the appearing spaces and operators.
L2[0, h)
UB × VB Rn ∼= VB
Bˇ
TB
(
0 B
)
Figure 3.4: Decomposition of Bˇ
In complete analogy we now define the subspaces and operators for
the decomposition of the output space as
L2[0, h) = Ran(Cˇ)
⊥ ⊕ Ran(Cˇ) =: UC ⊕ VC with dim(VC) <∞.
Again, this gives rise to the isometry TC : L2[0, h)→ UC × VC and the
embedding JC : UC × VC → L2[0, h). All results derived for TB hold
in an analogous fashion for TC and we infer
Cˇ∗Cˇ = C
∗
C = hCTC. (3.11)
Effect of the decomposition on z-transformed signals
The decomposition of Bˇ as in Figure 3.4 induces a decomposition of the
input signal wˆ as
TBwˆ(z) =
(
wˆi(z)
wˆf (z)
)
for |z| < 1
3.3. A class of sampled-data systems 47
since wˆ(z) ∈ L2[0, h) for z ∈ D. Note that wˆf takes finite dimensional
values. The analogue holds true for the output space, thus inducing the
decomposition of the transfer function as
TC Tˆ (z)T
−1
B =
(
0 0
0 T (z)
)
:
UB
×
VB
−→
UC
×
VC
for z ∈ T, (3.12)
where T (z) := C z(I − z Aˇ)−1B is a finite dimensional linear operator
for any z ∈ T. We can now take advantage of this decomposition for
the reduction of the FDI (3.10).
Reduction of the FDI
For a better illustration of the reduction process, we regard P as a
multiplication operator on a function space and express this by using 1
for the identity operator on an infinite dimensional space and I for that
on a finite dimensional space. Then, (3.10) takes the form(
Tˆ (z)
1
)∗(
p111 p121
p121 p221
)(
Tˆ (z)
1
)
≺ε 0 for all z ∈ T (3.13)
and can be reformulated as follows.
Theorem 3.8
With the previous definitions, the following statements are equivalent:
a) (3.13) holds.
b) p22 < 0 and for all z ∈ T, the FDI(
T (z)
I
)∗
P
(
T (z)
I
)
≺ε 0 (3.14)
holds with
P :=
(
p11I − p212/p22(J2C)∗(T 1B)∗T 1BJ2C p12T 2BJ2C
p12T
2
CJ
2
B p22I
)
. (3.15)
48 Chapter 3. A first generalization of the IQC theorem
Remark 3.9.
We emphasize that no approximation is needed; both criteria are equiv-
alent. This key step clears the way for efficient computations by LMI
techniques, as we can now verify Theorem 3.4.b) by checking the finite
dimensional FDI (3.14). The term p212/p22(J2C)
∗(T 1B)
∗T 1BJ
2
C , resulting
from a Schur complement, acts as a correction to the finite dimensional
part of the multiplier. In [16] (see also [115]) this correction is zero
since the multiplier for the H∞-norm bound is diagonal, i.e., p12 = 0. ?
Proof. Fix z ∈ T. Then
(
Tˆ (z)
1
)∗(
p111 p121
p121 p221
)(
Tˆ (z)
1
)
≺ε 0
⇐⇒ (?)∗(p111 p121
p211 p221
)(
T−1C TCCˇ z(I − z Aˇ)−1BˇT−1B TB
T−1B TB
)
≺ε 0
⇐⇒ (?)∗(p111 p121
p121 p221
)(
T−1C 0
0 T−1B
)
0 0
0 T (z)
1 0
0 I
 ≺ε 0
⇐⇒ (?)∗(p11TCT−1C p12TCT−1B
p12TBT
−1
C p22TBT
−1
B
)
0 0
0 T (z)
1 0
0 I
 ≺ε 0
⇐⇒ (?)∗

p111 0 p12T
1
CJ
1
B p12T
1
CJ
2
B
0 p11I p12T
2
CJ
1
B p12T
2
CJ
2
B
p12T
1
BJ
1
C p12T
1
BJ
2
C p221 0
p12T
2
BJ
1
C p12T
2
BJ
2
C 0 p22I


0 0
0 T (z)
1 0
0 I
 ≺ε 0
⇐⇒ (?)∗
 p221 p12T 1BJ2C 0p12T 2CJ1B p11I p12T 2CJ2B
0 p12T
2
BJ
2
C p22I

1 0
0
(
T (z)
I
) ≺ε 0.
(3.16)
3.3. A class of sampled-data systems 49
To eliminate the infinite dimensional part, we now employ the Schur
complement and obtain that (3.16) is equivalent to p22 < 0 and(
T (z)
I
)∗
P
(
T (z)
I
)
≺ε 0
with P given in (3.15). Since z ∈ T was arbitrary, this proves the
claim.
Matrix representations
Let us now compute matrix representations S22 and S∗12S12 of the oper-
ators T 2BJ
2
C : VC → VB and (J2C)∗(T 1B)∗T 1BJ2C : VC → VC occurring in
(3.15).
In order to determine S22 we solve(
T 2BJ
2
Cy1 · · · T 2BJ2Cyk
)
=
(
u1 · · · um
)
S22 (3.17)
with bases yj of VC and uj of VB . First, to construct yj , choose vectors
ξ1, . . . , ξk ∈ Rn such that Cξ1, . . . , Cξk forms a basis of Ran(C). Then,
yj = Cˇξj ∈ L2[0, h) and (yj)kj=1 is a basis of VC . In addition, we have
J2Cyj = yj for all j.
Now let (uj)mj=1 be a basis of VB . Since Bˇ is a bijective map between
the spaces VB and Ran(Bˇ), it follows that (Bˇuj)mj=1 is a basis of
Ran(Bˇ) = Ran(B,AB, . . . , An−1B) ⊂ Rn
and thus easy to compute. Together with BˇT 2B = Bˇ, (3.17) implies(
BˇCˇξ1 · · · BˇCˇξk
)
=
(
Bˇu1 · · · Bˇum
)
S22. (3.18)
As a basis matrix, F :=
(
Bˇu1 · · · Bˇum
)
has full column rank. By
recalling
BˇCˇξ =
∫ h
0
eA(h−τ)BC dτ ξ
we can thus compute S22 from (3.18).
50 Chapter 3. A first generalization of the IQC theorem
For the second matrix representation we solve(
(J2C)
∗(T 1B)
∗T 1BJ
2
Cy1 . . . (J
2
C)
∗(T 1B)
∗T 1BJ
2
Cyk
)
=
(
y1 . . . yk
)
S∗12S12
(3.19)
for S∗12S12. Since J2CCˇ = Cˇ and hence Cˇ
∗ = Cˇ∗(J2C)
∗, we get(
Cˇ∗(T 1B)
∗T 1BCˇξ1 . . . Cˇ
∗(T 1B)
∗T 1BCˇξk
)
=
(
Cˇ∗Cˇξ1 . . . Cˇ∗Cˇξk
)
S∗12S12.
(3.20)
To understand the operator on the left, let ξ ∈ Rn. Then T 1BCˇξ is the
projection of Cˇξ onto UB . Hence, using our basis of VB , this projection
may be written as
T 1BCˇξ = Cˇξ −
m∑
j=1
ujαj with αj ∈ R for j ∈ {1, . . . ,m}. (3.21)
The coefficients vector α =
(
α1 . . . αm
)T
can be computed from the
equation
0 = Bˇ
(
Cˇξ −
m∑
j=1
ujαj
)
= BˇCˇξ − (Bˇu1 . . . Bˇum)α = BˇCˇξ − Fα
as α = F+BˇCˇξ with the Moore-Penrose inverse F+. Finally, using
(3.21), we get
Cˇ∗(T 1B)
∗T 1BCˇξ = Cˇ
∗J1B
(
Cˇξ−
m∑
j=1
ujαj
)
= Cˇ∗Cˇξ−(Cˇ∗u1 . . . Cˇ∗u1)α.
Hence, with (3.11), we can compute S∗12S12 from (3.20).
Affine dependence on decision variables
The multiplier P in (3.15) depends fractionally on the original coeffi-
cients pij . For later computations, we show how to render the depen-
dence affine. Since S∗12S12 is positive semidefinite, it has a positive
semidefinite square root and(
T (z)
I
)∗
P
(
T (z)
I
)
≺ 0 for all z ∈ T (3.22)
3.4. Application to PWM feedback systems 51
is equivalent toT (z) 0I 0
0 I

∗ p11I p12S∗22 p12
√
S∗12S12
p12S22 p22I 0
p12
√
S∗12S12 0 p22I

︸ ︷︷ ︸
=:P e
T (z) 0I 0
0 I
 ≺ 0
(3.23)
for all z ∈ T. Equation (3.23) can be written more compactly as(
T e(z)
I
)∗
P e
(
T e(z)
I
)
≺ 0 for T e(z) :=
(
T (z) 0
)
. (3.24)
The transformation to an LMI and, hence, to a convex problem is now
standard (see Section 2.3).
3.4 Application to PWM feedback systems
Let us now apply the previously derived results to the stability analysis
of a PWM feedback interconnection. After having defined the overall
setup, we will illustrate two possibilities of incorporating PWM systems
into our framework. In Section 3.4.4 we compare these approaches to
the ones in [82] and [33] and show numerical results.
3.4.1 Definition of a PWM
The interconnection to be studied consists of a strictly proper and stable
plantM , realized by (A,B,C, 0), a PWM denoted as ∆, and an external
disturbance d ∈ML2 +NL2 (where N is a strictly proper and stable
transfer function such that ∆(Mw + d) is well defined). It is described
by the set of equations
z = Mw + d, w = ∆(z). (3.25)
52 Chapter 3. A first generalization of the IQC theorem
Given zmax, zmin ≥ 0 and the time span
τn =

0, |z(tn)| < zmin,
|z(tn)|
zmax
h, zmin ≤ |z(tn)| ≤ zmax,
h, |z(tn)| > zmax,
(3.26)
∆ is defined with µn := sgn(z(tn))zmax (see [69]) as
w(t) = ∆(z)(t) =
{
µn, t ∈ [tn, tn + τn),
0, t ∈ [tn + τn, tn+1).
(3.27)
Here zmin ≥ 0 is a constant threshold under which the PWM generates
no output pulse. We consider the cases zmin > 0 and zmin = 0 separately.
3.4.2 Direct approach to PWM analysis
The approach taken in this section heavily relies on zmin > 0, which
is required to derive a bound on ∆. In modeling physical processes
zmin > 0 is in fact a natural requirement, since it provides a positive
lower bound on the pulse length τn as
τn ≥ h zmin
zmax
. (3.28)
Embedding into our framework
By definition, the output of the PWM on the time interval In only
depends on the input at time instance tn = nh; hence ∆ = ∆S with
S = Sh. Following Section 3.3.1, we consider the interconnection of ∆
with SM (see Figure 3.3) represented by
SM : x˙ = Ax+Bw, x(0) = 0, w = ∆(z),
z = S(Cx) + Sd. (3.29)
At this point we apply the lifting formalism to the whole interconnection.
Well-posedness of the lifted interconnection (Figure 3.5) depends on the
3.4. Application to PWM feedback systems 53
domain E˜e of ∆˜. We choose Ee ⊂ L2e as the subspace of all functions
that are constant on each time interval In. Then Ee ⊂ PC[0,∞) and
(E˜e)N ⊂ E˜e for all N ∈ N0. Existence and uniqueness of a solution to
S˜M
∆˜
+S˜Mu˜+ S˜Nv˜
z˜ w˜
Figure 3.5: Lifted interconnection
(3.29) with ∆ replaced by τ∆, τ ∈ [0, 1] can be verified by existence
conditions for the defining differential equation: For n ∈ N0, t ∈ In we
have
z(t) = Cx(tn) + d(tn)
= CeAtnx0 + C
∫ tn
0
eA(tn−s)Bw(s)ds+ d(tn). (3.30)
The initial condition x0 leads to the initial output Cx0 that, together
with d(0), completely defines the output of τ∆ on I0. Since this output
is piecewise constant we get a unique solution on the interval I0. By
induction, we acquire a unique solution of (3.29) for t ∈ [0,∞) and ∆
replaced by τ∆. As we constructed this solution on each time interval,
existence and uniqueness are preserved during lifting. Moreover, z as
in (3.30) is constant on each time interval and hence z˜ ∈ E˜e with the
above defined domain E˜e. Consequently the feedback interconnection of
S˜M with ∆˜ is well-posed.
Now we show boundedness of ∆˜ on E˜e: Fix n ∈ N0, z ∈ E and
assume that |z(tn)| ≥ zmin (otherwise ∆(z)(.) = 0 on In). Then, with
γ = zmax/zmin and by recalling the definition of τn, we have (since z is
constant on In) that∫
In
[∆(z)(t)]2dt =
∫ tn+τn
tn
z2maxdt = zmax|z(tn)|h ≤ γ|z(tn)|2h
54 Chapter 3. A first generalization of the IQC theorem
= γ
∫
In
z(t)2dt. (3.31)
As lifting is isometric, this proves ‖∆˜(z˜)‖2 ≤ γ‖z˜‖2 for all z˜ ∈ E˜ . In
view of the boundedness of S˜M it remains to verify items b) and c) in
order to apply Theorem 3.4.
IQC description of the PWM
Since S˜ML˜2 + S˜NL˜2 ⊂ E˜ it suffices to derive IQC relations for signals
in E˜ . The output of the PWM does not change its sign in any time
interval (since it is either µn or zero) and thus 0 ≤ z(tn)∆(z)(tn). For
signals z ∈ E we hence have 0 ≤ z(t)∆(z)(t) for all t ∈ In, n ∈ N0. This
passivity property readily translates into the IQC∫ h
0
(
?
)∗(0 I
I 0
)(
z(tn + τ)
∆(z)(tn + τ)
)
dτ ≥ 0 for all z ∈ E , n ∈ N0.
(3.32)
Again, since (3.32) holds on each time interval, the IQC persists to hold
after lifting:〈(
?
)
,
(
0 1
1 0
)(
z˜
∆˜(z˜)
)〉
L˜2
=
∞∑
n=0
∫ h
0
(
?
)∗(0 I
I 0
)(
z(tn + τ)
∆(z)(tn + τ)
)
dτ
≥ 0.
In combination, the gain bound in (3.31) and the passivity property
above lead to the following set of valid multipliers:
Pκ :=
(
γI κI
κI −I
)
with γ =
zmax
zmin
and κ ≥ 0. (3.33)
Following our approach, we now define the extended multiplier Pe
according to (3.23) and arrive at the FDI (3.24). The validity of this
3.4. Application to PWM feedback systems 55
FDI for all z ∈ T is equivalent, via the KYP lemma (Lemma 2.12, [128]),
to the existence of some X = XT and κ ∈ [0,∞) for which(
I 0
Aˇ Be
)T (
X 0
0 −X
)(
I 0
Aˇ Be
)
+
(
C 0
0 I
)T
P e
(
C 0
0 I
)
≺ 0 (3.34)
is feasible with Be := (0, B). We show numerical results in Section 3.4.4.
3.4.3 Averaging approach to PWM
General setup
For zmin = 0 the PWM is unbounded even on Ee, which prevents us
from incorporating the PWM directly into our framework. To overcome
this difficulty, we follow the approach in [67, 69], based on averaging of
the modulator output. The underlying idea is related to the equivalent
area principle and works for general modulation laws; it relies on the
introduction of a so-called equivalent nonlinearity φ : R→ R with the
property ∫
In
φ(Sz)(t) dt =
∫
In
∆(z)(t) dt for all n ∈ N0. (3.35)
For our definition of the PWM, φ takes the form of a saturation [69]
with saturation level zmax:
φ(x) =
{
x, |x| ≤ zmax,
zmax sgn(x), |x| > zmax.
(3.36)
The key idea is to substitute the PWM by the sampled equivalent
nonlinearity ∆1 := Sφ = φS and absorb the resulting error, in integrated
form, into the second uncertainty as
∆2 =
∫
(∆−∆1), (3.37)
where
∫
denotes the map z(.) 7→ ∫ t
0
z(s)ds for z ∈ L1[0,∞). Hence,
we split the PWM into two uncertainties that, as we will see, both fit
56 Chapter 3. A first generalization of the IQC theorem
into our framework. If M has a relative degree of at least two, the
interconnection of M realized as (A,B,C, 0) and ∆ is equivalent to
the one of H with realization (A,
(
B AB
)
,
(
CT CT
)T
, 0) and ∆1 as
well as ∆2 [68] (Figure 3.6). Note that the assumption on the relative
H
∆2
Sφ
+
+
d2
d1
z2
z1
w2
w1
SH
∆2
φ
+
+
Sd2
Sd1
z2
z1
w2
w1
Figure 3.6: Averaging approach interconnections
degree ofM is not necessary for the transformation but ensures that the
transformed plant H is strictly proper and hence fits into our framework
[68, 33].
Averaging in the IQC framework
The method described in [68, 69] uses multipliers to capture the non-
linearities but relies on a Lyapunov argument for stability. Due to the
generalizations in Theorem 3.4 we are now able to incorporate this
approach into the IQC framework. Again it will be crucial to define the
uncertainties on an appropriate subspace Ee ⊂ L2e. For ∆2 as defined
in (3.37) the required property (3.6) holds: Since ∆2(z2) = ∆2(Sz2) as
shown above and φS = Sφ, we get
∆2(Sz2) =
∫
(∆− Sφ)(Sz2) =
∫ (
∆(Sz)− S2φ(z2)
)
= ∆2(z2).
Again, if we incorporate and move the sample-and-hold operator around
in the loop to obtain the interconnection on the right in Figure 3.6,
the first uncertainty channel now only contains the saturation φ, which
3.4. Application to PWM feedback systems 57
allows for standard descriptions with IQCs. By redefining ∆1 = φ, the
new system equations can be written as
SH : x˙ = Ax+Bw1 +ABw2, x(0) = 0, w1 = ∆1(z1),
z1 = S(Cx+ d1), w2 = ∆2(z2),
z2 = S(Cx+ d2). (3.38)
As before, we choose the space Ee of functions that are constant on
each time interval as the domain of definition for the uncertainties.
Well-posedness of interconnection (3.38) can then be verified with the
very same arguments as in Section 3.4.2.
Lifting of uncertainties
Since ∆1 is static, lifting is trivial. For ∆˜2 := L˜∆2L˜−1 we derive an
explicit expression: With (3.35), the definition of ∆2 reduces to
∆2(z)(t) =
∫ t
tn
(∆− Sφ)(z)(τ) dτ for t ∈ In and all z ∈ E .
Consequently, for t ∈ In and with τ := t− nh ∈ [0, h), we have(
∆˜2(z˜)
)
(n)(τ) =
(
L˜∆2(z)
)
(n)(τ) = ∆2(z)(nh+ τ)
=
∫ nh+τ
nh
(∆− Sφ)(z)(t) dt
=
∫ τ
0
(∆− Sφ)(z)(s+ nh) ds
=
∫ τ
0
(∆− Sφ)(z˜(n))(s) ds. (3.39)
This can now be used to derive IQCs in the lifted domain, as will be
discussed next.
IQC description
The uncertainty ∆1 is a so-called sector-bounded nonlinearity. We
capture this property by the multiplier corresponding to the circle
58 Chapter 3. A first generalization of the IQC theorem
criterion1 (with parameters α ≤ 0 ≤ β); the IQC in the lifted domain
then reads as〈(
z˜
∆˜1(z˜)
)
,
(−αβ1 α+β2 1
α+β
2 1 −1
)(
z˜
∆˜1(z˜)
)〉
L˜2
≥ 0 for z˜ ∈ E˜ . (3.40)
With φ given in (3.36) we can take α = 0, β = 1. Now consider the
second uncertainty. For z ∈ E and n ∈ N0, z˜(n) is constant and, with
(3.39), we immediately infer
z˜(n)(τ)∆˜2(z˜)(n)(τ) = z(nh)
∫ nh+τ
nh
(∆− Sφ)(z)(s) ds ≥ 0
for τ ∈ [0, h) and n ∈ N0. Nonnegativity follows directly with (3.35)
and is proven in [33]. This implies
∫ h
0
z˜(n)(τ)∆˜2(z˜)(n)(τ)dτ ≥ 0 and,
accordingly,〈(
z˜(n)
∆˜2(z˜)(n)
)
,
(
0 1
1 0
)(
z˜(n)
∆˜2(z˜)(n)
)〉
L2[0,h)
≥ 0 for all n ∈ N0, z˜ ∈ E˜ .
By taking the sum over all n, we arrive at the IQC〈(
z˜
∆˜2(z˜)
)
,
(
0 1
1 0
)(
z˜
∆˜2(z˜)
)〉
L˜2
≥ 0 for all z˜ ∈ E˜ ,
showing passivity of ∆˜2 on E˜ . In addition, from [69] we have the
following gain bound for z ∈ E and all n ∈ N0:∫ (n+1)h
nh
|(∆− Sφ)(z)(τ)|2 dτ ≤ Lh
2
3
∫ (n+1)h
nh
|z(τ)|2 dτ, (3.41)
where L denotes the Lipschitz constant of φ as defined in (3.36). Hence,
L = 1 and this trivially extends to a gain bound on L2[0, h) and, in
turn, on L˜2:
‖∆˜2(z˜)‖L˜2 ≤
h√
3
‖z˜‖
L˜2
for all z˜ ∈ E˜ .
1We discuss this and other criteria for sector-bounded nonlinearities in detail in
Chapter 5.
3.4. Application to PWM feedback systems 59
Conic combination of both IQCs for ∆˜2 leads to〈(
z˜
∆˜2(z˜)
)
,
(
h2
3 1 κ1
κ1 −1
)(
z˜
∆˜2(z˜)
)〉
L˜2
≥ 0 for all z˜ ∈ E˜ , κ ≥ 0.
With γ = h2/3 this implies in the frequency domain that
〈(
?
)
,

−αβ1 0 α+β2 1 0
0 κ1γ1 0 κ21
α+β
2 1 0 −1 0
0 κ21 0 −κ11


zˆ1
zˆ2
̂˜
∆1(z˜1)
̂˜
∆2(z˜2)

〉
H2
≥ 0 (3.42)
for all κ1, κ2 ≥ 0 and z˜1, z˜2 ∈ E˜ . In complete analogy to the case of
one uncertainty channel, the diagonally augmented structure of this
multiplier allows for a reduction to finite dimensions, as shown next.
Reduction to finite dimensional problem
If we apply Lemma 3.7 to SH given in (3.38), the operators defining
the lifted system are Aˇ = eAh,
Bˇψ=
(
Bˇ1 Bˇ2
)(ψ1
ψ2
)
=
∫ h
0
eA(h−τ)Bψ1(τ) dτ +
∫ h
0
eA(h−τ)ABψ2(τ) dτ
and
(Cˇξ)(τ) =
(
(Cˇ1ξ)
(Cˇ1ξ)
)
(τ) =
(
C
C
)
ξ for all τ ∈ [0, h).
Decomposition of the corresponding transfer function according to the
uncertainty channels results in
Tˆ (z) =
(
Cˇ1
Cˇ2
)
z(I − z Aˇ)−1 (Bˇ1 Bˇ2) =:
(
Tˆ11(z) Tˆ12(z)
Tˆ21(z) Tˆ22(z)
)
for z ∈ T.
Here the partitioning of Tˆ again induces the usual decomposition of input
and output spaces with the operators and embeddings for j ∈ {1, 2}:
TBj : L2[0, h)→ UBj × VBj , TCj : L2[0, h)→ UCj × VCj ,
60 Chapter 3. A first generalization of the IQC theorem
JBj : UBj × VBj → L2[0, h), JCj : UCj × VCj → L2[0, h).
Then, the FDI constraint on the LTI system corresponding to (3.42)
reads as
(
?
)

−αβ1 0 α+β2 1 0
0 κ1γ1 0 κ21
α+β
2 1 0 −1 0
0 κ21 0 −κ11


Tˆ11(z) Tˆ12(z)
Tˆ21(z) Tˆ22(z)
1 0
0 1
 ≺ε 0
for all z ∈ T, κ1, κ2 ≥ 0. Now fix z ∈ T. With
TCj Tˆij(z)T
−1
Bi =
(
0 0
0 T ij(z)
)
,
this leads first to
(
?
)∗

−αβI 0 α+β2 T 2C1J1B1 α+β2 T 2C1J2B1 0 0
? κ1γI 0 0 κ2T
2
C2J
1
B2 κ2T
2
C2J
2
B2
? ? −1 0 0 0
? ? 0 −I 0 0
? ? ? ? −κ11
? ? ? ? 0 −κ1I

×
×

0 T 11(z) 0 T 12(z)
0 T 21(z) 0 T 22(z)
1 0 0 0
0 I 0 0
0 0 1 0
0 0 0 I

≺ε 0
3.4. Application to PWM feedback systems 61
and then, by applying the Schur complement once again, to the finite
dimensional FDI
(
?
)∗

−αβI 0 α+β2
√
S1∗12S
1
12
α+β
2 S
1
22 0 0
? κ1γI 0 0 κ2
√
S2∗12S
2
12 κ2S
2
22
? ? −1 0 0 0
? ? 0 −I 0 0
? ? ? ? −κ1I
? ? ? ? 0 −κ1I

×
×

0 T 11(z) 0 T 12(z)
0 T 21(z) 0 T 22(z)
I 0 0 0
0 I 0 0
0 0 I 0
0 0 0 I

≺ 0,
where Sj22 and S
j∗
12S
j
12 denote the matrix representations of T
2
Bj
J2Cj and
(T 1BjJ
2
Cj
)∗T 1BjJ
2
Cj
, respectively, for j = 1, 2. For computations, we can
equivalently represent this FDI as an LMI according to (3.34).
3.4.4 Computational results
In order to compare our results with the analysis techniques in [82]
and [69, 33], we compute the largest sampling time h for which the
respective approaches still guarantee stability of the feedback intercon-
nection (3.25).
Direct approach
We first contrast our approach to the preliminary results in [33], where a
model for attitude control is considered based on the averaging approach
62 Chapter 3. A first generalization of the IQC theorem
in [69]. For zmin > 0 and zmax = 1 the equivalent nonlinearity (3.36)
takes the form
φ(x) =

0, |x| < zmin,
x, zmin ≤ |x| ≤ 1,
sgn(x), |x| > 1.
By (3.41) the jump of φ(.) at x = zmin renders ∆2 unbounded. To
model this behavior approximately, large Lipschitz constants L and
hence, with (3.41), large gain bounds are considered in [33]. This only
guarantees stability of the approximated system, while our multiplier
(3.33) does not depend on such a heuristic argument and hence guar-
antees stability for the original problem. We consider the two values
zmin,1 = 0.05 and zmin,2 = 0.01, corresponding to lower bounds on τn
as in (3.28) as well as gain bounds γ1, γ2 as in (3.33). Table 3.1 shows
the maximal sampling periods with guaranteed stability for the systems
M1(s) =
s+ 1
(s+ 2)(s2 + 10s+ 41)
and M2(s) =
1
5(s+ 10)2
,
obtained by applying the averaging method with the previously de-
scribed approximation for L ∈ {1, 25, 100} and the ones achieved by our
approach. Since the bounds on h derived in [69] are improved in [33],
we only compare our results with the latter.
Table 3.1: Maximal sampling times, zmin > 0
L = 1 L = 25 L = 100 IQC γ1 IQC γ2
M1 1.63 0.28 0.11 56.70 11.34
M2 3.68 0.73 0.36 5 · 103 1 · 103
For M1, the second row of Table 3.1 illustrates that even if we
disregard the jump completely (L = 1), the averaging approach leads to
3.4. Application to PWM feedback systems 63
significantly smaller sampling times if compared to the IQC approach for
both γ1, γ2. For better approximations, i.e., larger L, the improvement
achieved by the IQC approach becomes greater. This is even more
pronounced for M2.
Averaging approach
Let us also consider the case of zmin = 0. Hou and Michel propose a
method in [82] to translate the continuous-time interconnection of M
realized by (A,B,C, 0) and ∆ into an autonomous discrete-time system
with a dynamic matrix eAh(I − hWn). Here Wn depends on A,B,C
and the time constant τn. Since A is Hurwitz stable, there exists P > 0
such that (eAh)TPeAh−P = I. The basic idea is to define the quadratic
Lyapunov function V (x) = xTPx and derive bounds on the sampling
time such that V is also a Lyapunov function for the system
x(tn+1) = e
Ah(I − hWn)x(tn) (3.43)
when h > 0. Then it is shown that stability of this system implies
stability of the original interconnection. Since Wn depends on τn and
the chosen realization of G, in order to prove stability we need to
show that there exists some realization such that (3.43) is stable for all
τn ∈ [0, h). This is implemented using 1000 randomly generated state
coordinate transformations and a time grid on [0, h). The maximal
sampling period for which the system is stable is then approximated by
optimizing over different realizations and grid refinements. To compare
our results, we consider the family of LTI systems given by the transfer
functions
Ma(s) :=
1
(s+ a)2
with a ∈ {0.1, 0.3, 0.4, 0.5}.
Table 3.2 shows the maximal sampling period for which the respective
approaches prove stability for a given a. Since we could not establish
stability for h > 10−9 and any a with the averaging approach in [69],
64 Chapter 3. A first generalization of the IQC theorem
Table 3.2: Maximal sampling times, zmin = 0
a 0.1 0.3 0.4 0.5
Hou, Michel < 10−6 < 10−5 0.075 0.39
IQC approach 0.0012 0.032 0.075 0.15
this is not considered here. For a = 0.1 and a = 0.3 we were not able to
prove stability with [82] even for h = 10−6 and h = 10−5, respectively.
Here the IQC framework leads to much better results. For a = 0.4 we
obtain similar results and for a = 0.5 the maximal sampling periods
obtained by Hou and Michel are better than ours. However, as the
approach in [82] relies on randomized realizations, it is computationally
significantly more expensive than solving the LMI corresponding to
(3.43) and there is no guarantee of finding an optimal solution. Moreover,
the latter approach verifies stability only for the chosen grid points in
[0, h) and hence there is no certification for all τn ∈ [0, h).
3.5 Summary and possible extensions
In this chapter we present a first generalization of the IQC framework
that allows us to significantly broaden its range of applications, now also
including sampled-data type systems. We demonstrate how to apply
lifting and frequency-domain techniques in order to render the resulting
stability test computational without the need for approximations. The
effectiveness is illustrated for the particular case of a PWM feedback
interconnection that has so far been impossible to treat within IQC
theory. This permits to draw significantly less conservative stability
conclusions for such systems, if compared to state of the art techniques.
As is probably best seen from the realization of S˜M given in
Lemma 3.7, the problem considered in this chapter only slightly differs
from the question of deciding stability of an uncertain sampled-data
interconnection [50, 65, 16]. The main difference is that the sampling S
3.5. Summary and possible extensions 65
at the output of the linear system results in a lifted system with zero
direct feedthrough. For general uncertain sampled-data interconnections
this is not the case and, thus, the transfer matrix will be a genuine
infinite dimensional operator; a decomposition as in (3.12) will not
result in an essentially finite dimension mapping. However, we believe
that with a suitable approximation of the infinite dimensional operator,
the framework established in this chapter may be extended to comprise
uncertain sampled-data interconnections.
A further generalization worth investigating is the application of
more general multipliers for the equivalent nonlinearity. As discussed in
detail in Chapter 5, saturations allow for much tighter capturing if using
frequency dependent (and unstructured) multipliers. Yet, the reduction
to a finite dimensional LMI given herein relies on the use of static,
diagonally structured multipliers. Thus we expect a further improvement
of the obtained results for more sophisticated IQC descriptions of the
uncertainty.
66 Chapter 3. A first generalization of the IQC theorem
Chapter 4
A general framework for stability and per-
formance analysis based on dissipativity con-
straints
4.1 Introduction
In Chapter 2, we describe the emergence of very general stabilitycriteria from the early contributions of Zames and Yakubovich to
the later, more sophisticated ones by Safonov and Teel. In this line of
developments, the IQC framework as portrayed in Section 2.2 may be
seen as a step backwards, at least in terms of generality. However, in
conjunction with the celebrated KYP lemma (Lemma 2.12), it consti-
tutes the most effective stability test of the ones discussed above if it
comes to numerical verification.
This chapter is devoted to a further relaxation of the assumptions in
our preliminary stability result, Theorem 3.4, towards general function
spaces. Furthermore, we generalize the notion of truncation (Defini-
tion 2.1) towards a continuation based approach that allows for the
incorporation of signal spaces with additional regularity requirements.
We will highlight this advantage by considering the particular case of
Sobolev spaces that are, as demonstrated in the subsequent chapters, of
67
68 Chapter 4. A general analysis framework
great practical importance. Following the line of thought of Megretski
and Rantzer, we not only present an abstract stability result but a
whole framework that also comprises novel performance measures and
the means for verification of stability and a given performance criterion.
In contrast to what is typically seen in the literature, the proposed
separating functionals, both for stability and performance, are derived
from the underlying principle of dissipation [180, 181]. This direct
connection to dissipation theory is a key ingredient as it enables the
verification of stability and performance using LMIs by means of a
generalization of the KYP lemma that is valid on Sobolev spaces. We
illustrate the application of our framework for the example of an inter-
connection involving time-varying parametric uncertainties. For a clear
and concise presentation of the framework we move some proofs to the
appendix.
Finally, we emphasize that the present chapter has its roots in the
publications [58, 56] where stability and performance are considered
separately and from a different point of view. In the derivation at
hand, this artificial partitioning is now removed in order to emphasize
our unifying approach to global stability and performance analysis.
This necessitated some adaption to the statement of the main stability
theorem and its proof that now permit a natural transition from stability
to performance analysis. Still, apart from these changes, large parts of
the text overlap with [58, 56].
4.2 Function spaces, causality and boundedness
Let us begin by rigorously specifying the technical requirements on the
underlying function spaces that are imposed in the sequel. The following
construction is closely related to the one of resolution Hilbert spaces as
discussed in the exceptionally well-written monograph [54]. As we will
see, the introduced generalizations require slight modifications to the
definitions of causality, boundedness and well-posedness if compared to
the ones stated in Chapters 2 and 3.
4.2. Function spaces, causality and boundedness 69
Assumption 4.1.
Fix k ∈ N and let
a) (XT , ‖ · ‖XT )T>0 be a family of normed function spaces with ele-
ments XT 3 u : [0, T ]→ Rk;
b) Xe :=
{
u : [0,∞)→ Rk ∣∣ uT := u|[0,T ] ∈ XT for all T > 0} and
define the family of semi-norms ‖ · ‖Xe,T on Xe as ‖u‖Xe,T :=
‖uT ‖XT for all T > 0 and u ∈ Xe;
c) X := {u ∈ Xe | ‖u‖X := supT>0 ‖u‖Xe,T <∞};
d) there exist some KX ≥ 1 such that for every u ∈ Xe and T > 0
there is a function uT ∈ X with1
uT = (u
T )T and ‖uT ‖X ≤ KX ‖u‖Xe,T . (4.1)
?
We say that Xe satisfies Assumption 4.1 if there exists a family
of normed spaces as in 4.1 a), Xe and X are defined as in 4.1 b) and c),
respectively, and d) holds.
Remark 4.2.
Note that an essential requirement in the previous chapter concerns Ee,
the domain of definition of ∆, which is assumed to satisfy the property
(Ee)T ⊂ E (3.1). This assumption essentially dates back to the work
of Zames [195] with applications mainly focussing on Lp as underlying
function spaces. However, if Ee denotes a space consisting of smooth
signals u ∈ Ee, the instantaneous truncation PTu is typically not even
continuous. This poses a major motivation for Assumption 4.1.d), which
merely requires the existence of some continuation. By Assumption 4.1,
every element u ∈ Xe admits a continuation uT ∈ X that satisfies the
1Note that the definition of uT now differs from Chapters 2 and 3. The classical
(instantaneous) truncation at time T of some signal u : [0,∞)→ Rn is denoted by
PTu with the truncation operator PT defined in Definition 2.1.
70 Chapter 4. A general analysis framework
key property (4.1). Intuitively speaking, given some T > 0 and some
signal u, we allow, e.g., for the steering of u(T ) to zero – as long as
the resulting error if compared to the instantaneous truncation can be
bounded as stated. ?
Based on Assumption 4.1, it is now natural to define causality and
boundedness of operators as follows.
Definition 4.3.
Let Ue, Ze satisfy Assumption 4.1. An operator S : Ue → Ze is said
to be causal if
u˜T = uT =⇒ S(u)T = S(u˜)T for all T > 0 and u, u˜ ∈ Ue.
The gain ‖S‖ of S : Ue → Ze is the infimal γ ≥ 0 such that there
exists some γ0 ∈ R with
‖S(u)‖Z ≤ γ‖u‖U + γ0 for all u ∈ U .
S is bounded if ‖S‖ is finite. Recall that we may choose γ0 = 0 if S is
linear. ?
A sufficient condition for boundedness of S is given by
‖S(u)‖Ze,T ≤ γ˜‖u‖Ue,T + γ0 for all T > 0, u ∈ Ue. (4.2)
Indeed, u ∈ U implies u ∈ Ue and hence
‖S(u)‖Ze,T ≤ γ˜‖u‖Ue,T + γ0 ≤ γ˜ sup
T>0
‖u‖Ue,T + γ0 = γ˜‖u‖U + γ0;
the claim follows by taking the supremum over T > 0 on the left. The
fact that the converse holds if S is also causal will play a key role in
the derivation of our stability result.
Lemma 4.4
Let S : Ue → Ze be a causal operator satisfying ‖S(u)‖Z ≤ γ‖u‖U +γ0
for some pair γ ≥ 0, γ0 ∈ R and all u ∈ U . Then (4.2) holds with
γ˜ = γKU .
4.3. Fundamental stability result 71
Proof. Let T > 0 and u ∈ Ue. Since uT ∈ U and uT = (uT )T ,
causality of S implies
‖S(u)‖Ze,T = ‖S(u)T ‖ZT = ‖S(uT )T ‖ZT = ‖S(uT )‖Ze,T ≤ ‖S(uT )‖Z .
With boundedness of S and (4.1) we further obtain
‖S(u)‖Ze,T ≤ ‖S(uT )‖Z ≤ γ‖uT ‖U + γ0
≤ γKU ‖u‖Ue,T + γ0 = γ˜‖u‖Ue,T + γ0.
4.3 Fundamental stability result
Having established the basic properties of the function spaces and
operators under consideration, let us now turn to the interconnection
and our central stability result. With M : Ue → Ze, ∆ : Ze → We and
N : V → Z we consider the feedback interconnection
z = Mw +Mu+N(v), w = ∆(z) (4.3)
with external signals (u, v) ∈ Ue×V as in Figure 4.1 under the following
standing hypotheses.
Assumption 4.5.
a) Ue,We,Ze are function spaces satisfying Assumption 4.1 and the
compatibility condition We ⊂ Ue with the natural inclusion map
J : We → Ue, Jw = w being bounded.
b) M : Ue → Ze and ∆ : Ze → Ue are causal and bounded while M
is linear.
c) With a subset V of a normed space with norm ‖ · ‖V , the map
N : V → Z satisfies ‖N(v)‖Z ≤ γN‖v‖V for some γN ≥ 0 and
all v ∈ V . The possible choice V = {0} implies N(V ) = {0}.
72 Chapter 4. A general analysis framework
M
∆
+
+
N
u
Ue
V
v
Ze
z
We
w
Figure 4.1: Interconnection for main stability result.
?
This general setup necessitates a slight modification of the standard
definition of well-posedness that will be shown to pose no extra trouble
in subsequent applications (see Chapter 5).
Definition 4.6.
The interconnection (4.3) is well-posed if for each u ∈ Ue, v ∈ V ,
and τ ∈ [0, 1] there exists a unique z ∈ Ze satisfying z −Mτ∆(z) =
Mu+N(v) and such that the correspondingly defined response map
(u, v)→ z = Rτ (u, v) is causal in the first argument, i. e.,
Rτ (u, v)T = Rτ (u˜, v)T for all T > 0, v ∈ V
and u, u˜ ∈ Ue with uT = u˜T .
If Ze = L k2e, well-posedness is clearly a consequence of I − τM∆ :
L k2e → L k2e having a causal inverse for all τ ∈ [0, 1], since the response
map then equals Rτ (u, v) = (I −M∆)−1(Mu + N(v)). For V = L2
and N = id, this property is equivalent to the well-posedness defined in
Definition 2.5, which is from now on referred to as well-posedness in
the classical sense of (4.3). ?
In Theorem 3.4 we introduce a function δ0 : V → [0,∞) that
allowed to incorporate IQCs depending on the initial conditions of the
LTI system into the framework (see Theorem 3.4 b)). This was inspired
by the treatment of the Popov stability criterion in (2.24). Having
4.3. Fundamental stability result 73
the dependence on the initial condition in mind, we further generalize
this idea in the subsequent theorem and make use of a functional
l : Z → [0,∞) of which we require the following property:
∀z ∈ Ze, ∃c > 0 : sup
T>0
l(zT ) < c for all continuations zT of z. (4.4)
If l is bounded and only depends on the initial value z(0) (assuming
that the evaluation of z at t = 0 is well-defined), (4.4) is obviously
satisfied. As will become clear in the subsequent chapter, this change is
instrumental for the generalization of the Popov criterion towards only
proper LTI systems.
Theorem 4.7
In addition to Assumption 4.5 let
a) the feedback system (4.3) be well-posed;
b) Σ : Z × W → R be a map such that, for some σij ∈ R and all
(u, v, w) ∈ U × V ×W ,
Σ
(
Mw +Mu+N(v)
w
)
− Σ
(
Mw
w
)
≤
≤
‖w‖W‖u‖U
‖v‖V
T  0 σ12 σ13σ12 σ22 σ23
σ13 σ23 σ33
‖w‖W‖u‖U
‖v‖V
 ; (4.5)
c) there exist some function l : Z → [0,∞) satisfying (4.4) and such
that
Σ
(
z
τ∆(z)
)
≥ −l(z)2 (4.6)
for τ ∈ [0, 1], z = Mu+N(v) and (u, v) ∈ U × V ;
d) there exists some ε > 0 such that
Σ
(
Mw
w
)
≤ −ε‖w‖2W for all w ∈ W . (4.7)
74 Chapter 4. A general analysis framework
Then the interconnection (4.3) is stable, i. e., there exist some γ > 0,
γ0 ∈ R (only depending on J , M , N and Σ) such that z := R(u, v) ∈ Z
and
‖z‖Z ≤ γ(‖u‖U + ‖v‖V ) + γ0l(z) for all (u, v) ∈ U × V . (4.8)
Proof. A proof of Theorem 4.7 is found in Appendix C.3.1.
The most noticeable difference with Theorems 3.4 is certainly the
ability to work with general function spaces. Yet, as will become
apparent shortly, the concept of continuation of functions in the extended
space in conjunction with (4.1) is equally important.
Remark 4.8.
For Y := Z × W with norm |||(z, w)||| = √‖z‖2Z + ‖w‖2W , all maps
Σ in the subsequent chapters are defined as Σ(x) = 〈x, x〉, where
〈., .〉 : Y × Y → R is additive in the first and second argument and
continuous in the following sense: there exists a constant c with
|〈x, y〉| ≤ c|||x||||||y||| for all x, y ∈ Y.
It is easily seen that this together with the additivity in both arguments
implies Theorem 4.7 b). Indeed, let x = (Mw,w) and y = (Mu +
N(v), 0). Then 〈x+ y, x+ y〉−〈x, x〉 = 〈y, y〉+〈x, y〉+〈y, x〉 ≤ c(|||y|||2 +
2|||x||||||y|||) and thus we infer (4.5) with boundedness of M, N and J . ?
4.4 From stability to performance analysis
Apart from stability, our primary concern is the verification of some
performance criterion. Hence, consider the interconnection depicted in
Figure 4.2 as governed by the equations
z = Mw +Nd, w = ∆(z), d ∈ D (4.9)
e = N21w +N22d (4.10)
4.4. From stability to performance analysis 75
with linear systems N21 : U → E and N22 : D → E . Note that in the
performance setup the operator N : D → Z is assumed to be LTI and
we denote the (single) external disturbance by d. Yet as demonstrated
next, the previously developed stability results immediately apply.
M N
N21 N22
∆
de
wz
Figure 4.2: Performance interconnection
Suppose that the corresponding uncertainty loop (4.3) is well-posed
and assume that we are given a map Σp : E ×D → R satisfying
Σp
(
e
0
)
≥ 0 for all e ∈ E . (4.11)
Our objective is to verify the performance criterion
Σp
(
e
d
)
≤ −ε‖d‖2D + lp(z)2 for all d ∈ D , (4.12)
for some ε > 0, and some map lp : Z → [0,∞). As is standard in
dissipation theory, the function lp quantifies the price to pay for non-
zero initial conditions; as it is intertwined with the map l in (4.6), we
will keep it as a degree of freedom that will be specified later. Let
us now argue that stability and performance are guaranteed if there
exists ε > 0 and some map Σ satisfying (4.5) and (4.6) such that for all
(w, d) ∈ W ×D we have
Σ
(
Mw +Nd
w
)
+ Σp
(
N21w +N22d
d
)
≤ −ε(‖d‖2D + ‖w‖2W ). (4.13)
76 Chapter 4. A general analysis framework
M
∆
+
+
u
Nd
z
w
M N
∆
d
wz
Figure 4.3: From stability to performance
Indeed, for d = 0 we recover (4.7) from (4.11), (4.13) and thus both
interconnections depicted in Figure 4.3 are stable. Moreover, we may
use the loop equations and the fact that W ⊂ U (a consequence of
Assumption 4.5 a)) in order to infer that (4.13) implies
Σ
(
z
∆(z)
)
+ Σp
(
e
d
)
≤ −ε(‖d‖2D + ‖∆(z)‖2W )
for all (w, d) ∈ W ×D . This, in turn, implies with (4.6) that
Σp
(
e
d
)
≤ −ε‖d‖2D + l(z)2 (4.14)
holds for all (w, d) ∈ W ×D . In conclusion, we have proven the following
corollary.
Corollary 4.9
Let Σp satisfy (4.11) and let the assumptions of Theorem 4.7 with (4.7)
replaced by (4.13) be valid. Then there exist ε > 0 such that (4.12) holds
with lp = l.
Once stability of (4.3) has been verified and as the above arguments
reveal, performance is already guaranteed if the map Σ in (4.13) is
exchanged by any map Σ˜ satisfying Σ˜(z,∆(z)) ≥ −l(z)2 for all z =
Mw + Nd with (w, d) ∈ W × D . This observation opens the way
to guarantee stability and performance with different maps Σ and Σ˜,
which is often not emphasized in the literature but might be of practical
relevance in applications.
4.5. Application to Sobolev spaces 77
4.5 Application to Sobolev spaces
Let us now bring the above introduced generalizations to life by consid-
ering the specific example of LTI systems and quadratic forms defined
on Sobolev spaces H r for r ∈ N0 defined as follows.
Definition 4.10.
Let r ∈ N0. Then H r and H re denote the (extended) Sobolev spaces
of functions u : [0,∞) → Rk with ∂ju ∈ L2, or ∂ju ∈ L2e for
j ∈ {0, . . . , r}, respectively. H r is equipped with the norm ‖u‖2r =∑r
j=0 ‖∂ju‖2. ?
4.5.1 Sobolev spaces
We first show that the spaces H r satisfy all requirements on the
underlying function spaces. To this end, let H r[0, T ] denote the space
of functions u : [0, T ]→ Rk with ∂ju ∈ L2[0, T ] for j ∈ {0, . . . , r} and
equipped with the norm ‖u‖2H r[0,T ] =
∑r
j=0 ‖∂ju‖2L2[0,T ]. Then we
obtain H re and H r by proceeding in accordance with Assumption 4.1.
As already mentioned, PTu is typically not contained in H r if u ∈H r.
Still we can prove the existence of a sufficiently smooth continuation in
order to meet all requirements of Assumption 4.1. For clarity of display,
we define the differential operator
Drz := col(z, ∂z, . . . , ∂rz) on H r.
Lemma 4.11
Let r ∈ N0. Then H re and H r satisfy Assumption 4.1 d).
Proof. For r = 0 and uT := PTu we obtain (4.1) with KH 0 = 1.
Suppose r > 0. If T > 0 and u¯ ∈ H re fix ξ := (Dr−1u¯)(T ). We
then construct u := u¯T on [T,∞) with an optimal constant KH r by
minimizing the functional∫ ∞
T
r∑
j=0
‖∂ju(t)‖2 dt
78 Chapter 4. A general analysis framework
subject to u ∈H r, uT = u¯T and (Dr−1u)(T ) = ξ. This is achieved by
solving a linear quadratic optimal control problem with stability. Indeed,
with x := Dr−1u, w := ∂ru, our problem amounts to minimizing the
cost functional
∫∞
T
x(t)Tx(t) + w(t)Tw(t) dt over all trajectories of
x˙ =
(
0 I
0 0
)
x+
(
0
I
)
w, x(T ) = ξ (4.15)
with w ∈ L2 and x ∈ L2. Since the system in (4.15) is stabilizable
and with the special cost function, there exists a stabilizable solution P
of the corresponding LQ Riccati equation such that the optimal cost
equals ξTPξ. Now recall that Proposition 3 in [19] (see also [20]) implies
max
t∈[0,T ]
‖v(t)‖ ≤ ‖v‖H 1[0,T ] for all T > 0, v ∈H 1e . (4.16)
Thus, with the maximal eigenvalue of P , denoted by λmax(P ), we get
ξTPξ ≤ λmax(P )
r−1∑
l=0
‖∂lu¯(T )‖2 ≤ 2λmax(P )‖u¯‖2H r[0,T ].
This proves the claim with K
H r
≤ 1 +
√
2λmax(P ).
4.5.2 Systems and quadratic forms on H r
Let us now consider operators and quadratic forms onH r. In particular,
we assume that the linear system G =
(
M N
N21 N22
)
in (4.9), (4.10) is LTI
and realized as
x˙ = Ax+B1w +B2d, x(0) = 0,
z = C1x+D11w +D12d,
e = C2x+D21w +D22d (4.17)
with A ∈ Rn×n being Hurwitz. Then G defines a bounded operator on
H r ×H r for any r ∈ N0. With a pair of symmetric matrices Q1, Q2,
define the quadratic form
Σ(Q1,Q2)
(
z
w
)
:= σQ1
(
z
w
)
+ [(Dr−1w)(0)]TQ2[(Dr−1w)(0)] (4.18)
4.5. Application to Sobolev spaces 79
on H r ×H r, where
σQ1
(
z
w
)
:=
∫ ∞
0
(Drz(t)
Drw(t)
)T
Q1
(Drz(t)
Drw(t)
)
dt. (4.19)
For r = 0 the matrix Q2 is empty; for r > 0 it will be instrumental to
incorporate the initial values of w and its derivatives; still it is easily
seen that Σ(Q1,Q2) satisfies (4.5).
Also performance criteria are modeled with the map Σp in (4.12)
defined as
σP
(
e
d
)
:=
∫ ∞
0
(Dre(t)
Drd(t)
)T
P
(Dre(t)
Drd(t)
)
dt, (4.20)
where P is some specified symmetric matrix. This allows, e.g., to put spe-
cial emphasis on some derivatives or to consider weighted combinations
thereof. Moreover, we obtain a generalization of L2-gain to H r-gain
performance with bound γ by setting P = diag(I,−γ2I). However,
note that H r-gain bounds do not, in general, translate right away into
bounds on the L2-gain. The form σP as in (4.20) is sometimes called to
be a quadratic differential form as considered, e. g., in [186, 185] in the
context of behavioral systems. Yet, we are not aware of any references
in which such performance specifications have been considered in the
context of robustness analysis with (integral) quadratic constraints.
4.5.3 Verification of stability and performance
Let us now embed the stability and performance analysis for LTI systems
on Sobolev spaces into classical dissipation theory and thus open the
way for the verification of (4.13) using LMIs. Observe for u ∈H r with
r ≥ 1 that
Dr−1u(t) = Dr−1u(0) +
∫ t
0
Dr−1u˙(τ) dτ.
This implies that the initial values Dr−1u(0) and the highest derivative
∂ru ∈H 0 = L2 generate the whole signal Dr−1u. Now fix r ∈ N and
80 Chapter 4. A general analysis framework
apply the same reasoning to both inputs w, d ∈H r of the LTI system
(4.17). Then xe := col(x,Dr−1w,Dr−1d) is generated as the solution of
the extended system x˙e = Aexe +Beu defined by
x˙e =

A B1 0 B2 0
0 0 I 0 0
0 0 0 0 0
0 0 0 0 I
0 0 0 0 0
xe +

0 0
0 0
I 0
0 0
0 I

(
∂rw
∂rd
)
︸ ︷︷ ︸
=u
(4.21)
for xe(0) = col(0,Dr−1w(0),Dr−1d(0)). Moreover, with the abbrevia-
tions
Ki =

Ci
CiA
...
CiA
r
 , Lij =

Dij 0 . . . 0
CiBj Dij . . . 0
...
...
. . .
...
CiA
r−1Bj . . . . . . Dij
 , (4.22)
the derivatives of the inputs and outputs of G are related as
Drz
Drw
Dre
Drd
 =

K1 L11 L12
0 I 0
K2 L21 L22
0 0 I

 xDrw
Drd
 =: (T∆
Tp
) xe∂rw
∂rd
 . (4.23)
For notational compactness, we further introduce the abbreviation
Er(s) := col
(
1
sr
I, . . . ,
1
s
I, I
)
, F r(s) := srEr(s).
The main result of this section now gives a novel and precise extension
of the classical triple of equivalent conditions to Sobolev spaces, relating
the dissipativity constraint (4.13) to an FDI and an LMI.
Theorem 4.12
Fix r ∈ N, Q1 = QT1 and P = PT . Then the following statements are
equivalent.
4.5. Application to Sobolev spaces 81
a) There exist ε>0, R=RT such that for all w, d ∈H r:
σQ1
(
Mw +Nd
w
)
+ σP
(
N21w +N22d
d
)
≤
≤ −ε(‖w‖2r + ‖d‖2r)+ (?)T R(Dr−1w(0)Dr−1d(0)
)
. (4.24)
b) There exists X = XT with
(
?
)T ( 0 X
X 0
)(
I 0
Ae Be
)
+
(
?
)T (Q1 0
0 P
)(
T∆
Tp
)
≺ 0. (4.25)
c) There exists ε > 0 such that
(
?
)∗(Q1 0
0 P
)
ErM ErN
Er 0
ErN21 E
rN22
0 Er
 4 −εI on iR \ {0} (4.26)
and
(
?
)∗(Q1 0
0 P
)
F rM F rN
F r 0
F rN21 F
rN22
0 F r
 (0) ≺ 0 (4.27)
Proof. A proof is found in Appendix C.3.2
Remark 4.13.
Theorem 4.12 is a cornerstone of our analysis framework as it reduces
the verification of stability and performance of an interconnection, The-
orem 4.12 a), to the feasibility of an LMI, Theorem 4.12 b). Thus, in
contrast to [195, 136, 159], we not only formulate an abstract stability
result, but also provide the means for its efficient numerical verification.
As illustrated in detail in Chapter 5, this opens the way to a compre-
hensive treatment of one of the most important classes of nonlinearities
in control. ?
82 Chapter 4. A general analysis framework
Finally, let us briefly comment on the structure of the quadratic
form (4.18) and the use of the extended system (4.21) in the light of
Theorem 4.12. In contrast to [134, 118], where stability of a similarly
extended system interconnected with a nonlinearity is verified in order
to conclude stability of the original interconnection, we only need
the system (4.21) to establish the connection between the quadratic
constraint in Theorem 4.12 a) and the LMI in Theorem 4.12 b). As
visible from the proof of Theorem 4.12, both are related through a
dissipation inequality that naturally involves the initial condition of the
extended system. This, in turn, necessitates the inclusion of the term
[(Dr−1w)(0)]TQ2[(Dr−1w)(0)] in the definition of Σ in (4.18), and the
analogue for Σp.
4.5.4 Application to parametric uncertainties
Finally, let us highlight the main characteristics of our novel approach
by considering the concrete example of time-varying parametric uncer-
tainties. We analyze the following problem.
Let δ(.) be any sufficiently smooth time-varying parameter such that
some bounds ∂jδ(t) ∈ [αj , βj ] for j ∈ {0, . . . , r} are known. For any
such curve δ, the uncertainty ∆ : L2e → L2e in Figure 4.2 is defined as
∆δ(z) := δz. If, in addition, we assume z ∈H r then w = ∆δ(z) ∈H r
and the following relations hold:
w(t) = δ(t)z(t),
w˙(t) = δ˙(t)z(t) + δ(t)z˙(t),
w¨(t) = δ¨(t)z(t) + 2δ˙(t)z˙(t) + δ(t)z¨(t),
...
This can be expressed as Drw(t) = ∆(t)Drz(t) with ∆(t) ∈ ∆ for all
t ∈ [0,∞), where ∆ is some simple-to-specify compact polytope of
4.5. Application to Sobolev spaces 83
structured matrices depending on the bounds αj , βj . If ∆ext denotes
the set of extreme points of ∆ and with
Q =
{
Q = QT
∣∣∣∣(?)T Q( I∆
)
 0 for all ∆ ∈∆ext where Q22 ≺ 0
}
,
routine arguments2 show, for all Q1 ∈ Q and any parameter trajectory
δ, that ∆δ satisfies the following IQC:
σQ1
(
z
∆δ(z)
)
≥ 0 for all z ∈H r. (4.28)
By Theorem 4.12, feasibility of (4.25) for some Q1 ∈ Q guarantees
stability of (4.3) and the existence of ε > 0 with
σP
(
e
d
)
≤ −ε‖d‖2r for all d ∈H r0 (4.29)
along all trajectories of (4.9)-(4.10). This offers the possibility to exploit
bounds on higher derivatives of time-varying parametric uncertainties in
robustness analysis. Note that standard techniques relying on parameter-
dependent Lyapunov functions as outlined, e.g., in [90], only allow to
exploit bounds up to the first derivative. As yet another approach,
the so-called swapping lemma (see [160] and also [117, 144]) allows to
incorporate bounds on δ and δ˙ in the classical IQC setting [91, 102]. It
would be interesting to extend this technique to higher derivatives and
compare the obtained results to ours.
Finally, for a concrete numerical example we assume that G in (4.17)
is defined by
A =

−.9 1 0 0
0 −2 0.5 0
0 0 −0.4 4
0 0 −3.9 −0.4
 , (B1 B2) =

0.9 0
−2 0
1 −5
8 1
 ,
2See Section 6.3 for a detailed exposition.
84 Chapter 4. A general analysis framework
(
C1
C2
)
=
(−5 0 0 0
−1 0 0 0
)
, D = 0, (4.30)
and in feedback with ∆δ for r = 2 as in Figure 4.2. Since we are
interested in bounding the standard L2-gain of d → e, we take W =
Z = H 2, D = E = L2 and P = diag(1,−γ2). Moreover, δ(t), δ˙(t),
δ¨(t) for t ≥ 0 are assumed to be contained in
[α0, β0] = [0, 1.5], [α1, β1] = [0, 6], [α2, β2] = [0, 6],
respectively. Then ∆δ : H 2 → H 2 is causal and bounded. Since
D = 0, C1B2 = 0 and C1AB2 = 0, let us highlight at this point that it
is not required to work with a full system extension with respect to the
input d in (4.21) since d, d˙ and d¨ are not feed through to the uncertainty
output z, and d˙, d¨ are not involved in the performance specification.
We can hence work with the extended system
x˙w˙
w¨
 =
A B1 00 0 I
0 0 0
xw
w˙
+
0 B20 0
I 0
(w¨
d
)
.
By applying Theorem 4.12 for this extension and the IQC (4.28) we
obtain the following results.
If only exploiting the bound on δ(.), we are not able to verify stability
of the interconnection. If, in addition, we utilize the bound on δ˙(.) we
can guarantee stability with a certified L2-gain bound of γ1 = 14.89. By
also including the constraints on δ¨(.), the guaranteed bound improves
to γ2 = 1.72. It is important to emphasize that we do not require
X in (4.25) to be positive definite, as is usually required in standard
Lyapunov arguments. If we artificially impose such an extra constraint,
the performance bounds increase to the values of 16.38 > γ1 = 14.89
and 1.97 > γ2 = 1.72, respectively.
4.6. Summary 85
4.6 Summary
In this chapter, we derive a unified framework for stability and per-
formance analysis on general function spaces. The modifications may
be seen as an extension of the IQC framework towards the setting of
Safonov and Zames as discussed in Chapter 2. An important strength
of the present approach is that it naturally covers the special, yet
highly relevant, case of Sobolev spaces. Here, we not only give suffi-
cient conditions for stability and performance, but equivalently recast
their numerical verification as an LMI feasibility problem. Finally, we
illustrate how this general setup enfolds for the concrete example of
time-varying parametric uncertainties.
86 Chapter 4. A general analysis framework
Chapter 5
Full-block multipliers for repeated, slope-
restricted scalar nonlinearities –
continuous-time case
5.1 Introduction
In this chapter, we take advantage of the framework establishedin Chapter 4 by deriving a unified approach to the stability and
performance analysis of the feedback interconnection (4.3), where ∆ is
a static repeated nonlinearity defined through a scalar slope-restricted
function ϕ : R→ R. Even though this is a rather particular setting, it
is nevertheless of great practical relevance, as it comprises some of the
most important nonlinearities in control, such as multiple saturations
and dead-zones. Examples of such interconnections in engineering
applications typically stem from systems with actuator saturations (see,
e.g., [83, 157, 79, 125, 40]), but slope-restricted nonlinearities also arise
naturally in recurrent neural networks (see, e.g., [61, 95, 11] and [198]
for an overview) and in more mathematical applications such as the
analysis and design of optimization algorithms (see, e.g., [104, 53]).
As briefly touched upon in Chapter 2, the investigation of absolute
stability of such interconnections has a long standing history in control,
87
88 Chapter 5. Slope-restricted nonlinearities
probably starting with the works of Lurye and Postnikov [106], and is
hence also termed Lurye’s problem (see [10] for an excellent historical
overview). The main contributions to its solution date back to the
1960s with the works of Popov [126], Yakubovich [188], Zames [195]
and Zames and Falb [197]. Even today, the stability criteria associated
with these names remain the most effective analysis tools. As each of
them focusses on a different aspect of the nonlinearity ϕ, it is typically
desirable to apply several methods simultaneously in order to achieve
the best results.
The circle [195] and Popov criteria [126] both rely on sector bounds
for the nonlinearity. While the first criterion just exploits relations
between the input and the output of the nonlinearity, the second
one involves the derivative of the input. The method developed by
Yakubovich [188] proceeds further along this line of thought and uses
relations between the derivatives of both the input and the output if
the nonlinearities have a bounded derivative. In contrast, the result by
Zames and Falb [197] does not require the differentiability of signals
and is applicable for non-smooth slope-restricted nonlinearities as well.
As a major advantage, our general analysis framework permits the
direct application and combination of circle, Popov, Zames-Falb and
Yakubovich multipliers, even if the nonlinearities are not everywhere
differentiable and the underlying LTI system is not strictly proper.
Besides the modularity and the possibility for combined application of
different criteria, another aspect that separates the present approach
from existing ones in the literature is its focus on repeated nonlinearities,
as they are often emerging in practical applications. By treating each
nonlinearity individually, the scalar versions of the above discussed
multipliers can be easily combined to obtain stability tests for repeated
nonlinearities that involve structured diagonal multipliers (see, e. g.,
[137, 139, 74, 76, 37]). It is known how to employ unstructured full-block
multipliers for the circle [175] and the Zames-Falb criteria [42, 103, 166]
in order to potentially reduce conservatism in the stability analysis.
In a further contribution, we propose novel full-block multipliers for
5.1. Introduction 89
the Yakubovich stability criterion and suggest a parametrization of
the complete class of full-block Zames-Falb multipliers for effective
computations. Finally, our approach allows a seamless combination of
all four multiplier stability tests in a modular fashion, which leads to
computational stability tests in terms of LMIs.
Perhaps most closely related to our approach is the one proposed by
Altshuller using delay integral quadratic constraints [10]. As a distin-
guishing feature, this framework allows for the inclusion of (diagonal)
Yakubovich multipliers [9, 10], but at the expense of reduced flexibility
and by requiring individual proofs for each new multiplier. Moreover,
the emphasis of the present exposition lies on the derivation of full-block
criteria and their (combined) verification using LMIs; these aspects are
not touched upon or play a subordinate role in [10].
In the remainder of this chapter, we show how all the new ingredients
of Theorem 4.7 come to flourish even if applied to the special case when
M is an LTI system, ∆ is a repeated nonlinearity defined through a
sector-bounded and slope-restricted scalar function, and the underlying
function spaces are either L2e or H 1e (see Definition 4.10).
After precisely defining the operators, the signal spaces and the
considered quadratic constraints in Section 5.2, we carefully address
the issue of well-posedness even if the LTI system is not strictly proper.
Section 5.3 is devoted to a detailed presentation of stability analysis with
full-block circle and Zames-Falb as well as standard Popov multipliers.
For the circle criterion we generalize [175, 176] by using the so-called
Pólya relaxation and reveal new insights into the relation with the
classical circle criterion or other relaxation schemes; thus permitting to
systematically exploit the full power of this test. After briefly discussing
the incorporation of classical Popov multipliers into our framework,
we turn to full-block Zames-Falb multipliers, as described in [42] and
further generalized in [103], where we extend the parametrization in [37]
to repeated scalar nonlinearities and prove it to be asymptotically
exact. This new result enables us to tap the complete potential of the
Zames-Falb stability test in computations.
90 Chapter 5. Slope-restricted nonlinearities
Section 5.4 reveals that our framework allows for the inclusion of
the Yakubovich criterion [188, 46] with new full-block multipliers, and
it permits to drop the typically encountered restriction of Popov and
Yakubovich tests to strictly proper plants. The translation into LMIs
relies on Theorem 4.12. And finally, Section 5.5 serves to differentiate
our results from related ones in the literature, as supported by further
numerical examples in Section 5.6. We conclude by emphasizing that
the results in this chapter have already appeared, in parts even literally,
in [58, 56].
5.2 Application to slope-restricted nonlinearities
Let us now start by specializing the general framework of Chapter 4
to the particular setting featuring slope-restricted nonlinearities. As
already mentioned in the introduction, it suffices to consider derivatives
up to order one of the involved signals.
5.2.1 Slope-restricted nonlinearities
First, we specify the class of uncertainties under consideration.
Definition 5.1.
Let µ1 ≤ 0 ≤ µ2. Then ϕ : R → R is slope-restricted, in short
ϕ ∈ slope(µ1, µ2), if
ϕ(0) = 0 and µ1 ≤ ϕ(x)− ϕ(y)
x− y ≤ supx6=y
ϕ(x)− ϕ(y)
x− y < µ2
for all x, y ∈ R, x 6= y. (5.1)
If µ1 = 0 and the bound on the right is absent, ϕ is just monotone
and we write ϕ ∈ slope(0,∞). If there exist some α ≤ 0 ≤ β such that
ϕ satisfies
(ϕ(x)− αx)(βx− ϕ(x)) ≥ 0 for all x ∈ R, (5.2)
it is said to be sector-bounded which is expressed as ϕ ∈ sec[α, β]. ?
5.2. Application to slope-restricted nonlinearities 91
With such a nonlinearity ϕ let the map Φ : Rk → Rk be given
as Φ(x1, . . . , xk) =
(
ϕ(x1) . . . ϕ(xk)
)T
. In the sequel we restrict our
attention to the static (and obviously causal) operators defined as
∆ϕ(z)(t) := Φ(z(t)) for almost all t ∈ [0,∞) and all z ∈ L2e. (5.3)
We say that ∆ϕ ∈ slope(µ1, µ2)k and ∆ϕ ∈ sec[α, β]k if ϕ ∈ slope(µ1, µ2)
or ϕ ∈ sec[α, β], respectively. As an immediate consequence of Definition
5.1 we infer
∆ϕ ∈ slope(µ1, µ2)k =⇒ ∆ϕ ∈ sec[µ1, µ2]k. (5.4)
Thus finite slope restrictions always translate into finite sector bounds
with the same constants. However, often tighter sector bounds are
known, i. e., ϕ ∈ slope(µ1, µ2) ∩ sec[α, β] with β < µ2 or µ1 < α
implying ∆ϕ ∈ slope(µ1, µ2)k ∩ sec[α, β]k; in numerical examples we
will demonstrate that this additional information can be beneficially
exploited.
Remark 5.2.
Note that the assumptions 0 ∈ [α, β] and 0 ∈ [µ1, µ2] are not restrictive,
since we can always perform a loop transformation such that both
are met. As a benefit from 0 ∈ [α, β], we immediately infer that
∆ϕ ∈ sec[α, β]k implies τ∆ϕ ∈ sec[α, β]k for τ ∈ [0, 1] and the analog
holds true for slope restrictions. If applying Theorem 4.7, it hence
suffices to verify properties a) and c) for τ = 1 and all uncertainties ∆ϕ
in the respective class. ?
5.2.2 Signal spaces and operators
For the purpose of this chapter it suffices to consider the spaces L2 and
H 1. As the number of repetitions of ϕ in Φ plays an important role,
we include the dimension k of the signals in the signal space symbols,
i.e., L k2 and H 1,k. In order to distinguish the norm on L1 from the
norm on H 1,k, we denote the latter by ‖ · ‖H . By Lemma 4.11, L k2e
92 Chapter 5. Slope-restricted nonlinearities
and H 1,ke satisfy Assumption 4.1. We will only need the compatibility
property in Assumption 4.5 a) for H 1,ke ⊂ L k2e, which follows from the
obvious inequality ‖w‖L2e,T ≤ ‖w‖He,T for all T > 0 and w ∈ H 1,ke .
Following Section 4.5 and concerning the linear operator M , we restrict
our attention to stable LTI systems represented as
x˙ = Ax+Bw, x(0) = 0,
z = Cx+Dw, (5.5)
with A ∈ Rn×n being Hurwitz. Then both M and ∆ϕ are causal and
compatible with the considered spaces in the following sense. M :
L k2e → L k2e and M : H 1,ke → H 1,ke are bounded; in case of D = 0
also M : L k2e → H 1,ke is well-defined and bounded. Moreover, for
∆ϕ ∈ slope(µ1, µ2)k, all the maps ∆ϕ : L k2e → L k2e, ∆ϕ :H 1,ke → L k2e
and ∆ϕ : H 1,ke → H 1,ke are bounded. The second property is a
consequence of the first, while the third is stated next since it requires
a proof.
Lemma 5.3
If ϕ ∈ slope(µ1, µ2) then ∆ϕ : H 1,ke → H 1,ke is well-defined and
bounded.
Proof. A proof is found in Appendix C.4.3.
Finally, N : V → L k2 or N : V → H 1,k are assumed to obey the
properties in Assumption 4.5 c).
5.2.3 Well-posedness
Let us briefly discuss the issue of well-posedness in this setting and
based on the following result.
Lemma 5.4
Suppose ϕ ∈ slope(µ1, µ2) and let
Θ({µ1, µ2}, k) =
{
diag(δ1, . . . , δk) ∈ Rk×k
∣∣ δi ∈ {µ1, µ2}} .
5.2. Application to slope-restricted nonlinearities 93
Then I − DΦ : Rk → Rk is invertible and (I − DΦ)−1 : Rk → Rk is
globally Lipschitz if and only if
det(I −D∆) > 0 for all ∆ ∈ Θ({µ1, µ2}, k). (5.6)
Proof. The proof of necessity is an adaption of [86, Proof of Claim 1]
in the context of saturated systems. For sufficiency observe that det(I−
D∆) > 0 for all ∆ ∈ Θ({µ1, µ2}, k) implies det(I − D∆) > 0 for
all ∆ ∈ Θ([µ1, µ2], k) =
{
diag(δ1, . . . , δk) ∈ Rk×k
∣∣ δi ∈ [µ1, µ2]} since
det(I −D∆) is a multi-affine function in (δ1, . . . , δk). Then the claim
follows from Proposition 2 in [193].
Lemma 5.5
Property (5.6) implies that I −M∆ϕ : L k2e → L k2e has a causal inverse
for all ∆ϕ ∈ slope(µ1, µ2)k.
Proof. The map which takes u ∈ L k2e into y = u −M∆ϕ(u) ∈ L k2e
is described by x˙ = Ax+BΦ(u), y = u−DΦ(u)− Cx with x(0) = 0.
Since I −DΦ is invertible, this is equivalent to
x˙ = Ax+BΦ(I −DΦ)−1(Cx+ y), x(0) = 0,
u = (I −DΦ)−1(Cx+ y). (5.7)
Since Φ(I − DΦ)−1 is also globally Lipschitz, standard ODE theory
implies that for each y ∈ L k2e there exists a unique response u ∈ L k2e
with (5.7) that depends causally on y.
For repeated slope-restricted nonlinearities, we have thus identified
the easily verifiable condition (5.6) that guarantees well-posedness of
(4.3) in the classical sense. If ϕ ∈ sec[α, β], other arguments are required
to show well-posedness; for example, if D = 0 and ϕ is locally Lipschitz
continuous, again standard ODE arguments guarantee this classical
well-posedness property.
94 Chapter 5. Slope-restricted nonlinearities
5.2.4 Quadratic forms and verification of constraints
In Section 5.3 we put Theorem 4.7 to use by employing quadratic forms
Σ of the structure already discussed in Section 2.2:
ΣΠ
(
z
w
)
:=
∫ ∞
−∞
(
?
)T
Π(iω)
(
zˆ(iω)
wˆ(iω)
)
dω with Π =
(
Q S
S∗ R
)
. (5.8)
However, in contrast to (2.16), we only assume that the multiplier Π is
a hermitian valued and measurable function defined on the imaginary
axis. By omitting the (essential) boundedness assumption originally
introduced by Megretski and Rantzer, we may seamlessly incorporate
(unbounded) Popov multipliers (2.20). For reasons of compactness, we
occasionally write ΣΠ(z, w).
In case that Π is indeed essentially bounded on the imaginary axis,
we note that ΣΠ is defined as described in Remark 4.8 and, hence, does
satisfy the technical property Theorem 4.7 b). Moreover, with (5.8) and
M as in (5.5), Theorem 4.7 d) reads for W = L k2 as follows:
∃ε > 0, ∀w ∈ L k2 :∫ ∞
−∞
(
M(iω)wˆ(iω)
wˆ(iω)
)∗
Π(iω)
(
M(iω)wˆ(iω)
wˆ(iω)
)
dω ≤ −ε‖w‖2. (5.9)
It is well-known that this is equivalent to FDI:
∃ε > 0 : (?)∗Π(iω)(M(iω)
I
)
4 −εI for almost all ω ∈ R. (5.10)
If the left hand side of the inequality in (5.10) is also rational, the
standard KYP lemma, Lemma 2.12, can be applied to computationally
verify this property through the solution of an LMI feasibility problem.
In Section 5.4 we encounter more general scenarios and derive the related
LMI by means of Theorem 4.12.
5.3. Derivation and application of multipliers 95
5.2.5 Sketch of procedure
The general procedure for stability analysis now enfolds along the pattern
outlined in Chapter 2. Well-posedness, Theorem 4.7 a), is verified
separately. The key step is to capture the properties of the nonlinearity
∆ϕ with a whole class of multipliers Π such that Theorem 4.7 b) and
in particular 4.7 c) hold with ΣΠ and for all Π ∈ Π. For guaranteeing
stability, it then remains to verify the existence of one multiplier Π ∈ Π
such that ΣΠ also satisfies Theorem 4.7 d); since all classes Π in the
following Section 5.3 are convex cones, this latter search of Π ∈ Π boils
down to solving a convex optimization problem as addressed in more
detail in Section 5.3.4.
5.3 Derivation and application of multipliers
This and the next section are devoted to a comprehensive collection of
stability tests for slope-restricted nonlinearities. A particular focus lies
on the extension of standard criteria to facilitate the application of the
more powerful full-block multipliers. Moreover, we present a framework
that allows to combine all introduced multipliers in a modular fashion.
5.3.1 Full-block multipliers for the circle criterion
We start by considering the static (frequency independent) class of
multipliers for the circle criterion and for nonlinearities that are merely
sector-bounded. The property ∆ϕ ∈ sec[α, β]k is traditionally captured
using multipliers that share the structure of the nonlinearity [139, 79, 21].
This set of diagonally repeated circle criterion multipliers is given by
Πdr[α, β]
k =
{
Π ∈ S2k
∣∣∣∣Π =(−αβ diag(λ) α+β2 diag(λ)α+β
2 diag(λ) −diag(λ)
)
, λ ∈ Rk+
}
.
(5.11)
96 Chapter 5. Slope-restricted nonlinearities
As will be seen, unstructured multipliers offer more freedom. Let us
hence introduce the class of full-block circle criterion multipliers [176,
Section 5.8.2.] adapted to the diagonally repeated case. With Ω ⊂ R,
Θ(Ω, k) := {∆ = diag(δ1, . . . , δk) | δj ∈ Ω}
and
FΠ(∆) :=
(
I
∆
)T
Π
(
I
∆
)
these are given by
Π[α, β]k =
{
Π ∈ S2k ∣∣ FΠ(∆)  0 ∀∆ ∈ Θ([α, β], k)} . (5.12)
With (5.12) we arrive at the following IQC.
Lemma 5.6 (Full-block circle IQC)
Let Π ∈ Π[α, β]k and ΣΠ be taken as in (5.8). Then
ΣΠ
(
z
∆ϕ(z)
)
≥ 0 for all ∆ϕ ∈ sec[α, β]k and z ∈ L k2 . (5.13)
With Theorem 4.7 this IQC immediately translates into the following
robust stability test.
Theorem 5.7 (Full-block circle criterion)
Let N : V → L k2 and suppose that the interconnection (4.3) is well-
posed in the classical sense for all ∆ϕ ∈ sec[α, β]k. If there exists some
Π ∈ Π[α, β]k with (5.10), then (4.3) is robustly stable: There exists
γ > 0 such that
‖z‖ ≤ γ(‖u‖+ ‖v‖V ) and all (u, v) ∈ L2 × V , ∆ϕ ∈ sec[α, β]k.
Remark 5.8.
Note that V = L k2 and N = I recovers the standard setting of [110]
with two free inputs as in Figure 2.3. Robust stability then implies that
(I −M∆ϕ)−1 maps L k2 into L k2 . ?
5.3. Derivation and application of multipliers 97
Proof. As the IQC (5.13) is valid onL k2 , we takeWe = Ze = Ue = L k2e.
In view of Section 5.2, all requirements of Theorem 4.7 are fulfilled for
ΣΠ with Π ∈ Π[α, β]k and l = 0.
Since Π[α, β]k is defined through infinitely many constraints, the
application of Theorem 5.7 requires approximations in order to render
this criterion computational. As it is also at the heart of the derivation
of our novel full-block Yakubovich criterion, we summarize the most
important relaxation schemes in some detail and give new insights
regarding their interrelation. With the partition of Π as in (5.8), two
nested inner approximations Πc[α, β]k ⊂ Πpc[α, β]k ⊂ Π[α, β]k are
given by the so-called convex and partially convex relaxations
Πc[α, β]
k =
{
Π ∈ S2k ∣∣R ≺ 0 and FΠ(∆)  0 ∀∆ ∈ Θ({α, β}, k)}
(5.14)
and
Πpc[α, β]
k=
{
Π ∈ S2k ∣∣Rii<0 and FΠ(∆)  0 ∀∆ ∈ Θ({α, β}, k)} ,
(5.15)
respectively. Since Πc or Πpc are described by a finite number of LMIs
they can be substituted for Π in Theorem 5.7 in order to arrive at
a computationally tractable stability test. This substitution causes
conservatism whose degree cannot be judged a priori. Based on a
matrix version of a classical theorem of Pólya [127], this motivated the
introduction of an asymptotically exact parameterization of Π in [148]
as follows. Let ∆j denote the K := 2k matrices in Θ({α, β}, k) and
define the Hermitian-valued polynomial matrix
Λd(λ,Π, [α, β]
k) := (λ1+. . .+λK)
d
[
K∑
l=1
λl
(
I
∆l
)]T
Π
[
K∑
m=1
λm
(
I
∆m
)]
98 Chapter 5. Slope-restricted nonlinearities
in λ on the standard simplex S := {(λ1, . . . , λK) | λj ≥ 0,
∑
λj = 1} ⊂
RK . This polynomial is homogenous of degree d+2 and can be expressed
with the standard multi-index notation as
Λd(λ,Π, [α, β]
k) =
∑
κ∈NK0 , |κ|=d+2
Cd,κ(Π, [α, β]
k)λκ. (5.16)
Clearly Π ∈ Π[α, β]k is equivalent to Λ0(λ,Π, [α, β]k)  0 for all λ ∈ S.
If d ∈ N0 and because of
∑
λj = 1 for λ ∈ S, this is trivially equivalent
to
Λd(Π, λ, [α, β]
k)  0 for all λ ∈ S. (5.17)
Since λ ∈ S implies λκ ≥ 0 for all multi-indices with |κ| = d+ 2, and
since the inequality is strict for at least one of them, Cd,κ(Π, [α, β]k)  0
for all κ ∈ NK0 with |κ| = d + 2 does imply (5.17). This motivates to
define the d-th order Pólya relaxation as
ΠPold [α, β]
k :=
{
Π ∈ S2k∣∣Cd,κ(Π, [α, β]k)  0
for all κ ∈ NK0 with |κ| = d+ 2
}
. (5.18)
We conclude ΠPold [α, β]k ⊂ Π[α, β]k for all d ∈ N0. As a first insight, let
us establish that the family ΠPold [α, β]k is nondecreasing with increasing
d and recall the known fact that it contains any element of Π[α, β]k
for d → ∞, as formulated next. Note that we argue in terms of set
inclusions and thus we recover the results in [120, Lemma 5] for a specific
performance index as a special case.
Lemma 5.9
a) Suppose that Π ∈ Π[α, β]k. Then there exists an integer d ∈ N0
such that Π ∈ ΠPold [α, β]k.
b) If d1 < d2 then ΠPold1 [α, β]
k ⊂ ΠPold2 [α, β]k.
Proof. Statement a) is a special case of Theorem 7.1 in [148]. For
b) it suffices to show that ΠPold [α, β]k ⊂ ΠPold+1[α, β]k holds for all
5.3. Derivation and application of multipliers 99
d ∈ N0. Assume that Cd,κ(Π, [α, β]k)  0 for all multi-indices with |κ| =
2 + d. Since Λd+1(λ,Π, [α, β]k) = (λ1 + . . .+ λK)Λd(Π, λ, [α, β]k) and
due to (5.16), every Cd+1,κ˜(Π, [α, β]k) is a sum of suitable coefficients
Cd,κ(Π, [α, β]
k) and thus positive definite.
Together with Πc[α, β]k ⊂ ΠPol0 [α, β]k as shown in [102], Lemma
5.9 gives rise to the following chain of inclusions:
Πdr[α, β]
k ⊂ Πc[α, β]k ⊂ ΠPol0 [α, β]k ⊂ ΠPol1 [α, β]k ⊂ . . . ⊂ Π[α, β]k.
(5.19)
It is possible to also consider the subset Πdc[α, β]k of Πc[α, β]k (or of
Πpc[α, β]
k) where we restrict the blocks Q, S and R in (5.8), (5.14) to
be diagonal. Despite the fact that Πdc[α, β]k is larger than Πdr[α, β]k,
it is yet another new insight that its use does not provide any advantage
over the classical circle criterion with diagonally repeated multipliers.
Lemma 5.10
With Πdc[α, β]k as just defined, (5.10) holds for some Π ∈ Πdc[α, β]k if
and only if there exists Π ∈ Πdr[α, β]k with (5.10).
Proof. A proof is found in Appendix C.4.1.
Numerical examples (see Example 5.14) reveal that the extra freedom
offered by the larger classes in (5.19) or by Πpc[α, β]k over Πdr[α, β]k
can lead to a substantial reduction in conservatism. Due to their
simple implementation and cheap computations (since involving only
few constraints), all examples in the sequel tacitly employ the partially
convex relaxation.
5.3.2 Classical Popov criterion
In general, stability criteria with frequency dependent multipliers are
superior if compared to static ones. Historically, the most important
dynamic stability test is the Popov criterion. Its incorporation into the
IQC framework was proposed by Jönsson [93] and relies on a customized
100 Chapter 5. Slope-restricted nonlinearities
version of the IQC result in [110] (see Section 2.2.2). It is one of the
major benefits of our formulation of the IQC stability Theorem 4.7
that the Popov criterion can be incorporated without any modifications.
Even if there is no known full-block version of the Popov multipliers for
slope-restricted nonlinearities in continuous time (see Chapter 6 for a
discrete-time result), we state the result for diagonally structured ones
for completeness. Translated into our setting, the Popov IQC on the
Sobolev space H 1,k reads as follows.
Theorem 5.11 (Popov IQC)
Let Λ be a diagonal k × k matrix1, i.e., Λ ∈ Dk, and define
ΠΛ(iω) :=
(
0 −iωΛ
iωΛ 0
)
for all ω ∈ R. (5.20)
Then there exists some δ ≥ 0 such that for all τ ∈ [0, 1] the following
IQC holds:
ΣΠΛ
(
z
τ∆ϕ(z)
)
≥ −δ‖z(0)‖2
for all z ∈H 1,k and all ∆ϕ ∈ sec[α, β]k. (5.21)
Proof. Let Iϕ(x) :=
∫ x
0
ϕ(s) ds. With η := max{|α| , β}, elementary
calculations show that
|Iϕ(x)| ≤ η
2
x2,
d
dx
Iϕ(x) = ϕ(x) and Iϕ(x) ≥ 0 for all x ∈ R.
(5.22)
Now let τ ∈ [0, 1] and z ∈H 1,k. Using Plancherel’s theorem, we arrive
at ∫ ∞
0
[τΦ(z(t))]
T
Λz˙(t) dt =
k∑
i=1
τΛii
∫ ∞
0
ϕ(zi(t))z˙i(t) dt. (5.23)
1We denote with Dk the diagonal matrices in Rk×k. Note that this is a slight
abuse of notation as D already denotes the unit disc in C. However, the meaning
will always be clear from the context.
5.3. Derivation and application of multipliers 101
With (5.22) and since limt→∞ zj(t) = 0 for all j ∈ {1, . . . , k} we obtain
k∑
i=1
τΛii
∫ ∞
0
ϕ(zi(t))z˙i(t) dt = −
k∑
i=1
τΛiiIϕ(zi(0))
≥ −
k∑
i=1
η
2
|Λii| ‖z(0)‖2∞,
where ‖ · ‖∞ denotes the maximum norm on Rk. Finally, since all norms
are equivalent on Rk, we obtain (5.21).
For the next theorem we require D = 0 in (5.5) in order to ensure
z = Mw ∈ H 1,k for all w ∈ L k2 ; together with the assumption
N : V →H 1,k this permits the use of the Popov IQC in order to prove
stability.
Theorem 5.12 (Popov criterion, strictly proper systems)
Assume that the interconnection (4.3), with strictly proper M and N :
V →H 1,k, is well-posed in the classical sense. Moreover, let Λ ∈ Dk.
If (5.10) holds with Π = ΠΛ, then there exist constants γ > 0 and γ0
such that,
‖z‖ ≤ γ(‖u‖+ ‖v‖V ) + γ0‖z(0)‖ for all (u, v) ∈ L k2 × V ,
and all ∆ϕ ∈ sec[α, β]k. (5.24)
Proof. We apply Theorem 4.7 based on the uncertainty IQC in Theorem
5.11 which only holds on H 1,k. This motivates the choice Ze =H 1,ke .
Since M is strictly proper we know that M : L k2e → H 1,ke and ∆ϕ :
H 1,ke → L k2e are bounded (see Section 5.2) and we can take Ue =
We = L k2e. From classical well-posedness and due to ML k2e ⊂ H 1,ke
and N(V ) ⊂ H 1,k, we can infer well-posedness of (4.3) in the sense
of Definition 4.6. If observing that l(z) =
√
δ‖z(0)‖ satisfies (4.4),
Theorem 5.11 implies the validity of (4.6). Again since M is strictly
102 Chapter 5. Slope-restricted nonlinearities
proper, Theorem 4.7 d) follows from (5.10), where we factorize Π such
that the FDI reads as(
iωM(iω)
I
)∗(
0 I
I 0
)(
iωM(iω)
I
)
4 −εI on C∞0 .
This obviously renders the resulting (passivity) multiplier bounded,
while maintaining properness of the outer factors.
It remains to verify that ΣΠΛ satisfies (4.5) despite the fact that
neither ΣΠΛ (as a quadratic form) nor ΠΛ (as a function on the imaginary
axis) are bounded. Indeed, for w, u ∈ L k2 and v ∈ V we have
ΣΠΛ
(
M(w + u) +N(v)
w
)
− ΣΠΛ
(
Mw
w
)
= ΣΠΛ
(
Mu+N(v)
w
)
.
(5.25)
With f = N(v), the response z = Mu+N(v) = Mu+ f satisfiesx˙z
z˙
 =
 A BC 0
CA CB
(x
u
)
+
0f
f˙
 , x(0) = 0. (5.26)
With γ1 := ‖(sI − A)−1B‖∞ and ‖f˙‖ ≤ ‖N(v)‖H ≤ γN‖v‖V , we
obtain
‖z˙‖ ≤ ‖CA‖‖x‖+ ‖CB‖‖u‖+ ‖f˙‖
≤ (γ1‖CA‖+ ‖CB‖) ‖u‖+ γN‖v‖V .
Hence, with z as above, (5.25) can be bounded as
ΣΠΛ
(
z
w
)
= 2
∫ ∞
0
w(t)TΛz˙(t) dt ≤
≤ 2‖Λ‖‖w‖[(γ1‖CA‖+ ‖CB‖)‖u‖+ γN‖v‖V ]
and the right-hand side is a quadratic form in ‖w‖, ‖u‖, ‖v‖V not
depending on ‖w‖2.
5.3. Derivation and application of multipliers 103
Remark 5.13.
In case of a nonzero initial condition x0 as in (2.22), we set V :=
{eA•x0} ⊂H 1,k and N = C. In analogy to Theorem 2.10, the conclu-
sion of Theorem 5.12 can then be formulated as follows: There exist
γ > 0 and γ0 such that ‖z‖ ≤ γ‖u‖ + γ0‖x0‖ for all u ∈ L k2 and all
∆ϕ ∈ sec[α, β]k. ?
It is important to note that (5.10) with Π replaced by ΠΛ implies
(CB)TΛ+ΛCB ≺ 0 for ω →∞; hence the Popov criterion as formulated
cannot be applied directly if, e. g, CB is singular. For this reason, Popov
multipliers should always be used in combination with those for the
circle criterion, as in the following example which compares different
classes of circle with Popov multipliers.
Example 5.14.
In order to compare diagonally repeated circle multipliers to unstruc-
tured ones and those for the Popov criterion, we consider interconnection
(4.3) with M as in (5.5) and
A =
−4 −3 02 0 0
−1 −1 −2
 , B =
 0 4 1 32 0 3 1
1 0 3 1
 , C =

−0.1 −0.2 1
−1 −0.3 0.1
−0.2 0.1 1
0.1 −0.2 0.2
 ,
as well as D = 0. For ∆ϕ ∈ sec[0, β]4, the goal is to estimate the maxi-
mal value of β ≥ 0 such that the interconnection (4.3) for V = {0} is
stable. In anticipation of the computational procedure in Section 5.3.4,
we actually determine for various fixed values of β ≥ 0 the infimal value
of γ > 0 for which the stability conditions of Theorems 5.7/5.12 can
be assured. The dotted curve in Figure 5.1 plots the results for the
diagonally structured circle criterion (5.11), whereas for obtaining the
dashed and solid ones we employed multipliers for the full-block circle
criterion (5.12) and their combination with Popov multipliers (5.20),
respectively. Already the unstructured circle multipliers lead to consid-
erable improvements, since the computed gains only diverge for much
104 Chapter 5. Slope-restricted nonlinearities
larger values of β if compared to the dotted curve. The combination
with Popov multipliers even allows for further increased values of β
until stability is lost.
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6


0
10
20
30
40
50

drCC
fbCC
fbCC+P
Figure 5.1: Comparison of L2-gain estimates for different approaches
?
5.3.3 Full-block Zames-Falb criterion
In the literature on slope-restricted nonlinearities, the Zames-Falb stabil-
ity test is often labeled as the least conservative of all available criteria.
However, from a computational point of view, it is also the most expen-
sive one since it relies on the approximation of an infinite dimensional
space of multipliers. After stating the Zames-Falb stability criterion, we
give a new approximation family of full-block Zames-Falb multipliers
and prove its asymptotic exactness.
Full-block Zames-Falb multipliers
The following Theorem is a combination of the main results in [42]
and [103] that completely describes the class of full-block Zames-Falb
multipliers.
5.3. Derivation and application of multipliers 105
Theorem 5.15 (Full-block Zames-Falb IQCs)
Let H ∈ L1(−∞,∞)k×k and G ∈ Rk×k satisfy
Gii >
k∑
j=1
j 6=i
|Gij |+
k∑
j=1
‖Hij‖1 for all i = 1, . . . , k. (5.27)
and
Gii >
k∑
j=1
j 6=i
|Gji|+
k∑
j=1
‖Hji‖1 for all i = 1, . . . , k. (5.28)
Assume either that ϕ is odd or that Gij ≤ 0 for i 6= j and H(t) ≥ 0 for
almost all t ∈ R. Then the following IQCs hold:
a) ΣΠZF,∞(z,∆ϕ(z)) ≥ 0 for all z ∈ L k2 and ∆ϕ ∈ slope(0,∞)k ∩
sec[0, β]k with
ΠZF,∞(iω) :=
(
0 GT − Hˆ(iω)∗
G− Hˆ(iω) 0
)
. (5.29)
b) ΣΠZF (z,∆ϕ(z)) ≥ 0 for all z ∈ L k2 and ∆ϕ ∈ slope(µ1, µ2)k if
ΠZF (iω) :=
(
?
)T ( 0 GT − Hˆ(iω)∗
G− Hˆ(iω) 0
)(
µ2I −I
−µ1I I
)
.
(5.30)
Remark 5.16.
Note that the multipliers in Theorem 5.15 a) and b) are related via
a loop transformation. It is one of the essential advantages of IQC
theory over classical multiplier theory that such loop transformations
can be incorporated into the multipliers and thus need not be carried
out explicitly. ?
In case of ∆ϕ ∈ slope(0, µ)k, both IQCs in Theorem 5.15 hold
simultaneously with µ1 = 0 and µ2 = µ. However, for stability analysis
106 Chapter 5. Slope-restricted nonlinearities
based on verifying the FDI (5.10), let us show next that (5.30) defines
the stronger class in the following sense. For this reason we continue to
work with (5.30) only.
Lemma 5.17
Let ∆ϕ ∈ slope(0, µ)k for µ > 0. Suppose (5.10) holds for Π = ΠZF,∞+
ΠZF with (5.29), (5.30) defined through (G,H), (G1, H1). Then Π in
(5.30) satisfies (5.10) for G2 := 1µG+G1 and H2 :=
1
µH +H1.
Proof. With Z := G − H and Z1 = G1 − H1, the FDI (5.10) for
Π = ΠZF,∞ + ΠZF reads as
He[(Z + µZ1)M ]−He[Z1] 4 −εI on iR. (5.31)
Now note that |Hij(iω)| ≤ ‖Hij‖1 for all i, j ∈ {1, . . . , k} and all ω ∈ R.
Using Geršgorin’s theorem ([81, Theorem 6.1.1]), the constraints (5.27),
(5.28) hence guarantee that the eigenvalues of He[Z(iω)] are contained
in the open right half complex plane, i. e., He[Z(iω)]  0 for all ω ∈ R.
We infer
He[µ(
1
µ
Z + Z1)M ]−He[ 1
µ
Z + Z1] 4 −εI
and hence
He[µZ2M ]−He[Z2] 4 −εI on iR
for Z2 := 1µZ +Z1, which is the FDI (5.10) for (5.30) with (G2, H2). It
remains to observe that (G2, H2) also obeys (5.27), (5.28).
Parametrization
In order to use Zames-Falb multipliers for computational stability anal-
ysis, we need to optimize over functions H ∈ L1(−∞,∞)k×k. In [42]
this is done in a non-systematic way by fixing a small number of basis
functions and optimizing in the resulting spanned subspace. Obviously,
more freedom in the choice of H, i. e., an increase in the dimension of
the considered subspaces, leads, in general, to improved results but also
causes higher computational complexity. In the sequel, we present an
5.3. Derivation and application of multipliers 107
approach that allows to balance both by extending the ideas for the
scalar case [37, 176] which rely on a family of dense subspaces of L1.
Fix a real pole ρ < 0. With ν ∈ N define
Aν :=

ρ 0 . . . . . . 0
1
. . . . . . . . .
...
0
. . . . . . . . .
...
...
. . . . . . . . . 0
0 . . . 0 1 ρ

∈ Rν×ν , Bν :=

1
0
...
...
0

∈ Rν×1,
and
Qν(t) := e
AνtBν = e
ρtϕν(t),
where
ϕν(t) = diag(0!, 1!, . . . , (ν − 1)!) col(1, t, . . . , tν−1).
This choice is motivated by the well-known fact that eρtp(t) with some
polynomial p can approximate functions in L1 and L2 arbitrary closely
(see, e. g., [155]). With coefficient matrices C1, C2, C3, C4 ∈ Rk×kν
and the identity matrix Ik ∈ Rk×k we now select the function Hν ∈
L1(−∞,∞)k×k as
Hν(t) = Hν,1(t)−Hν,2(t) = (C1 − C2)(ϕν(−t)⊗ Ik)e−ρt for t < 0,
Hν(t) = Hν,3(t)−Hν,4(t) = (C3 − C4)(ϕν(t)⊗ Ik)eρt for t ≥ 0;
(5.32)
if ϕ is odd, we impose the constraint Hν,l(t) > 0 for l = 1, . . . , 4 and all t
in the respective domains, which still defines a function Hν without sign-
constraint; if ϕ is not odd, we take C2 = C4 = 0 and Hν,1, Hν,3 to be
positive. Note that we use the same basis functions for all components
of Hν . With
ψν(iω) :=
(
1 1iω−ρ . . .
1
(iω−ρ)ν
)T
realized as ψν =
Aν Bν0 1
I 0
 ,
108 Chapter 5. Slope-restricted nonlinearities
the equations (5.32) give
Hˆν(iω) = (ψν(iω)⊗ Ik)∗
(
0 C1 − C2
)T
+
(
0 C3 − C4
)
(ψν(iω)⊗ Ik).
This leads to the multiplier
Πν(iω) =
(
?
)T ( 0 GT − Hˆν(iω)∗
G− Hˆν(iω) 0
)(
µ2I −I
−µ1I I
)
=
(
?
)∗
PZF
(
µ2I −I
−µ1I I
)(
ψν(iω)⊗ Ik 0
0 ψν(iω)⊗ Ik
)
, (5.33)
where
PZF =
(
0 PT12
P12 0
)
with P12 =
(
G C4 − C3
CT2 − CT1 0
)
. (5.34)
For ν = 0 we choose Hν(t) = 0 for all t ∈ R and define Π0 in the same
fashion with ψν = 1 and empty coefficient matrices C1, C2, C3, C4. All
this leads to the following computational stability test by combining
Theorems 5.15 and 4.7.
Theorem 5.18 (Full-block Zames-Falb criterion)
Let N : V → L k2 and suppose that the interconnection (4.3) is well-posed
in the classical sense for all ∆ϕ ∈ slope(µ1, µ2)k. With a pole location
ρ < 0 and an expansion length ν ∈ N0, the feedback interconnection
(4.3) is robustly stable for ∆ϕ ∈ slope(µ1, µ2)k if
a) Cl ∈ Rk×kν satisfy
Cl(ϕν(t)⊗ I) > 0 for all t ≥ 0 and l = 1, . . . , 4; (5.35)
b) for all i = 1, . . . , k the matrix G ∈ Rk×k satisfies
k∑
j=1
‖(Hν)ij‖1 +
k∑
j=1
j 6=i
|Gij | < Gii and
5.3. Derivation and application of multipliers 109
k∑
j=1
‖(Hν)ji‖1 +
k∑
j=1
j 6=i
|Gji| < Gii; (5.36)
c) either ϕ is odd or Gij ≤ 0 for i 6= j, (5.35) holds for l = 1, 3 and
C2 = C4 = 0.
d) the FDI (5.10) is valid with Π = Πν given in (5.33), (5.34).
Proof. Since Πν is bounded on the imaginary axis and no particular
signal regularity requirements are needed, the result is an immediate
consequence of Theorem 5.15 and Theorem 4.7 with all extended spaces
taken as L2e and l = 0.
Note that both (5.35) and (5.36) can be easily turned into standard
finite dimensional LMI constraints, along the same lines as for the scalar
case in [37, 176]. This allows for a straightforward implementation of
the Zames-Falb stability test, which is also applicable to nonlinearities
ϕ that are not odd, in contrast to the one proposed in [163].
Remark 5.19.
For ν = 0 the Zames-Falb multiplier is static (and independent from
ρ). In [169, Lemma 3] a similar multiplier (with the unnecessary ad-
ditional constraint G = GT ) is, slightly misleadingly, introduced as a
less conservative substitute for a circle criterion multiplier in certain
cases. As a consequence of our exposition (see also [42]), this multiplier
neither requires a separate proof for its validity nor does it serve as a
replacement for circle criterion multipliers. Instead, applying them both
is computationally inexpensive and often beneficial, as demonstrated by
Example 5.20. ?
Example 5.20.
In order to highlight the advantages of Zames-Falb multipliers over
110 Chapter 5. Slope-restricted nonlinearities
circle and Popov ones, we consider interconnection (4.3) with M as in
(5.5) defined by
A =
−10 −2.5 −2.53 −1 0
0 2 0
 , B =
 1 1 21 0 1
1.5 0.5 0
 , C =
 1 1 −10 −1 1
−1 0 0
 ,
and D=0. Let ∆ϕ ∈ slope(0, 1) with ϕ being odd and U = L2 as
well as V = {0}. Table 5.1 shows bounds on the L2-gain from u to z
computed with different criteria. As can be seen, neither the full-block
circle criterion (fbCC) nor its combination with Popov (fbCC+P) can
guarantee stability of the interconnection. However, a combination
Table 5.1: L2-gain estimates
fbCC fbCC+P ν = 0 ν = 1 ν = 3 ν = 5
∞ ∞ 25.0 11.25 7.98 7.55
of multipliers for the circle criterion even with only static (full-block)
Zames-Falb multipliers (ν = 0) allows to verify stability. If increasing
the order of the Zames-Falb multipliers with pole ρ = −1 up to ν = 5,
we obtain improved bounds for the L2-gain that approach the open
loop gain γol = 7.52 quite fast. ?
Example 5.21.
As made precise in [24, 25], Popov multipliers may be thought of as
Zames-Falb multipliers of order one with a pole at infinity (Note that this
has been pointed out on numerous occasions; see, e.g., [140, 195, 181]).
In order to illustrate this effect, we choose an example that is particularly
5.3. Derivation and application of multipliers 111
Table 5.2: Maximal values of β
ρ −1 −10 −102 −103 −104 fbCC+P
βmax 22 40 160 360 407 434
well suited for the analysis with Popov multipliers (see Example 3 in
[123]). Here M is described with A = −diag(1, 4, 6, 2, 9, 8, 3, 10, 12),
BT = −
1 0 0 0 1 0 0 1 00 1 0 1 0 0 0 0 1
0 0 1 0 0 1 1 0 0
 , C =
1 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 0
0 0 0 0 0 0 1 1 1
 , D = 0.
For ∆ϕ ∈ sec[0, β]3 ∩ slope(0, 2β)3 with ϕ being odd, the goal is to
estimate the maximal value of β ≥ 0 such that the interconnection (4.3)
for V = {0} is stable. We analyze this interconnection by combining a
full-block Zames-Falb multiplier for decreasing values of ρ and ν = 1
with a full-block multiplier for the circle criterion which leads to the first
five stability margins in Table 5.2. Indeed, these margins improve for
larger negative values of ρ, i. e., for better approximations of the Popov
multiplier. However, even for ρ = −10000 we are still more conservative
than the margin obtained by combining full-block Circle and Popov
criteria (last value in Table 5.2) which is all the more astonishing as
both only exploit the sector constraint. Yet, it is important to note that
this example works with tighter sector bounds than slope restrictions,
which tips the scale towards the circle and Popov criteria. ?
The parametrization of Zames-Falb multipliers in [37], which is also
at the heart of ours, is often criticized for being difficult to implement
and computationally expensive (see, e. g., [123, 28, 29]). Far from
that and in view of the possibility to combine them for different poles
(see Section 5.3.4), this class allows for a flexible trade-off between
computational load and accuracy, which is all the more important for
larger values of k. Moreover, in contrast to what is often claimed, it
112 Chapter 5. Slope-restricted nonlinearities
does not require a line search over the pole location as we will clarify in
the next section.
Asymptotic exactness of the parametrization
As one of the distinguishing features of the above proposed parametriza-
tion, if compared for example to the one in [163] (extended to the
full-block case in [169]), even for a fixed choice of ρ < 0 it can be proven
to be asymptotically exact.
Theorem 5.22
Fix ρ < 0 and a stable transfer matrix M of dimension k × k. Then
there exist H ∈ L1(−∞,∞)k×k and G ∈ Rk×k that satisfy the FDI
(5.10) with ΠZF as in (5.27), (5.30) if and only if there exists ν ∈ N0
and C1, C2, C3, C4 ∈ Rk×kν such that (5.35), (5.36) hold and (5.10) is
satisfied with Πν in (5.33), (5.34).
This result holds (with a simpler proof) in the same fashion if
including the extra constraints that are required in case the nonlinearity
ϕ is not odd. The following technical fact (Lemma A.1 in [175], due
to Jonathan R. Partington) provides the foundation for our proof of
Theorem 5.22.
Lemma 5.23
Let ρ < 0 and h ∈ L1[0,∞) be nonnegative. Then for all ε > 0 there
exists a real polynomial p such that q(t) = eρtp(t) satisfies ‖h− q‖1 < ε
and p(t) > 0 for all t ≥ 0.
Proof of Theorem 5.22. Since Πν is a Zames-Falb multiplier, one
implication is trivial. Now assume that (5.10) holds with ΠZF . Since
the inequalities (5.36) are strict, there exists some δ > 0 such that (5.10)
persists to hold for all K ∈ L1(−∞,∞)k×k with
‖Hˆij − Kˆij‖∞ ≤ δ for all i, j ∈ {1, . . . , k}. (5.37)
We split Hij into two nonnegative functions in L1(−∞,∞) as
H+,ij(t) := max{Hij(t), 0} and H−,ij(t) := −min{Hij(t), 0}.
5.3. Derivation and application of multipliers 113
Let us further choose ε ∈ (0, δ/4) with
Gii >
k∑
j=1
j 6=i
|Gij |+
k∑
j=1
‖Hij‖1 + 4kε for all i = 1, . . . , k;
Gii >
k∑
j=1
j 6=i
|Gji|+
k∑
j=1
‖Hji‖1 + 4kε for all i = 1, . . . , k. (5.38)
Since H+,ij , H−,ij are nonnegative, by Lemma 5.23, there exists some
ν ∈ N and coefficient vectors cl,ij ∈ R1×ν for l = 1, . . . , 4 with (5.35)
and such that the L1-norms of
H+,ij(t)− c1,ijQν(−t), H−,ij(t)− c2,ijQν(−t) for t ∈ (−∞, 0),
H+,ij(t)− c3,ijQν(t), H−,ij(t)− c4,ijQν(t) for t ∈ [0,∞)
are smaller than ε for all i, j = 1, . . . , k. In view of (5.32) we now define
the components of Hν as
Hij,ν(t) := (c1,ij − c2,ij)Qν(−t) for t < 0
and
Hij,ν(t) := (c3,ij − c4,ij)Qν(t) for t ≥ 0.
The triangle inequality implies ‖Hij,ν −Hij‖1 < 4ε, and hence the L∞-
norm of Hˆij,ν−Hˆij is bounded by 4ε < δ for all i, j = 1, . . . , k. Therefore,
by our choice of δ in (5.37), the FDI (5.10) still holds for Hˆν . Finally,
we also have ‖Hij,ν‖1 ≤ ‖Hij,ν −Hij‖1 + ‖Hij,ν‖1 ≤ 4ε+ ‖Hij,ν‖1 for
all i, j = 1, . . . , k which implies by (5.38) that (5.36) is true as well.
5.3.4 Combination of multipliers in the frequency
domain
Let us now address in detail how to implement combinations of the
developed stability tests for ∆ϕ ∈ slope(µ1, µ2)k ∩ sec[α, β]k if ϕ is odd
and under the well-posedness assumption that
114 Chapter 5. Slope-restricted nonlinearities
I −M∆ϕ : L k2e → L k2e has a causal inverse
for all ∆ϕ ∈ slope(µ1, µ2)k ∩ sec[α, β]k. (5.39)
As seen in Example 5.21 (see also [173]), for small lengths ν the pole
location ρ influences the stability guarantees achieved with the Zames-
Falb multiplier Πν in (5.33). This motivates to choose several pole
locations 0 > ρ1 > . . . > ρL with lengths ν1, . . . , νL ∈ N0 and to merge
the corresponding multipliers as follows. Each individual one reads as
Πνl,ρl = Ψ
∗
νl,ρl
PZF,lΨνl,ρl
where
Ψνl,ρl =
(
µ2I −I
−µ1I I
)(
ψνl,ρl ⊗ Ik 0
0 ψνl,ρl ⊗ Ik
)
and
PZF,l as in (5.34)− (5.36)
are defined for νl and ρl. With PCC ∈ Πpc[α, β]k, the sum PCC +∑L
l=1 Πνl,ρl is a valid IQC multiplier for ∆ϕ due to Lemma 5.6 and
Theorem 5.15. With P := diag(PCC , PZF,1, . . . , PZF,L), and Ψ :=
col(I2k,Ψν1,ρ1 , . . . ,ΨνL,ρL), this sum is described as
Π(P ) := PCC +
L∑
l=1
Πνl,ρl = Ψ
∗P Ψ (5.40)
with a fixed stable dynamic part Ψ and a real symmetric matrix
P varying in some set P; in view of (5.15) and as emphasized for
PZF,1, . . . , PZF,L satisfying the constraints (5.34)-(5.36) in Section 5.3.3,
the set P is a convex cone with an LMI description. If (A,B, C,D) is a
minimal realization of Ψ col(M, Ik) and by Lemma 2.12, (5.10) holds
for Π(P ) and some P ∈ P if and only if there exist X = XT and P ∈ P
such that the following LMI is satisfied: I 0A B
C D
T  0 X 0X 0 0
0 0 P
 I 0A B
C D
 ≺ 0. (5.41)
5.3. Derivation and application of multipliers 115
In this fashion, the robust stability test with multiplier class (5.40) for
P ∈ P boils down to solving a standard LMI feasibility problem.
If M(∞) = 0, we use the stable transfer matrix H(s) = sM(s) to
include the Popov multiplier ΠΛ from Theorem 5.11 as follows. We
extend Ψ as ΨPop := diag(I2k,Ψ) and now take a minimal realiza-
tion (A,B, C,D) of ΨPop col(H, Ik,M, Ik). Testing robust stability then
amounts to checking the feasibility of (5.41) with P ∈ P replaced by
diag
((
0 Λ
Λ 0
)
, P
)
for Λ ∈ Dk and P ∈ P.
As explained in more detail, for example, in [176], let us finally
address how the performance setting illustrated in Section 4.4 specializes
to the computation of guaranteed bounds on the L2-gain of d→ e in the
uncertain interconnection in Figure 5.2 for stable transfer matrices M ,
N , N21, N22 (of compatible dimension) under the assumption (5.39).
M N
N21 N22
∆ϕ
de
wz
Figure 5.2: Performance setting
For γ > 0 suppose there exist some P ∈ P and ε > 0 such that the
following FDI is valid:
(
?
)∗
Π(P )
(
M M12
I 0
)
+
(
?
)∗(I 0
0 −γ2I
)(
N21 N22
0 I
)
4 −εI on iR.
(5.42)
Since the left upper block implies (5.10), our stability results imply that
(I −M∆ϕ)−1 maps L k2 into L k2 for all considered uncertainties ∆ϕ
116 Chapter 5. Slope-restricted nonlinearities
(Remark 5.8). It is then elementary to show that (5.42) guarantees the
following robust performance property:
sup
d∈L2\{0}
‖e‖
‖d‖ < γ for all ∆ϕ ∈ slope(µ1, µ2)
k ∩ sec[α, β]k.
With the KYP lemma, the FDI (5.42) is easily turned into an LMI, and
it is even possible to find the smallest possible value of γ for which the
FDI is satisfied with some P ∈ P and ε > 0. If M and N are strictly
proper, the inclusion of Popov multipliers proceeds as for stability.
It remains to note that (4.3) for V = {0} (Figure 4.1) can be
subsumed into the interconnection in Figure 5.2 with the choices e = z,
d = u and N = N21 = N22 = M . This is the way how all the optimal
L2-gain bounds of u→ z and for a chosen multiplier class have been
computed in the present chapter.
Finally, let us highlight the necessity to combine stability multipliers
for different criteria with the following example.
Example 5.24.
For single input single output (SISO) nonlinearities it has been shown
in [24, 25] that the inclusion of Popov multipliers cannot improve
the Zames-Falb stability test for a particular system class and under
the assumption [α, β] = [µ1, µ2]. In general, this is not true for a
computationally tractable finite dimensional approximation of Zames-
Falb multipliers or if [α, β] 6= [µ1, µ2], as [167] suggests. Let us illustrate
this effect by continuing with Example 5.14, for which we now impose the
slope restriction ∆ϕ ∈ slope(0, 4β)4 in addition to the sector constraint
∆ϕ ∈ sec[0, β]4. Naturally, this does not affect the stability margins
obtained by using circle and Popov multipliers as recalled in Table 5.3
(CCP). If combining full-block circle and Zames-Falb multipliers for
ρ = −1000, we can compute an increased margin for ν = 1 that cannot
be improved for larger values of ν (CCZF in Table 5.3). However, by
further adding a Popov multiplier (CCZFP), this improvement gets
much more pronounced even for small basis lengths. ?
5.4. General Popov and Yakubovich criteria 117
Table 5.3: Maximal values of β for different multipliers
CCP CCZF (ν = 1) CCZF (ν = 5) CCZFP (ν = 1)
βmax 2.66 3.30 3.30 9.48
5.4 General Popov and Yakubovich criteria
In the previous section we saw that the incorporation of a Popov
multiplier may reduce conservativeness significantly. However, the given
derivation is limited to strictly proper LTI systemsM due to the required
filtering property. Yakubovich introduced a stability test [188] that is
based on a relation between the derivative of both the input z and the
output w of the uncertainty ∆ϕ. Here again we fail in applying classical
IQC theory due to higher regularity properties needed for the involved
signals, and not even strict properness of M serves as a remedy. In the
present section we further exploit Theorem 4.7 in order to overcome
these problems and derive stability results in both cases.
The general idea of both Popov and Yakubovich stability tests is to
capture the operation of ∆ϕ by exploiting relations between the signals
z, w, z˙ and w˙. This can easily be realized using the quadratic forms
in (4.18) that, specialized to the present setting, read as
Σ(P,P0)
(
z
w
)
=
∫ ∞
0
(
?
)T
P

z(t)
w(t)
z˙(t)
w˙(t)
 dt+ w(0)TP0w(0) (5.43)
for (z, w) ∈H 1,k×H 1,k and with P ∈ S4k and P0 ∈ Sk. Note that we
rearranged the signals if compared to (4.18) which will permit a straight-
forward application of the already derived criteria. Theorem 4.7 d) then
requires to certify
∃ε > 0 : Σ(P,P0)
(
Mw
w
)
≤ −ε‖w‖2H for all w ∈H 1,k. (5.44)
118 Chapter 5. Slope-restricted nonlinearities
From Theorem 4.12, we immediately deduce the following corollary
that relates (5.44) to an LMI and an FDI. We use the notations and
definitions introduced in Section 4.5.3.
Corollary 5.25
Let P ∈ S4k. Then the following statements are equivalent:
a) There exists some P0 ∈ Sk such that (5.44) holds.
b) There exist X = XT with(
I 0
Ae Be
)T (
0 X
X 0
)(
I 0
Ae Be
)
+ TT∆PT∆ ≺ 0. (5.45)
c) There exists ε > 0 such that for all ω ∈ R \ {0}
(
?
)∗
P

1
iωM(iω)
1
iω I
M(iω)
I
 4 −εI and (?)T P

M(0)
I
0
0
 ≺ 0. (5.46)
5.4.1 Full-block Yakubovich criterion
Let us first turn to the Yakubovich stability criterion [46, 188] (see also
[123, 133, 134, 10]), which has originally been formulated for differen-
tiable nonlinearities ϕ : R→ R that satisfy the inequalities
0 ≤ ϕ(x)/x ≤ κ and − κ1 ≤ ϕ′(x) ≤ κ2 for all x ∈ R \ {0}
with κ1 ≥ 0 and κ2 ≥ κ. Since some practically relevant nonlinearities,
such as the saturation and the dead-zone function, are not differentiable,
we need the following lemma for their rigorous treatment within our
framework.
Lemma 5.26 (Yakubovich IQC)
Let ϕ ∈ slope(µ1, µ2) and define w(t) = ϕ(z(t)) for t ∈ [0,∞) and
5.4. General Popov and Yakubovich criteria 119
z ∈ H 1,1e . Then for almost every t ∈ [0,∞) the following inequality
holds:
(w˙(t)− µ1z˙(t))(µ2z˙(t)− w˙(t)) ≥ 0. (5.47)
Proof. See Appendix C.4.2.
Observe that (5.47) is just a sector constraint on the derivative of
both the input and the output of the nonlinearity ∆ϕ. Hence, for
∆ϕ ∈ slope(µ1, µ2)k we can employ PY ∈ Π[µ1, µ2]k (in complete
analogy to the full-block circle criterion) in order to arrive at the
following IQC:
∫ ∞
0
(
z˙(t)
w˙(t)
)T
PY
(
z˙(t)
w˙(t)
)
dt ≥ 0 for all z ∈H 1,k
where w = ∆ϕ(z). (5.48)
As a distinguishing feature of our stability theorem, this Yakubovich
IQC may be seamlessly included into our framework. We differentiate
between strictly proper or just proper LTI systems M . In the first case,
a specialization of the general quadratic form (5.43) to the one in (5.48)
immediately results in a novel full-block generalization of Yakubovich’s
stability theorem [188].
Theorem 5.27 (Full-block Yakubovich criterion, strictly proper sys-
tems)
Consider the interconnection (4.3) with strictly proper M and N : V →
H 1,k. If there exist PY ∈ Π[µ1, µ2]k and P0 ∈ Sk such that (5.44)
holds with P = diag(0, PY ), then there exits γ ≥ 0 and γ0 with
‖z‖H ≤ γ(‖u‖2 + ‖v‖V ) + γ0‖z(0)‖ for all (u, v) ∈ L k2 × V
and all ∆ϕ ∈ slope(µ1, µ2)k.
Like in the Popov criterion for D = 0, this is again a guarantee for
disturbances u in the full space L k2 , in contrast to what is often seen
120 Chapter 5. Slope-restricted nonlinearities
in the literature [46, 134]. Example 5.32 reveals the great benefit of
combining Yakubovich multipliers with those from Section 5.3.
Proof. Well-posedness follows as in Theorem 5.12. We apply The-
orem 4.7 for Ue = L k2e and We=Ze=H 1,ke . Recall that Σ(P,P0)
satisfies Theorem 4.7 b). Due to (5.44), also d) holds. For δ0 :=
‖P0‖max{|µ1|, |µ2|}2 we next note that
[∆ϕ(z)(0)]
TP0[∆ϕ(z)(0)] ≥ −‖Φ(z(0))‖2‖P0‖ ≥ −δ0‖z(0)‖2
which means
Σ(0,P0)(z,∆ϕ(z)) ≥ −δ0‖z(0)‖2 for all z ∈H 1,k (5.49)
and all ∆ϕ ∈ slope(µ1, µ2)k. If z ∈ MU + N(V ) ⊂ H 1,k and ∆ϕ ∈
slope(µ1, µ2)
k, (5.48) reads as
Σ(P,0)(z,∆ϕ(z)) ≥ 0
and together with (5.49) we get
Σ(P,P0)(z,∆ϕ(z)) ≥ −δ0‖z(0)‖2 for all z ∈MU +N(V ); (5.50)
which, with Remark 5.2, implies Theorem 4.7 c) with
l(z)2 = δ0‖z(0)‖2.
In case of M(∞) 6= 0, we only need to confine the disturbance set Ue
to H 1,ke . We emphasize that well-posedness is part of the conclusion in
the next result.
Theorem 5.28 (Full-block Yakubovich criterion, general case)
Let N : V → H 1,k. If there exist PY ∈ Π[µ1, µ2]k and P0 ∈ Sk such
that (5.44) holds with P = diag(0, PY ), then there exist γ ≥ 0 and γ0
with
‖z‖H ≤ γ(‖u‖H + ‖v‖V ) + γ0‖z(0)‖ for all (u, v) ∈H 1,k × V
and all ∆ϕ ∈ slope(µ1, µ2)k.
5.4. General Popov and Yakubovich criteria 121
Proof. We first show that (5.44) guarantees well-posedness in the
classical sense. Indeed, due to (5.46) for ω → ∞, (5.44) implies
(DI )
T
PY (DI ) ≺ 0. Further, since ( I∆ )T PY ( I∆ )  0 for all ∆ ∈
Θ([µ1, µ2], k), we infer that det(I −D∆) > 0 for all ∆ ∈ Θ([µ1, µ2], k);
then the claim follows from Lemma 5.5. The remaining proof proceeds
as for Theorem 5.27 but with Ue = We = Ze =H 1,ke .
5.4.2 Popov criterion for D 6= 0
Let us finally provide a stability result based on combining full-block
circle and Yakubovich with Popov multipliers for the interconnection
(4.3) with ∆ϕ ∈ slope(µ1, µ2)k ∩ sec[α, β]k, ϕ odd, and a general proper
M . We use (5.43) with P0 ∈ Sk and
P =

Q1 S1 0 0
ST1 R1 Λ 0
0 Λ Q2 S2
0 0 ST2 R2
 , (5.51)
where Λ ∈ Dk,(
Q1 S1
ST1 R1
)
∈ Π[α, β]k, and
(
Q2 S2
ST2 R2
)
∈ Π[µ1, µ2]k. (5.52)
Remark 5.29.
We further generalize the multiplier P in (5.51) to a completely unstruc-
tured version in the subsequent chapter. The results derived there, for
the discrete-time case, immediately carry over to the present setting. ?
Theorem 5.30 (Popov criterion, general case)
Let N : V → H 1,k. If there exist P with (5.51), (5.52) and P0 ∈ Sk
such that (5.44) holds, then (4.3) is robustly stable in the sense of
Theorem 5.28 for all ∆ϕ ∈ slope(µ1, µ2)k ∩ sec[α, β]k with ϕ being odd.
Proof. The proof proceeds as for Theorem 5.28. With the IQCs
from Lemma 5.6, Theorem 5.11 and (5.48) we have Σ(P,0)(z,∆ϕ(z)) ≥
122 Chapter 5. Slope-restricted nonlinearities
−δ‖z(0)‖2 for all considered uncertainties and z ∈H 1,k; together with
(5.49) we now get Σ(P,P0)(z,∆ϕ(z)) ≥ −(δ+ δ0)‖z(0)‖2 replacing (5.50);
the remainder stays unchanged.
To conclude this section let us link, for scalar nonlinearities ϕ ∈
slope(µ1, µ2) ∩ sec[0, β] (with β > 0) and strictly proper LTI systems,
Theorem 5.30 to the classical frequency-domain inequality as, e.g.,
stated in [46]; yet, with much more limiting regularity requirements.
We need to restrict the class of multipliers by substituting Π[0, β],
Π[µ1, µ2] through Πdr[0, β], Πdr[µ1, µ2] in (5.52), respectively. An
analogous reasoning as in the proof of Theorem 4.12 then reveals that
the existence of such a pair (P, P0) in Theorem 5.30 is equivalent to the
existence of positive scalars λ, κ and some real Λ with
(
?
)∗ [λ(0 β2β
2 −1
)
+ Λ
(
0 iω
−iω 0
)
+ κω2
(−µ1µ2 µ1+µ22
? −1
)](
M(iω)
1
)
≺ 0
(5.53)
for all ω ≥ 0 and(
M(∞)
1
)T [
κ
(−µ1µ2 µ1+µ22
? −1
)](
M(∞)
1
)
≺ 0. (5.54)
By homogeneity we can set λ = 1/β in (5.53), while (5.54) trivially
holds due to M(∞) = 0; then (5.54) indeed just boils down to the
inequality (6) in [46]. Note that this relation is also confirmed by the
numerical results in Example 5.31.
5.4.3 Combination of multipliers in the state space
Let us show how to incorporate Zames-Falb multipliers in Theorem 5.30
and how to render the resulting - even stronger - stability test compu-
tational on the basis of Corollary 5.25. The inclusion of Zames-Falb
multipliers is indeed possible since the corresponding IQCs persist to
hold on the subspace H 1,k of L k2 as well.
5.4. General Popov and Yakubovich criteria 123
Suppose Π(Pzf) = Ψ∗PzfΨ with Pzf ∈ P is a family of Zames-Falb
multipliers for various poles and lengths as described in Section 5.3.4
(without PCC). To guarantee stability it is required to verify
∃ε > 0 : ΣΠ(Pzf )
(
Mw
w
)
+ Σ(P,P0)
(
Mw
w
)
≤ −ε‖w‖2H
for all w ∈H 1,k (5.55)
with suitable Pzf ∈ P and P , P0 as in Theorem 5.30.
We proceed in the state-space by minimally realizing the stable
transfer matrix Ψ as (AΨ, BΨ, CΨ, DΨ). For w ∈ H 1,k and z = Mw
we infer that zΨ = Ψ col(z, w) is the output of
ξ˙ = AΨξ +BΨ,1z +BΨ,2w, ξ(0) = 0,
zΨ = CΨξ +DΨ,1z +DΨ,2w.
In view of (4.21), (4.23) all relevant trajectories in (5.55) hence satisfy
the differential equation
ξ˙
x˙
w˙
zΨ
z
w
z˙
w˙

=

Aψ BΨ,1C BΨ,1D +BΨ,2 0
0 A B 0
0 0 0 I
CΨ DΨ,1C DΨ,1D +DΨ,2 0
0 C D 0
0 0 I 0
0 CA CB D
0 0 0 I


ξ
x
w
w˙
 =:
 A BC1 D1
C2 D2


ξ
x
w
w˙

with initial conditions ξ(0) = 0, x(0) = 0 and w(0) specified by w ∈
H 1,k. This allows us to simply apply Corollary 5.25 in order to infer
that (5.55) holds if and only if there exists X = XT such that
I 0
A B
C1 D1
C2 D2

T 
0 X 0 0
X 0 0 0
0 0 Pzf 0
0 0 0 P


I 0
A B
C1 D1
C2 D2
 ≺ 0. (5.56)
124 Chapter 5. Slope-restricted nonlinearities
In this fashion we end up with an LMI stability test that incorporates
all multipliers of this chapter. The introduction of performance criteria
proceeds in complete analogy to Section 5.3.4 and with the help of
Theorem 4.12.
We illustrate the advantages with the next three examples. The first
one is based on a combination of two systems in [46] and demonstrates
the effectiveness of Yakubovich multipliers over Zames-Falb multipliers
in terms of computational effort. The second example is a slightly
modified version of one given in [46] that nicely exhibits the benefits
from the combination of all stability criteria. Finally, in the third one
we impose an H 1,k-gain performance criterion and compare the result
obtained by a combination of Zames-Falb and circle multipliers with
our comprehensive approach.
Example 5.31.
Let the 2 × 2 system M be defined with the strictly proper elements
M12(s) = M21(s) = (s+ 1)
−1,
M11(s) =
−s2 − 1
(s2 + δs+ 1)(s+ 10)
and
M22(s) :=
−40
(s+ δ)(s+ 1)(s2 + 0.8s+ 16)
(5.57)
for δ = 0.0001. We examine stability of the interconnection (4.3) with
∆ϕ ∈ slope(0, µ)2 and U = L 22 . By combining circle, Popov and
Yakubovich multipliers, we may ensure stability of (4.3) for up to
µ = 0.73. If replacing the Yakubovich by a Zames-Falb multiplier with
ρ = −1, we need to choose a length ν ≥ 7 to also conclude stability for
µ = 0.73. The computation time is considerably smaller for the first
approach since it only involves 89 decision variables, in contrast to at
least 898 for the second. ?
Example 5.32.
Consider (4.3) for ∆ϕ ∈ slope(0, µ) and M = M22 from (5.57). For
5.4. General Popov and Yakubovich criteria 125
δ = 0 it is shown in [46] that the combination of circle and Popov criteria
guarantees stability up to µ = 0.65, while the addition of a Yakubovich
multiplier results in a maximal value of µ = 1.43. Note that these
conclusions were drawn under the assumption that ϕ is differentiable
and that u, u˙ and u¨ are all contained in L2. If using Theorem 5.27
in conjunction with circle and Popov multipliers for V = {0}, we
obtain the very same results for all nonlinearities ∆ϕ ∈ slope(0, µ) and
all u ∈ L2. If adding Zames-Falb multipliers with (ν1, ν2) = (8, 1),
(ρ1, ρ2) = (−2,−0.5), the guaranteed margin increases to µ = 1.68. ?
Example 5.33.
For the interconnection in Figure 5.2 with ∆ϕ ∈ slope(0, 1)3 let N =
N21 = N22 = M be realized as
A =
−10 −2.5 −2.53 −1 0
0 2 0
 , B =
 1 1 21 0 1
1.5 0.5 0
 , C =
 1 1 −10 −1 1
−1 0 0
 ,
and D = 0.5I. Then the inverse of I − DΦ exists and is Lipschitz,
which guarantees well-posedness for all considered nonlinearities. For
our application, we choose W = Z = H 1,3 and, for simplicity of
the presentation, D =H 1,30 , that is, the subspace of H
1,3 containing
functions with zero initial condition. The goal is to bound the H 10 -gain
of d→ e, i. e., to estimate the smallest γ > 0 such that ‖e‖21 ≤ γ2‖d‖21
for all d ∈ D by simply choosing P = diag(I,−γ2I) for r = 1 in (4.20).
Table 5.4 shows the obtained estimates for these gain bounds if
only using a combination of Zames-Falb and circle criterion multipliers
(ΠCZF ) and if employing the combination (5.55). Clearly, the possibility
of including Popov and Yakubovich multipliers allows for a significant
improvement even for small values of ν, while the classical approach
using the circle and Zames-Falb multipliers does not even guarantee
stability. This is especially important since the number of decision
variables increases substantially with ν, which makes it desirable to keep
ν small; in our example, the computation of the bound for the multiplier
126 Chapter 5. Slope-restricted nonlinearities
Table 5.4: H 10 -gain bounds
ν 0 1 2 3 4
ΠCZF ∞ ∞ ∞ 22.98 22.77
Π in (5.40) 7.41 7.36 7.36 7.36 7.36
combination and ν = 0 involves 109 decision variables, whereas the
one for ΠCZF and ν = 4 requires 667 variables and still achieves worse
results. ?
5.5 Related stability tests
Let us discuss in this brief section some related stability tests in the
literature and highlight the connection between these and our results.
5.5.1 Multiplier proposed by Park
The stability test in [123] relies on a Lurye-Postnikov type Lya-
punov function and is an LMI representation of the circle, Popov and
Yakubovich stability criteria as proposed much earlier in [188, 46].
However, both [188] and [46] require additional constraints for their
Lyapunov based proof. In contrast to what seems to be implied by [123],
this approach only captures first order Zames-Falb multipliers as shown
in [24, 25]. Moreover, all proposed criteria are confined to diagonally
structured multipliers. Since our approach allows a search in the full
class of Zames-Falb multipliers in combination with Circle, Popov and
Yakubovich tests, it is at least as strong as (and often much stronger
than) Park’s method, as illustrated in the subsequent examples.
5.5. Related stability tests 127
5.5.2 Stability criterion by Hu et al.
The approach by Hu et al. (see, e. g., [51, 84] and [83] for an overview) is
specifically designed for the local stability analysis of saturated systems.
This means that it can incorporate bounds dmax on the disturbance
energy ‖d‖ (or its amplitude) in the performance setting as described in
Section 5.3.4, which typically improves the obtained results especially
for small values of dmax. One can show that, in the limit dmax → ∞,
the obtained results correspond to those achievable with the full-block
circle criterion as discussed in Section 5.3.1. For global stability and
performance analysis, this implies that our method typically outperforms
the one proposed by Hu et al., sometimes even substantially as illustrated
in Examples 5.34 and 5.35.
5.5.3 Zames-Falb implementation by Turner et al.
As mentioned before, there exist other ways of parameterizing Zames-
Falb multipliers (see, e. g., [163, 167] and [29] for an overview). These
methods for k = 1 offer the advantage that the multiplier poles do not
have to be chosen a priori which allows to optimize over their location as
well. However, this comes at the expense of various significant drawbacks.
As probably the most important one, the occurring L1-norm constraint
(5.27) is enforced by using LMIs that involve intrinsic conservatism and a
line-search over some parameter. In addition, the algorithm is based on a
common Lyapunov function for both the L1-constraint and the stability
LMI. Furthermore, the original approach is limited to multipliers of the
same degree as the system that are either causal or anti-causal and,
more severely, only apply to odd nonlinearities. In a series of papers (see,
e. g., the non-exhaustive list [26, 165, 163, 167, 27, 28, 164, 170]) various
authors tried to reduce several of the non-intrinsic drawbacks of this
method, for example, by combining the obtained Zames-Falb multipliers
with those corresponding to the Popov criterion or by allowing for higher
order approximations. Still, for k-fold repeated nonlinearities, numerical
tractability suffers since a non-convex search must be performed for
128 Chapter 5. Slope-restricted nonlinearities
a k-dimensional parameter [168, 166, 169]; also the use of a common
Lyapunov function for all L1-constraints and stability LMIs might cause
additional conservatism for increasing values of k. Our approach avoids
these troubles and is guaranteed to achieve no worse results, since it
freely combines multiple pole Zames-Falb multipliers (each based on
asymptotically exact parameterizations) with those from other criteria
for repeated nonlinearities.
5.6 Numerical examples
In this section we compare the results achieved within our framework
with some related stability test addressed in the previous section.
Example 5.34.
The canonical application for slope-restricted nonlinearities arises from
systems with saturations or dead-zones. Although a great number of
papers focus on so-called local stability and performance issues (see,
e. g., [157, 83] and references therein for an overview), global tests are
also discussed (see, e. g., [40], [85]). We adopt Example 1 from [85] and
compute an L2-gain estimate for the channel d→ e in Figure 5.2 with
∆ϕ ∈ slope(0, 1)2 for the unit saturation function ϕ = sat and
(
M N
N21 N22
)
=
 A B1 B2C1 D11 D12
C2 D21 D22
 =

0 0 −1 1 0 0 1
1 0 −2 0 1 1 0
0 1 −3 1 −1 1 1
1 0 1 −3 −1.3 1 −1
0 1 0 −2.3 −4 0 1
0 1 0 1 0 −1 0
0 0 1 0 1 0 −1

.
We compare the results achieved in [85] to those obtained by the full-
block circle criterion (fbCC), its combination with full-block Zames-Falb
(fbZF) as well as a diagonally repeated combination of Circle and Zames-
Falb (drZF, [37, 176]) criteria. Note that the techniques in [166] and
5.6. Numerical examples 129
[123] are not applicable because they require M to be strictly proper.
Since the approach by Hu et al. [85] can take a bound dmax on the
energy of d into account, we plot the computed L2-gain bounds γ
over dmax in Figure 5.3 (with numerical values in the first column of
Table 5.5). Figure 5.3 shows that [85] slightly outperforms all other
10
0
10
2
10
4
d
max
0
50
100
150


Hu et al.
fbCC
drZF
fbZF
Figure 5.3: L2-gains for Example 7.15
techniques for small values of dmax. However, the guaranteed L2-gains
quickly increase substantially above those levels as globally guaranteed
by the Zames-Falb based techniques and approach the ones for the
full-block circle criterion if dmax →∞ (see Section 5.5.2). ?
130 Chapter 5. Slope-restricted nonlinearities
Example 5.35.
In [40] two variants of the above example are considered where D#111 :=
D11 is exchanged with
D#211 =
(−3 −1
−2 −4
)
and D#311 =
(−3 −2
−2 −4
)
and multiple approaches for global L2-gain estimation using different
Lyapunov function based techniques are compared. The paper ultimately
proposes a non-convex search relying on piecewise quadratic Lyapunov
functions that leads to the best results for all three examples, with the
related values appearing in the first row of Table 5.5. Only recently,
these results were further refined in [171] where a convex relaxation of
the approach in [40] is proposed using sum of squares (SOS) methods
(see second row of Table 5.5). All Lyapunov function based techniques
Table 5.5: Global L2-gain estimates with different techniques
case #1 #2 #3
piecewise (pw.) quadratic [40] 17.19 15.13 25.86
pw. quadratic using SOS relaxation [171] 12.39 12.04 17.79
quadratic [85] 170.15 38.96 ∞
drZF (ν = 4, ρ = −1) 8.47 8.47 9.81
fbZF (ν = 2, ρ = −1) 6.92 7.09 9.43
fbZF (ν = 4, ρ = −1) 6.86 7.02 9.43
are, however, clearly outperformed by diagonally structured Zames-Falb
multipliers even for small lengths ν, with further improvements for
full-block versions. ?
Example 5.36.
The next example only features one nonlinearity and serves to disprove
5.6. Numerical examples 131
the claims in [123] and [163] that Park’s method [123] is less conservative
than the one by Chen and Wen [37].2 In addition, we also aim at
demonstrating the advantage of being able to employ both causal and
non-causal multipliers simultaneously. In order to do so, we adopt three
examples from [28] defined by the transfer functions
M1(s) =
s2
s4 + 0.2s3 + 6s2 + 0.1s+ 1
M2(s) = −M1(s)
M3(s) =
s2
s4 + 5.001s3 + 7.005s2 + 5.006s+ 6
.
As both [123] and [28] only consider stability and not performance, we
apply the same stability criterion as in [28]. We compute the maximal
value of µ for which (4.3) remains stable for all ∆ϕ ∈ slope(0, µ) where ϕ
is odd. The values in the first three columns of Table 5.6 are copied from
[28] and correspond to the approaches by Park and a combination of
anticausal (causal) Zames-Falb and Popov multipliers, denoted as ACP
(CP). The last column displays the maximal slope restrictions under
which we can guarantee stability using a combination of Circle, Popov
and two Zames-Falb multipliers (fbZFP) for lengths (ν1, ν2) = (6; 1)
Table 5.6: Maximal slope constraints obtained with different techniques
Transfer function Park [123] ACP [28] CP [167] fbZFP
M1 0.79 1.45 0.78 1.75
M2 0.71 0.72 1.08 1.21
M3 26.01 91.09 13.78 3510
and pole locations (ρ1, ρ2) = (−3;−10). Higher values of ν further
2In fact, as [37] proposes an asymptotically exact parameterization of SISO
Zames-Falb multipliers (for nonlinearities that are not odd) and Park emploies circle,
Popov and Yakubovich ones, a sensible comparison is impossible.
132 Chapter 5. Slope-restricted nonlinearities
improve the results at the expense of higher computational effort. Park’s
approach is always outperformed by Zames-Falb methods, either for
causal or anticausal multipliers. For the two systems M1 and M2,
only slight improvements are gained from our approach, while for M3
(designed to display the key features of [28]), the improvements are
rather significant. ?
Example 5.37.
Finally, we compare our results to those achieved by [166] for repeated
nonlinearities. To this end we revisit Example 5.31 and, in view of
the limitation of [166], we restrict our attention to odd nonlinearities.
Table 5.7 displays the maximal values of β achieved by the different
approaches. In our implementation of the method in [166] we chose
λ1 = λ2 = λ and performed a line search over λ ∈ [0,∞) with 1000
points which leads to a maximal tolerable value of β = 0.26. As the
approach by Turner et al. (cZFP) already includes a Popov multiplier,
we first compare it to the same combination of multipliers (CCZFP)
for ν = 8 and ρ = −1 and later, as in Example 5.31, to one where
the Zames-Falb multiplier is exchanged with a Yakubovich one. We
conclude that the approach by Turner et al. (cZFP) is significantly
more conservative, and the combination of all multipliers (in the last
two columns of Table 5.7) even allows us to guarantee stability up to
the Nyquist value of βN = 0.767.
Table 5.7: Maximal values of β for different approaches
cZFP [166] CCZFP(ν= 8) CCYP CCZFYP(2) CCZFYP(8)
0.26 0.73 0.73 0.75 0.766
?
5.7. Summary and recommendations 133
5.7 Summary and recommendations
This chapter presents a novel and flexibly applicable framework, which
is used to obtain the least conservative results available both inside and
outside the IQC framework for the global analysis of feedback systems
featuring repeated slope-restricted scalar nonlinearities. The extensions
relative to previous works are manifold.
For the circle criterion, we give new relations between different com-
putationally tractable subsets of the set of full-block multipliers, and
prove that these genuinely extend the classical diagonally structured
versions. In order to make the full potential of the Zames-Falb stability
test accessible for computations, we propose a new tractable family
of full-block Zames-Falb multipliers that is asymptotically exact, thus
solving an open problem postulated in [29]. In addition, we rigorously
include Yakubovich multipliers into IQC theory which makes it possible
to propose a new full-block version thereof. We fully exploit the gen-
erality of our analysis framework which permits a tractable modular
combination of the circle, Popov, Yakubovich and Zames-Falb tests
even for non-proper systems. Note in particular that the assumptions for
the Yakubovich criterion vary significantly in the literature [46, 188, 10],
while the ones presented here are the least restrictive.
Yet many problems still need to be addressed. Of primary concern is
the application of the developed tests for robust estimator or controller
design [175]. A particularly interesting application would be anti-windup
synthesis, which has received much attention in the literature, see
e.g., [87, 66, 158, 100]. Yet, this topic is closely linked to that of the
analysis of systems that are only locally stable, which is one of the
major assets of Lyapunov theory. We will present an approach that
allows for the effective merging of both IQC and Lyapunov theory in
Chapter 7 and focus, in particular, on the class of Zames-Falb multipliers
in Chapter 8.
Furthermore, as mentioned several times in this chapter, the applica-
tion of full-block Zames-Falb multipliers can be very expensive in terms
134 Chapter 5. Slope-restricted nonlinearities
of computational effort. One remedy could be the choice of more effec-
tive basis functions that, for optimal performance, should be adapted to
the problem at hand. This would pave the way for much more accurate
stability test even for large scale (real world) applications.
Chapter 6
Absolute stability analysis of discrete-time
feedback interconnections
6.1 Introduction
Let us revisit the feedback interconnection of an LTI system M anda nonlinear operator ∆ defined via a slope-restricted or sector-
bounded nonlinearity. Complementary to the previous chapter, we
now focus on the discrete-time case. Our main goal is to highlight the
fundamental principles behind the stability results and, in this way,
establish the connection to Chapter 5.
In contrast to the continuous-time case, we do not exploit a dis-
tinction between the inputs u and v affection the interconnection (4.3).
Thus it suffices to consider the simplified interconnection depicted in Fig-
ure 6.1, where the external disturbance d is square summable. In order
to provide additional insights if compare to the continuous-time deriva-
tion, we also discuss non-repeated nonlinearities in the present chapter,
that, as will be revealed, also admit full-block multiplier descriptions.
Stability analysis of such interconnections also has a long standing
history, probably starting with the works of Tsypkin [162] and Jury
and Lee [96]. Both approaches employed ideas developed by Popov
for continuous-time systems [126]. However, in contrast to the Popov
135
136 Chapter 6. Absolute stability of discrete-time interconnections
criterion that only requires the existence of some sector bound on the
nonlinearity, it became apparent that the discrete-time counterpart also
necessitated the assumption of monotonicity.
M
∆
+
d
z w
Figure 6.1: Feedback interconnection
Under the additional hypothesis that the derivative of the nonlin-
earity ϕ : R → R is bounded, O’Shea and Younis [121] proposed a
discrete-time counterpart to the celebrated Zames-Falb stability crite-
rion [197] that was later generalized by Willems and Brockett [184].
O’Shea and Younis already claim that their criterion is less restrictive
than the one proposed in [96], which was the most effective test at that
time.
Following these early results, many researchers have contributed
to this field of study and, in particular, extended the above described
stability tests to multi-variable nonlinearities (see, e.g, [122, 75, 103]).
Recently, there seems to be renewed interest in the subject with several
publications proposing seemingly different yet closely related criteria
(see, e.g., [5, 7, 2, 3, 4, 177, 3, 17]).
The wealth of earlier and more recent results makes it rather difficult
to judge which of the proposed stability tests is the most effective one,
in the sense that it leads to the least conservative estimates for stability
margins. In part, this is due to the fact that most proofs are based on
Lurye-type Lyapunov function arguments by adding extra degrees of
freedom whose generating principles often remain somewhat obscure.
Moreover, the resulting tests are formulated in terms of linear matrix
inequalities whose intricate structures prevent insightful comparisons.
6.1. Introduction 137
In case of just one scalar nonlinearity, the result obtained by [177],
which is based on Zames-Falb multipliers, is shown to subsume all
earlier stability tests and thus leads to the least conservative estimates
for stability margins (see also [184]). However, as will be revealed
in the present chapter, for multiple nonlinearities it is beneficial to
combine Zames-Falb multipliers with those corresponding to other
stability criteria.
As one of the contributions of this chapter, we highlight the fun-
damental principles underlying all above mentioned stability tests and
provide insights into their interrelation. On the one hand, this allows
us to show how even the most recent versions can actually be derived
from the ones proposed in [121], [184] and [103]. On the other hand,
we may seamlessly extend the classical results in order to arrive at
less conservative stability tests for multiple nonlinearities. Moreover,
it is then easy to reveal that (at least for scalar nonlinearities) both
discrete-time counterparts of the Popov and the Yakubovich [188, 46]
stability criteria are already included in the one based on Zames-Falb
multipliers. This should be contrasted with the situation in the previous
chapter, where both tests may only be approximately handled by using
Zames-Falb multipliers [140, 24, 25].
Apart from the work of [103], all the above discussed stability tests
employ diagonally structured multipliers even if considering repeated
multi-variable nonlinearities. As another contribution of this chapter we
demonstrate how unstructured full-block multipliers may be combined
with diagonal (full-block) Zames-Falb multipliers for non-repeated (re-
peated) nonlinearities in order to generate more powerful novel tests, as
shown by numerical examples.
The chapter is structured as follows. After setting the stage in
Section 6.2 we derive in Section 6.3 our full-block stability multipliers
and give their relation to previous ones in Section 6.4. Subsequently,
we discuss the implementation of these multipliers in Section 6.5 and
close with numerical examples in Section 6.6. We emphasize that a
138 Chapter 6. Absolute stability of discrete-time interconnections
condensed version of this chapter is accepted for presentation at the
IFAC World Congress 2017 [57].
6.2 Preliminaries
Assume that we are given k ∈ N nonlinearities ϕ1, . . . , ϕk that are
sector-bounded or slope-restricted according to Definition 5.1. With
such nonlinearities, let Φ : Rk → Rk be given as Φ(x1, . . . , xk) =(
ϕ1(x1) . . . ϕk(xk)
)T
and let the operator ∆Φ be defined as
∆Φ(z)(t) := Φ(z(t)) for all t ∈ N0, z ∈ `k2e. (6.1)
In contrast to (5.3) we now allow for different nonlinearities with varying
sector and slope constraints. In order to take this into account, we write
∆Φ ∈ slope(µ, ν) or ∆Φ ∈ sec[α, β] (as well as Φ ∈ slope(µ, ν) or Φ ∈
sec[α, β]) if ϕj ∈ slope(µj , νj) or ϕj ∈ sec[αj , βj ] for all j ∈ {1, . . . , k}
and with µ = diag(µj), ν = diag(νj), α = diag(αj), β = diag(βj),
respectively. In a natural extension of this notation, we write [α, β] for
the following set of diagonal matrices:
[α, β] :=
{
∆ = diag(δ1, . . . , δk) ∈ Dk
∣∣αi ≤ δi ≤ βi
for all i ∈ {1, . . . , n}}.
For the special case when all nonlinearities coincide, i.e., ϕj = ϕ for
all j, we say that Φ is a repeated nonlinearity and indicate this by
∆Φ ∈ slope(µI, νI) or ∆Φ ∈ sec[αI, βI] for the operator.
Given such a nonlinearity ∆Φ, we consider its feedback interconnec-
tion (see Figure 6.1) with a stable LTI system M described through a
state-space realization as follows:
x(t+ 1) = Ax(t) +Bw(t), x(0) = 0, w = ∆Φ(z),
z(t) = Cx(t) +Dw(t) + d(t), (6.2)
for t ∈ N0. Here we assume that A ∈ Rn×n is Schur stable, i.e.,
eig(A) ⊂ D, and that the external disturbance d is square summable,
i.e., d ∈ `k2 . We denote the standard norm on `k2 by ‖ · ‖.
6.2. Preliminaries 139
In complete analogy to the previous chapter, we define well-posedness
and stability as follows.
Definition 6.1.
The interconnection (6.2) is said to be well-posed if for each d ∈ `k2
and each τ ∈ [0, 1] there exists a unique response z ∈ `k2e of (6.2) with
∆Φ replaced by τ∆Φ which depends causally on d. Moreover, (6.2) is
stable if there exists some γ > 0 such that
‖z‖ ≤ γ‖d‖ for all d ∈ `k2 . (6.3)
?
In discrete time, the quadratic form (5.8) translates into
ΣΠ
(
z
w
)
=
∫ 2pi
0
(
zˆ(eiω)
wˆ(eiω)
)∗
Π(eiω)
(
zˆ(eiω)
wˆ(eiω)
)
dω,
where Π is measurable, bounded and Hermitian valued on T and zˆ as
well as wˆ denote the z-transforms of the `k2 signals z and w, respectively.
As before, a causal operator ∆ : `k2 → `k2 satisfies the IQC imposed by
Π in case that
ΣΠ
(
z
∆(z)
)
≥ 0 for all z ∈ `k2 . (6.4)
We end this section by stating a particular version of Theorem 3.4
adapted to our special configuration (see also [97]).
Theorem 6.2
Assume that the interconnection (6.2) with ∆Φ as in (6.1) is well-posed.
Then (6.2) is stable if
a) τ∆Φ satisfies the IQC defined by Π for all τ ∈ [0, 1];
b) the following FDI holds:(
M(eiω)
I
)∗
Π(e iω)
(
M(eiω)
I
)
≺ 0 for all ω ∈ [0, 2pi]. (6.5)
140 Chapter 6. Absolute stability of discrete-time interconnections
The verification of well-posedness for ∆Φ ∈ sec[α, β] or ∆Φ ∈
sec(µ, ν) in Section 5.2.3 literally carries over to discrete-time inter-
connections; in the sequel we tacitly assume that (6.2) is well-posed.
6.3 Principles of stability multipliers
Complementarily to Chapter 5, we divide this section according to the
generating principles of the derived stability criteria, which allows for
a straightforward categorization of all multiplier based results in the
literature. As will become apparent, the stability multipliers employed
in the papers cited in the introduction either rely on a subgradient
argument or on polytopic bounding for the creation of inequalities.
Another focus of this section is the formulation of stability test using
full-block multipliers even if the nonlinearities are not repeated. Aiming
at a self-contained exposition of stability criteria we briefly summarize
the results from the previous chapter if necessary.
6.3.1 Methods based on polytopic bounding
Let ∆Φ ∈ sec[α, β]. Conceptually, the circle criterion exploits the simple
fact that w(t) = ∆Φ(z)(t) = Φ(z(t)) for z ∈ `k2 can be expressed, due to
(5.2), as
w(t) = ∆(t)z(t) for all t ∈ N0 (6.6)
with ∆(t) ∈ [α, β]; indeed we can take ∆(t) = diag(δj(t)) and δj(t) =
ϕj(zj(t))/zj(t) if zj(t) 6= 0 or δj(t) = 0 if zj(t) = 0 for all j ∈ {1, . . . , k}.
If we now choose any element Π in the following class of full-block
multipliers
Π[α, β] =
{
Π ∈ S2k
∣∣∣∣∣
(
I
∆
)T
Π
(
I
∆
)
 0 for all ∆ ∈ [α, β]
}
(6.7)
6.3. Principles of stability multipliers 141
(which seamlessly extends the definition in (5.12)), we obviously infer(
z(t)
w(t)
)T
Π
(
z(t)
w(t)
)
= z(t)T
(
I
∆(t)
)T
Π
(
I
∆(t)
)
z(t) ≥ 0
for all t ∈ N0. By summation we conclude that ∆Φ satisfies the IQC
ΣΠ
(
z
∆Φ(z)
)
≥ 0 for all z ∈ `k2 . (6.8)
Since the class Π[α, β] was originally defined to handle time-varying
parametric uncertainties in polytopes [89, 147], these multipliers are
said to be generated by polytopic bounding.
It is now straightforward to derive a full-block circle and a full-block
Yakubovich stability criterion in our setting.
Circle criterion
Let ∆Φ ∈ sec[α, β]. Since (6.8) holds for all Π ∈ Π[α, β], Theorem 6.2
implies the following result.
Corollary 6.3 (Circle criterion)
The interconnection (6.2) with ∆Φ ∈ sec[α, β] is stable if there exists
some Π ∈ Π[α, β] with1(
M(z)
I
)∗
Π
(
M(z)
I
)
≺ 0 for all z ∈ T. (6.9)
Note that Corollary 6.3 is actually a generalization of Lemma 5.6
in the discrete-time setting, as it allows for non-repeated nonlinearities
while still employing full-block multipliers. As emphasized above, all
discrete-time circle criteria for stability in the literature restrict the
search of Π to the subclass Πdr[α, β] ⊂ Π[α, β] of diagonally structured
multipliers. Therefore, full-block multipliers will not be worse than
1As above, we use the symbol z to distinguish the frequency domain variable z
form the signal z.
142 Chapter 6. Absolute stability of discrete-time interconnections
the conventional ones (see Lemma 5.10), and it can be concluded from
numerical examples that they typically reduce conservatism significantly
(see Example 5.14).
Yakubovich criterion
Let us now turn to the discrete-time analogue of the Yakubovich criterion
for ∆Φ ∈ slope(µ, ν). In order to demonstrate the underlying ideas, we
first assume that the nonlinearities ϕj are continuously differentiable for
j ∈ {1, . . . , k}. The general case is merely more technical but proceeds
in similar fashion.
Choose z ∈ `k2 and let w := ∆Φ(z). By the mean value theorem
there exist ξtj (depending on t ∈ N0) such that
wj(t+ 1)− wj(t) = ϕj(zj(t+ 1))− ϕj(zj(t))
= ϕ′j(ξ
t
j)(zj(t+ 1)− zj(t)).
Since the slope restriction Φ ∈ slope(µ, ν) translates into
µj ≤ ϕ′j(ξ) ≤ νj for all j ∈ {1, . . . , k} and all ξ ∈ R,
we infer, in complete analogy to the circle criterion, that there exist
∆(t) ∈ [µ, ν] with
w(t+ 1)− w(t) = ∆(t)(z(t+ 1)− z(t)) (6.10)
for all t ∈ N0. Thus, for ΠY ∈ Π[µ, ν] we obtain(
z(t+ 1)− z(t)
w(t+ 1)− w(t)
)T
ΠY
(
z(t+ 1)− z(t)
w(t+ 1)− w(t)
)
≥ 0 (6.11)
for all t ∈ N0. As the time shift in the outer factors of (6.11) gives rise
to a multiplication with z−1 in the frequency domain, we obtain, by
summation and with Parseval’s theorem, the IQC ΣΠ(z, w) ≥ 0 for the
dynamic (z-dependent) multiplier
Π(z) :=
(
(z−1)I 0
0 (z−1)I
)∗
ΠY
(
(z−1)I 0
0 (z−1)I
)
= | z−1|2ΠY .
(6.12)
6.3. Principles of stability multipliers 143
We deal with the general case, where the functions ϕj are only
differentiable almost everywhere, in the following Lemma.
Lemma 6.4
With the definitions above, let Φ ∈ slope(µ, ν). Then there exist ∆(t) ∈
[µ, ν] with (6.10) for all t ∈ N0.
Proof. The proof employs Clarke’s generalized derivatives and the mean
value theorem of Lebourg (see [39]) and is found in Appendix C.5.1.
In summary, with Theorem 6.2, we arrive at the following result.
Corollary 6.5 (Yakubovich criterion)
The interconnection (6.2) with ∆Φ ∈ slope(µ, ν) is stable if there exists
some ΠY ∈ Π[µ, ν] with(
(z−1)M(z)
(z−1)I
)∗
ΠY
(
(z−1)M(z)
(z−1)I
)
≺ 0 for all z ∈ T.
Combined polytopic criterion
Let us now discuss how we may combine the circle and Yakubovich
multipliers and embed the combination into a more general class of com-
pletely unstructured multipliers. The following discussion immediately
translates to the continuous-time case, also resulting in a larger class of
suitable multipliers.
Assume that ∆Φ ∈ sec[α, β]∩slope(µ, ν). Of course, the most simple
way of exploiting both constraints simultaneously is to just add up the
according multipliers. The corresponding FDI then reads as
M(z)
I
(z−1)M(z)
(z−1)I

∗(
ΠC 0
0 ΠY
)
M(z)
I
(z−1)M(z)
(z−1)I
 ≺ 0 for all z ∈ T
(6.13)
144 Chapter 6. Absolute stability of discrete-time interconnections
with some ΠC ∈ Π[α, β] and ΠY ∈ Π[µ, ν]. Yet, this obviously results
in a potentially conservative block diagonal structure. Based on the
above described generating principle, the generalization to unstructured
multipliers is simple. Indeed, for w = ∆Φ(z) and z ∈ `k2 we have(
w(t)
w(t+ 1)− w(t)
)
=
(
∆C(t) 0
0 ∆Y (t)
)(
z(t)
z(t+ 1)− z(t)
)
with suitable ∆C(t) ∈ [α, β], ∆Y (t) ∈ [µ, ν] and for all t ∈ N0. Since
diag(∆C(t),∆Y (t)) ∈ [diag(α, µ),diag(β, ν)],
we infer for any ΠCY ∈ Π[diag(α, µ),diag(β, ν)] that
z(t)
z(t+ 1)− z(t)
w(t)
w(t+ 1)− w(t)

T
ΠCY

z(t)
z(t+ 1)− z(t)
w(t)
w(t+ 1)− w(t)
 ≥ 0 for all t ∈ N0.
In exactly the same fashion as described above this leads to a stability
test that is formulated with the FDI
M(z)
(z−1)M(z)
I
(z−1)I

∗
ΠCY

M(z)
(z−1)M(z)
I
(z−1)I
 ≺ 0 for all z ∈ T. (6.14)
We arrive at the following general full-block stability test.
Corollary 6.6
The interconnection (6.2) with ∆Φ ∈ sec[α, β] ∩ slope(µ, ν) is stable if
there exists some ΠCY ∈ Π[diag(α, µ),diag(β, ν)] with (6.14).
6.3.2 Subgradient based arguments
We start this subsection by giving a direct convexity proof for full-block
finite impulse response (FIR) Zames-Falb IQCs as originally proposed
by Willems and Brockett [184]. The derivation will serve as a foundation
for the subsequent comparison of multipliers in the literature.
6.3. Principles of stability multipliers 145
Full-block FIR Zames-Falb multipliers
Let us recall some definitions introduced in [184].
Definition 6.7.
Let L = (Lij) ∈ Rk×k. Then L is a Z-matrix if Lij ≤ 0 for i 6= j.
Moreover, L is doubly hyperdominant if it is a Z-matrix and if, in
addition,
Le ≥ 0 and eTL ≥ 0.
It is said to be doubly dominant if, for all i ∈ {1, . . . , n},
Lii ≥
n∑
j=1
j 6=i
|Lij | and Lii ≥
n∑
j=1
j 6=i
|Lji| .
?
Remark 6.8.
Any doubly dominant matrix L can be decomposed as L = Ld +
Lod where Ld and Lod contain the diagonal and off-diagonal elements,
respectively; then Ld − |Lod| is doubly hyperdominant if we take the
absolute value element-wise. ?
The following lemma provides the foundation for discrete-time Zames-
Falb multipliers. We formulate it for repeated nonlinearities that com-
prise scalar ones as a special case.
Lemma 6.9
Let Φ ∈ slope(0I,∞I). If L ∈ Rk×k is doubly hyperdominant then
Φ(x)TLx ≥ 0 for all x ∈ Rk.
In case that, in addition, ϕ is odd, this holds for any doubly dominant
matrix L.
This is a matrix version of a result in [184]; our direct proof highlights
the role of the underlying principles, namely convexity and permutation
invariance.
146 Chapter 6. Absolute stability of discrete-time interconnections
Proof. Suppose that ϕ is not necessarily odd and choose the convex
primitive Iϕ satisfying Iϕ(0) = 0. Define the convex function
Ψ(x) := Iϕ(x1) + · · ·+ Iϕ(xk).
Since ∇Ψ(x) = col(ϕ(x1), . . . , ϕ(xk)) = Φ(x) we infer by convexity that
Φ(x)T (x− y) ≥ Ψ(x)−Ψ(y) for all x, y ∈ Rk. (6.15)
We now exploit that Φ is repeated by observing Ψ(Px) = Ψ(x) for any
permutation matrix P . Thus (6.15) implies
Φ(x)T (x− Px) ≥ 0 for all x ∈ Rk.
By the Birkhoff-von Neumann theorem [81, Theorem 8.7.1] we infer for
all doubly stochastic matrices S that
Φ(x)T (I − S)x ≥ 0 for all x ∈ Rk.
For the given Z-matrix L with Le ≥ 0 and eTL ≥ 0 it is now clearly
possible to choose r > 0 small enough such that I − rL ≥ 0 and
1− reTLe ≥ 0. Thus
S :=
(
I − rL rLe
reTL 1− reTLe
)
≥ 0
and S is obviously doubly stochastic. As just seen, we can conclude
that
Φ(x)TLx =
1
r
(
Φ(x)
0
)T
(I − S)
(
x
0
)
≥ 0 for all x ∈ Rk.
If ϕ is also odd, the result follows from |Φ(x)| = Φ(|x|) for all x ∈ R
and with L = Ld+Lod in Remark 6.8. Indeed, since Ld−|Lod| is doubly
hyperdominant, we get
Φ(x)TLx = Φ(x)TLdx+ Φ(x)
TLodx
6.3. Principles of stability multipliers 147
≥ |Φ(x)|TLd|x| − |Φ(x)|T |Lod||x|
= Φ(|x|)T (Ld − |Lod|)|x| ≥ 0 for all x ∈ Rk.
It is now standard to extend Lemma 6.9 from monotone to slope-
restricted nonlinearities (see, e.g., [197], [42]). For later reference, we
state the result in terms of a quadratic form as follows.
Corollary 6.10
Let Φ ∈ slope(µI, νI) with µ ≤ 0 ≤ ν and assume that L is doubly
hyperdominant or that ϕ is odd and L is doubly dominant. Then(
x
Φ(x)
)T (
νI −I
−µI I
)T (
0 LT
L 0
)(
νI −I
−µI I
)(
x
Φ(x)
)
≥ 0
for all x ∈ Rk.
The extension to infinite block matrices defining operators on `k2
follows as in [184]. Let L = (Lij)i,j∈Z be an infinite block matrix with
Lij ∈ Rk×k such that there exists some b ≥ 0 with∑
i∈Z
‖Lji‖ ≤ b and
∑
i∈Z
‖Lij‖ ≤ b for all j ∈ Z. (6.16)
It is then well-known that
L : `k2 → `k2 , (Lx)i :=
∑
j∈Z
Lijxj
defines a bounded linear operator. Now suppose that L is a Z-matrix.
Due to (6.16) and if e∞ ∈ `k2e is the sequence of all-ones (column)
vectors then Le∞ and eT∞L are well-defined sequences in `k2e. Let us
assume that, in addition, Le∞ ≥ 0 and eT∞L ≥ 0 element-wise. Then
we obtain the following result as a consequence of Corollary 6.10.
148 Chapter 6. Absolute stability of discrete-time interconnections
Corollary 6.11 (Zames-Falb IQC)
With µ ≤ 0 ≤ ν let Φ ∈ slope(µI, νI) and assume that L with (6.16) is
either an (infinite) doubly hyperdominant matrix or that ϕ is odd and L
is doubly dominant. Then(
?
)T (0 LT
L 0
)(
ν1 −1
−µ1 1
)(
z
∆Φ(z)
)
≥ 0 for all z ∈ `k2 . (6.17)
For the subsequent discussion it suffices to restrict the attention to
block Toeplitz matrices with the structure
L =

. . . . . . . . . . . . . . . . . . . . .
. . . 0 Ll+ . . . L0 . . . L−l− 0 . . .
. . . 0 Ll+ . . . L0 . . . L−l− 0 . . .
. . . 0Ll+ . . . L0 . . . L−l− 0 . . .
. . . . . . . . . . . . . . . . . . . . .

(6.18)
for some chosen l± ∈ N0; it is then required that L0 is a Z-matrix,
L−j ≤ 0 for j ∈ {1, . . . , l−}, and Lj ≤ 0 for j ∈ {1, . . . , l+} as well as
eT
( l+∑
j=−l−
Lj
)
≥ 0 and
( l+∑
j=−l−
Lj
)
e ≥ 0.
For y ∈ `k2 we then infer
L̂y(z) =
( l+∑
j=−l−
Lj z
j
)
yˆ(z) =: HL(z)yˆ(z)
and, due to the structure of L,
L̂T y(z) =
( l+∑
j=−l−
LTj
1
zj
)
yˆ(z) = HL(1/ z)
T yˆ(z).
Based on (6.18) let us now define the class of FIR Zames-Falb multipliers
as the set
ΠL(µI, νI) =
{
Π
∣∣∣∣ Π = (?)T ( 0 H∗LHL 0
)(
νI −I
−µI I
)}
(6.19)
6.3. Principles of stability multipliers 149
where HL(z)∗ = HL(1/ z)T and
HL(z) =
l+∑
j=−l−
Lj z
j . (6.20)
If ∆Φ ∈ slope(µI, νI) then (6.17) implies (via Parseval’s theorem) for
all Π ∈ ΠL(µI, νI) that
ΣΠ
(
z
∆Φ(z)
)
≥ 0 for all z ∈ `k2 . (6.21)
In the light of the parametrization of continuous-time Zames-Falb mul-
tipliers in (5.33), the choices (6.19) and (6.20) correspond to basis func-
tions with pole location zero. In discrete time, it remains an interesting
research topic if nonzero pole locations also admit a parameterization as
in (6.19), (6.20) and whether different poles would be computationally
beneficial.
The multiplier classes corresponding to (6.19) for ν = ∞ and µ =
−∞ are derived analogously and take the form
ΠL(µI,∞I) =
{
Π
∣∣∣∣ Π = (−µ(H∗L +HL) H∗LHL 0
)}
(6.22)
and
ΠL(µI,∞I) =
{
Π
∣∣∣∣ Π = (ν(H∗L +HL) −HL−H∗L 0
)}
, (6.23)
respectively. This leads us to the following stability result, as a conse-
quence of (6.21) and Theorem 6.2.
Corollary 6.12 (FIR Zames-Falb criterion)
Let µ ≤ 0 ≤ ν, ∆Φ ∈ slope(µI, νI) and assume that L in (6.18) is either
a doubly hyperdominant matrix or that ϕ is odd and L is doubly dominant.
Then the interconnection (6.2) is stable if there exists Π ∈ ΠL(µI, νI)
such that (
M(z)
I
)∗
Π(z)
(
M(z)
I
)
≺ 0 for all z ∈ T. (6.24)
150 Chapter 6. Absolute stability of discrete-time interconnections
Finally, multipliers for some non-repeated uncertainty ∆Φ ∈
slope(µ, ν) may be obtained by choosing scalar functions HL,j for
each ϕj and combining them diagonally, which just amounts to the
restriction that all Lj are diagonal; we denote the respective multiplier
class by ΠL(µ, ν).
6.4 Relation to multipliers in the literature
First note that the complete class of full-block Zames-Falb multipliers
was already described in [103]. The multipliers in (6.19), (6.20) are the
full-block versions of the FIR Zames-Falb multipliers as suggested for
the scalar case in [177] and can be easily implemented numerically; this
renders the results in [103] computational.
6.4.1 Zames-Falb multipliers of order one
In this section we prove that the criteria proposed in [162] and in [96] as
well as all later derivatives thereof (see, e.g., [75, 5, 7, 2, 6, 4]) are special
cases of (6.19), (6.20) for l− = l+ = 1. This reveals that all variants
of the discrete-time counterpart to the Popov criterion are rendered
obsolete by a Zames-Falb stability test using (6.19), (6.20) with l± ≥ 1.
Let us hence assume Φ ∈ slope(µ, ν), choose l− = l+ = 1, and select
L with the zeroth block row(
. . . 0 L1 L0 L−1 0 . . .
)
where L0 ≥ 0, L−1 ≤ 0, L1 ≤ 0 are diagonal and satisfy
(L−1 + L0 + L1)e ≥ 0, eT (L−1 + L0 + L1) ≥ 0.
These requirements are obviously fulfilled for the more special (and
potentially restrictive; see Example 6.15) choices
L−1 = −Λ˜, L0 = Λ + Λ˜, L1 = −Λ.
6.4. Relation to multipliers in the literature 151
with diagonal Λ ≥ 0 and Λ˜ ≥ 0. We denote the resulting multiplier
classes corresponding to (6.19), (6.22) and (6.23) by
Π(Λ,Λ˜)(µ, ν), Π(Λ,Λ˜)(µ,∞) and Π(Λ,Λ˜)(−∞, ν), (6.25)
respectively.
Note that all multiplier classes in the present an previous chapter are
convex cones and their combination is just obtained by summing them
up. Therefore, both Π(Λ1,Λ˜1)(µ,∞)+Π(Λ2,Λ˜2)(−∞, ν) and Π(Λ,Λ˜)(µ, ν)
(with varying Λi, Λ˜i and Λ, Λ˜ as described above) define valid multiplier
classes for ∆Φ ∈ slope(µ, ν).
Table 6.1 illustrates how various stability tests proposed in the
literature relate to Corollary 6.12 with (6.25). In the second column we
state the uncertainty class under consideration in the respective paper
as listed in the first column, while the third one gives the employed
multiplier combination. Even if considering uncertainties in slope(µ, ν),
several papers just only exploit the fact that they are contained in
either slope(µ,∞) or slope(−∞, ν). Yet some use, e.g., the information
∆Φ ∈ slope(0, ν) = slope(−∞, ν) ∩ slope(0,∞) by additively combining
multipliers for slope(−∞, ν) and slope(0,∞), respectively. We devote
the subsequent section to the proof that this is not beneficial if compared
to using the dedicated multipliers for the class slope(0, ν) directly.
Several more aspects are worth pointing out in Table 6.1. All cited
papers employ a combination of diagonally structured circle criterion
multipliers (Πdr[α, β]) and first order Zames-Falb multipliers. [6] also
includes one of Yakubovich type (ΠY,dr[µ, ν], see (6.27) below), but this
is also covered by Zames-Falb multipliers as shown later. Further note
that several approaches either take Λ = 0 or Λ˜ = 0 which, of course,
increases conservativeness if computing stability margins. Thus, a com-
bination of the multipliers proposed in the present paper is guaranteed
to lead to the same or improved stability estimates.
We can further conclude that both the multipliers proposed by [162]
and [96] (as well as the later proposed derivatives thereof) are special
cases of Zames-Falb multipliers of order one. Hence, ΠL(µ, ν) with
152 Chapter 6. Absolute stability of discrete-time interconnections
Table 6.1: Overview of some multipliers employed in the literature
Reference Uncertainty class Multiplier class combination
[162] sec[0, β] ∩ slope(0,∞) Πdr[0, β] + Π(0,Λ˜)(0,∞)
[96] sec[0, β] ∩ slope(−ν, ν) Πdr[0, β] + Π(0,Λ˜)(−∞, ν)
[75] sec[0, β] ∩ slope(0, ν) Πdr[0, β] + Π(0,Λ˜)(−∞, ν)
[5] sec[0, β] ∩ slope(0, ν) Πdr[0, β] + Π(0,Λ˜)(−∞, ν) +
Π(Λ,0)(0,∞)
[7] sec[0, β] ∩ slope(0, ν) Πdr[0, β] + Π(0,Λ˜)(−∞, ν) +
Π(Λ,0)(0,∞)
[2] sec[0, ν] ∩ slope(0, ν) Πdr[0, ν] + Π(Λ,Λ˜)(0, ν)
[6] sec[0, β] ∩ slope(0, ν) Πdr[0, β] + ΠY,dr[µ, ν] +
Π(0,Λ˜1)(−∞, ν)+Π(Λ2,Λ˜2)(0,∞)
[4] sec[0, ν] ∩ slope(0, ν) Πdr[0, ν] + Π(Λ,Λ˜)(0, ν)
L as in (6.20) and l± = 1 could be seen as a full-block generalization
of Tsypkin multipliers that, to the best of the authors knowledge, has
not been described anywhere in the literature. As a side-remark, we
emphasize that our approach does not require the LTI system in the
loop to be strictly proper, as is typically encountered in the literature.
6.4.2 Redundant multiplier combinations
We have seen that a large number of papers handle ϕ ∈ slope(µ, ν) by
combining Zames-Falb multipliers for slope(µ,∞) and slope(−∞, ν); let
us now settle that it is more beneficial to work with the single class of
dedicated multipliers ΠL(µ, ν).
6.5. Implementation 153
Lemma 6.13
Let Π ∈ ΠL(µ, ν), Π1 ∈ ΠL(µ,∞), Π2 ∈ ΠL(−∞, ν) be three given
Zames-Falb multipliers. Then there exists another Zames-Falb multiplier
Π˜ ∈ ΠL(µ, ν) such that Π˜ 4 Π + Π1 + Π2 on T.
Proof. A proof is found in the Appendix C.5.2.
In summary, we can just work with the tightest slope restriction in
Corollary 6.12, i.e., ϕ ∈ slope(µ, ν), since the validity of (6.24) for a
combination of multipliers as in Lemma 6.13 implies the existence of
some multiplier in ΠL(µ, ν) also satisfying (6.24).
In case of a single nonlinearity, let us finally stress that Yakubovich
multipliers for ∆Φ ∈ slope(µ, ν) are also covered by first order Zames-
Falb multipliers. Indeed if k = 1, Lemma 5.10 shows that Π[µ, ν] can
be parameterized as
Π[µ, ν] =
{
Π
∣∣∣∣ Π = λ(−2µν µ+ νµ+ ν −2
)
, λ > 0
}
(6.26)
The claim then follows by the simple observation that, with (6.26),
|z−1|2 Π[µ, ν] = Π(λ,λ)(µ, ν) as in (6.25). Also for k > 1 this shows
that diagonally structured Yakubovich multipliers
ΠY,dr[µ, ν] := |z−1|2 Πdr[µ, ν] (6.27)
offer no benefit if combined with Zames-Falb multipliers. Yet, this no
longer holds true for the full-block versions (see Example 6.16).
6.5 Implementation
In order to keep the derivation and comparison of multipliers as in-
sightful as possible, we relied on the formulation of our tests in terms
of inequalities in the frequency domain. Still, the translation to LMIs
154 Chapter 6. Absolute stability of discrete-time interconnections
with an insightful structure is routine. Indeed, via multiplication with
1 = 1/(z z) on T, observe that (6.14) holds on T if and only if we have
(
?
)∗
ΠCY

1
z 0
1− 1z 0
0 1z
0 1− 1z

︸ ︷︷ ︸
ΨCY (z)
(
M(z)
I
)
≺ 0 for all z ∈ T; (6.28)
clearly, ΨCY is a proper and stable transfer function.
Let us now sketch how to render Corollary 6.12 computational for
some pair l = (l−, l+). For brevity of display, consider the case l+ ≥ l−
and define
ψl+ =
(
I 1z I . . .
1
zl+
I
)T
, Ψl+ = diag(ψl+ , ψl+)
as well as the square matrix Pl ∈ R(l++1)k×(l++1)k given by
Pl =

L0 L−1 . . . L−l− 0 . . . 0
L1 0 . . . 0 0 . . . 0
...
...
. . .
...
...
...
...
Ll+ 0 . . . 0 0 . . . 0
 . (6.29)
Then the multiplier (6.19), (6.20) may be expressed as
(?)∗ΠlTΨl+ =
(
?
)∗( 0 PTl
Pl 0
)(
νI −I
−µI I
)
Ψl+ ; (6.30)
again ΨZF := TΨl+ is proper and stable. The case of l− > l+ is treated
analogously.
In this way we obtain the following combined stability test.
Corollary 6.14
Let ∆Φ ∈ sec[αI, βI] ∩ slope(µI, νI) and fix l−, l+ ≥ 0. Suppose there
exists some ΠCY ∈ Π[diag(αI, µI),diag(βI, νI)] and some L as in
6.5. Implementation 155
(6.18) that is either doubly hyperdominant or doubly dominant (if ϕ is
odd) with
(
?
)∗(ΠCY 0
0 Πl
)(
ΨCY
ΨZF
)(
M
I
)
≺ 0 on T. (6.31)
Then the interconnection (6.2) is stable
This also holds for ∆Φ ∈ sec[α, β] ∩ slope(µ, ν) if just restricting
all matrices Lj in (6.29) to be diagonal and assuming that ΠCY ∈
Π[diag(α, µ),diag(β, ν)].
It is now routine to turn the verification of (6.31) (for some ΠCY ,
Πl satisfying the respective constraints) into an LMI; just choose a
realization (A,B, C,D) of (
ΨCY
ΨZF
)(
M
I
)
and apply the discrete-time KYP lemma ([128]); then (6.31) holds iff
there exist some X = XT with I 0A B
C D

T

X
−X
ΠCY
Πl

 I 0A B
C D
 ≺ 0. (6.32)
This should be compared to the continuous-time analogue, (5.41), where
the structure of the LMI is identical, apart from the fact that the left
upper block of the middle matrix takes the form ( 0 XX 0 ). In summary,
the IQC framework allows us to state and combine stability criteria
in an intuitive fashion, as demonstrated for (6.31), and to generate
the corresponding LMI conditions (6.32) in a structured, insightful
and routine way. Unfortunately, in the literature, this structure is
usually not as apparent, which renders the task of comparing different
approaches and extracting the underlying generating principles for the
employed criteria unnecessarily tedious.
156 Chapter 6. Absolute stability of discrete-time interconnections
6.6 Examples
Let us finally provide some numerical illustrations.
Example 6.15.
First, we adopt an example from [2], where M is given by
M(z) =
(
0.2
z−0.98
−0.2
z−0.92
0.3
z−0.97
0.1
z−0.91
)
.
Our goal is to estimate the largest r > 0 such that the feedback intercon-
nection (6.2) remains stable for all ∆Φ ∈ slope(0, ν) with ν = diag(r, r).
We first assume that the nonlinearities are non-identical. As can be
inferred from Table 6.1, the stability test proposed in [2] is the least
conservative of all discussed approaches. The maximal r estimated
therein is r = 3.626. Using diagonally structured Zames-Falb multipli-
ers in (6.32) (with l± = 1), we can still improve on that and obtain
r = 3.808 which is already very close to the Nyquist value rN = 3.85.
This supports the fact that diagonally structured first order Zames-Falb
multipliers may already lead to improved estimates if compared to the
Popov tests in the literature. If we further assume that the nonlinear-
ities are repeated, stability can be guaranteed up to rN by means of
full-block multipliers. ?
Example 6.16.
Let the LTI system M in (7.2) be defined by
A =

0.74 −0.3 0 0 −0.1
0 0.98 0 0 0
0 0 0.97 0 0
0 0 0 0.72 0
0 0.1 0.31 0 0.9
 , B =

2.2 0.2
0.3 0
0.5 0
−0.5 0.5
0 0.05
 ,
C =
(−0.21 −0.4 −0.01 0.40 0
0.3 0.3 −0.3 0 −0.36
)
, D = 0,
6.7. Summary 157
and assume, for simplicity, that ∆Φ ∈ slope(µ, ν) with µ = 0 and ν = I,
yet with non-identical functions ϕj . Let us now compare the standard
approach from the literature, namely a combination of diagonally struc-
tured Zames-Falb and circle multipliers, to Corollary 6.14. In order
to contrast both approaches, we compute `2-gain estimates, i.e., the
smallest γ > 0 such that (6.3) holds. This is easily achieved by following
the procedure outlined, for the continuous-time case, in Section 5.3.4.
Table 6.2: `2-gain estimates for Example 6.16
multiplier l± = 1 l± = 2 l± = 3 l± = 4
ZF+Πdr 148.43 104.35 89.39 82.08
Corollary 6.14 95.23 76.47 69.99 66.58
The results depicted in Table 6.2 nicely illustrate that, unlike the
diagonally structured circle and Yakubovich ones, the combined poly-
topic multipliers also provide additional benefit if employed together
with those for the Zames-Falb criterion. ?
6.7 Summary
In this chapter, we present a comprehensive approach to the problem
of absolute stability of feedback interconnections in the discrete-time
setting. We reveal that all multiplier based stability criteria in the
literature can be subsumed into one general framework that even allows
for seamless extension and generalisation of the previously proposed
stability tests. This leads to the derivation of novel and completely
unstructured full-block multipliers that are, in numerical examples,
shown to lead to less conservative stability estimates if compared to
existing criteria. Moreover, the present chapter serves to illustrate
158 Chapter 6. Absolute stability of discrete-time interconnections
that, if formulated correctly, the translation of stability results from
continuous time to discrete time is rather straightforward.
Part II
From input-output properties
to the analysis of internal behavior
159

Introduction to Part II
Let us briefly take one step back and focus on the conceptual de-velopment of the IQC stability results presented so far; from the
classical ones discussed in Chapter 2, their first extension in Chapter 3,
and, finally, the general framework derived in Chapter 4.
All the above mentioned results share the key common theme that
stability is inferred solely from input-output properties of the systems
composing the interconnection. These systems are given maps defined on
the full underlying signal spaces and the constraints that form the basis
of our stability proof are assumed to hold for all signals in the respective
space. This global perspective is one of the reasons for the flexibility
of the derived framework and, furthermore, allows us to entirely avoid
the internal dynamics in stability considerations. However, we make
fundamental use of the state-space description of the LTI system for
the verification of stability based on LMIs.
On the downside, the exclusive focus on the input-output behavior
also comes at the expense of limited applicability. So far, this is probably
best visible by reviewing the generalizations made possible by the
framework derived in Chapter 4. While in Theorem 3.4 we relax the
assumption on the domain of definition of the uncertainty and its IQC
description, we further proceed along this line of thought in Theorem 4.7
by carefully addressing the external inputs. Nevertheless, the restriction
to arbitrary input sets or to IQC only valid for signals in a certain
subset of the underlying signal space could not be achieved. As most
161
162
severe consequences, this prevents us from locally analyzing feedback
interconnections, and here in particular the analysis of only locally
stable systems or the verification of state and output constraints.
It will be the main contribution of the two subsequent chapters
to establish a natural link between the general input-output theory
based on IQCs and the local analysis of feedback interconnections as is
standard in Lyapunov or dissipation theory. In Chapter 7 we present a
comprehensive framework that, under rather mild conditions, allows to
derive a Lyapunov function from the standard FDI condition in IQC
theory. We furthermore illustrate that this puts us in the position to
proceed with local analysis using identical arguments as in standard
Lyapunov theory. Building on the framework established in Chapter 7,
we propose in Chapter 8 a unified approach where the uncertainty is
described with both hard and soft IQCs. On the one hand, this further
widens the applicability of our framework, while, one the other hand,
it effectively reduces the conservativeness in stability analysis. Beyond
that, a particular focus in Chapter 8 lies on the class of Zames-Falb
multipliers, where both the subclasses of causal and anti-causal Zames-
Falb multipliers are shown to admit lossless incorporation into the
framework developed in Chapter 7.
Finally, we emphasize that a slightly different version of Chapter 7
is provisionally accepted for publication in the IEEE Transactions on
Automatic Control [60]. Furthermore, the results derived in Chapter 8
are, in a self contained form, accepted for publication in the IEEE
Control Systems Letters [59].
Chapter 7
Invariance with dynamic multipliers
7.1 Introduction
As already indicated in Chapter 2, robust or absolute stabilityanalysis of feedback interconnections composed of an LTI system
and an uncertain component ∆, as shown in Figure 7.1, can essentially
be divided into two still not fully connected fields. In fact, trajectory
oriented time-domain techniques such as Lyapunov function methods
(see, e.g., [106, 188, 118]) are considered in parallel to operator based
functional analytic approaches. The latter may be further divided into
multiplier theory [195, 196, 197, 44] and graph separation techniques
[138, 139, 72, 159, 89], formalized, for example, in the framework of IQCs
(see Section 2.2). In this thesis we have so far restricted our attention to
the IQC approach that has been shown to offer the additional advantages
over classical multiplier results that there is generally no need for loop
transformations and that noncausal multipliers may be incorporated
with ease (see [112, 63] for relations among [195, 196, 136, 138] and
[110]).
Lyapunov techniques permit to exploit properties of signals as they
are generated within the interconnection, which often allows to reduce
conservativeness by taking the specifics of the interconnection structure
163
164 Chapter 7. Invariance with dynamic multipliers
M N12
N21 N22
∆
de
wz
Figure 7.1: Performance setting
or the external stimuli into account. As a major asset, Lyapunov
methods facilitate regional analysis1, such as the verification of hard
state and output constraints in the time domain, and, therefore, even
allow for the analysis of interconnections that are not globally stable.
This is often achieved by restricting the input signals to a certain
(bounded) subset of the encompassing signal space and by imposing
suitable additional constraints on the Lyapunov function.
In contrast, stability conditions obtained from the operator based
approach are usually formulated in the frequency domain using multi-
pliers. While Lyapunov arguments typically require an individual proof
if additional aspects of the uncertain component are taken into account,
refined uncertainty descriptions in terms of added multipliers may be
easily incorporated into the operator framework. This allows for a
simple reduction of conservativeness by capturing the uncertainty more
accurately. However, in the configuration of Figure 7.1, interconnection
stability boils down to guaranteeing global boundedness of the inverse
(I −M∆)−1 on the underlying function space, which prevents regional
analysis with operator approaches.
Already this cursory exposition reveals that merging both rather
complementary techniques may drastically improve stability and perfor-
mance estimates and simultaneously widen the range of applications.
1In the literature, the term regional is sometimes used to distinguish the analysis
on fixed, often large set form local results that merely hold in some neighborhood.
In the sequel, we will use both terms synonymously.
7.1. Introduction 165
However, even though both have their origins around the 1960s, their
effective combination at a high level of generality is still largely open.
Following an earlier result by Safonov and Athans [138], which
establishes the link between a classical theorem of Zames on conic
sectors [195] and quadratic Lyapunov functions, there have been multiple
attempts to bridge the gap between Lyapunov theory and IQCs (see,
e.g., [74, 79, 125, 13, 124, 153, 32] and [174, 150]). A well-known tool
in linking the FDIs arising in IQC analysis with the LMIs underlying
the Lyapunov approach is the KYP lemma [128, 187]. This result
equivalently characterizes the validity of an FDI by the feasibility of a
related LMI; the corresponding LMI solution is a so-called certificate
for the FDI. Since such certificates are, in general, indefinite, it is a
pivotal question how they can be used to construct positive definite
Lyapunov functions in order to enable the inclusion of extra constraints
for regional analysis.
The aforementioned papers all address this issue from various angles.
In fact, in [79, 125] the circle [195] and Popov [126] criteria are employed
in the analysis of regionally stable saturated systems. For the circle
criterion, these results were strengthened in [124] by also exploiting
information regarding the derivative of the output of the LTI system.
Not necessarily globally stable interconnections are also the subject
of study in [153, 32], where hard IQCs (2.14) are employed in order
to verify regional stability. Finally, in [52], an attempt was made to
first prove contractive invariance of an ellipsoid and regional stability of
the interconnection using general soft IQCs (2.16). Unfortunately, the
authors incorrectly claim that the validity of an IQC on some invariant
set suffices to conclude stability by applying the standard IQC theorem
[110, Thm. 1]. The critical stumbling block in [52] is also at the very
heart of the present chapter, namely the fact that the stability proof
given in [110] fundamentally requires the IQCs to hold on the full space
L2 of square integrable functions. As shown in Chapters 3 and 4 this
can indeed be relaxed to subspaces of L2, but not to arbitrary subsets
thereof.
166 Chapter 7. Invariance with dynamic multipliers
The paper by Balakrishnan [13] stands out in this list, since it
does connect dynamic IQC techniques with the Lyapunov approach
and exhibits the possibility for regional analysis. However, the paper
focusses on structured LTI uncertainties that are handled with so-called
dynamic D-scalings, and the proposed method (including the proofs)
heavily relies on these particularities.
In conclusion, the cited papers either focus on rather specific classes
of uncertainties or employ a pretty limiting set of IQC multipliers.
By contrast, in [174] and later in [150]2 first attempts were made to
overcome these restrictions, by giving dissipation based proofs of the
IQC stability theorem in [110] for a rich class of multipliers. It was
hoped that these would permit the inclusion of time-domain constraints
into general (frequency-domain) IQC theory, and vice versa. Yet, as a
major obstacle, both derivations rely on the solution of an indefinite
algebraic Riccati equation (ARE) that acts as a shift to the KYP
certificate in order to obtain a positive definite Lyapunov function.
Unfortunately, a simultaneous search over the class of multipliers and
the corresponding shift matrix is impossible, since the indefinite ARE
cannot be turned into an LMI constraint. In addition, even though the
proofs in [174, 150] are both trajectory based in nature, the derived
stability results require globally valid IQC descriptions of the uncertainty
which prevents regional analysis.
As the main contributions of this chapter we present a framework
that allows to overcome both limitations of the approaches in [174, 150],
even under fewer assumptions. We propose a novel, multiplier depen-
dent shift of the KYP certificate that is described by LMI constraints
and enables the formulation of a regional stability result within IQC
theory and based on solving LMIs. As an additional benefit, we can
consider uncertainties, IQCs and external disturbances that are only
defined on general subsets of L2. We demonstrate how our extension
of the standard IQC framework permits the merging of multiplier de-
scriptions of uncertainties with concepts from Lyapunov and dissipation
2A correction of the paper [151] that contains a technical glitch.
7.1. Introduction 167
theory [182, 183]. In particular, we reveal how to guarantee state as well
as output constraints, and how these can be systematically exploited
in order to tighten IQCs for the reduction of conservatism. A selection
of numerical examples illustrate that our approach can substantially
improve on existing ones, some of which are even tailored to specific
scenarios.
The chapter is structured as follows. After setting the stage by
adapting the IQC framework to our regional scenario and stating the
relation between the emerging FDIs and dissipation inequalities in
Section 7.2, we formulate and prove our central local IQC stability result
in Section 7.3. Furthermore, we discuss its novel features if compared
to related approaches in the literature in Section 7.4, and highlight how
to derive several concrete regional stability and performance criteria
in Section 7.5. Section 7.6 concludes the chapter with a selection of
numerical examples that demonstrate the benefit of our approach if
compared to various state of the art methods in the literature for
different scenarios. Again, we emphasise the fact that the results from
this chapter are submitted for journal publication [60].
For convenience, we remind the reader of the following facts from
Chapter 2 that are particularly relevant for the present chapter. If Ψ is
a stable transfer matrix realized by (A,B,C,D) and P = PT , we say
that X = XT is a certificate for the FDI Ψ∗PΨ ≺ 0 on C∞0 in case
that X solves the (KYP-) LMI
 I 0A B
C D
T  0 X 0X 0 0
0 0 P
 I 0A B
C D
 ≺ 0. (7.1)
Moreover, for H < 0 and η ≥ 0 we use
E(H, η) := {x ∈ Rn ∣∣ xTHx ≤ η} .
168 Chapter 7. Invariance with dynamic multipliers
7.2 Preliminaries
This section serves to introduce the setup for this paper and to discuss
the distinctions to the classical IQC framework (see Section 2.2.1) made
necessary by our focus on regional analysis. Furthermore, we discuss
the connection between IQCs and dissipation theory [182, 183] in some
detail.
7.2.1 Setup
As depicted in Figure 7.1, we consider the interconnection(
z
e
)
=
(
M N12
N21 N22
)
︸ ︷︷ ︸
N
(
w
d
)
, w = ∆(z), (7.2)
involving the stable LTI system N , as realized by
x˙ = Ax+B1w +B2d, x(0) = 0,
z = C1x+D11w +D12d,
e = C2x+D21w +D22d (7.3)
with A ∈ Rn×n being Hurwitz, in feedback with some uncertainty ∆.
As the first deviation from the assumptions imposed in Part I of this
thesis, it is not required that ∆ is defined globally. Instead, its domain
of definition is some subset
Ze ⊂ L2e satisfying {zT | z ∈ Ze} ⊂ Z for all T > 0, (7.4)
where Z := Ze ∩L2 denotes the set of finite energy signals in Ze.
Remark 7.1.
Unfortunately, the assumptions on Ze are closer to the ones in Chapter 3
than the more general ones in Chapter 4. As a main consequence,
the requirement (Ze)T ⊂ Z in (7.4) will prevent us from employing
multipliers with additional regularity assumptions in regional analysis.
?
7.2. Preliminaries 169
As in all previous chapters, uncertainties are causal maps
∆ : Ze → L2e but now satisfying only ∆(Z ) ⊂ L2. (7.5)
The latter inclusion is interpreted as a weak form of stability of ∆,
while causality means, as usual, that ∆(z)T = ∆(zT )T holds3 for all
z ∈ Ze and T > 0. Additionally, in (7.2) we also allow for the external
disturbance d to be confined to some set D ⊂ L2. The definitions of
well-posedness and stability of (7.2) are adapted to the current setting
as follows.
Definition 7.2.
The feedback interconnection (7.2) is well-posed on D if, for each
d ∈ D , there exists a unique response z ∈ Ze such that the map d→ z
is causal. It is stable on D if
there exists γ > 0 with ‖z‖ ≤ γ‖d‖ for all d ∈ D . (7.6)
?
Remark 7.3.
For the choices Ze = L2e, D = L2 and N12 = I, well-posedness of
(7.2) translates into I−M∆ : L2e → L2e having a causal inverse, while
stability means that, in addition, (I −M∆)−1 maps L2 into L2 and
is bounded; therefore, this setting generalizes the classical notions of
well-posedness from Definition 2.5 insofar as we do not require it for all
∆ replaced by τ∆ and τ ∈ [0, 1]. ?
In deviating from our general framework developed in Chapter 4,
we restrict our attention in the present chapter to classical IQCs (2.16).
Two signals z, w ∈ L2 with Fourier transforms zˆ, wˆ are said to satisfy
the IQC defined by a Hermitian valued multiplier Π ∈ RL∞ if
ΣΠ
(
z
w
)
=
∫ ∞
−∞
(
zˆ(iω)
wˆ(iω)
)∗
Π(iω)
(
zˆ(iω)
wˆ(iω)
)
dω ≥ 0.
3(·)T denotes the standard truncation on L2e, see Definition 2.1.
170 Chapter 7. Invariance with dynamic multipliers
Definition 7.4.
The uncertainty (7.5) satisfies the (soft) local IQC imposed by Π ∈ RL∞,
in short ∆ ∈ IQCZ (Π), if
ΣΠ
(
z
∆(z)
)
≥ 0 for all z ∈ Z . (7.7)
?
As a specifically relevant feature of our framework, we can and will
beneficially exploit the ability to work with uncertainty IQCs that are
only satisfied for signals in a genuine subset Z of L2. In this sense,
(7.7) is a local IQC (despite the fact that Z might be “large”) and, in
the sequel, our stability and performance guarantees are addressed as
local IQC results.
Locality not only allows us to deal with uncertainties that are un-
bounded or even undefined on the full space L2e (see Example 7.16
in Section 7.6), but it also permits us to construct stronger IQCs for
reducing the conservatism of global results, as illustrated with several
applications of our main theorem in Section 7.5. Even if ∆ is defined
on L2e, we emphasize that it might still be beneficial to consider its
restriction to some subset Ze on which it exhibits a more desirable
behavior.
Let us now specify the classes of multipliers considered in this chapter
in more detail. As noted in Section 2.3, any Π = Π∗ ∈ RL∞ can be
factorized as
Π = Ψ∗PΨ with a real P = PT and Ψ ∈ RH∞; (7.8)
this relation will be denoted by Π ∼ (P,Ψ). For extensive lists of
multipliers that satisfy (7.8) and capture the behavior of practically
relevant uncertainties, we refer the reader to [110, 176].
In the sequel we tacitly assume that Π = (Πij) and Ψ = (Ψ1 Ψ2)
are partitioned according to the dimensions of the signals z and w;
moreover, Ψ is supposed to be realized as
ξ˙ = AΨξ +BΨ,1z +BΨ,2w, ξ(0) = 0,
7.2. Preliminaries 171
zΨ = CΨξ +DΨ,1z +DΨ,2w, (7.9)
where AΨ is Hurwitz. This allows us to emphasize the first relevance
of multiplier factorizations, since they provide a means for translating
IQCs in the frequency domain into their corresponding time-domain
versions. Indeed, using (7.8), (7.9) and w = ∆(z), we infer from
Parseval’s theorem that the IQC (7.7) is equivalent to the (infinite
horizon) time-domain constraint∫ ∞
0
zΨ(t)
TPzΨ(t) dt ≥ 0 for all z ∈ Z . (7.10)
We indicate the fact that (7.10) depends on the factorization Π ∼ (P,Ψ)
by abbreviating it as ∆ ∈ IQCZ(P,Ψ).
All this motivates to work with classes of multipliers that are pa-
rameterized as in (7.8) with a fixed stable (usually tall) outer factor Ψ
and with a variable symmetric matrix P that is constrained by (7.10).
Using the notation introduced above, P thus varies in the set{
P = PT
∣∣ ∆ ∈ IQCZ(Ψ∗PΨ)} .
The fact that we allow for arbitrary stable Ψ in (7.8) is an important
strength of the present approach and allows to readily employ, e.g.,
Zames-Falb multipliers with tall outer factors exactly as in Chapter 5.
This is in contrast to the works in [150, 174] that require J-spectral
factorizations which may only be obtained from (7.8) in a non-convex
additional step.
7.2.2 Performance specifications
Let us now discuss performance specification and their verification
using LMIs as these provide a connection between IQCs and dissipation
inequalities, thus enabling regional analysis. Suppose that (7.2) is
well-posed and has been shown to be stable. It is then standard to
characterize a certain desired behavior of the interconnection (7.2) by
172 Chapter 7. Invariance with dynamic multipliers
imposing quadratic performance criteria defined through a symmetric
matrix Pp and expressed in the time domain as∫ ∞
0
(
e(t)
d(t)
)T
Pp
(
e(t)
d(t)
)
dt ≤ 0 (7.11)
for all trajectories of (7.2) in response to d ∈ D . Let us recapitulate the
well-known fact that one can guarantee this performance specification
in terms of the FDI
(
?
)∗(P 0
0 Pp
)(
Ψ 0
0 I
)
M N12
I 0
N21 N22
0 I
 ≺ 0 on C∞0 , (7.12)
as made precise in the subsequent Theorem.
Remark 7.5.
Note that we could incorporate dynamic performance criteria defined
by a multiplier Πp ∼ (Pp,Ψp) in complete analogy to the stability
multiplier Π. This only results in an additional filter Ψp instead of the
identity matrix in (7.12). However, it remains to be explored if the
enlargement of the certificate for (7.12), resulting from the additional
dynamics, will introduce conservatism in the subsequent derivation. ?
Theorem 7.6
Assume that the interconnection (7.2) is well-posed and stable on D.
Moreover, suppose there exists some P = PT with the following two
properties
a) ∆ ∈ IQCZ (P,Ψ);
b) P satisfies the performance FDI (7.12).
Then (7.11) holds for all trajectories of (7.2) with d ∈ D .
7.2. Preliminaries 173
Proof. Choose d ∈ D . By well-posedness of (7.2) on D , the response
of (7.2) satisfies z ∈ Ze and by stability we also get z ∈ L2. This
implies z ∈ Z and, due to (7.5), also w = ∆(z) ∈ L2; then stability
of N guarantees e ∈ L2. Right- and left-multiplying the inequality in
(7.12) with col(wˆ(iω), dˆ(iω)) and its conjugate transpose, respectively,
leads to
(
zˆ(iω)
wˆ(iω)
)∗
Ψ(iω)∗PΨ(iω)
(
zˆ(iω)
wˆ(iω)
)
+
(
eˆ(iω)
dˆ(iω)
)∗
Pp
(
eˆ(iω)
dˆ(iω)
)
≤ 0
for almost all ω ∈ R. After integration over frequency and exploiting
wˆ = ∆̂(z), we get
ΣΨ∗PΨ
(
z
∆(z)
)
+
∫ ∞
−∞
(
eˆ(iω)
dˆ(iω)
)∗
Pp
(
eˆ(iω)
dˆ(iω)
)
dω ≤ 0.
Using a), (7.7) and Parseval’s theorem finishes the proof.
In addition to their role in the translation from frequency-domain to
time-domain constraints, and as a second relevance of the factorization
(7.8), we recall how to characterize the validity of (7.12) in terms of an
LMI feasibility test. In fact we just need to take a realization of
Ψ1 Ψ2 0 00 0 I 0
0 0 0 I


M N12
I 0
N21 N22
0 I

and apply the KYP lemma [128] to justify the following fact.
174 Chapter 7. Invariance with dynamic multipliers
Lemma 7.7
For Π ∼ (P,Ψ) described as in (7.8), the FDI (7.12) is equivalent to
the existence of a solution X = XT of the LMI
(
?
)T

0 X 0 0
X 0 0 0
0 0 P 0
0 0 0 Pp


I 0 0 0
0 I 0 0
AΨ BΨ,1C1 BΨ,1D11 +BΨ,2 BΨ,1D12
0 A B1 B2
CΨ DΨ,1C1 DΨ,1D11 +DΨ,2 DΨ,1D12
0 C2 D21 D22
0 0 0 I

≺ 0.
(7.13)
In order to provide a link between the frequency-domain constraints
within the IQC framework and the time-domain descriptions used in
Lyapunov arguments, we rely on dissipation theory. Let us hence recall
how the relation of FDIs and LMIs in Lemma 7.7 allows us to extract
the corresponding dissipation inequality. In the sequel we tacitly assume
that X = (Xij) in (7.13) is partitioned according to the dimensions of
AΨ and A, respectively.
Lemma 7.8
For Π ∼ (P,Ψ) described as in (7.8) suppose that (7.13) holds. Then,
for any w, d ∈ L2e and a nonzero initial condition x(0) = x0 of N ,
the trajectory of the nominal system (7.3) interconnected with the filter
(7.9) satisfies
(
ξ(T )
x(T )
)T (
X11 X12
X21 X22
)(
ξ(T )
x(T )
)
+
∫ T
0
zΨ(t)
TPzΨ(t) dt+
+
∫ T
0
(
e(t)
d(t)
)T
Pp
(
e(t)
d(t)
)
dt− xT0 X22x0 ≤ 0 (7.14)
for all T > 0.
7.2. Preliminaries 175
Let us now establish the connection between Lemma 7.8 and classical
Lyapunov and dissipation theory while simultaneously highlighting the
key challenges if using soft IQCs.
7.2.3 Technical motivation of contributions
Suppose we take Pp = diag(I,−γ2I) and assume that the system N is
structured as (N21 N22) = (M N12); then the gain inequality in (7.6) is
equivalent to (7.11). Let us pinpoint why, even in this particular case,
the hypotheses in Theorem 7.6 do not allow to conclude stability. Indeed,
along the response of (7.2) to d ∈ D , the dissipation inequality (7.14)
reads as(
ξ(T )
x(T )
)T
X
(
ξ(T )
x(T )
)
+
∫ T
0
zΨ(t)
TPzΨ(t) dt+
+
∫ T
0
‖z(t)‖2 dt ≤ γ2
∫ T
0
‖d(t)‖2 dt. (7.15)
If trying to assure that
∫ T
0
‖z(t)‖2 dt is bounded for T →∞ (in order to
guarantee z ∈ L2), we need to argue that the sum of the first two terms
in (7.15) is bounded from below for T →∞. However, in general, the
solution X of the LMI (7.13) will not be positive (semi)definite. More-
over, since it is not clear whether z is contained in Z , we cannot directly
apply (7.10) in order to draw the conclusion that
∫ T
0
zΨ(t)
TPzΨ(t) dt
is bounded from below for T → ∞. Yet, if assuming X  0 and the
validity of the so-called hard IQC∫ T
0
zΨ(t)
TPzΨ(t) dt ≥ 0 for all T > 0, (7.16)
it is straightforward to infer z ∈ L2 and ‖z‖ ≤ γ‖d‖ from (7.15); the
latter two properties are standard hypothesis in robust stability proofs
based on dissipation arguments that often appear in the literature, with
[182, 138] being early references.
176 Chapter 7. Invariance with dynamic multipliers
However, if working with the more general soft IQC (7.10) and
without imposing a positivity constraint on X, this direct reasoning
fails [174, 150]. In the following section, we will use (7.10) in order to
derive a computationally tractable lower bound on the finite horizon
integral in (7.16) which does allow to conclude robust stability.
7.3 Main result
Let us now present the key technical local IQC stability theorem of
this chapter. Besides being of independent interest, it provides the
foundation for all subsequent concrete novel regional stability and
performance analysis results based on local IQCs.
Along the lines of standard IQC theory, stability of (7.2) will be
characterized solely in terms of M and involves the FDI(
M
I
)∗
Ψ∗PΨ
(
M
I
)
≺ 0 on C∞0 (7.17)
as certified by some symmetric matrix Xs satisfying
(
?
)T 0 Xs 0Xs 0 0
0 0 P


I 0 0
0 I 0
AΨ BΨ,1C1 BΨ,1D11 +BΨ,2
0 A B1
CΨ DΨ,1C1 DΨ,1D11 +DΨ,2
 ≺ 0. (7.18)
We need to restrict our attention to multipliers (7.8) that satisfy the
additional property
Π22 = Ψ
∗
2PΨ2 ≺ 0 on C∞0 (7.19)
which is certified by some Y22 = Y T22 solving the LMI(
ATΨY22 + Y22AΨ Y22BΨ,2
BTΨ,2Y22 0
)
+
(
CΨ DΨ,2
)T
P
(
CΨ DΨ,2
) ≺ 0. (7.20)
7.3. Main result 177
We emphasize that this property is automatically satisfied for most of
the multiplier classes proposed in the literature (as can be extracted
from the non-exhaustive survey in [176]). However, note that, e.g., the
Pólya relaxation presented in Section 5.3.1 does not lead to multipliers
satisfying (7.19). We will discuss the incorporation of these multipliers
in the subsequent chapter.
Theorem 7.9
Suppose that the interconnection (7.2) of the stable LTI system N and
the causal uncertainty ∆, satisfying (7.5), is well-posed on D. Then
(7.2) is stable on D if there exists some P = PT with the following two
properties
a) ∆ ∈ IQCZ (P,Ψ);
b) there exists a certificate Xs of (7.17) and a certificate Y22 of
(7.19) which are coupled as(
Xs11 − Y22 Xs12
Xs21 X
s
22
)
 0. (7.21)
This provides a generalization of [110, Thm. 1] towards stability
analysis of the interconnection (7.2) on the basis of local (soft) integral
quadratic constraints and general disturbance sets D . In contrast
to [110], we require well-posedness and the validity of the uncertainty
IQC (7.7) to only hold for ∆ and not for the whole set of all τ∆ with
τ ∈ [0, 1]. If contrasted with the stability results in [174, 150, 13]
(involving global IQCs), ours only requires the a priori restriction (7.19)
on the multipliers and is, most importantly, not depending on any
special features of the factorization of Π ∼ (P,Ψ); in particular, there
is no need to work with a J-spectral factorization, which is the essential
aspect that renders our results computational. As the price to pay,
we require to enforce the coupling (7.21) between the certificates for
the stability FDI (7.17) and (7.19). It comes to our benefit that all
conditions impose LMI constraints on P .
178 Chapter 7. Invariance with dynamic multipliers
Proof. Step 1. We start by proving the following fact: There exists
some γ > 0 such that for all trajectories of (7.3) with d ∈ D , w ∈ L2e
and for all T > 0 one has∫ T
0
1
γ
‖z(t)‖2−γ‖d(t)‖2 dt ≤ −
∫ T
0
zΨ(t)
TPzΨ(t) dt−
(
?
)T
Xs
(
ξ(T )
x(T )
)
.
(7.22)
Let us consider (7.13) for the given P and the choices (C2 D21 D22) :=
(C1 D11 D12), X := Xs, and Pp := diag(γ−1I,−γI) with γ > 0; then,
the performance measure (7.11) implies stability as in (7.6). If we denote
the left-hand side of (7.18) by K11, a simple computation shows that
the left-hand side of (7.13) can then be expressed as
(
K11 K12
K21 K22
)
+
(
1
γL11
1
γL12
1
γL21
1
γL22 − γI
)
, (7.23)
where Kij , Lij are matrix blocks that do not depend on γ. Since
K11 ≺ 0, there does exist some (large) γ > 0 such that (7.23) is negative
definite. Now apply Lemma 7.8 with x0 = 0. Due to our choice of the
matrices in (7.13), the dissipation inequality (7.14) is identical to (7.22),
which proves the claim.
Step 2. In this step we derive a bound on the integral on the right-
hand side of (7.22) in terms of the solution to a suitable LQ problem.
We employ similar arguments as in [150] but extend these to only locally
defined uncertainties and local IQCs. For inputs z, w ∈ L2e and the
response zΨ of (7.9) let us use the abbreviation
FΨ(z, w) := z
T
ΨPzΨ =
[
Ψ
(
z
w
)]T
P
[
Ψ
(
z
w
)]
.
The assumption ∆ ∈ IQCZ (P,Ψ) then translates into∫ ∞
0
FΨ(z,∆(z))(t) dt ≥ 0 for all z ∈ Z , (7.24)
7.3. Main result 179
and we immediately extract the trivial lower bound∫ T
0
FΨ(z,∆(z))(t) dt ≥ −
∫ ∞
T
FΨ(z,∆(z))(t) dt (7.25)
for the finite horizon integral in (7.22) and all T > 0, z ∈ Z .
Note, however, that this inequality cannot be directly applied for
the response of (7.2) to d ∈ D , since well-posedness merely guarantees
that the truncated signal zT (and not z ∈ Ze itself) is contained in Z .
As a remedy, we construct a finite energy signal z˜ ∈ Z that coincides
with z ∈ Ze on [0, T ], but not necessarily on (T,∞). Then we use the
resulting freedom in order to arrive at a lower bound in (7.25) that can
be related to Y22.
To do so, fix any d ∈ D and the resulting response z of (7.2) as well as
any T > 0. We concatenate zT |[0,T ] with another signal zf ∈ L2(T,∞)
in such a way that the concatenation z˜, defined as
z˜(t) := (zT ∧
T
zf )(t) :=
{
zT (t), t ∈ [0, T ],
zf (t), t > T,
(7.26)
is contained in Z . This is always possible since zf = 0|(T,∞) is a valid
choice due to (Ze)T ⊂ Z . Let w˜ := ∆(z˜). Causality of ∆ then implies
w˜T = ∆(z˜)T = ∆(z˜T )T = ∆(zT )T = ∆(z)T = wT .
Hence the signals z˜, w˜ coincide with the actual system trajectories
z, w = ∆(z) on [0, T ] and we infer from (7.24) and (7.25) applied to
z˜ ∈ Z that∫ T
0
FΨ(z,∆(z))(t) dt ≥ −
∫ ∞
T
FΨ(z˜,∆(z˜))(t) dt. (7.27)
We can tightly bound the left hand side from below as∫ T
0
FΨ(z,∆(z))(t) dt ≥ J(ξ(T )) (7.28)
180 Chapter 7. Invariance with dynamic multipliers
if we define J(ξ(T )) as the value of the optimization problem
J(ξ(T )) := sup
zf∈L2(T,∞)
z˜∈Z
−
∫ ∞
T
FΨ(z˜, w˜)(t) dt
with z˜ as in (7.26), w˜ = ∆(z˜) and subject to
ξ˙ = AΨξ +BΨ,1z˜ +BΨ,2w˜, ξ(0) = 0,
z˜Ψ = CΨξ +DΨ,1z˜ +DΨ,2w˜. (7.29)
Since z˜, w˜ coincide with z, w on [0, T ], respectively, the state ξ(T ) to
which ξ(.) has evolved at time T is independent of zf ∈ L2(T,∞). Fur-
thermore, due to z˜|(T,∞) = zf , the response z˜Ψ of (7.29) also coincides
on (T,∞) with the one of
˙˜
ξ = AΨξ˜ +BΨ,1zf +BΨ,2wf , ξ˜(T ) = ξ(T ),
z˜Ψ = CΨξ˜ +DΨ,1zf +DΨ,2wf (7.30)
in case that wf = w˜|(T,∞). Due to z˜ ∈ Z and by (7.5) we infer
w˜ = ∆(z˜) ∈ L2 and thus w˜|(T,∞) ∈ L2(T,∞). This implies
−
∫ ∞
T
FΨ(z˜, w˜)(t) dt ≥ inf
wf∈L2(T,∞)
−
∫ ∞
T
z˜Ψ(t)
TP z˜Ψ(t) dt
under the constraints (7.30). Always subject to (7.30) we conclude
sup
zf∈L2(T,∞)
z˜∈Z
−
∫ ∞
T
FΨ(z˜, w˜)(t) dt
≥ sup
zf∈L2(T,∞)
z˜∈Z
inf
wf∈L2(T,∞)
−
∫ ∞
T
z˜Ψ(t)
TP z˜Ψ(t) dt (7.31)
≥ sup
zf=0
inf
wf∈L2(T,∞)
−
∫ ∞
T
z˜Ψ(t)
TP z˜Ψ(t) dt. (7.32)
7.3. Main result 181
Clearly (7.30), (7.32) describes a standard LQ problem on (T,∞);
it is worth noting that the final estimate completely eliminates the
dependence of this problem on the set Z .
Step 3. If J˜(ξ(T )) denotes the value of
inf
w∈L2
∫ ∞
0
y(t)T (−P )y(t) dt (7.33)
subject to the dynamics
x˙ = AΨx+BΨ,2w, x(0) = ξ(T ),
y = CΨx+DΨ,2w, (7.34)
we hence infer from Step 2. that J(ξ(T )) ≥ J˜(ξ(T )). With(
Q S
ST R
)
:=
(
CΨ DΨ,2
)T
(−P ) (CΨ DΨ,2)
the ARE corresponding to the LQ problem (7.33), (7.34) reads according
to [180, 116] as
ATΨZ22 + Z22AΨ +Q− (Z22BΨ,2 + S)R−1(?)T = 0. (7.35)
We now exploit −(7.20) in order to infer R  0 and, by taking the
Schur complement with respect to this block, that −Y22 satisfies the
corresponding strict algebraic Riccati inequality
−ATΨY22 − Y22AΨ +Q− (−Y22BΨ,2 + S)R−1(?)T  0.
Standard results from Riccati theory (as formulated, e.g., in [145, The-
orem 2.23]) then show, on the one hand, that the stabilizing solu-
tion Z22,− of (7.35) does exist and, on the other hand, that it sat-
isfies Z22,−  −Y22. Furthermore, we infer that the optimal value
J˜(ξ(T )) equals ξ(T )TZ−ξ(T ) and can, hence, be estimated as J˜(ξ(T )) ≥
ξ(T )T (−Y22)ξ(T ). In summary, we have shown
J(ξ(T )) ≥ ξ(T )T (−Y22)ξ(T ). (7.36)
182 Chapter 7. Invariance with dynamic multipliers
Step 4. For any d ∈ D and the responses z, w of (7.2) driving (7.9),
we can combine (7.28) with (7.36) to conclude∫ T
0
zΨ(t)
TPzΨ(t) dt ≥ ξ(T )T (−Y22)ξ(T ) (7.37)
for all T > 0. In combination with (7.22) we end up with
1
γ
∫ T
0
‖z(t)‖2 dt ≤ γ
∫ T
0
‖d(t)‖2 dt− (?)T (Xs11 − Y22 Xs12
Xs21 X
s
22
)(
ξ(T )
x(T )
)
for all T > 0. If finally using (7.21), we infer∫ T
0
‖z(t)‖2 dt ≤ γ2
∫ T
0
‖d(t)‖2 dt ≤ γ2‖d‖2 <∞
for all T > 0. This shows z ∈ L2 and ‖z‖ ≤ γ‖d‖ by taking T → ∞.
Since d ∈ D was arbitrary, the proof is finished.
7.4 Relation to existing local IQC results
As pointed out in the introduction, several articles have appeared in the
literature that employ dynamic IQC descriptions of uncertainties for
local analysis in particular settings. In order to emphasize the general
applicability of our novel approach, we use this section to illustrate
its connections to previous ones. Following up on the discussion in
the introduction and with Theorem 7.9 at hand, we may now contrast
our approach to the above cited works that also employ (dynamic)
multipliers in regional analysis.
The first part of the proof of Theorem 7.9 was inspired by the
derivation in [150] for Z = L2 and multipliers satisfying
Π11 = Ψ
∗
1PΨ1  0 and Π22 = Ψ∗2PΨ2 ≺ 0 on C∞0 .
Seiler proves in [150] that the value of (7.31) subject to (7.30) equals
ξ(T )TYcξ(T ) where Yc is the stabilizing solution of the indefinite ARE
ATΨYc + YcAΨ +Qc − (YcBΨ + Sc)R−1c (YcBΨ + Sc)T = 0 (7.38)
7.4. Relation to existing local IQC results 183
with (
Qc Sc
STc Rc
)
=
(
CΨ DΨ
)T
P
(
CΨ DΨ
)
.
Moreover, he shows that all certificates of (7.17) satisfy
(
Xs11 − Yc Xs12
Xs21 X
s
22
)
 0, (7.39)
which permits to apply Lyapunov arguments to prove stability. Yet, as
a major stumbling block and due to the non-convex constraint (7.38),
there seems to be no possibility to incorporate Yc in the convex search
for multipliers to verify stability. On the one hand, this prevents
the use of Yc for merging IQC theory with Lyapunov techniques for
computational regional stability and performance analysis as addressed
in the remainder of this paper. And, on the other hand, this motivates
us to work with the lower bound −Yc < −Y22, which can be included
in computations since Y22 is coupled to the multipliers by the convex
LMI constraint (7.20). In the sequel, it will be a key additional benefit
over [150] that the uncertainty IQCs are not required to be valid on the
whole space L2.
In [153, 32] the authors work with hard IQCs and assume positivity
of the certificates for the stability and performance FDIs. As discussed
in Section 7.2.3, these restrictions simplify proofs considerably and
provide the main motivation for the current work. Indeed, it is to be
expected from global analysis (see Chapter 5) that the possibility to
employ soft IQCs can substantially reduce conservativeness in local
analysis as well; this will be illustrated in Examples 7.15 and 7.16 of
Section 7.6.
Balakrishnan focusses in [13] on diagonally repeated LTI uncertain-
ties ∆ captured by dynamic D-scalings. Consider the special case of a sin-
gle k-times repeated uncertainty, i.e., ∆ defined by wˆ(iω) = δ(iω)Ikzˆ(iω)
184 Chapter 7. Invariance with dynamic multipliers
with δ ∈ RH∞ satisfying ‖δ‖∞ ≤ 1. Then [13] uses the multipliers (7.8)
with Ψ := diag(ψ,ψ) for a fixed ψ ∈ RHν×k∞ and ν ∈ N as well as
P ∈ P :=
{(
P11 0
0 −P11
) ∣∣∣∣ ψ∗P11ψ  0 on C∞0 } .
As main technical results in [13], it is shown that the feasibility of (7.18)
with P ∈ P is equivalent to the feasibility of (7.18) where both P11 and
Xs are positive definite; again this allows to infer robust stability with
hard IQC arguments. However, the proofs heavily rely on the special
structure of Ψ∗PΨ, including a classical commutation property with the
uncertainties and a particular choice of ψ; this limits the applicability of
[13] to a rather specific setting. We demonstrate the enhanced flexibility
of our general approach for parametric uncertainties with D/G scalings
in Example 7.14 of Section 7.6.
Finally, in [52] a generalized sector condition [157, 83] is employed
in the verification of contractive invariance of an ellipsoid. Theorem
7.9 now justifies the conclusion in [52] that it suffices to guarantee the
validity of a local (static) IQC defined through this ellipsoid in order to
conclude interconnection stability.
7.5 Application to regional performance criteria
Let us now support our claim that local IQCs and the choice of bounded
disturbance input sets D significantly widen the applicability of the IQC
framework. This is done by demonstrating how to verify several regional
invariance properties using dynamic IQCs with only minor modifications.
The subsequent list is by no means complete, but the extension to other
questions (as shown, e.g., in [13]) is pretty straightforward.
7.5. Application to regional performance criteria 185
7.5.1 Invariance with general dynamic IQC multi-
pliers
As a first application, we consider one of the most common examples
for regional analysis, namely the computation of invariant sets of the
state-space
d ∈ Dα := {d ∈ L2 | ‖d‖ ≤ α} for some fixed α > 0. (7.40)
Theorem 7.10
Suppose that the stable LTI system N and the causal ∆ : Ze → L2e
with (7.5) are interconnected as in (7.2). Further assume that (7.2) is
well-posed on Dα and that there exists some P = PT such that
a) ∆ ∈ IQCZ (P,Ψ);
b) there exists a certificate X of the FDI (7.12) with Pp = diag(0,−I),
a certificate Y22 of (7.19), and some H = HT withH 0 I0 X11 − Y22 X12
I XT12 X22
  0. (7.41)
Then (7.2) is stable on Dα and, for any d ∈ Dα, the state trajectories
x(.) of (7.2) (starting at the origin) satisfy
x(t) ∈ E(H−1, α2) for all t ≥ 0. (7.42)
For the formulation of this result it is relevant to emphasize that,
due to Theorem 7.9, we are in the position to proceed exactly as is
often done in Lyapunov theory. In order to verify some desired extra
properties for the trajectories of the interconnection (7.2), we solely
impose appropriate additional LMI constraints on the matrix in (7.21)
that plays the role of defining a suitable Lyapunov function. This
reasoning will also be at the heart of all subsequent results.
186 Chapter 7. Invariance with dynamic multipliers
Proof. Stability of (7.2) on Dα follows immediately from Theorem 7.9.
To see this, we cancel the last block row/column of (7.13); due to the
structure of Pp we infer that (7.18) holds for Xs replaced by X; since
(7.41) for Xs = X implies (7.21), all hypothesis in Theorem 7.9 are
satisfied.
To show invariance we fix any d ∈ Dα and consider the response
of (7.2) driving (7.9). Lemma 7.8 with Pp = diag(0,−I) and x0 = 0
leads to(
ξ(T )
x(T )
)T
X
(
ξ(T )
x(T )
)
+
∫ T
0
zΨ(t)
TPzΨ(t) dt ≤
∫ T
0
‖d(t)‖2 dt (7.43)
for all T > 0. We then combine (7.37), as shown in the proof of
Theorem 7.9, with (7.43) and ‖d‖ ≤ α to infer(
ξ(T )
x(T )
)T (
X11 − Y22 X12
XT12 X22
)(
ξ(T )
x(T )
)
≤ α2 (7.44)
for all T > 0. By taking the Schur complement, (7.41) leads to(
0
I
)
H−1
(
0 I
) ≺ (X11 − Y22 X12
XT12 X22
)
. (7.45)
A combination of (7.44) and (7.45) then shows x(T )TH−1x(T ) ≤ α2
for all T > 0, which proves (7.42).
In practical applications one is often interested in bounds on the
individual components zj of the output signal z. For j ∈ {1, . . . , k} let
C1,j denote the rows of C1 in (7.3). Then we get the following result for
the interconnection (7.2) under the further assumption that
(
M N12
)
is strictly proper.
Corollary 7.11
In addition to the assumptions of Theorem 7.10 let D11 = D12 = 0 and
replace b) by the following hypothesis:
7.5. Application to regional performance criteria 187
b’) there exists a certificate X of the FDI (7.12) with Pp = diag(0,−I),
a certificate Y22 of (7.19), and some γj > 0 such that γj 0 C1,j0 X11 − Y22 X12
CT1,j X
T
12 X22
  0 for all j ∈ {1, . . . , k}. (7.46)
Then (7.2) is stable on Dα and, for all d ∈ Dα, the components of the
response z of (7.2) are bounded as
|zj(t)| ≤ √γjα for all t ≥ 0, j ∈ {1, . . . , k}. (7.47)
Proof. The changes in our assumptions necessitate only minor alter-
ations to the proof of Theorem 7.10. By the Schur complement formula
and for any j ∈ {1, . . . , k}, (7.46) implies(
0
CT1,j
)(
0 C1,j
) ≺ γj (X11 − Y22 X12
XT12 X22
)
. (7.48)
Hence, for d ∈ Dα and T > 0 we conclude with (7.44) that
|zj(T )|2 =
(
ξ(T )
x(T )
)T (
0
CT1,j
)(
0 C1,j
)( ξ(T )
x(T )
)
≤
≤ γj
(
ξ(T )
x(T )
)T (
X11 − Y22 X12
XT12 X22
)(
ξ(T )
x(T )
)
≤ γjα2.
This result can be interpreted as providing guaranteed bounds on the
energy to peak gain for the channel d→ z of the uncertain interconnec-
tion (7.2) and based on soft local dynamic IQCs. A mere substitution
of matrices leads to a similar result for the performance channel d→ e.
Our focus on d → z is motivated by the next section, in which these
bounds are exploited in order to improve stability tests with local IQCs.
188 Chapter 7. Invariance with dynamic multipliers
7.5.2 Invariance using regionally valid IQCs
In the previous section we derived bounds on the states and outputs of
the LTI systemN in (7.2), depending on the maximal disturbance energy
α. Let us now demonstrate how these bounds may be used in order
to tighten the uncertainty description and thus reducing conservatism
by adapting the domain of local IQCs accordingly. A similar idea also
appears in the analysis of regionally stable saturated systems (see, e.g.,
[79, 125, 124] and the monographs [157], [83]). In order to do so, we
distinguish between Ze, the domain of definition of the uncertainty, and
subsets Ve ⊂ Ze such that the IQC for ∆ is satisfied on V := Ve ∩L2
only.
Specifically, we work with the amplitude bounded sets
VR,e := {z ∈ L2e | |zj(t)| ≤ Rj for all j ∈ {1, . . . , k}, t ≥ 0}
(7.49)
and VR := VR,e ∩L2 parameterized by R = (R1, . . . , Rk) ∈ Rk+.
Theorem 7.12
Suppose that the stable LTI system N with D11 = D12 = 0 and the
causal ∆ : Ze → L2e with (7.5) are interconnected as in (7.2). With
R ∈ Rk+ and (7.49) let VR,e ⊂ Ze and VR := VR,e∩L2. Further assume
that (7.2) is well-posed on Dα and that there exists some P = PT such
that
a) ∆ ∈ IQCVR(P,Ψ);
b) there exist certificates X of the FDI (7.12) with Pp = diag(0,−I)
and Y22 of (7.19) such that (7.46) is valid for γj := R2j/α2 and
all j ∈ {1, . . . , k}.
Then (7.2) is stable on Dα and (7.47) holds for all d ∈ Dα.
Proof. Fix d ∈ Dα and consider the response of (7.2) driving (7.9).
Since D11 = D12 = 0, z ∈ Ze satisfies z(0) = 0 and is continuous.
Hence it makes sense to define
T := sup {T > 0 | zT ∈ Ve} ∈ (0,∞].
7.5. Application to regional performance criteria 189
We then infer |zj(t)| ≤ Rj for all j ∈ {1, . . . , k} and all t ∈ [0, T ), which
shows zT ∈ VR for all T ∈ (0, T ). For T ∈ (0, T ), this allows us to follow
Steps 2. and 3. of the proof of Theorem 7.9 with z˜ = zT ∧
T
zf where
zf is now chosen in such a way that z˜ ∈ VR. We can hence exploit
∆ ∈ IQCVR(P,Ψ) and conclude as before that∫ T
0
zΨ(t)
TPzΨ(t) dt ≥ ξ(T )T (−Y22)ξ(T ) for all T ∈ (0, T ).
(7.50)
As in the proof of Theorem 7.10 we have (7.43) for all T ∈ (0, T ). This
can be combined with (7.50) to infer (7.44) for all T ∈ (0, T ).
Let us now assume that T < ∞. On the one hand, by continuity,
(7.44) then also holds for T = T and with (7.48) as well as γj = R2j/α2
we infer
|zj(T )|2 < R2j for all j ∈ {1, . . . , k}. (7.51)
On the other hand and again by continuity, we have |zj(T )| ≤ Rj for
all j ∈ {1, . . . , k} and, due to the definition of T , there must exist some
index j0 ∈ {1, . . . , k} with
|zj0(T )| = Rj0 . (7.52)
This contradiction to (7.51) allows to infer T = ∞, and thus also
|zj(t)| ≤ √γjα for all t ≥ 0 and all j ∈ {1, . . . , k}.
In summary, we have shown z ∈ VR,e for the response of (7.2) to
any d ∈ Dα. We may hence restrict ∆ to the smaller set VR,e ⊂ Ze
while still maintaining well-posedness of the resulting interconnection
(7.2) on Dα. This permits us to apply Corollary 7.11 for Z replaced by
VR; hence (7.2) is stable on Dα.
7.5.3 Excitation through nonzero initial conditions
As a final topic, let us consider the interconnection (7.2) without per-
formance channel as in
z = Mw, w = ∆(z). (7.53)
190 Chapter 7. Invariance with dynamic multipliers
The only excitation is given by the nonzero initial condition of the LTI
system M with the realization
x˙ = Ax+B1w, x(0) = x0 ∈ Rn,
z = C1x. (7.54)
As is standard [93, 58], we say that (7.53) is well-posed for all initial con-
ditions if (7.2) is well-posed on the subspace D :=
{
C1e
A•x0
∣∣ x0 ∈ Rn}
with N12 = I.
Theorem 7.13
Suppose that the stable LTI system M (7.54) and the causal ∆ : Ze →
L2e with (7.5) are interconnected as in (7.53). Let R, VR,e and VR be
defined as in Theorem 7.12. Further, suppose that (7.53) is well-posed
for all initial conditions and that there exists some P = PT such that
a) ∆ ∈ IQCVR(P,Ψ);
b) there exist a certificate X of (7.17), a certificate Y22 of (7.19),
and some H = HT such that (7.41) holds;
c) for all j ∈ {1, . . . , k} the matrix H also satisfies(
H HCT1,j
C1,jH R
2
j
)
 0.
Then the state trajectories x of (7.53), (7.54) with initial conditions
x0 ∈ E(X22, 1) satisfy
x(t) ∈ E(H−1, 1) for all t ≥ 0. (7.55)
Proof. We follow the proof of Theorem 7.12 and consider the response
of (7.53) to some fixed x0 ∈ E(X22, 1). Using the Schur complement,
assumptions b) and c) imply (7.45) and
CT1,jC1,j ≺ R2jH−1 for all j ∈ {1, . . . , k}, (7.56)
7.6. A selection of concrete applications 191
respectively. Since (7.56) is strict, we can choose some ε > 0 such that
(7.45) with (7.56) imply
|zj(T )|2 ≤ (Rj − ε)2
(
ξ(T )
x(T )
)T (
X11 − Y22 X12
XT12 X22
)(
ξ(T )
x(T )
)
(7.57)
for all j ∈ {1, . . . , k} and T ≥ 0.
For T = 0, we infer with ξ(0) = 0 and xT0 X22x0 ≤ 1 that
|zj(0)|2 < R2j for all j ∈ {1, . . . , k}.
Thus we can define T ∈ (0,∞] as in the proof of Theorem 7.12.
By Lemma 7.7 and 7.8 applied to (7.17), the dissipation inequality(
ξ(T )
x(T )
)T
X
(
ξ(T )
x(T )
)
+
∫ T
0
zΨ(t)
TPzΨ(t) dt ≤ xT0 X22x0
holds for all T > 0. In complete analogy to previous arguments, we first
obtain (7.50) and, with xT0 X22x0 ≤ 1, also(
ξ(T )
x(T )
)T (
X11 − Y22 X12
XT12 X22
)(
ξ(T )
x(T )
)
≤ 1 for all T ∈ (0, T ). (7.58)
In combination with (7.57) we finally arrive at
|zj(T )|2 ≤ (Rj − ε)2 for all j ∈ {1, . . . , k}, T ∈ (0, T ). (7.59)
Now suppose T < ∞. Since z is continuous, we can argue again
that there exists some j0 ∈ {1, . . . , k} with (7.52), while (7.59) shows
|zj0(T )| < Rj0 , a contradiction.
Consequently, T = ∞ and z ∈ VR,e. Moreover, (7.58) and (7.45)
clearly show (7.55). Finally, z ∈ L2 is proven as for Theorem 7.12.
7.6 A selection of concrete applications
The great practical relevance of the stability and performance analysis
problems discussed in the previous sections has sparked the development
192 Chapter 7. Invariance with dynamic multipliers
of many specialized approaches for in itself relevant subproblems. In
this section we illustrate the benefits of our novel framework, in that it
is not only generally applicable to a large variety of such specializations
but even leads to often much less conservative results if compared to
several recently developed alternative techniques.
7.6.1 Real parametric uncertainties
Let us start by considering real parametric uncertainties where ∆ is
defined as ∆(z) = δIkz for z ∈ L k2e and with δ ∈ R satisfying |δ| ≤ κ.
IQCs for such uncertainties may be described by dynamic D or D/G
scalings as discussed in detail, e.g., in [176, Sec. 5.3.1]. Dynamic D
scalings were already employed in [13] in order to perform regional
analysis. As one of the main motivation for the current paper, we
emphasize that the techniques developed in [13] do not extend to other
multiplier classes for parametric uncertainties or to general IQCs. Let us
hence demonstrate that our approach opens the way for regional stability
analysis with D/G scalings, thus allowing for significant reduction of
conservatism.
Example 7.14.
Specifically, consider the interconnection (7.2) with k = 1 where N is
realized by (A,B,C,D) given as
A =
−2 −1 −11 0 0.1
0 1 0
 , B =
 1 20 −0.1
−0.1 0.2
 ,
C =
(−1.1 0.5 0.1) , D = (0 1) .
Our goal is to determine an ellipsoid of smallest size that bounds the
state of N for any d ∈ Dα using Theorem 7.10. As a measure for the
size of the ellipsoid in (7.42), E(H−1, α2), we choose the trace of H.
7.6. A selection of concrete applications 193
With a vector of stable basis functions ψν ∈ RHν+1∞ of McMillan degree
ν ≥ 0 and free matrix variables P11 = PT11, P12 consider the multiplier
ΠDG :=
(
κψν 0
0 ψν
)∗(
P11 P12
PT12 −P11
)(
κψν 0
0 ψν
)
. (7.60)
If P11 satisfies ψ∗νP11ψν  0 and P12 is skew symmetric, one easily
checks ∆ ∈ IQCL2(ΠDG) [176]. We obtain the conditions for D-scalings,
as derived in [13], by setting P12 = 0, assuming P11  0, and dropping
Y22 in the LMI (7.41). In order to compare our results to [13], we also
choose ψν as
ψν(s) =
(
1 1(s−p) . . .
1
(s−p)ν
)T
with p < 0, (7.61)
and consider the two parameter bounds κ = 0.4 and κ = 0.8.
The computed values for the optimal trace of H as achieved with
the pole location p = −1 and with different basis lengths ν are shown
in Table 7.1 (for κ = 0.4) and Table 7.2 (for κ = 0.8).
If ν = 0, the skew symmetric matrix P12 ∈ R vanishes and, as
expected, both multiplier classes lead to the same sizes of the invariant
ellipsoid. By contrast, for increasing McMillan degrees of the basis
functions, the use of ΠDG instead of ΠD leads to much tighter ellipsoidal
bounds as seen in Table 7.1.
We may further improve these results as follows. Using the optimal
multiplier Πopt ∼ (Popt,Ψ) as obtained from the optimization conducted
in the second row of Table 7.1, we compute the shift Yc by solving the
ARE (7.38). By fixing P = Popt and Y22 = Yc and again optimizing
trace(H) now subject to X and H, we achieve the slightly improved
values of trace(H) as stated in the last row of Table 7.1. Note that this
two step procedure will always lead to the same or improved results due
to the relation −Yc < −Y22.
If we increase the parameter bound to κ = 0.8, the LMIs from [13]
involving ΠD are infeasible for values of ν up to 8, while the use of ΠDG
leads to the ellipsoidal bounds as shown in the second row of Table 7.2.
194 Chapter 7. Invariance with dynamic multipliers
Table 7.1: Bounds on trace(H) for κ = 0.4
ν 0 1 2 4 8
ΠD [13] 285 265.2 235.2 234.9 234.9
ΠDG 285 15.1 14.1 14.03 14.02
ΠDG, improved 285 13.77 13.35 13.39 13.38
Table 7.2: Bounds on trace(H) for κ = 0.8
ν 0 1 2 4 8
ΠDG ∞ 32.7 27.1 26.1 25.9
ΠDG, improved ∞ 25.6 24.0 23.9 24.0
Moreover, applying the two step procedure discussed above, we infer
the improved estimates stated in the last row. These results nicely
illustrate the benefit of extra dynamics in the D/G scalings (increased
lengths ν) to improve the bounds; we emphasize again that no other
multiplier-based technique in the literature is able to provide these
guarantees.
?
7.6.2 Locally stable saturated systems
As one of the major driving forces behind regional stability analysis, we
now consider open loop unstable systems with a stabilizing saturated
unity output feedback controller. Specifically, let the interconnection
be given by
x˙ = Ax+B1w +B2d, x(0) = 0, w = sat(z),
z = C1x (7.62)
7.6. A selection of concrete applications 195
with the unit saturation function sat : R→ R. Here we assume that A+
B1C1 is Hurwitz (but not necessarily so is A). With the standard dead-
zone nonlinearity dz = id− sat, (7.62) may be equivalently expressed
as
x˙ = (A+B1C1)x−B1w +B2d, x(0) = 0, w = dz(z),
z = C1x. (7.63)
Let us briefly recap from [79] how, adopted to our framework, one
may capture local properties of the dead-zone nonlinearity in order to
exploit the classical circle criterion for regional stability analysis. Fix
some R ≥ 1. Then dz(.) satisfies the local sector constraint
dz(x)((1− 1/R)x− dz(x)) ≥ 0 for all x ∈ [−R,R];
in short dz ∈ secR[0, 1− 1/R]. With
P :=
{
λ
(
0 1− 1R
1− 1R −2
)
: λ > 0
}
, Ψ := I2,
this implies for any P ∈ P that(
x
dz(x)
)T
Ψ∗PΨ
(
x
dz(x)
)
≥ 0 for x ∈ [−R,R].
With (7.49) (k = 1) and VR := VR,e ∩L2 we immediately infer
dz ∈ IQCVR(P,Ψ) for all P ∈ P.
This puts us in the position to apply Theorem 7.12 to (7.63) for Ze :=
L2e and draw the following conclusion: If there exists some P ∈ P such
the assumption b) is valid (which amounts to solving an LMI feasibility
problem) then (7.63) is stable on Dα; with an LMI optimization problem
one can directly determine the supremal value of α > 0 for which this
is true. Note that α depends on the chosen R ≥ 1, and either a plot of
α over R or a line-search finally allows us to compute the largest value
196 Chapter 7. Invariance with dynamic multipliers
α > 0 such that interconnection stability on Dα is guaranteed by the
chosen class of multipliers.
All this has been proposed in the literature and extended to multi-
variable saturations with diagonal multipliers for the circle criterion. If
only using static multipliers, we point out that Y22 is an empty matrix
and, thus, our approach recovers these results as special cases.
In global stability analysis, the benefit of, e.g., Zames-Falb multipliers
has been often emphasized in the literature. This is also visible from
our detailed exposition of multiplier implementations based on (7.61)
in Chapter 5.
Our new approach now opens the way to exploit this superiority for
regional analysis with ease. We just employ any valid class of multipliers
for the dead-zone nonlinearity and apply Theorem 7.12 in exactly the
same fashion as for the circle criterion. It is as well possible to exploit
the extra information that dz is odd and its slope is confined to [0, 1]
with soft IQCs, for which the introduction of Y22 is instrumental.
Let us now compare our results to the ones achieved by Lyapunov
function techniques in [51] which are specifically designed to deal with
the saturation or dead-zone. In contrast to our approach that extends
[79] to dynamic IQCs, the method proposed by Fang et al. [51] relies
on a reformulation of the dead-zone nonlinearity as a time dependent
parametric uncertainty and does not involve any line search.
Example 7.15.
Let the system in (7.62) be defined by
A =
0.05 1 20 −0.4 −2
0 1 −0.7
 , B1 =
 12−0.2
−1
 , B2 =
 0.2−0.1
0.5
 ,
and C1 =
(−.1 −1.5 −1). Then A is obviously not Hurwitz but
A+B1C1 is. Hence we can locally analyze the interconnection in (7.63).
If R ∈ [0, 1] we note that the system operates in the so-called linear
region where the output of the dead-zone is zero. This is the case for
7.6. A selection of concrete applications 197
5 10 15 20
R
0
5
10
,
CC
8=3
8=5
8=7
Figure 7.2: Maximal disturbance energy over R
disturbance energies smaller than α = 1.41. The approaches in [79] and
[51] permit an increase of the allowed energy level without endangering
stability to α = 1.45 and α = 2.07, respectively.
As seen next, dynamic multipliers certify levels of α = 9.04 which
amounts to a reduction of conservativeness by more than a factor of four
if compared to [51]. Figure 7.2 shows estimates of the maximal tolerable
energy level α plotted versus R for the local circle criterion (CC) [79]
and a combination of circle criterion and Zames-Falb multipliers for
different basis lengths ν and p = −1 in (7.61) (as detailed in Chapter 5).
We observe that the LMIs remain feasible for radii up to R = 23.6 and
all three depicted basis lengths ν. Moreover, dynamic multipliers allow
for a significant increase of the maximal tolerable disturbance energy,
with improvements that get more pronounced for larger values of ν;
indeed, we reach α = 9.04 for ν = 7.
Let us now augment the interconnection (7.63) with a performance
output e =
(
1 1 1
)
x and estimate the local L2-gain γ from d to e
for α ∈ [0, 9.04]; since interconnection stability is guaranteed, we can
use Theorem 7.6 with Pp = diag(I,−γ2I) for computations. As the
approach by Fang et al. in [51] outperforms the local circle criterion for
198 Chapter 7. Invariance with dynamic multipliers
1 2 3 4 5 6 7 8 9 10
,
0
100
200
300
400
.
ZF, 8=7
Fang et al.
Figure 7.3: Local L2-gain γ plotted over disturbance energy bound α.
the present example, we only compare our method for ν = 7, p = −1
to theirs. Figure 7.3 depicts the computed values for α ≥ 1.41. For
energy bounds close to α = 1.41, both methods return the L2-gain of
the underlying linear system. If α approaches the maximal tolerable
values for the the respective procedure, the estimated L2-gains tend to
infinity; with the technique in [51] this happens for considerably lower
values of α.
Thus we conclude that dynamic multipliers allow for a much more
accurate description of the nonlinear effects of the dead-zone, which
permits the system to enter further into the nonlinear regime as driven
by higher disturbance energies and without loosing stability. In addition,
this also translates into much less conservative L2-gain estimates. The
price to pay, however, is higher computational complexity. This is
illustrated in Table 7.3 which displays the number of decision variables
and the computation times of our implementation (not optimized for
computational performance) for a fixed value of R, together with the
maximal disturbance energy α obtained for different basis lengths.
Although Zames-Falb multipliers for ν = 3 significantly outperform the
circle criterion ones, the computation time is only slightly increased.
7.6. A selection of concrete applications 199
Table 7.3: Computational burden in Example 7.15
ν 0 3 5 7
Maximal disturbance energy α 1.45 6.59 8.39 9.04
Number of decision variables 10 87 195 351
Computation time [s] 0.25 0.35 0.83 3.67
For larger basis lengths ν, however, the increase in computational effort
intensifies. ?
7.6.3 Unbounded nonlinearities
In our last example, let us consider unbounded nonlinearities as dis-
cussed in [153], but for hard IQCs only. It is well worth to illustrate
how unbounded nonlinearities may be easily incorporated into our soft
multiplier framework. Consider (7.2) with ∆ϕ : L2e → L2e defined by
ϕ(x) = x3 as
∆ϕ(z)(t) = ϕ(z(t)) for almost all t ≥ 0 (7.64)
and z∈L2e. If the response of the loop is guaranteed to satisfy z ∈ VR,e
with VR,e from (7.49), we may modify ϕ(x) = x3 to
ϕ˜(x) :=
{
ϕ(x), |x| ≤ R,
sgn(x)R3, |x| > R,
and in turn replace ∆ϕ by ∆ϕ˜ while still maintaining the same loop
dynamics. Moreover, ∆ϕ˜ is globally bounded on L2e and ϕ˜ satisfies
the local sector constraint ϕ˜ ∈ secR[0, R2], while its slope is locally
restricted by 3R2. All this permits to perform a regional stability and
performance analysis on the basis of soft local IQCs on VR exactly as
in the previous paragraph.
200 Chapter 7. Invariance with dynamic multipliers
Before we consider a concrete example, let us briefly point out a
special feature of nonlinearities of the form (7.64). If θ ∈ R+ then
θ3ϕ(θ−1x) = ϕ(x) for all x ∈ R. Thus, for θ = (θ1, . . . , θk)T ∈ Rk+ and
Tθ : L2e → L2e defined by (Tθz)(t) = diag(θ1, . . . , θk)z(t) for t ∈ [0,∞)
we infer
Tθ3∆ϕTθ−1 = ∆ϕ on L2e. (7.65)
Hence, for all θ ∈ Rk+, the interconnection (7.2) with ∆ = ∆ϕ and
d ∈ Dα remains invariant if we shift the transformation in (7.65) to the
LTI system, i.e., consider N → Nθ with
Nθ =
(
Tθ−1 0
0 id
)(
M N12
N21 N22
)(
Tθ3 0
0 id
)
. (7.66)
In the sequel we apply our results to this family of interconnections by
optimizing over θ ∈ Rk+. Note that, due to the symmetry of ϕ and Dα,
it suffices to consider positive θj ’s only.
Example 7.16.
Let us adopt an example from [153], where the interconnection depicted
in Figure 7.4 is studied. Here Γ = −1.05 is a static gain and each
subsystem S is given by the upper feedback interconnection of the
nonlinearity ∆ϕ in (7.64) with the LTI system N˜ ∼ (A˜, B˜, C˜, D˜) for
A˜ = −1, B˜ = (1 1) , C˜ = B˜T , D˜ = 0.
It is straightforward to redescribe the interconnection of Figure 7.4 as
in (7.2) where ∆ is defined through ϕ or ϕ˜ being 3-times repeated. The
goal is to estimate the maximal disturbance energy bound α > 0 such
that the interconnection in Figure 7.4 remains stable for all d ∈ Dα and,
subsequently, to compute the L2-gain from d to e.
In [153] two different dissipation approaches are compared. First,
novel hard IQCs are used to regionally capture the effect of ∆ϕ in (7.64).
Second, sum-of-squares (SOS) algorithms from [156] with polynomial
7.6. A selection of concrete applications 201
S S SΓ +
d e
Figure 7.4: Feedback interconnection in [153]
storage functions of different degrees are applied. The first three graphs
in Figure 7.5 are readily estimated from a plot in [153]. As can be seen,
the maximal disturbance energies as well as the corresponding L2-gains
obtained using hard IQCs are significantly more conservative than those
obtained using polynomial storage functions of degree two (SOS (2)) or
four (SOS (4)), respectively. Yet, as emphasized in [153], one should
note that SOS techniques are computationally much more expensive
than those with IQCs.
We apply Theorem 7.12 to the interconnection (7.2) with N replaced
by Nθ, and for fixed Rj = 0.5 for all j using a gridding approach over θ.
The search over θ rather than R is equivalent in terms of computational
burden but offers the additional advantage that the three nonlinearities
ϕ˜ remain identical, as they are truncated at the same value of R. This
enables the application of the larger class of full-block multipliers for the
Zames-Falb criterion [58]. Our goal is to minimize the linear functional∑3
j=1 γj/R
2 using full-block circle and Zames-Falb multipliers with pole
p = −3 and ν = 1; if the optimal value is vopt, the estimated energy
level then equals 1/√vopt.
In the dashed plot in Figure 7.5 we only show the results obtained
using the local (full-block) circle criterion, as Zames-Falb multipliers
offer no additional improvement for the present case. Yet, already the
application of the static multipliers outperforms the dynamic hard IQC
multipliers proposed in [153] both in terms of maximal disturbance
energy and local L2-gain. For a wide range of disturbance energies,
our IQC approach also improves on the L2-gain estimates obtained
using SOS techniques. However, the derived estimates of the maximal
202 Chapter 7. Invariance with dynamic multipliers
0 0.2 0.4 0.6 0.8 1
,
1
1.5
2
.
hard IQC
SOS (2)
SOS (4)
iqc
0.32 0.68
Figure 7.5: Local L2-gain γ plotted over disturbance energy bound α.
disturbance energy for which the interconnection remains stable are
more conservative.
The lack of improvement if using dynamic multipliers can probably
be attributed to the overly simple dynamics of N˜ . Let us hence modify
N˜ to
A˜ =
(−3.5 −6
0.1 −1
)
, B˜ =
(−2 2
1 −1
)
, C˜ =
(
0.4 0.6
0.7 0.1
)
and D˜ = 0. Even though this is a minor change, we now computed
α = 22.3 for the circle criterion and more than α = 105 if including a
full-block Zames-Falb multiplier (with ν = 1, p = −3). If only using
diagonally repeated Zames-Falb multipliers the obtained bounds on the
maximal energy drop by about a factor of four. Moreover, note that
for moderate basis lengths ν, the IQC analysis is computationally very
cheap (requiring only 0.8 s for their solution with fixed Rj) while even
for such academic examples with a small number of states, the SOS
analysis quickly becomes intractable. ?
7.7. Summary and recommendations 203
7.7 Summary and recommendations
We present a new local stability theorem that allows for uncertainty
descriptions with general soft dynamic integral quadratic constraints.
Based on this result, we develop a framework that extends classical
IQC theory and allows to merge Lyapunov techniques with multiplier
approaches. This enables the verification of several performance specifi-
cations, such as invariance of sets in the state and output space, just by
imposing additional LMI constraints. As our novel approach allows for a
very broad class of IQC multipliers, the theory developed is immediately
applicable to a large variety of different problems. This fact is illustrated
by means of various examples and comparisons with related techniques.
Yet, we only touch upon the wealth of possibilities for local analysis
offered by our local IQC stability result. Many further results, similar
to the ones listed in [13] can be derived in exactly the same fashion as
outlined in the present chapter, namely by imposing additional LMI
constraints on the Lyapunov matrix. In addition, and beyond the
analysis of feedback interconnections with LTI systems in the forward
path, we believe that our results allow for straightforward extensions
to linear parameter varying (LPV) systems, which generalizes [178] to
regional analysis with local IQCs. Furthermore, it is expected that our
analysis approach provides the foundation for controller synthesis as in
[149, 175].
However, there are also some obvious limitations of the present
approach. First, we require an additional definiteness property from
the multiplier (Π22 ≺ 0 on C∞0 ) if compared to standard IQC theory
that is often a priori satisfied but also violated in certain interesting
cases. Furthermore, and most importantly, we have to artificially enforce
positivity of the Lyapunov matrix X − diag(Y22, 0). This will typically
result in added conservatism. By contrast, the approach proposed in [13]
allows to work with Y22 = 0 while even guaranteeing positivity of the
certificate X. In the subsequent chapter, we illustrate how to avoid these
204 Chapter 7. Invariance with dynamic multipliers
limitations of the present approach for some concrete example resulting
from multipliers corresponding to slope-restricted nonlinearities.
Chapter 8
Hard Zames-Falb factorizations for invari-
ance
8.1 Introduction
In Chapter 7 we developed a rather general extension of classicalIQC theory towards local analysis, i.e., the analysis of locally stable
systems or the verification of state and output constraints under the
assumption that the input signals are restricted to a certain (bounded)
subset of L2. In the sequel, we recapitulate the setting and main
stability theorem of Chapter 7 in order to highlight several limitations
that sparked the development of the present, more general approach.
Given an uncertain operator ∆ : Ze → L2e, defined on a set Ze
satisfying (Ze)T ⊂ Z , and a stable LTI system N =
(
M N12
N21 N22
)
, realized
as
x˙ = Ax+B1w +B2d, x(0) = 0,
z = C1x+D11w +D12d,
e = C2x+D21w +D22d (8.1)
205
206 Chapter 8. Hard Zames-Falb factorizations for invariance
M N12
N21 N22
∆
de
wz
Figure 8.1: Performance setting
with A ∈ Rn×n being Hurwitz, we consider the standard performance
interconnection (see Figure 8.1)
z = Mw +N12d w = ∆(z), (8.2a)
e = N21w +N22d. (8.2b)
Here, the disturbance d is assumed to be contained in some subset D
of L2 and the uncertainty is required to satisfy the weak boundedness
condition ∆(Z ) ⊂ L2. Well-posedness of (8.2a) on D is then simply
defined by the existence of a unique response z ∈ Ze for each d ∈ D
such that the map d 7→ z is causal.
One of the key contributions of Chapter 7 was the transformation of
general soft IQCs into hard ones in such a way that the residual convexly
depends on the multiplier data. In particular, let Π ∈ RL∞ be factorized
as Ψ∗PΨ with real symmetric P and Ψ ∈ RH∞ (see Section 2.3).
Moreover, let Πij be structured according to the dimensions of the
signals z, w and satisfy the FDI
Π22 ≺ 0 on C∞0 . (8.3)
Then, ∆ ∈ IQCZ (P,Ψ), i.e., ∆ satisfying the soft IQC∫ ∞
0
[
Ψ
(
z
∆(z)
)
(t)
]T
PΨ
(
z
∆(z)
)
(t) dt ≥ 0 for all z ∈ Z (8.4)
8.1. Introduction 207
implies the shifted hard IQC∫ T
0
(
?
)T
PΨ
(
z
∆(z)
)
(t) dt ≥ ξ(T )T (−Y22)ξ(T ) for all z ∈ Z ,
(8.5)
where Y22 is a certificate for (8.3). We can now use this bound together
with the FDI (
M
I
)∗
Ψ∗PΨ
(
M
I
)
≺ 0 on C∞0 (8.6)
in order to arrive at a local IQC stability result using general dynamic
multipliers. For completeness, we restate Theorem 7.9 below.
Theorem 8.1
Suppose that the interconnection (8.2) of the stable LTI system N and
the causal uncertainty ∆, satisfying (7.5), is well-posed on D. Then
(8.2) is stable on D if there exists some P = PT with the following two
properties
a) ∆ ∈ IQCZ (P,Ψ);
b) there exists a certificate Xs of (8.6) and a certificate Y22 of (8.3)
which are coupled as(
Xs11 − Y22 Xs12
Xs21 X
s
22
)
 0. (8.7)
If compared to the classical global result, Theorem 2.6, we imme-
diately infer that the price to pay for locality (apart form Π22 ≺ 0
on C∞0 ) is the additional constraint (8.7). As Y22 is empty for static
multipliers Π, the shifting of Xs in (8.7) is only necessary for dynamic
IQCs. However, for hard IQCs the right hand side in (8.5) is zero.
Hence, one would expect that general hard IQCs can be incorporated
without relying on Y22.
This brings us to the main motivation for the present chapter. In
the first part, we illustrate how hard and soft IQCs can be effectively
208 Chapter 8. Hard Zames-Falb factorizations for invariance
combined in order to avoid unnecessary conservatism introduced by the
lower bound in (8.5). In the second part, we focus on the particularly
interesting example of Zames-Falb multipliers that, as illustrated in
Chapter 5 allow for significant improvements in global stability analysis
if compared to static multipliers. By considering causal and anti-causal
Zames-Falb multipliers separately, another contribution of this chapter
is to prove that both can be losslessly incorporated into the framework
presented in Chapter 7. In particular, for causal and anti-causal Zames-
Falb multipliers the additional constraint (8.7) is a priori satisfied.
Consequently, these multiplier classes admit local analysis with identical
hypothesis if compared to the global case (Theorem 2.6). As supported
by a concrete numerical example, this leads to significantly improved
stability margins.
Finally, we emphasize again that a self-contained version of the
results developed in the present chapter is accepted for publication [59].
8.2 Local analysis with hard and soft IQCs
We lay the foundation for the combination of hard and soft IQCs by
immediately turning our focus to the case where we have two available
IQCs for the same uncertainty ∆ that are both valid on Z and imposed
by Π(k) ∼ (P (k),Ψ(k)) for k = 1, 2. Using the factorization in (7.8)
for both Π(k), we obtain a structured factorization of the sum Π :=
Π(1) + Π(2) as
Π ∼ (P,Ψ) =
((
P (1) 0
0 P (2)
)
,
(
Ψ(1)
Ψ(2)
))
, (8.8)
and infer from (8.4) that ∆ ∈ IQCZ (P,Ψ) holds. In the sequel we tacitly
assume that Π(k) = (Π(k)ij ) and Ψ
(k) = (Ψ
(k)
1 Ψ
(k)
2 ) are partitioned
according to the dimensions of z and w in (8.4); moreover, Ψ(k) is
supposed to be realized as
ξ˙(k) = A
(k)
Ψ ξ
(k) +B
(k)
Ψ,1z +B
(k)
Ψ,2w, ξ
(k)(0) = 0,
8.2. Local analysis with hard and soft IQCs 209
z
(k)
Ψ = C
(k)
Ψ ξ
(k) +D
(k)
Ψ,1z +D
(k)
Ψ,2w (8.9)
with A(k)Ψ being Hurwitz and k = 1, 2.
As will become clear in the sequel, it is indeed beneficial to distinguish
in this way between multipliers defining hard (subsumed in Π(1)) and
soft IQCs (in Π(2)). Thus, we further require the validity of the hard
IQC constraint∫ T
0
z
(1)
Ψ (t)
TP (1)z
(1)
Ψ (t) dt ≥ 0 for all T > 0, z ∈ Z , (8.10)
imposed by the first multiplier Π(1) ∼ (P (1),Ψ(1)). We abbreviate (8.10)
as ∆ ∈ HIQCZ (P (1),Ψ(1)) in the sequel.
In deviating from the setup in the previous chapter, we require the
additional property
Π
(2)
22 = (Ψ
(2)
2 )
∗P (2)Ψ(2)2 ≺ 0 on C∞0 , (8.11)
which is certified by the LMI(
(A
(2)
Ψ )
TY
(2)
22 + Y
(2)
22 A
(2)
Ψ Y
(2)
22 B
(2)
Ψ,2
(B
(2)
Ψ,2)
TY
(2)
22 0
)
+
(
?
)T
P (2)
(
C
(2)
Ψ D
(2)
Ψ,2
)
≺ 0,
(8.12)
only for the second multiplier. Apart from the fact that (8.11) is, e.g.,
not valid for certain multipliers capturing sector-bounded nonlinearities
(see Chapter 5), it will be of crucial importance for our non-conservative
incorporation of Zames-Falb multipliers into local analysis that we
require (8.11) only for Π(2).
8.2.1 Two local IQC results
Naturally, is possible to embed all local stability and performance
theorems derived in Chapter 7 into the present setting. We illustrate
this exemplarily for two results.
210 Chapter 8. Hard Zames-Falb factorizations for invariance
Let us start with a reformulation of Theorem 8.1 that allows for the
simultaneous use of hard and soft IQCs. The key ingredients are the
lower bound (8.5) for the second (soft) multiplier∫ T
0
z
(2)
Ψ (t)
TP (2)z
(2)
Ψ (t) dt ≥ ξ(2)(T )T (−Y (2)22 )ξ(2)(T ) for all T > 0,
(8.13)
and the following stability FDI, adjusted to the split in hard and soft
constraints:(
?
)∗(P (1) 0
0 P (2)
)(
Ψ
(1)
1 Ψ
(1)
2
Ψ
(2)
1 Ψ
(2)
2
)(
M
I
)
≺ 0 on C∞0 . (8.14)
Theorem 8.2
Suppose that the interconnection (8.2a) is well-posed on D . Then (8.2a)
is stable on D if the following conditions hold:
a) ∆ ∈ HIQCZ (P (1),Ψ(1));
b) ∆ ∈ IQCZ (P (2),Ψ(2)) and (8.11) is satisfied;
c) there exists a certificate Xs of (8.14) and a certificate Y (2)22 of
(8.11) which are coupled asXs11 Xs12 Xs13Xs21 Xs22 − Y (2)22 Xs23
Xs31 X
s
32 X
s
33
  0. (8.15)
If either Π(1) or Π(2) is empty, Theorem 8.2 remains valid if we merely
drop the respective assumption a) or b) and cancel the corresponding
block rows and columns in (8.15).
Proof. Since the proof proceeds in exactly the same fashion as the one
of Theorem 7.9, we only state the necessary alterations here.
The sole use of the FDI (8.14) is to guarantee, for the trajectories of
the LTI system in (8.2) with (N21 N22) = (M N12) driving (8.9) and
with ξ = col(ξ1, ξ2), the existence of some γ > 0 with
8.2. Local analysis with hard and soft IQCs 211
∫ T
0
1
γ
‖z(t)‖2 − γ‖d(t)‖2 dt ≤ −
(
ξ(T )
x(T )
)T
Xs
(
ξ(T )
x(T )
)
−
−
∫ T
0
(
z
(1)
Ψ (t)
z
(2)
Ψ (t)
)T (
P (1) 0
0 P (2)
)(
z
(1)
Ψ (t)
z
(2)
Ψ (t)
)
dt (8.16)
for all d ∈ D , w ∈ L2e, and T > 0. By exploiting
∆ ∈ HIQCZ (P (1),Ψ(1)) ∩ IQCZ (P (2),Ψ(2)),
P (2) satisfying (8.11), and (8.13), we deduce (8.5) with (ξ, Y ) replaced
by (ξ(2), Y (2)22 ); here we make crucial use of
∫ T
0
z
(1)
Ψ (t)
TP (1)z
(1)
Ψ (t) dt ≥ 0
for all T > 0 and all z ∈ Z . If combining with (8.15), we directly infer
stability of (8.2a).
Theorem 8.2 offers two main advantages over Theorem 7.9, as it
allows for the incorporation of general hard IQCs into Π(1), irrespective
of the way by which they are generated and even if they violate (8.11).
As already mentioned in Chapter 7, Balakrishnan proves in [13] that
dynamic D-scaling multipliers (7.60) admit hard factorizations. His
arguments only rely on a special choice of Ψ in (7.8) and a classical
commutation property. Even though derived in a fundamentally different
way, we may now incorporate the resulting hard factorized multipliers
into our framework and thus merge them with others for a more accurate
local analysis without any modifications.
A second advantage of our novel formulation of Theorem 8.2 arises
from the fact that it allows to consider the most effective full-block
multipliers corresponding to the celebrated circle criterion (see Sec-
tion 5.3.1), namely those based on Pólya relaxations. Although these do
not satisfy (8.11), they can still be employed in the realm of Theorem 8.2
since defining hard IQCs.
As a concrete application of Theorem 8.2, let us consider the scenario
introduced in Section 7.5.2 that is mainly motivated by the analysis
of only locally stable saturated systems. The underlying idea is to
capture the action of the uncertainty ∆ on some set of input signals
212 Chapter 8. Hard Zames-Falb factorizations for invariance
while simultaneously guaranteeing invariance of a related set in the
state-space (see [79] for an early reference).
In Section 7.5.2 we specifically work with amplitude bounded signals
in
VR,e := {z ∈ L2e | ‖z‖∞ ≤ R for all t ≥ 0} (8.17)
and VR := VR,e ∩L2 where R > 0. Moreover, we restrict the inputs d
to the set Dα := {d ∈ L2 | ‖d‖2 ≤ α}. The following corollary of Theo-
rem 8.2 can be established along the lines of the proof of Theorem 7.12.
Corollary 8.3
Suppose that N in (8.1) is realized with D11 = D12 = 0. With R > 0
and (8.17) let VR,e ⊂ Ze. Further assume that (8.2a) is well-posed on
Dα and that
a) ∆ ∈ HIQCVR(P (1),Ψ(1));
b) ∆ ∈ IQCVR(P (2),Ψ(2)) and (8.11) is satisfied;
c) there exist certificates X and Y (2)22 of the FDIs (7.12) with Pp=
diag(0,−I) and (8.11), respectively, such that
R2/α2 0 0 C1
0 X11 X12 X13
0 X21 X22 − Y (2)22 X23
CT1 X31 X32 X33
  0. (8.18)
Then (8.2a) is stable on Dα and the response z of (8.2a) is bounded as
‖z(t)‖ ≤ R for all t ≥ 0 and all d ∈ Dα.
Remark 8.4.
For fixed R > 0, we emphasize that (8.18) is an LMI in X, Y (2)22 and
1/α2. By a line search over R we can thus determine the maximal
admissible disturbance energy α such that (8.2a) remains stable. ?
After having established two local analysis results, we focus in
the subsequent section on a rather specific yet very relevant class of
8.3. Zames-Falb multipliers 213
multipliers in order to illustrate how the above derived setup enfolds in
a concrete scenario.
8.3 Zames-Falb multipliers
The claim that Zames-Falb multipliers define one of the most effective
classes of soft IQCs for global analysis was supported by many com-
parisons and examples in Part I of the present thesis. Furthermore, in
Chapter 7, we demonstrated that Zames-Falb multipliers also play an
important role in the verification of local criteria. The present section is
devoted to the proof that both the subclasses of causal and anticausal
Zames-Falb multipliers can, individually, be incorporated into our local
analysis framework without any conservatism. This is based on two con-
tributions. First we reveal that simple factorizations of both subclasses,
as used for parameterization and subsequent numerical computations,
lead to hard IQC constraints. Based on these factorizations, we prove in
a second step that all certificates Xs of (8.14) with empty (P 2,Ψ2) are
positive definite. Thus we may drop the assumption b) in Theorem 8.2
and infer that (8.15) does not cause extra limitations. In summary, we
may proceed with local analysis as in Corollary 8.3 without any added
conservatism.
For simplicity of presentation, we restrict the class of uncertainties
under consideration if compared to Chapter 5 as follows. Let ϕ : R→ R
and assume that ϕ(0) = 0 and
0 ≤ ϕ(x)− ϕ(y)
x− y ≤ supx 6=y
ϕ(x)− ϕ(y)
x− y < µ (8.19)
for all x, y ∈ R with x 6= y. If the bound on the right is absent but ϕ
remains locally Lipschitz continuous and bounded as in |ϕ(x)| ≤ k |x|,
it is said to be bounded and monotone. As before, we define the
nonlinear operator ∆Φ as
∆Φ(z)(t) = ϕ(z(t)) for all z ∈ Z (8.20)
214 Chapter 8. Hard Zames-Falb factorizations for invariance
and almost all t ≥ 0. The following fundamental theorem from [197]
provides the basis for Zames-Falb multipliers.
Theorem 8.5
Let ∆Φ be as in (8.20) with a bounded and monotone function ϕ. More-
over, let h ∈ L1(−∞,∞) be nonnegative and satisfy ‖h‖1 < g for some
g > 0. Then (8.4) holds on Z = L2 with
Π =
(
0 g − hˆ∗
g − hˆ 0
)
. (8.21)
If ϕ further satisfies (8.19), then (8.4) is valid on L2 with
Π[0,µ] :=
(
µ −1
0 1
)T
Π
(
µ −1
0 1
)
. (8.22)
Remark 8.6.
Note that Theorem 8.5 is merely a specialization of Theorem 5.15 for a
single nonlinearity and under the assumptions on ϕ stated above. ?
Now observe that any h ∈ L1(−∞,∞) can be split up as h = h−+h+
with h− and h+ supported on (−∞, 0] and [0,∞). By setting either
h− or h+ to zero we obtain causal or anticausal Zames-Falb multipliers
that may be factorized according to (7.8) as
Π+ ∼ (P+,Ψ+) :=
((
0 1
1 0
)
,
(
g − hˆ+ 0
0 1
))
, (8.23)
or
Π− ∼ (P−,Ψ−) :=
((
0 1
1 0
)
,
(
1 0
0 g − hˆ∗−
))
, (8.24)
respectively. Let us show next that both define hard IQCs.
Theorem 8.7
Let ϕ be bounded and monotone. Then ∆Φ ∈ HIQC(P+,Ψ+) as well as
∆Φ ∈ HIQC(P−,Ψ−).
8.3. Zames-Falb multipliers 215
The following proofs of both claims in Theorem 8.7 will be fun-
damentally different. We show ∆Φ ∈ HIQC(P+,Ψ+) from scratch by
tracing the multiplier back to its generating principle, namely convexity.
This allows to justify the claim without resorting to Theorem 8.5. By
contrast, we derive the hard IQC ∆Φ ∈ HIQC(P−,Ψ−) form the corre-
sponding soft one in Theorem 8.5. This is achieved by truncating the
input to the uncertainty at time T , in complete analogy to the case of
general IQCs discussed in the proof of Theorem 7.9.
Proof. Let us first give an elementary proof for the validity of a hard
IQC with causal Zames-Falb multipliers. The assumed bound on ϕ
implies
|ϕ(x)|2 ≤ k2|x|2 for all x ∈ R. (8.25)
Since sϕ(s) ≥ 0 for s ∈ R, the C1-function χ(x) = ∫ x
0
ϕ(s) ds satisfies
χ(x) ≥ 0 and χ′(x) = ϕ(x) for all x ≥ 0. By monotonicity of the
derivative, χ is convex and hence, by the subgradient inequality,
χ(x)− χ(y) ≤ ϕ(x)(x− y) for all x, y ∈ R. (8.26)
For y = 0 and with χ(0) = 0 we infer χ(x) ≤ ϕ(x)x ≤ |ϕ(x)||x| and
thus
0 ≤ χ(x) ≤ k|x|2 for all x ∈ R. (8.27)
Now fix z ∈ L2 and τ ≥ 0. From (8.25) we have ϕ(z(.)) ∈ L2;
moreover, as z(.− τ) ∈ L2 = L2[0,∞) and by (8.27) also χ(z(.− τ)) ∈
L1[0,∞). Using (8.26) we arrive at
χ(z(t))− χ(z(t− τ)) ≤ ϕ(z(t))(z(t)− z(t− τ)) (8.28)
for almost all t ≥ 0. Since χ(z(.−τ)) = 0 on [0, τ ] we note for 0 ≤ T ≤ τ
that ∫ T
0
χ(z(t))− χ(z(t− τ)) dt =
∫ T
0
χ(z(t)) dt ≥ 0 (8.29)
216 Chapter 8. Hard Zames-Falb factorizations for invariance
and, similarly for T > τ , that∫ T
0
χ(z(t))− χ(z(t− τ)) dt =
∫ T
T−τ
χ(z(t)) dt ≥ 0. (8.30)
For all τ ≥ 0, (8.29), (8.30) in combination with (8.28) imply the validity
of the hard IQC
∫ T
0
ϕ(z(t))[z(t) − z(t − τ)] dt ≥ 0 for all T ≥ 0. If
h ∈ L1[0,∞) is non-negative, multiplication with h(τ) and integration
over τ ∈ [0,∞) shows nonnegativity of∫ T
0
ϕ(z(t))
[
z(t)
∫ ∞
0
h(τ) dτ −
∫ ∞
0
z(t− τ)h(τ) dτ
]
dt
for all T ≥ 0. It remains to note for t ≥ 0 and z ∈ L2 that∫ ∞
0
z(t− τ)h(τ) dτ =
∫ t
0
h(t− σ)z(σ) dσ
in order to arrive at∫ T
0
ϕ(z(t))
[
‖h‖1z(t)−
∫ t
0
h(t− τ)z(τ) dτ
]
dt ≥ 0
for all T ≥ 0. The claim follows by noting g > ‖h‖1 and simple
application of the Fourier transform.
Let us now turn to the anticausal case. Consequently, let h ∈
L1(−∞, 0] be non-negative and define f(t) = h(−t) for almost all
t ∈ [0,∞). Then f ∈ L1[0,∞), hˆ∗ = fˆ and the soft IQC defined by
(P−,Ψ−) reads as∫ ∞
0
z(t)
[
gϕ(z(t))−
∫ t
0
f(t− τ)ϕ(z(τ)) dτ
]
dt ≥ 0
for all z ∈ L2. In particular, for zT ∈ L2 we infer∫ T
0
zT (t)
[
gϕ(zT (t))−
∫ t
0
f(t− τ)ϕ(zT (τ)) dτ
]
dt ≥ 0.
8.3. Zames-Falb multipliers 217
As z and zT coincide on [0, T ] and ϕ is static, this implies∫ T
0
z(t)
[
gϕ(z(t))−
∫ t
0
f(t− τ)ϕ(z(τ)) dτ
]
dt ≥ 0
for all z ∈ L2.
With arguments given in [197, Ch. 7], one can likewise show
that the transformed multipliers corresponding to (8.22), namely
Π+,[0,µ] ∼(P+,Ψ+
(
µ −1
0 1
)
) and Π−,[0,µ] ∼ (P−,Ψ−
(
µ −1
0 1
)
), define hard
IQCs for ∆Φ’s with (8.19).
We proceed by introducing the following parameterization of Π[0,µ]
that comprises the one proposed in Section 5.3.3 for the case of a single
nonlinearity. Given a strictly proper and stable column vector φ ∈ RH l∞,
let h in (8.21)-(8.22) be defined through
hˆ = hˆ− + hˆ+ = φ∗λ− + λT+φ with λ−, λ+ ∈ Rl (8.31)
to obtain
Π[0,µ] =
(
µ −1
0 1
)T (
0 g − λT−φ− φ∗λ+
g − φ∗λ− − λT+φ 0
)(
µ −1
0 1
)
.
(8.32)
Let us now prove that both the factorizations (8.23) and (8.24),
if employed individually, imply that all certificates Xs of (8.14) are
a priori positive definite. We first consider causal multipliers and
take (P (1),Ψ(1)) := (P+,Ψ+
(
µ −1
0 1
)
) while leaving (P (2),Ψ(2)) empty
in Theorem 8.2. Then (8.14) is
(
?
)∗(0 1
1 0
)(
g − λT+φ 0
0 1
)(
µM − 1
1
)
≺ 0 on C∞0 . (8.33)
Due to the passivity structure of P+ and the stability of µM − 1, it is a
matter of direct verification that all solutions Xs of the LMI related
to (8.33) are positive definite. Indeed, let (A˜, B˜, C˜, D˜) be an minimal
218 Chapter 8. Hard Zames-Falb factorizations for invariance
realization of the stable LTI system (g − λT+φ)(µM − 1). Then, (8.33)
is equivalent to the existence of some symmetric Xs satisfying(
I 0
A˜ B˜
)T (
0 Xs
Xs 0
)(
I 0
A˜ B˜
)
+
(
C˜ D˜
0 1
)T (
0 1
1 0
)(
C˜ D˜
0 1
)
≺ 0.
By inspection of the left upper block, we infer that Xs satisfies A˜TXs +
XsA˜ ≺ 0. As A˜ was Hurwitz, the claim follows.
For anticausal multipliers we choose (P (1),Ψ(1)) := (P−,Ψ−
(
µ −1
0 1
)
).
With (8.31) and a minimal realization (Aφ, Bφ, Cφ, 0) of φ, (8.14) is
then certified by Xs satisfying
(
?
)T ( 0 Xs
Xs 0
)
I 0 0
0 I 0
Aφ 0 Bφ
0 A B
+ (?)T
(
0 I
I 0
)(
0 µC −1
−λT−Cφ 0 g
)
≺ 0.
After a congruence transformation eliminating −λT−Cφ, the left upper
block reads as
(?)TXs +Xs
(
A˜φ 0
? A
)
≺ 0 with A˜φ := Aφ + g−1λT−CφBφ.
It remains to show that A˜φ is Hurwitz in order to conclude that Xs is
positive definite. Indeed, using the L1-norm constraint in Theorem 8.5,
a close inspection of the proof of Lemma 5.17 reveals that −(g−λT−φ) is
strictly negative real; this translates into the existence of some K = KT
with(
?
)T ( 0 K
K 0
)(
I 0
Aφ Bφ
)
+
(
?
)T (0 1
1 0
)(−λT−Cφ g
0 −1
)
≺ 0, (8.34)
whose left upper block is ATφK + KAφ ≺ 0; since Aφ is Hurwitz, we
obtain K  0; if applying the same congruence transformation as before
to (8.34), we infer
(
?
)T ( 0 K
K 0
)(
I 0
A˜φ Bφ
)
+
(
?
)T (0 1
1 0
)(
0 g
? −1
)
≺ 0 (8.35)
8.3. Zames-Falb multipliers 219
with left upper block A˜TφK + KA˜φ ≺ 0; hence A˜φ is Hurwitz due to
K  0. Consequently, Xs  0 holds in both cases.
Since Π−,[0,µ] satisfies Π22 ≺ 0 on C∞0 , we could alternatively select
(P (2),Ψ(2)) := (P−,Ψ−
(
µ −1
0 1
)
), while leaving (P (1),Ψ(1)) empty in
Theorem 8.2. The set of certificates Xs of (8.14) remains identical to
that for the previous choice. However, we now need to introduce the
shift Y (2)22 in (8.15) which satisfies (8.12). Since (8.12) is in fact identical
to (8.34) with Y (2)22 replacing K, we infer Y
(2)
22  0 and conclude that
(8.15) now imposes more stringent constraints on the certificates if com-
pared to those in the previous paragraph, which is a severe disadvantage.
The same conclusion can be drawn for causal multipliers Π+,[0,µ].
To sum up, both parameterizations of causal and anticausal Zames-
Falb multipliers, if employed individually, can be and should be treated
as hard IQCs; since the related KYP certificates Xs are all positive
definite, we conclude that (8.15) then imposes no extra limitation
and may be applied in local analysis without introducing any further
conservatism.
If working with the full Π[0,µ] in (8.32) the picture is slightly more
complicated. As Π[0,µ] defines a genuine soft IQC, we are led to incor-
porate it into Π(2) and cannot expect all certificates Xs to be positive
definite. Still, also in this case we have Y (2)22  0; in fact, if factorizing
Π[0,µ] in (8.32) as
(
P (2),
(
Ψ
(2)
1 Ψ
(2)
2
))
=

(
0 I
I 0
)
,

µ −1
µφ −φ
0 g − λT−φ
0 −λ+

 ,
the resulting realization of Ψ(2)2 is given by (AΨ, BΨ2 , CΨ, DΨ2) equal to(Aφ 00 Aφ
)
,
(−Bφ
Bφ
)
,

0 0
Cφ 0
0 −λT−Cφ
0 0
 ,

−1
0
g
−λ+

 ;
220 Chapter 8. Hard Zames-Falb factorizations for invariance
we extract CTΨP
(2)CΨ = 0, and (8.12) shows again Y
(2)
22  0.
Most importantly, also in this case we can avoid the need to work
with Y (2)22 and thus reduce conservatism. We only need to decompose
Π[0,µ] = Π−,[0,µ] + Π+,[0,µ] into its anticausal and causal parts as above
and consider Π−,[0,µ], Π+,[0,µ] as defining hard IQC constraints by
incorporating both of them into Π(1). The significant advantages of
this approach over the former is illustrated in the subsequent section
by means of a numerical example.
8.4 Concrete numerical example
Let us now demonstrate the benefit gained by our refined approach
for the very specific application of saturated systems that are of great
practical importance and, thus, have been extensively researched (see,
e.g., [157, 83] and references therein). One of the associated analysis
problems may be stated as follows: Given an exponentially unstable
LTI system that is locally (but not globally) stabilized by saturated
state feedback, what is the maximal admissible input energy such that
the feedback interconnection remains stable?
As addressed in detail in Section 7.6.2 (see also [157, 83]), a standard
loop transformation reduces the described problem to the stability anal-
ysis of the interconnection (8.2), with an uncertainty ∆Φ and ϕ = dz,
the unit dead-zone function. In addition to the Zames-Falb multipliers
discussed in Section 8.3, we may also use multipliers corresponding to
the celebrated circle criterion. Following [79], we capture the restriction
of ∆dz to the amplitude bounded set (8.17) by means of local sectors.
Furthermore, we combine the resulting local circle multipliers with
those for the Zames-Falb stability criterion, which allows us to also take
the slope restriction into account.
8.4. Concrete numerical example 221
Specifically, let us revisit Example 7.15, where the linear system is
described after the loop transformation by
A =
−1.15 −17 −100.02 −0.1 −1.8
0.1 2.5 0.3
 , B1 =
−120.2
1
 , B2 =
 0.2−0.1
0.5

and C1 =
(−0.1 −1.5 −1) with D = Dα. We leave the output e void,
since we are only interested in stability. As the dead-zone is globally
Lipschitz continuous, the interconnection is well-posed for Ze = L2e.
We apply Corollary 8.3 by performing a line search over R > 0. For
each R we adapt our combination of multipliers corresponding to the
circle and Zames-Falb criteria [58] such that (8.4) holds for Z = VR
defined in (8.17).
Let us briefly recap the results derived in Example 7.15. For
R ∈ [0, 1], the interconnection operates in the so-called linear region
such that the output of the dead-zone nonlinearity is zero; thus stability
is guaranteed. As stated in Table 8.1, this is the case for α ≤ 1.41.
Using a local version of the circle criterion, stability is proven in [79] for
values of α up to 1.45. This was further increased by Fang et al. in [51]
guaranteeing stability up until α = 2.07. The derivation in the previous
section builds on the one in [79] by adding soft factorized Zames-Falb
multipliers to the local circle criterion. This leads to a significant
push of the threshold to 9.04. Finally, our novel approach based on
a combination of hard factorized causal and anticausal Zames-Falb
multipliers as described in Section 8.3 allows for a further increase of
the admissible energies up to the bound α = 12.03. This amounts to
nearly six times the energy level obtained with [51].
Table 8.1: Maximal disturbance energy levels α
lin. region [79] [51] [60] novel approach
≤ 1.41 1.45 2.07 9.04 12.03
222 Chapter 8. Hard Zames-Falb factorizations for invariance
8.5 Summary
The contribution in this chapter is twofold. First, we provide a com-
prehensive framework that allows to optimally combine hard and soft
IQCs for local analysis of feedback interconnections. This allows for
the nonconservative incorporation of hard IQC regardless of generating
principles. It is expected that this lays the foundation for the merging of
dissipation based results in the literature with the framework outlined
in Chapter 7. Second, we prove that both the subclasses of causal and
anticausal Zames-Falb multipliers may easily be factorized such that
they, individually, impose hard IQC constraints and may be losslessly in-
corporated into our local framework. In combination both contributions
allow, on the one hand, to significantly reduce the conservativeness in the
approach presented in Chapter 7. On the other hand, the possibility for
adding general hard IQCs considerably widens the are of applications.
Chapter 9
Concluding remarks
This thesis provides several crucial steps towards its main goal ofa comprehensive robustness analysis theory that provides the
framework for the computational verification of global as well as local
stability and performance criteria.
In case of global robust stability and performance analysis, we
present a general theorem that merges ideas from the abstract graph
separation results proposed by Safonov and Teel with those from
classical IQC theory established by Megretski and Rantzer. As one of
the key features of the first part of the present thesis, this allows us to
develop a unified framework for global robustness analysis on Sobolev
spaces that is shown to subsume and extend all multiplier based results
for the classical problem of absolute stability. Consequently, and as
supported by numerous examples, the results presented in this thesis
define the least conservative robust stability and performance estimates
available in the literature for this essential problem.
The second part of this thesis is devoted to local analysis of feedback
structures. For interconnections consisting of an LTI system and an
uncertain component, we demonstrate that standard soft IQCs can
be incorporated into classical dissipation theory in order to guarantee
state and output constraints, or even to handle only locally bounded
223
224 Chapter 9. Concluding remarks
nonlinearities within the IQC framework. The full power of our novel
approach is illustrated for the particular case of Zames-Falb multipliers.
Here it is shown that the subclasses of causal and anticausal Zames-Falb
multipliers can be employed in local analysis without any conservatism.
However, in line with the title of the present thesis which does
not claim completeness of our comprehensive framework, there remain
several issues that we believe should be addressed in the future. As
we discussed individual recommendations already at the end of each
chapter, we only highlight some major general questions in the sequel.
The formulation of our main global stability result on Sobolev spaces
allows to incorporate constraints and also performance specification
involving higher order derivatives with ease. This is hoped to provide
the basis for more accurate descriptions of nonlinearities and also for
more sophisticated performance criteria. However, apart from our
treatment of time-varying parametric uncertainties, the generation of
novel stability criteria based on higher order derivatives of the involved
signals remains an open problem.
Moreover, as indicated in Chapter 7, our approach to local robustness
analysis of feedback interconnections based on IQCs is still subject to
certain limitations. First and foremost, we are as of yet unable to include
multipliers that require additional signal regularity into the framework.
Although one would expect that all Lyapunov based criteria are also ap-
plicable in our setting, Popov multipliers remain, so far, out of reach. In
addition, the ability to exploit general soft IQCs comes at the expense of
added conservatism. It remains unclear whether the overall approach can
be refined in order to improve the obtained stability estimates or, as illus-
trated in Chapter 8, whether it is more beneficial to enhance the present
framework by exploiting individual properties of given multiplier classes.
Part III
Appendices
225

Appendix A
Explanation of symbols
A.1 Sets and matrices
N, N0 positive, nonnegative integers
R, C real and complex numbers
R+ set of positive real numbers, i.e., (0,∞).
C0, C∞0 = C0 ∪
{∞}
imaginary axis and extension thereof
D, T open unit disc in the complex plane and its
boundary
sgn(a) sign of a real scalar, i.e., sgn(a) = a |a|
Rn×m real valued matrices of dimension n×m
Sn,Dn subspace of symmetric, diagonal matrices of
Rn×n
A∗, AT conjugate transpose and transpose of matrix
A
A > (≥) 0 used for A ∈ Rn×m if Aij > (≥) 0 for all i, j
A ≺ (4) B used for A, B ∈ Sn if B −A is positive (semi-)
definite
A⊗B Kronecker product of matrices A and B
227
228 Appendix A. Explanation of symbols
det(A) determinant of A
trace(A) trace of A
Ran(A), Ker(A) range space and kernel of A (also if A is a
bounded operator between Hilbert spaces)
diag(A1, . . . , An) block diagonal matrix with matrices Aj on
diagonal
col(A1, . . . , An) for matrices Aj with appropriate dimensions
[α, β] for α = diag(αi), β = diag(βi) ∈ Dn, the set
of diagonal matrices {diag(δ1, . . . , δk) : αi ≤
δi ≤ βi for all i ∈ {1, . . . , n}}.
eig(A) set of eigenvalues of A
e the all ones vector in Rn
A.2 Function spaces and signals
In the following table, Ω denotes a measurable subset of R. Note that
for function spaces, we typically omit the superscript indicating the
dimension k of the contained signals.
L k2 (Ω), L k2e(Ω) space of square integrable, locally square inte-
grable functions mapping Ω ⊂ R into Rk with
k ∈ N
L k1 (Ω), L k∞(Ω) absolute integrable, essentially bounded func-
tions mapping Ω ⊂ R into Rk
‖ · ‖p norm on L kp (Ω) for p = 1, 2,∞
`k2 , (`k2e) the space of (locally) square summable func-
tions mapping N0 into Rk; `k2 is equipped with
the norm ‖u‖2 = ∑∞j=0 ‖u(j)‖2
A.2. Function spaces and signals 229
H r,k Sobolev space of functions u : [0,∞) → Rk
such that u and its distributional derivatives
∂ju for j ∈ {1, . . . , r} are all contained in
L2; it is equipped with the norm ‖u‖2r :=∑r
j=0 ‖∂ju‖20
C[0,∞)
(PC[0,∞))
set of (piecewise) continuous functions mapping
[0,∞) into Rk
RL∞ space of real-rational and proper matrix func-
tions without poles on C0
RH∞ ⊂ RL∞ subspace of RL∞ containing proper and stable
transfer matrices
? convolution operator and objects that can be
inferred by symmetry
uˆ Fourier transform of a signal u in L k2 or L k1
G(s),
(A,B,C,D)
transfer matrix and its realization, i.e., G(s) =
C(sI −A)−1B +D
uT denotes either the restriction of a signal u :
[0,∞)→ Rk to [0, T ], i.e., uT = u|[0,T ], or its
truncation, i.e., u(t) = u on [0, T ] and zero
otherwise
uT extension of u : [0,∞) → Rk that coincides
with u on [0, T ], i.e., uT |[0,T ] = u|[0,T ]
D±, D± lower and upper Dini derivatives
230 Appendix A. Explanation of symbols
Appendix B
List of terms
ARE algebraic Riccati equation.
FDI frequency-domain inequality.
IQC integral quadratic constraint.
KYP Kalman Yakubovich Popov.
LMI linear matrix inequality.
LTI linear time-invariant.
PWM pulse-width modulator.
SISO single input single output.
SOS sum of squares.
231
232 Appendix B. List of terms
Appendix C
Some additional proofs
C.1 For Chapter 2
C.1.1 Proof of Theorem 2.2
Step 1. Since M is bounded, there exist some γ˜, γ˜0 ≥ 0 with
‖M(w)T ‖2 ≤ γ˜2‖wT ‖2 + γ˜20 and thus∥∥∥∥∥
(
M(w)T
wT
)∥∥∥∥∥ ≤
√
(1 + γ˜2)‖wT ‖2 + γ˜20 ≤
√
(1 + γ˜2)‖wT ‖+ γ˜0
for all T > 0 and w ∈ L k2e. Applying (2.10) to u = col(M(w)T , wT )
and v = col(dT , 0) hence leads to
Σ
(
M(w)T + dT
wT
)
− Σ
(
M(w)T
wT
)
≤ 2c
(√
(1 + γ˜2)‖wT ‖+ γ˜0
)
‖dT ‖+ c‖dT ‖2
=

1
‖wT ‖
‖dT ‖

T 
0 0 σ13
0 0 σ23
σ13 σ23 σ33


1
‖wT ‖
‖dT ‖
 (C.1)
233
234 Appendix C. Some additional proofs
for all T > 0 and (w, d) ∈ L k2e ×L l2e, with σij only depending on M
and Σ. If M is linear one can choose γ˜0 = 0 which implies σ13 = 0.
Step 2. In this crucial step we show that there exist γ > 0 and γˆ0
such that
Σ
(
M(w)T + dT
wT
)
+
1
γ
‖M(wT )T + dT ‖2 − γ‖dT ‖2 ≤ γˆ0 (C.2)
for all T > 0 and (w, d) ∈ L k2e ×L l2e as follows. Add (C.1) and (2.11)
to get
Σ
(
M(w)T + dT
wT
)
≤

1
‖wT ‖
‖dT ‖

T 
m0 0 σ13
0 −ε σ23
σ31 σ32 σ33


1
‖wT ‖
‖dT ‖
 (C.3)
for all T > 0 and (w, d) ∈ L k2e ×L l2e. With ‖M(w)T ‖ ≤ γ˜‖wT ‖+ γ˜0
and the triangle inequality we infer for all γ > 0 that
1
γ
‖M(w)T + dT ‖2 − γ‖dT ‖2 ≤ 1
γ
(γ˜‖wT ‖+ γ˜0 + ‖dT ‖)2 − γ‖dT ‖2 =
=

1
‖wT ‖
‖dT ‖

T 
m11/γ m12/γ m13/γ
m12/γ m22/γ m23/γ
m13/γ m23/γ m33/γ − γ


1
‖wT ‖
‖dT ‖
 (C.4)
for all T > 0 and (w, d) ∈ L k2e ×L l2e, where mij ∈ R do not depend
upon γ. For any γˆ0 > m0 observe that there exists some (sufficiently
large) γ > 0 for which
m0 0 σ13
0 −ε σ23
σ31 σ32 σ33
+

m11/γ m12/γ m13/γ
m12/γ m22/γ m23/γ
m13/γ m23/γ m33/γ − γ
 4

γˆ0 0 0
0 0 0
0 0 0
 . (C.5)
If we add (C.3) and (C.4), we can exploit (C.5) to arrive at (C.2). If
M is linear we can choose γ˜0 = 0 which implies m11 = m12 = m13 = 0.
We can then take γˆ0 = m0.
C.2. For Chapter 3 235
Step 3. To finish the proof choose d ∈ D and a correspond-
ing response z ∈ L l2e of (2.9). Then wT = ∆(z)T and zT =
M(w)T + dT . Now observe that the inequality in (2.12) holds for
v := M(wT ) + d ∈ M(L k2 ) + D . On the other hand, by causality we
have vT = M(wT )T + dT = M(w)T + dT = zT and ∆(v)T = ∆(vT )T =
∆(zT )T = ∆(z)T = wT . This allows to combine (2.12) with (C.2) in
order to infer (2.13) for γ0 = γˆ0 +δ0 (which equalsm0 +δ0 ifM is linear).
C.2 For Chapter 3
C.2.1 Proof of Theorem 3.4
We use the abbreviations D := `2(H k)× V and E := Ee ∩ `2(H l).
Step 1. Only for proving the following key fact we make use of the
properties of Σ and the constraints b), c): There exists a (τ -independent)
γ > 0 such that
τ ∈ [0, 1] and Rτ (D) ⊂ E imply
‖Rτ (u, v)‖2 ≤ γ2
(‖u‖2 + ‖v‖2)+ γδ0(v) for all (u, v) ∈ D . (C.6)
Observe for all γ > 0 and all w ∈ `2(H k), (u, v) ∈ D that
1
γ
‖Mw +Mu+Nv‖2 − γ (‖u‖2 + ‖v‖2)
≤ 1
γ
(‖M‖‖w‖+ ‖M‖‖u‖+ ‖N‖‖v‖)2 − γ (‖u‖2 + ‖v‖2)
=

‖w‖
‖u‖
‖v‖

T 
m11/γ m12/γ m13/γ
m12/γ m22/γ − γ m23/γ
m13/γ m23/γ m33/γ − γ


‖w‖
‖u‖
‖v‖
 , (C.7)
where the mij only depend on ‖M‖ and ‖N‖. Now add (3.2), b), and
(C.7) to get
Σ
(
Mw +Mu+Nv
w
)
+
1
γ
‖Mw +Mu+Nv‖2 − γ(‖u‖2 + ‖v‖2)
236 Appendix C. Some additional proofs
≤
(
?
)T 
−ε+m11/γ m12/γ + σ12 m13/γ + σ13
m12/γ + σ12 m22/γ − γ + σ22 m23/γ + σ23
m13/γ + σ13 m23/γ + σ23 m33/γ − γ + σ33


‖w‖
‖u‖
‖v‖

for (u, v) ∈ D and w ∈ `2(H k). Since ε > 0 there exists some large
γ > 0 such that
Σ
(
Mw +Mu+Nv
w
)
+
1
γ
‖Mw +Mu+Nv‖2 − γ (‖u‖2 + ‖v‖2) ≤ 0
(C.8)
for all w ∈ `2(H k), (u, v) ∈ D . Now fix any (u, v) ∈ D and τ ∈ [0, 1].
Due to the hypothesis in (C.6), we infer that z := Rτ (u, v) ∈ E and
thus, since ∆ was assumed to be bounded, also w := τ∆(z) ∈ `2(H k).
Moreover, with the loop equation z = Mw + Mu + Nv by (3.3), we
can exploit (C.8) to get
Σ
(
z
τ∆(z)
)
+
1
γ
‖Rτ (u, v)‖2 − γ
(‖u‖2 + ‖v‖2) ≤ 0. (C.9)
Since w+u ∈ `2(H k) and v ∈ V it remains to use c) in order to obtain
from (C.9) that 1γ ‖Rτ (u, v)‖2 ≤ γ
(‖u‖2 + ‖v‖2) + δ0(v) as was to be
shown.
Step 2. Since ∆ is bounded, there exist δˆ > 0, δˆ0 ≥ 0 such that
‖∆(z)T ‖ ≤ δˆ‖zT ‖ + δˆ0 for all T ∈ N0, z ∈ Ee. With γ > 0 from Step
1 we now fix any ρ0 > 0 with γρ0δˆ < 1. In this step we show that
τ ∈ [0, 1], τ + ρ ∈ [0, 1], |ρ| ≤ ρ0, Rτ (D) ⊂ E imply Rτ+ρ(D) ⊂ E .
(C.10)
Choose ρ and τ as in (C.10) and any (u, v) ∈ D . We have to show
that z = Rτ+ρ(u, v) ∈ `2(H l). Observe that z − τM∆(z)− ρM∆(z) =
Mu+Nv can be written as
z − τM∆(z) = M(ρ∆(z) + u) +Nv ⇐⇒ z = Rτ (ρ∆(z) + u, v).
C.3. For Chapter 4 237
As in [110, 93], the key idea is to just employ a small-gain argument
based on 1−γρ0δˆ > 0 as follows. We know that z ∈ Ee. The hypothesis
in (C.10) allows us to exploit Step 1; since Rτ is causal in the first
argument, we infer with γ0 =
√
γδ0(v) for T ∈ N0 that
‖zT ‖ = ‖Rτ (ρ∆(z)T + uT , v)T ‖ ≤ ‖Rτ (ρ∆(z)T + uT , v)‖
≤ γ‖ρ∆(z)T + uT ‖+ γ‖v‖+ γ0
≤ (γρ0δˆ)‖zT ‖+ γ‖u‖+ γ‖v‖+ γρ0δˆ0 + γ0.
Hence
(1− γρ0δˆ)‖zT ‖ ≤ γ‖u‖+ γ‖v‖+ γρ0δˆ0 + γ0 for all T ∈ N0,
which implies z ∈ Ee ∩ `2(H l) = E .
Step 3. Due to boundedness of M, N and assumption a) we have
R0(D) ⊂ E . Since ρ0 in Step 2 does not depend on τ , we can inductively
apply (C.10) in order to infer Rτ (D) ⊂ E for τ ∈ [0, νρ0]∩ [0, 1] and all
ν = 1, 2, . . . and thus in particular also for τ = 1. Then (C.6) implies
(3.4).
C.3 For Chapter 4
C.3.1 Proof of Theorem 4.7
In view of item a) and for τ ∈ [0, 1], we can introduce the nota-
tion z = Rτ (u, v) for the response of the interconnection (4.3) if
(u, v) ∈ Ue × V and if replacing ∆ with τ∆.
Step 1. Only for proving the following fact we make use of the
properties of Σ and and the quadratic constraints in c) and d): There
exist (τ -independent) γ > 0, γ0 such that
τ ∈ [0, 1] and Rτ (U × V ) ⊂ Z imply
‖Rτ (u, v)‖Z ≤ γ(‖u‖U + ‖v‖V ) + γ0l(Rτ (u, v))
for all (u, v) ∈ U × V . (C.11)
238 Appendix C. Some additional proofs
Indeed, by exploiting boundedness of J : W → U , M : U → Z and
N : V → Z , observe for all γ¯ > 0 and all (u, v, w) ∈ U × V ×W that
1
γ¯
‖Mw +Mu+N(v)‖2Z − γ¯
(‖u‖2U + ‖v‖2V )
≤ 1
γ¯
(‖M‖‖J‖‖w‖W + ‖M‖‖u‖U + γN‖v‖V )2 − γ¯
(‖u‖2U + ‖v‖2V )
=

‖w‖W
‖u‖U
‖v‖V

T 
m11/γ¯ m12/γ¯ m13/γ¯
m12/γ¯ m22/γ¯ − γ¯ m23/γ¯
m13/γ¯ m23/γ¯ m33/γ¯ − γ¯


‖w‖W
‖u‖U
‖v‖V
 , (C.12)
where the constants mij only depend on J , M and N . If we add (4.5)
and (4.7) we infer
Σ
(
Mw +Mu+N(v)
w
)
≤
(
?
)T 
−ε σ12 σ13
σ12 σ22 σ23
σ13 σ23 σ33


‖w‖W
‖u‖U
‖v‖V
(C.13)
for all (u, v, w) ∈ U ×V ×W . We can clearly fix some sufficiently large
γ¯ > 0 with
m11/γ¯ m12/γ¯ m13/γ¯
m12/γ¯ m22/γ¯ − γ¯ m23/γ¯
m13/γ¯ m23/γ¯ m33/γ¯ − γ¯
+

−ε σ12 σ13
σ12 σ22 σ23
σ13 σ23 σ33
 ≺ 0.
Thus, by adding (C.12) and (C.13) we get
Σ
(
Mw +Mu+N(v)
w
)
+
1
γ¯
‖Mw +Mu+N(v)‖2Z−
− γ¯ (‖u‖2U + ‖v‖2V ) ≤ 0 for all (u, v, w) ∈ U × V ×W . (C.14)
Now fix any (u, v) ∈ U × V . Due to the hypothesis in (C.11), we
infer that z := Rτ (u, v) satisfies z ∈ Z and thus, since the bounded
C.3. For Chapter 4 239
uncertainty ∆ maps Z into W , also w := τ∆(z) ∈ W . From the loop
equation z = Mw +Mu+N(v) we conclude with (C.14) that
‖Rτ (u, v)‖2Z ≤ γ¯2
(‖u‖2U + ‖v‖2V )− γ¯Σ
(
z
τ∆(z)
)
. (C.15)
Since Assumption 4.5.a) implies W ⊂ U , we get w + u ∈ U and with
v ∈ V we conclude z ∈ MU + N(V ). Hence, we can exploit (4.6) in
order to obtain from (C.15) the inequality
‖Rτ (u, v)‖2Z ≤ γ¯2
(‖u‖2U + ‖v‖2V )+ γ¯l(Rτ (u, v))2,
and thus (C.11).
Step 2. By Lemma 4.4 there exist δ > 0, δ0 ≥ 0 with ‖∆(z)‖Ue,T ≤
δ‖z‖Ze,T + δ0 for all T > 0, z ∈ Ze. With the constant KU for the
space Ue as in (4.1) and with γ from the first step, let us fix any ρ0 > 0
satisfying ρ0γδKU < 1. In this second step we show that
τ ∈ [0, 1], τ + ρ ∈ [0, 1], |ρ| ≤ ρ0, Rτ (U × V ) ⊂ Z
imply Rτ+ρ(U × V ) ⊂ Z . (C.16)
Choose ρ and τ as in (C.16) and take any (u, v) ∈ U × V . Then
z = Rτ+ρ(u, v) is known to be contained in Ze; due to loop equation,
we have z − τM∆(z)− ρM∆(z) = Mu+N(v) which is equivalent to
z− τM∆(z) = M(ρ∆(z) +u) +N(v) and hence also to z = Rτ (ρ∆(z) +
u, v). Fix any T > 0. Since ρ∆(z) ∈ We ⊂ Ue and thus ρ∆(z) + u ∈ Ue
we get ((ρ∆(z) + u)T , v) ∈ U × V , which implies by the assumption
in (C.16) that Rτ ((ρ∆(z) + u)T , v) ∈ Z . Hence, due to (C.11),
‖Rτ ((ρ∆(z) + u)T , v)‖Z ≤ γ‖(ρ∆(z) + u)T ‖U + γ‖v‖V +
+γ0l(Rτ ((ρ∆(z) + u)
T , v)). (C.17)
Since Rτ is causal in the first argument, we get zT = Rτ (ρ∆(z) +
u, v)T = Rτ ((ρ∆(z) + u)
T , v)T and thus ‖z‖Ze,T = ‖zT ‖ZT =
‖Rτ ((ρ∆(z) + u)T , v)T ‖ZT ≤ ‖Rτ ((ρ∆(z) + u)T , v)‖Z . Now note
240 Appendix C. Some additional proofs
that zT = Rτ ((ρ∆(z) + u)T , v) is actually a valid extension of z at time
T . If we combine with (C.17) and exploit (4.1), we obtain
‖z‖Ze,T ≤ γ‖(ρ∆(z) + u)T ‖U + γ‖v‖V + γ0l(zT )
≤ γKU ‖ρ∆(z) + u‖Ue,T + γ‖v‖V + γ0l(zT )
≤ γKU (|ρ|δ‖z‖Ze,T + |ρ|δ0 + ‖u‖Ue,T ) + γ‖v‖V + γ0l(zT )
≤ (ρ0γδKU )‖z‖Ze,T + γKU (ρ0δ0 + ‖u‖U ) + γ‖v‖V + γ0l(zT ).
This implies with (4.4) that (1 − ρ0γδKU )‖z‖Ze,T ≤ γKU (ρ0δ0 +
‖u‖U ) + γ‖v‖V + γ0c. Since T > 0 was arbitrary and the right-hand
side does not depend on T , we can exploit 1 − ρ0γδKU > 0 to infer
supT>0 ‖z‖Ze,T <∞ and thus z ∈ Z as was to be shown.
Step 3. Clearly R0(U ×V ) ⊂ Z . Since ρ0 in Step 2 does not depend
on τ , we can inductively apply (C.16) in order to infer Rτ (U ×V ) ⊂ Z
for τ ∈ [0, νρ0] ∩ [0, 1] and all ν = 1, 2, . . ., and thus this inclusion holds
in particular also for τ = 1. Then (C.11) for τ = 1 implies (4.8).
C.3.2 Proof of Theorem 4.12
b) ⇒ a): Choose w, d ∈ H r and let u := col(∂rw, ∂rd). By right-
and left-multiplying a perturbed version of (4.25) with a trajectory
col(xe(t), u(t)) of (4.21) and its transpose, we infer for some ε > 0 and
almost all t ≥ 0 that
(
?
)T ( 0 X
X 0
)(
xe(t)
x˙e(t)
)
+
(
?
)T (Q1 0
0 P
)(
T∆
Tp
)
xe(t)
∂rw(t)
∂rd(t)

≤ −ε
r∑
l=0
(‖∂lw(t)‖2 + ‖∂ld(t)‖2).
Since xe vanishes at infinity and x(0) = 0, integration over [0, T ] and
taking the limit T →∞ results in
C.3. For Chapter 4 241
σQ1
(
Mw +Nd
w
)
−
(
?
)T (X11 X12
X21 X22
)
0
Dr−1w(0)
Dr−1d(0)
+
+ σP
(
N21w +N22d
d
)
≤ −ε(‖w‖2r + ‖d‖2r).
The statement follows by setting R = −X22.
a)⇒ b): With the real matrix
H :=
(
T∆
Tp
)T (
Q1 0
0 P
)(
T∆
Tp
)
define
qη(x, w˜, d˜, u) :=
(
?
)T
[H + ηI] col(x, w˜, d˜, u) for η ∈ R.
By (4.24) there exists some ε > 0 such that the trajectories of (4.17)
with w, d ∈H r satisfy∫ ∞
0
q0
(
x(t),Dr−1w(t),Dr−1d(t), u(t)) dt ≤ −ε(‖w‖2r + ‖d‖2r)+ r0(w, d)
with
r0(w, d) :=
(
Dr−1w(0)
Dr−1d(0)
)T
R
(
Dr−1w(0)
Dr−1d(0)
)
.
Since A is Hurwitz, we have (with ‖ · ‖ denoting the norm on L2)
‖x‖ ≤ γ(‖w‖+ ‖d‖) ≤ γ(‖w‖r + ‖d‖r)
with γ := ‖(sI −A)−1
(
B1 B2
)
‖∞ along all the above trajectories. For
some sufficiently small ε˜ > 0 this implies∫ ∞
0
q2ε˜
(
x(t),Dr−1w(t),Dr−1d(t), u(t)) dt
242 Appendix C. Some additional proofs
≤ −ε
2
(‖w‖2r + ‖d‖2r)+ r0(w, d).
Now consider trajectories of the system in (4.17) for w, d ∈ H r and
x(0) = ξ; the state response is xξ := x + v if x is the response with
x(0) = 0 and v = eA•ξ. Note that there exists some Qv = QTv with
‖v‖2 = ξTQvξ. Moreover, there also exists some (large) γ˜ > 0 such that
TT [H + ε˜I]T 4
(
γ˜I 0
0 H + 2ε˜I
)
for T :=

I I 0 0
0 0 I 0
0 0 0 I
 .
This implies
qε˜
(
x(t),Dr−1w(t),Dr−1d(t), u(t)) ≤
≤ γ˜v(t)T v(t) + q2ε˜
(
x(t),Dr−1w(t),Dr−1d(t), u(t))
and thus, by combining with what we have derived so far,∫ ∞
0
qε˜
(
x(t),Dr−1w(t),Dr−1d(t), u(t)) dt ≤ γ˜ξTQvξ−
− ε
2
(‖w‖2r + ‖d‖2r)+ r0(w, d). (C.18)
Now we use the fact that the above considered trajectories are also trajec-
tories of the system in (4.21) with xe(0) = col(ξ,Dr−1w(0),Dr−1d(0)).
Let U(ξe) denote the set of all control functions u ∈ L2 ×L2 such
that the response of x˙e = Aexe +Beu with xe(0) = ξe satisfies xe ∈ L2.
With ε˜ from above introduce
V (ξe) := inf
u∈U(ξe)
{∫ ∞
0
−qε˜
(
xξ,Dr−1w,Dr−1d, u
)
dt
∣∣∣∣
x˙e(t) = Aexe(t) +Beu(t), xe(0) = ξe
}
.
Then (C.18) implies that V (ξe) > −∞ for all ξe ∈ Rn+rk. The re-
maining part of this proof is now an application of classical dissipation
C.3. For Chapter 4 243
arguments [180, 116]. We infer by Theorem 1 in [116] and controllability
of (Ae, Be) that V satisfies the dissipation inequality
V (xe(t1)) ≤
∫ t2
t1
qε˜
(
xe(t), w˙(t)
)
dt+ V (xe(t2)) (C.19)
for any trajectory of (4.21) and any 0 ≤ t1 ≤ t2. Finally, ε˜ > 0 and
Theorems 2 and 3 in [180] imply the existence of a symmetric solution
of the strict inequality (4.25).
b)⇔c): By the generalized KYP lemma, Lemma 2.11 (see also [14,
Lemma 1]), the LMI (4.25) is feasible if and only if (by setting 1/∞ = 0)
(
?
)∗(Q1 0
0 P
)
ErM ErN
Er 0
ErN21 E
rN22
0 Er
 (∞) ≺ 0 (C.20)
and(
xe
u
)
︸ ︷︷ ︸
6=0
∈ Ker
(
Ae − iωI Be
)
=⇒
(
?
)T
PTM
(
xe
u
)
≺ 0 for all ω ∈ R. (C.21)
With a tedious yet elementary calculation we can equivalently reformu-
late (C.21) as
(
?
)∗(Q1 0
0 P
)
ErM ErN
Er 0
ErN21 E
rN22
0 Er
 ≺ 0 on iR \ {0} (C.22)
and (4.27).
244 Appendix C. Some additional proofs
If (4.26) holds, then obviously both (C.20) and (C.22) are satisfied.
The proof of the converse implication relies on a continuity argument
and proceeds in complete analogy to the one given in [58].
C.4 For Chapter 5
C.4.1 Proof of Lemma 5.10
It suffices to prove the statement for k = 1 and α 6= β (since otherwise
we have α = 0 = β and ϕ = 0). As Πdr[α, β]k ⊂ Πdc[α, β]k, the “if”
statement is trivial. To prove “only if” along the lines of an argument in
[45], assume that we found a multiplier Π ∈ Πdc[α, β] with (5.10). The
constraints R ≺ 0 and FΠ(∆)  0 for all ∆ ∈ Θ({α, β}, k) translate
into r < 0, m1 > 0 and m2 > 0 for(
m1 n
n m2
)
:=
(
1 1
α β
)T (
q s
s r
)(
1 1
α β
)
=
(
1 1
α β
)T
Π
(
1 1
α β
)
.
Since α 6= β, the equation(
1 1
α β
)T
Πτ
(
1 1
α β
)
=
(
1 1
α β
)T (
qτ sτ
sτ rτ
)(
1 1
α β
)
=
(
τm1 n
n τm2
)
has a unique solution Πτ for every τ ∈ [0, 1]. Moreover, because the
derivative of the right hand side with respect to τ is positive definite,
we infer Π˙τ  0 for all τ ∈ [0, 1] and thus Π0 ≺ Π1 = Π. Hence (5.10)
also holds with Π replaced by Π0. Moreover, we obtain
r0 < r < 0 and
(
1
δ
)T (
q0 s0
s0 r0
)(
1
δ
)
= 0 for δ ∈ {α, β}. (C.23)
With q˜ = q0/ |r0|, s˜ = s0/ |r0| we get from (C.23) that q˜ + 2δs˜− δ2 = 0
for δ ∈ {α, β}, which implies q˜ = −αβ and s˜ = α+β2 . Hence
Π0 = |r0|
(
−αβ α+β2
α+β
2 −1
)
∈ Πdr[α, β] which completes the proof.
C.4. For Chapter 5 245
C.4.2 Proof of Lemma 5.26
The following proof uses Dini derivatives and some properties of abso-
lutely continuous functions. A comprehensive treatment of the concepts
relevant for our purpose is for example given in [131].
Set µ := max{|µ1| , µ2} and suppose that z is differentiable at
t (which is the case for almost every t ∈ [0,∞)). If z˙(t) = 0 then
w(.) := ϕ(z(.)) is differentiable with derivative zero at t: Indeed, with
(5.1) we infer
|ϕ(z(t+ h))− ϕ(z(t))|
|h| ≤
µ|z(t+ h)− z(t)|
|h| → 0 for h→ 0.
Now suppose z˙(t) 6= 0. We can then choose δ > 0 such that for
0 < |h| < δ we have
∣∣∣ z(t+h)−z(t)h − z˙(t)∣∣∣ ≤ 12 |z˙(t)|. This implies
z(t+ h)− z(t) 6= 0 and hence the right-hand side in
ϕ(z(t+ h))− ϕ(z(t))
h
=
ϕ(z(t+ h))− ϕ(z(t))
z(t+ h)− z(t)
z(t+ h)− z(t)
h
is well-defined for 0 < |h| < δ.
Let us now consider the limit h ↘ 0. First, if z˙(t) > 0 then
z(t+ h)− z(t) > 0 for h ∈ (0, δ). Suppose D+ϕ, D+w and D+ϕ, D+w
are the right upper and right lower Dini derivatives of ϕ, w respectively.
We can then choose hν ∈ (0, δ) with w(t+hν)−w(t)hν → D+w(t). With
zν := z(t+ hν)− z(t) > 0 we infer that
lim sup
ν→∞
ϕ(z(t) + zν)− ϕ(z(t))
zν
≤
≤ lim sup
h↘0
ϕ(z(t) + h)− ϕ(z(t))
h
= D+ϕ(z(t)).
Note that zν ↘ 0 for ν →∞. This finally implies
D+w(t) = lim sup
ν→∞
[
ϕ(z(t) + zν)− ϕ(z(t))
zν
z(t+ hν)− z(t)
hν
]
246 Appendix C. Some additional proofs
≤ D+ϕ(z(t))z˙(t).
We argue for D+w(t) in a similar fashion to get
D+ϕ(z(t))z˙(t) ≤ D+w(t) ≤ D+w(t) ≤ D+ϕ(z(t))z˙(t).
Now consider the case z˙(t) < 0. Then zν = z(t+hν)−z(t) < 0 and thus
lim inf
ν→∞
ϕ(z(t) + zν)− ϕ(z(t))
zν
≥
≥ lim inf
h↗0
ϕ(z(t) + h)− ϕ(z(t))
h
= D−ϕ(z(t))
with D−, D− denoting left Dini derivatives. Then
D+w(t) = lim sup
ν→∞
[
ϕ(z(t) + zν)− ϕ(z(t))
zν
z(t+ hν)− z(t)
hν
]
=
= lim inf
ν→∞
[
ϕ(z(t) + zν)− ϕ(z(t))
zν
]
z˙(t) ≤ D−ϕ(z(t))z˙(t).
By a similar argument for D+w(t) we infer
D−ϕ(z(t))z˙(t) ≤ D+w(t) ≤ D+w(t) ≤ D−ϕ(z(t))z˙(t).
If both z and w are differentiable at t we conclude
D+ϕ(z(t))z˙(t) ≤ w˙(t) ≤ D+ϕ(z(t))z˙(t) if z˙(t) ≥ 0,
D−ϕ(z(t))z˙(t) ≤ w˙(t) ≤ D−ϕ(z(t))z˙(t) if z˙(t) < 0.
Now note that ϕ ∈ slope(µ1, µ2) implies
µ1 ≤ D−ϕ(z) ≤ D−ϕ(z) ≤ µ2 and
µ1 ≤ D+ϕ(z) ≤ D+ϕ(z) ≤ µ2 for all z ∈ R.
For z ∈ H 1,1e , w = ϕ(z) and t ≥ 0 such that both z and w are
differentiable at t (which is true for almost all such points), we infer
µ1z˙(t) ≤ w˙(t) ≤ µ2z˙(t) and µ2z˙(t) ≤ w˙(t) ≤ µ1z˙(t) (C.24)
for z˙(t) ≥ 0 and z˙(t) < 0, respectively. Consequently, irrespective of
the sign of z˙(t), we get (5.47).
C.5. For Chapter 6 247
C.4.3 Proof of Lemma 5.3
Let z ∈ H 1,k and set µ := max{|µ1| , µ2}. Then z is locally abso-
lutely continuous. As ∆Φ : L k2 → L k2 is bounded with ‖∆Φ‖ ≤ µ
and ϕ is Lipschitz continuous, w = ∆(z) is square integrable with
‖w‖2 ≤ µ2‖z‖2 and locally absolutely continuous. Hence w˙(t) exists
for almost every t ∈ [0,∞) and, by (C.24), ‖w˙(t)‖ ≤ µ‖z˙(t)‖ for al-
most every t ∈ [0,∞). Thus z˙ ∈ L k2 implies w˙ ∈ L k2 and ‖w‖2H =
‖w‖2 + ‖w˙‖2 ≤ µ2(‖z‖2 + ‖z˙‖2) = µ2‖z‖2H . This proves the claim.
C.5 For Chapter 6
C.5.1 Proof of Lemma 6.4
It suffices to prove the claim for a scalar nonlinearity ϕ : R→ R satisfy-
ing ϕ ∈ slope(µ, ν) for some µ ≤ 0 ≤ ν. We use the notation from [39].
Let x ∈ R and v > 0. Then we have for the generalized directional
derivative
ϕ◦(x; v) = lim sup
y→x
t↘0
ϕ(y + tv)− ϕ(y)
t
= v lim sup
y→x
t↘0
ϕ(y + tv)− ϕ(y)
tv
≥ µv
With the same reasoning, we obtain for v < 0 that ϕ◦(x; v) ≥ νv. In
conclusion, we obtain
µ ≤ ϕ
◦(x; v)
v
≤ ν
Finally, with ϕ◦(x; 0) ≥ 0 and the definition of the generalized gradient
∂ϕ, we arrive at
∂ϕ(x) ⊂ [µ, ν] for all x ∈ R.
Now let w(t) = ϕ(z(t)) for z ∈ `k2 and t ∈ N0. Then the statement
follows form Lebourg’s mean value theorem ([39, Theorem 2.3.7]).
248 Appendix C. Some additional proofs
C.5.2 Proof of Lemma 6.13
The proof relies on an argument similarly to one made in [45], which
is also used in the proof of Lemma 5.10. Let Π, Π1 and Π2 be param-
eterized by L, L1 and L2, respectively. A short computation reveals
that the combined multiplier
Π := Π + Π1 + Π2
satisfies(
1 1
µ ν
)T
Π
(
1 1
µ ν
)
−
(
0 (ν − µ)H∗L
(ν − µ)HL 0
)
=
=
(
0 (ν − µ)H∗L1
(ν − µ)HL1 (ν − µ)(H∗L1 +HL1)
)
+
+
(
(ν − µ)(H∗L2 +HL2) (ν − µ)H∗L2
(ν − µ)HL2 0
)
.
Define (
Q S∗
S R
)
:=
(
1 1
µ ν
)T
Π
(
1 1
µ ν
)
to infer Q = (ν − µ)(H∗L2 +HL2) and R = (ν − µ)(H∗L1 +HL1) which
are both positive semi-definite on T. Since µ < ν, the equation(
τQ S∗
S τR
)
:=
(
1 1
µ ν
)T
Πτ
(
1 1
µ ν
)
has a unique solution Πτ for all τ ∈ [0, 1]. Moreover, the derivative of
the left hand side with respect to τ is positive semi-definite. Hence, we
infer Π˙τ  0 and thus Π0 4 Π1 = Π on T. Now note that
S =
(
I νI
)
Π0
(
I
µI
)
=
(
I νI
)
Π
(
I
µI
)
C.5. For Chapter 6 249
= (ν − µ)(HL +HL1 +HL2) =: (ν − µ)HL3
where HL3 satisfies all requirements of a Zames-Falb multiplier (since
the constraints define convex cones). We finally get
Π0 = (ν − µ)
(
1 1
µ ν
)−T (
0 H∗L3
HL3 0
)(
1 1
µ ν
)−1
=
1
ν − µ
(
νI −I
−µI I
)T (
0 H∗3
H3 0
)(
νI −I
−µI I
)
.
Thus the lemma follows with HL˜ = (ν − µ)−1HL3 , which defines again
a Zames-Falb multiplier.
250 Appendix C. Some additional proofs
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Declaration
I hereby certify that this thesis has been composed by myself, and
describes my own work, unless otherwise acknowledged in the text. All
references and verbatim extracts have been quoted, and all sources of
information have been specifically acknowledged.
Stuttgart, June 2017
Matthias Fetzer
273